We will continue the all in squeeze play series and start with the same framework we used in the 2x pot squeeze play facing 10% hands, only this time we will use 20.36% hands or
22+,A9+,KJ+,QJ,A2s+,K9s+,Q9s+,J9s+,T8s+,98s
Since last time 50% of hands worked out to be close to equilibrium, we will start there.
50% of this range is 66+,AJ+,A9s+,KTs+,QTs+,JTs.
Once again, we need to win 1/3rd of the time we are called to have a profitable shove.
That means shove:66+,A4,A5,A7+,K9+,QT+,A2s+,K3s+,Q8s+,J9s+,T9s.
The weird relationship with A6 vs A4 and A5 means A4 and A5 because of wheel possibilities is actually better to shove. A6 is not a profitable shove bu A4 and A5 is.
Well this shove range wins .33325x pots on average per shove with 26.4% of hands yielding 0.088 pots or about .66 big blinds per hand dealt in this situation.
I don't know if this is optimal, but right now I don't feel like doing all the work. Shoving with any two I approximated the solution as 6% of hands. So if opponent calls with 99+,AQ+,ATs+,KQs or 88+,AQ+,AJs+,KQs or something to that extent or tighter you can shove with any two.
Approximate Optimal Shove*:66+,A4,A5,A7+,K9+,QT+,A2s+,K3s+,Q8s+,J9s+,T9s.
Approximate Optimal Call*:is 66+,AJ+,A9s+,KTs+,QTs+,JTs.
Any two if opponents Call ranges are this tight or tighter:99+,AQ+,ATs+,KQs
Assuming it's close enough until I do the work to prove otherwise which I don't feel like because the advantage for a better strategy if there is one will probably be pretty small. It is far worth my more on my time to get to the other situations of 30%, 40%, 50%, any two and then repeat at 3x, 4x and 5x pot than to make sure these strategies are all perfect. Afterall, your opponent probably won't be perfect anyways, and you probably won't remember all of these anyways, so since it's just to get a "feel" for how to shove in a few spots so you have more accurate poker intuition, perfection is not the goal.
Tuesday, June 30, 2015
Squeeze Play All In 2x pot 10% raise, 10% caller
The squeeze play has different forms. You can squeeze opponents for say 15 big blinds when you have over 100. If you only had say 80 big blinds your opponents can shove wider and force you to fold without an easy recourse other thanthe equity of your hand, thus the actual hands you squeeze with for 80 big blinds and less need to be less than with over 100 assuming your opponents play close to "optimally".
Then there's the squeeze play all in for 5 times the pot and less. This is probably around 40 big blinds and less.
Today we are focusing specifically on the 2x pot shove. So if you have 15 big blinds and opponent in hijack makes it 2.5 and cutoff calls plus blinds and antes put 7.5 in the pot, you shove from the button for 15 big blinds.
We need to analyze this in a lot of depth. First, we need to know opponent's range of opening and calling range. Secondly we need to identify given your shove range how strong must opponent be to call. Then we must adjust both your shove range and opponent's call range until neither side can improve the EV by much.
To simplify, if one person calls the other's hand is dead and instantly mucks and it goes to a 1-on-1 all in showdown.
So let's start with the 10% raise and 10% caller.
10% of hands we will represent by
66+,AJ,A9s,+KTs+,QTs+,JTs
We will start by assuming one opponent calls with 50% of this range. That is 99+,AQ. Since there are two opponents, the chance that both opponents fold is 25%. .5*.5.
When you get folds, you pick up 1x the pot.
When you get called, you get 3x the pot when you win.
When you get called, you lose 2x the pot when you lose.
25% of the time you get 1x. This translates to .25 in value.
You must win 1/3rd of the time vs this range to shove.
Your shoving range vs this calling range are those hands which have 33.33% equity or better which translates into TT+,AQ+,AJs+,KTs+.
On average our equity of this range vs opponents range is 49.86% when called which translates to 0.6197 times the pot per shove. Since we shove with this range, we will only pick up .6197 pots 5.8823% of the time. This is important because a wider range that picks up a smaller amount more often may actually be better. so .6197*.058823=.036453x the pot per hand dealt If there are 7.5 big blinds in the pot that's 0.2734 big blinds per hand dealt.
What if opponent calls less? What if opponent calls more? What we are looking for now is a condition in which opponent can improve expected value vs this pushing range, and then we would in turn adjust our pushing range. If we are unable to reach a condition as profitable we're moving in the right direction. We'd then optimize pushing range vs that calling range, then adjust the calling range vs that pushing range and repeat until we can't really improve much.
We also want to find the point by which pushing with 27o and 23o are profitable so we can identify how overly tight opponents have to be until we can shove with any two. Hopefully, by identifying this spot and seeing a spot or two in between as well as finding the optimal shoving range, we can get a feel for how rapidly we can widen pushing range as opponents are less optimal.
What if opponent calls less? Calling range 99+,AK. Both fold 37.4471% of the time. We only need to be 28.03% to win so we can widen our range. This range is a bit strange since there is no AQ so Q8s is better than K9s. Also, Ax suited is good because it has a single overcard to a pair often enough to be profitable and since AQ is not in the mix it isn't as likely to be dominated by a premium ace.
Our shoving range is 22+,AT+,KJ+,QT+,A2s+,KTs+,Q8s+,J8s+
That translates into .212459x pots per shove but because we shove on 19.76% of hands in this spot, we actually gain .042 per hand dealt given this set up which is more than..0365 from looser calling ranges.
As a rule of thumb, usually when you can widen your profitable shove range, your opponent did something wrong. Usually either that means opponent makes shoving more hands profitable, because he folds too much, or more hands have value over opponent's calling range. A too tight opponent is the only situation where you can shove any two as the too loose one you still have to have equity over a random hand after pot odds.
So if 2 opponents who raise and call with 66+,AJ,A9s,+KTs+,QTs+,JTs only call a 2x pot shove with TT+,AKs, that means they are playing 34/134 combinations of hands or 25.4%. 2 opponents folding 74.62% means 55.7% chance both fold. With this information, we can see we only need to win 14.86% of the time.
If opponent tightens calling range up to TT+,AKs we can shove with any two and although 27o and 38 doesn't quite profit, it is within less than 1% of being profitable and 23o does and 28o does and 26o does so it's close enough for me.
So we found the tightest boundary. Now the question is, would our opponent loosening up their calling range from 99+,AQ force us to make less profits? If so, we are not quite at equilibrium.
Let's say opponent calls 88+,AQ+,AJs+,KQs. Both opponents fold only 15.06% of the time. This means we will have very little value in pushing with a poor hand. We need to be 36.46% to win to shove. I can already tell it's going to be close because before we couldn't push with 99, but now we can, but we can no longer push with KTs. The exact range is 99+,AQ+,AJs+,KJs+ which is pretty close to the same as the original.
We pick up .6505 pots when we shove on average. But we only shove on 6.03% of hands. The total of 0.039243917x pots per shove or about .294 big blinds per hand dealt per situation that comes up is more than our original esimation of 50% of hands.
Therefore we can conclude a shove range with 2x pot vs 10% raiser and 10% caller of TT+,AQ+,AJs+,KTs+ is approximately optimal with opponent's approximately optimal calling range of 99+AQ. The calling range should actually be slightly tighter for the initial raiser unless his sole goal is making us lose since he still has to worry about the opponent left to act. This means our shoving range may be slightly looser, but likely not by much if at all since there is also the possibility that both opponents call which wasn't calculated.
If opponent tightens up only slightly to 99+,AK our shoving range can widen to 22+,AT+,KJ+,QT+,A2s+,KTs+,Q8s+,J8s+
Shoving with any two becomes profitable when opponents in this spot only call your shove with TT+,AKs.
This should give us a better feel of how we can widen our shoving range as opponents tighten up their calling range
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notes:
However, always keep in mind that if opponents don't call optimally, shoving optimally is still more profitable than if opponents did not call optimally. You just have the opportunitiy for even greater gains if you can accurately speculate on what range opponents will call with. Typically people are going to be surprised at how opponents play and are not going to know how strong they will call. Plus, you already are making an assumption about your opponent's calling range which can also be incorrect. However, if you choose somewhere between "optimal" and exploitative" you have given yourself a margin of error if your assumptions are wrong. For example, if I thought opponent would call 99+,AK and he actually calls with AQ, I would make less that shoving the optimal range. But if I'm right, I'll make more shoving the exploitative range. By doing somewhere between, I can make a little more when I'm right and lose less, or have less missed opportunity.
So I might aim for something like 77+,AQ+,KQ,QJ,A9s+,KJs+,QTs+,JTs. That allows me more profits than just shoving the optimal amount if I'm right, and I lose less if I'm wrong than shoving the exploitable amount and because I'm probably right more often than not, but not to a large enough degree where it would be dangerous to shove with too wide of range, overall I should approximately optimiize my profits. Another alternative is having an "advertisment play. If you think opponent is playing tooo close to optimal, you can try to show either a crazy bluff or a KK,AA type hand so they are more likely to make a mistake later and adjust their range as a reaction to you.
Now we have to basically repeat this exercise for 3x,4x and 5x pot and also for hand ranges of initial raiser and caller of 20%,30%,40%,50%, 50% and 100% and any two for both. This means we have to do 15 times the work we just did, which will be exhausting, particularly as the range widens and there are more hands individually to run through to determine if they are okay to shove.
Nevertheless, doing work like this gives you so much better of a feel for poker than actually playing even 10,000 hands. So it actually happens to be a MAJOR shortcut to poker excellence.
Then there's the squeeze play all in for 5 times the pot and less. This is probably around 40 big blinds and less.
Today we are focusing specifically on the 2x pot shove. So if you have 15 big blinds and opponent in hijack makes it 2.5 and cutoff calls plus blinds and antes put 7.5 in the pot, you shove from the button for 15 big blinds.
We need to analyze this in a lot of depth. First, we need to know opponent's range of opening and calling range. Secondly we need to identify given your shove range how strong must opponent be to call. Then we must adjust both your shove range and opponent's call range until neither side can improve the EV by much.
To simplify, if one person calls the other's hand is dead and instantly mucks and it goes to a 1-on-1 all in showdown.
So let's start with the 10% raise and 10% caller.
10% of hands we will represent by
66+,AJ,A9s,+KTs+,QTs+,JTs
We will start by assuming one opponent calls with 50% of this range. That is 99+,AQ. Since there are two opponents, the chance that both opponents fold is 25%. .5*.5.
When you get folds, you pick up 1x the pot.
When you get called, you get 3x the pot when you win.
When you get called, you lose 2x the pot when you lose.
25% of the time you get 1x. This translates to .25 in value.
You must win 1/3rd of the time vs this range to shove.
Your shoving range vs this calling range are those hands which have 33.33% equity or better which translates into TT+,AQ+,AJs+,KTs+.
On average our equity of this range vs opponents range is 49.86% when called which translates to 0.6197 times the pot per shove. Since we shove with this range, we will only pick up .6197 pots 5.8823% of the time. This is important because a wider range that picks up a smaller amount more often may actually be better. so .6197*.058823=.036453x the pot per hand dealt If there are 7.5 big blinds in the pot that's 0.2734 big blinds per hand dealt.
What if opponent calls less? What if opponent calls more? What we are looking for now is a condition in which opponent can improve expected value vs this pushing range, and then we would in turn adjust our pushing range. If we are unable to reach a condition as profitable we're moving in the right direction. We'd then optimize pushing range vs that calling range, then adjust the calling range vs that pushing range and repeat until we can't really improve much.
We also want to find the point by which pushing with 27o and 23o are profitable so we can identify how overly tight opponents have to be until we can shove with any two. Hopefully, by identifying this spot and seeing a spot or two in between as well as finding the optimal shoving range, we can get a feel for how rapidly we can widen pushing range as opponents are less optimal.
What if opponent calls less? Calling range 99+,AK. Both fold 37.4471% of the time. We only need to be 28.03% to win so we can widen our range. This range is a bit strange since there is no AQ so Q8s is better than K9s. Also, Ax suited is good because it has a single overcard to a pair often enough to be profitable and since AQ is not in the mix it isn't as likely to be dominated by a premium ace.
Our shoving range is 22+,AT+,KJ+,QT+,A2s+,KTs+,Q8s+,J8s+
That translates into .212459x pots per shove but because we shove on 19.76% of hands in this spot, we actually gain .042 per hand dealt given this set up which is more than..0365 from looser calling ranges.
As a rule of thumb, usually when you can widen your profitable shove range, your opponent did something wrong. Usually either that means opponent makes shoving more hands profitable, because he folds too much, or more hands have value over opponent's calling range. A too tight opponent is the only situation where you can shove any two as the too loose one you still have to have equity over a random hand after pot odds.
So if 2 opponents who raise and call with 66+,AJ,A9s,+KTs+,QTs+,JTs only call a 2x pot shove with TT+,AKs, that means they are playing 34/134 combinations of hands or 25.4%. 2 opponents folding 74.62% means 55.7% chance both fold. With this information, we can see we only need to win 14.86% of the time.
If opponent tightens calling range up to TT+,AKs we can shove with any two and although 27o and 38 doesn't quite profit, it is within less than 1% of being profitable and 23o does and 28o does and 26o does so it's close enough for me.
So we found the tightest boundary. Now the question is, would our opponent loosening up their calling range from 99+,AQ force us to make less profits? If so, we are not quite at equilibrium.
Let's say opponent calls 88+,AQ+,AJs+,KQs. Both opponents fold only 15.06% of the time. This means we will have very little value in pushing with a poor hand. We need to be 36.46% to win to shove. I can already tell it's going to be close because before we couldn't push with 99, but now we can, but we can no longer push with KTs. The exact range is 99+,AQ+,AJs+,KJs+ which is pretty close to the same as the original.
We pick up .6505 pots when we shove on average. But we only shove on 6.03% of hands. The total of 0.039243917x pots per shove or about .294 big blinds per hand dealt per situation that comes up is more than our original esimation of 50% of hands.
Therefore we can conclude a shove range with 2x pot vs 10% raiser and 10% caller of TT+,AQ+,AJs+,KTs+ is approximately optimal with opponent's approximately optimal calling range of 99+AQ. The calling range should actually be slightly tighter for the initial raiser unless his sole goal is making us lose since he still has to worry about the opponent left to act. This means our shoving range may be slightly looser, but likely not by much if at all since there is also the possibility that both opponents call which wasn't calculated.
If opponent tightens up only slightly to 99+,AK our shoving range can widen to 22+,AT+,KJ+,QT+,A2s+,KTs+,Q8s+,J8s+
Shoving with any two becomes profitable when opponents in this spot only call your shove with TT+,AKs.
This should give us a better feel of how we can widen our shoving range as opponents tighten up their calling range
-------------------------------------------------------------------------------------------------------------------------
notes:
However, always keep in mind that if opponents don't call optimally, shoving optimally is still more profitable than if opponents did not call optimally. You just have the opportunitiy for even greater gains if you can accurately speculate on what range opponents will call with. Typically people are going to be surprised at how opponents play and are not going to know how strong they will call. Plus, you already are making an assumption about your opponent's calling range which can also be incorrect. However, if you choose somewhere between "optimal" and exploitative" you have given yourself a margin of error if your assumptions are wrong. For example, if I thought opponent would call 99+,AK and he actually calls with AQ, I would make less that shoving the optimal range. But if I'm right, I'll make more shoving the exploitative range. By doing somewhere between, I can make a little more when I'm right and lose less, or have less missed opportunity.
So I might aim for something like 77+,AQ+,KQ,QJ,A9s+,KJs+,QTs+,JTs. That allows me more profits than just shoving the optimal amount if I'm right, and I lose less if I'm wrong than shoving the exploitable amount and because I'm probably right more often than not, but not to a large enough degree where it would be dangerous to shove with too wide of range, overall I should approximately optimiize my profits. Another alternative is having an "advertisment play. If you think opponent is playing tooo close to optimal, you can try to show either a crazy bluff or a KK,AA type hand so they are more likely to make a mistake later and adjust their range as a reaction to you.
Now we have to basically repeat this exercise for 3x,4x and 5x pot and also for hand ranges of initial raiser and caller of 20%,30%,40%,50%, 50% and 100% and any two for both. This means we have to do 15 times the work we just did, which will be exhausting, particularly as the range widens and there are more hands individually to run through to determine if they are okay to shove.
Nevertheless, doing work like this gives you so much better of a feel for poker than actually playing even 10,000 hands. So it actually happens to be a MAJOR shortcut to poker excellence.
Saturday, June 27, 2015
The Goal Of Poker - Money Is Only The Side Effect
Many people think the goal of poker is to make money. That's not correct. To some it's just a lotto ticket and it's just an escape to have a little fun and have a time to dream of money and "see what happens". Sorry to be blunt, but that's the mindset of losers that you must avoid by any means necessary or you are better off just taking out a loan and sending me your money.... and your house and your wife and kids while you're at it...
If you want to make money in poker it is possible, but it should never be the goal. Money is only the possible result. It is only a probable side effect of making good decisions. Having fun is a side effect of playing well and making money. Therefore the goal of poker is to make correct decisions. Money is the probable side effect of correct decisions. Let me repeat for emphasis. The goal of poker is to make the correct decisions. If you succeed in this goal, the probable side effect will be more money.
So if opponent is tight and plays JJ+,AQ+ in a spot, your Ace King or QQ is no good, fold. If opponent plays 88+,AQ in a certain spot your AQ suited is no good and your ace king suited is a coinflip and ace King offsuit is actually slightly behind. On the other hand, these hands can still be called if you are getting the right price so you win say twice the amount you risk. But each of these decisions require a lot of knowledge about what's good in certain spots or at least a good feel based upon doing the math. Experience at the table will never get you enough repetition to know with as much precision as those who do put in the work.
If you flip a coin and you lose $1 when you win, and lose $1 when you lose the expected value is ZERO. This can be calculated as probability times result plus probability times result.
(.50*1)+(.50*-1)=(.50)+(-.50)=0
Positive expected value calculations determine when the probable side effect over the long run is more money assuming good bankroll management. It can be calculated by the sum of the probability of each result multiplied by the corresponding result.
So let's say you have AQo and opponent only has one of a few cards. A pocket pair of 88 or better or AQ and he always will play any card in this range. You are 39.55% to win vs this range. Intuition should at least get you to estimate close to this number if you have done the work. say 40%. If your opponent has moved all in for $40 into a pot of $10 after you raised $4 and everyone else folds, should you call? You have to call $30 more. You win $50 when you are right, lose $30 when you lose. You are getting 5:3 or 1.667 to 1. This means you need to win at least 1/2.667 or 37.5% to win after rake in order to call.
Your expected value is about:
(.40*50)+(.60*-30)=$2. Assuming your raise attempt to risk $3 was a break even decision, the result of both raising and then calling profits $1 less than it would in situations when you get callers or opponents to fold but it profits $2 more than folding. In some cases, you make an unprofitable decision to raise in the first place, so the EV calculation of +$2 is actually only gaining back some of what you lose from raising. In other words if you always are raised and usually fold, you usually lose $3, but in this one instance, you are able to gain $2 of that $3 you usually lose back. So just because this is "+EV" or better than folding, doesn't mean the decision to get into the spot was necessarily profitable*.
This point of emphasis is made so you understand that the EV of any particular decision is often connected with the prior decisions. You can make a correct decision in one spot, but be incorrect to get into that spot in the first place.
*(To be clear, it is unlikely that raising Ace Queen is ever a mistake but if a hand like A2 against a reckless opponent yielded $2 to call their all in while the individual mistake after you raised may make money in most cases you usually will fold to everyone else's raise and usually you lose money so that $2 may not be enough to make up for where you lose a portion of that $3 you risk initially).
Every decision should be analyzed including what hands you play preflop.
Over the very long run with consistent bankroll management, your bankroll is the starting bankroll (and any additions or subtractions to it) plus or minus the sum of the EV of all decisions you make.
If 50 hands yield $1 each but 11 hands after that yields -$5, you may be capable of being a winning player, but overall you will not be because of tilt being the biggest leak in your game.
Unfortunately, break even decisions while in "expected value" are neutral, in terms of bankroll decisions, they are negative. Even positive decisions with poor bankroll management can lead to the probable outcome over a long period of time being losing money. The goal therefore cannot be solely based upon "expected value". It is vital to understand why.
The reason break even decisions lose money is because any bet has both the possibility of having a positive result and a negative result. However, several negative results strung together require a disproportional number of positive results to break even. To illustrate this start with $100 and imagine losing 50% then gaining 50%. Think you are back to even? Think again. A 50% loss from $100 takes you to $50. A 50% gain from $50 only takes you to $75. So a break even proposition has actually cost you 25% of your bankroll given this amount of volatility over this amount of time.
You may incorrectly deduce that you'll just increase the amount as you lose. But this doesn't solve the underlying problem is that you now have the same probability of an even greater loss, such that the gain once again is not enough to offset the loss. The only possible solution other than getting money from other sources (and even then sometimes it's not enough) is risking less to reduce volatility or to dramatically increase your edge so that type of drawdown doesn't happen and the gains are skewed to the upside large enough to overcome them. It's very uncommon to have that great of edge, and those with a great edge tend to be overconfident even more so and risk their bankroll more liberally than they should to their own detriment.
Greater volatility over longer periods of time will cost a fixed bankroll to go to zero, and a growing bankroll (such as a bankroll derived from excess income at a job) will continue to over time decline. Certainly it could go the other way. You COULD gain 50% first to $150 and then lose 50% to $75. The result is eventually the same over infinite time horizon. Timing it, or varying your bet size after you lose isn't going to compensate for a losing philosophy of playing a losing game.
This is what happens if you risk over 2 times the "Kelly"
Ruin becomes certain over a long enough period of time, but the rate at which it occurs accelerates. with an uncertain edge, this favors risking FAR less than the full Kelly even with an infinite time horizon.
However, respecting that you don't have an infinite time horizon is more reason to bet less if you are looking for the probable outcome of more money. Over a fixed time horizon the supposed "optimal" amount to bet is actually way too much unless you want a "lottery type" distribution where a very very very small minority ends up with a very large return and a large majority end up at a poor or negative return, while the upper 25% may have satisfactory returns.
Betting less instead would produce a satisfactory return maybe 40% of the time but without the absurdly unlikely outliers that will never actually occur to any one individual in the history of the next 100 generations to any one of billions attempting it. I don't think people realize that you can pretty much rule out the possibility of the "positive outlier event". The length and impact of the "infinite time horizon" cannot be overstated enough. Having the duration of time of the entire universe and all life forms ever in history of the universe would be less significant to infinity than the size of an atom to the size of the universe.
Even though it's a stock trading website, there is plenty of useful information in understanding modeling risk with a massive fixed, known edge at various risk levels produces distributions of results over a fixed time period. More about Risk Management here.
At the extreme, not even a single pixel can represent the difference between the number of people achieving outlier results and zero. At 10% most result in losing over 90% of their bankroll maybe 0.5% are outliers and 0.01% produce an average of an amazing return. At 20% risk 999.99 go effectively busto. Dream on.
You can see that with starting bankroll of $1000, larger amounts of risk produces a larger and larger percentage of people that actually lose money over 300 bets. It's difficult to really see what the actual percentage is, but the actual numbers wouldn't apply anyways. The concept that nearly ALL results effectively go bankrupt when they are betting closer to the supposed "optimal" amount to maximize long term growth goes to show you that even betting 1/3rd of this amount is risky.
In theory you might be able to eventually get your 10 cent bankroll up to a million when the 1/1000 outlier actually hits a series of several wins and eventually all those major outliers on either ide even out to the average. But we are talking about "quattroduodecatrillions" of hands until that happens and we are also talking about playing at stakes using fractions of a penny which isn't reality. Or if you have an income of $1,000, you are just going to reload and lose it all every single month or so and maybe over 1000 months or perhaps 4000 before you finally hit that one outlier month in which you pay for all the months before it and then some. The problem is a human life span is only 960 months, and only probably 480 of it if you lived and breathed poker could be spend playing poker. And how boring of life is that to be broke for most of your life?
What's worse, is you can never really retire with this strategy or play poker for a living with a high amount of risk if you don't have some income elsewhere. In order to really play poker for a living, you're going to have to play at absurdly low risk percentages such as 1/30th of the "optimal bet size" at risk at any one time (we're talking like 0.20% of your bankroll at risk) and have so much excess separate from your bankroll saved on the side that your probable result is centered right around break even anyways and most likely you are just withdrawing from 2 years worth of savings, but the overall "average" with 0.20% of your bankroll is theoretically enough to live off of if you made it each and every month with certainty.
So if you made $50 an hour at $2/5 live games buying in for $500, you'd need a bankroll of more than $250,000. Well, maybe earning $10 an hour is enough at $0.50/$1.00. You'd need $50,000. net worth. If you always had a way to make ends meet if you absolutely had to and were okay with having a risk of ruin and living life in fear of going broke, you could use the standard measurements of risk management. Maybe you are okay with knowing that you can always sell your house or take out a mortgage if you have your home paid off. Either way there are very few people in the world with
1)The skill to play poker
2)The understanding of bankroll management
3)The willingness to grind at less conservative stakes as a side job until capital has been built up
4)The discipline to stick to insanely conservative bankroll management strategies
5)The passion for the game to play for thousands of hours each year.
6)The discipline to not spend their earnings at at a higher rate than their "expected" earnings.
Realistically there are plenty of people that can earn professionally if they have sources to be staked and endorsement deals and/or business income, or book deals and poker coaching income, but the days of easy endorsements and free capital in the poker community is gone and in a bad economy your edge will likely go from positive to negative and the difficulty of games and easy money floating around will evaporate.
Furthermore, if you're an online player, you'd have a higher average and lower volatility multitabling at 2% risk with 5 tables than 10% risk at one assuming the edge was the same so risking more has no practical advantage in any normal form.
The principal of volatility hurting more than it helps is true even with an edge if you do not manage bankroll properly. While some people may chime in about compound interest and how several wins in a row will compound as well, that may be true to a very small degree, but the benefit is smaller than the drawback of risking more. It is not worth it and you will not benefit to a strong enough degree to compensate for the cost of volatility if you risk too much. The amount you can risk with a break even proposition to have positive results is zero. Not to mention it's not worth your time.
The casino will always have a larger bankroll than you, and if they don't, they will kick you out because they don't want that kind of risk. Phil Ivey was kicked out for playing craps, a game in which he had the worst of it from very small casinos because he was placing very large 100k bets. The casinos don't gamble, they play mathematical certainties over enough times so that they make money, and they didn't want to allow to put their entire operation at a risk of ruin. They will turn down your "business" if either they know you have the best of it at a small stakes or you are playing at such large stakes that their "probable outcome" is not more money over a very long term.
For similar reasons, decisions with positive EV can still have negative long term expected growth rates. That may sound like a contradiction, but it's not. The "Kelly criterion" is like "EV" but for your bankroll over an unlimited time horizon. It already compensates for the fact that a 20% drawdown in your account requires a 25% growth rate to get back to even and a 50% drawdown requires a 100% gain. But realize that over an infinite time horizon a 50% loss can eventually be recovered, even if for all practical purposes it cannot easily be in real life in a finite timeframe.
While an individual decision repeated for the same bet amount an infinite amount of times may be positive, repeated as a percentage of bankroll or within the realistic parameters of actually having a limited bankroll, the long term result is losing money.
However, the odds of going 6 tournaments in a row without a win are greater than 50% .9^6=53.14% If this player risks 100% of his bankroll he has a 90% chance of going bust. If he risks 50% of his bankroll, EVEN if he adjusts his bet every tournament so he only bets 50% of his remaining bankroll after each loss, he has an over 50% chance of having less than 2% of his bankroll before he wins. When he finally does win, he will be doing so at such small stakes that 20 buy in victory won't even take him remotely close to where he was. As such, he has a very good chance of not even being able to afford a low stakes buy in with this plan.
The theory of "adjusting downward" as you lose is reasonable, but only if you start low enough to begin with. And a fixed percentage at an amount between the "adjusting downward" amount and the starting amount probably has the same impact of adjusting downward with less volatility and more consistent gains. A "fixed percentage" actually does adjust upwards and downwards since it's a percent of your bankroll.
The theory of increasing your risk as you lose is unreasonable as this increases your chances of losing such a greater amount that you won't even have enough to continue even if you have Bill Gate's credit line, and if you did, you'd have to start at such a small bet that it wouldn't even be worth your time and you'd still put unreasonable risk of losing everything. Take a coinflip. Risk a $5 and double the risk 35 times or so and you'd lose all $85 billion of Bill Gate's bankroll. Gain $5 35 times and you've gained $175. Is $175 worth risking $85 billion? Is it worth the time? This strategy is guaranteed to lose eventually it is not nearly as logical as you'd like to wish it was.
You cannot change how you "time" the amount that you bet or to turn a profitable decision or unprofitable decision to be more profitable or less unprofitable. You can only influence the probability of a gain at the expense of the downside or probability of a huge loss. The best decision is to sit with a fixed percentage of your bankroll or lower than that fixed amount, and to determine which "fxied amount" suits your goals.
Fortunately there is a solution to increase the results if you are a winning player, and it involves understanding the nature of risk.
There is an amount to bet which over bets that approach an infinite amount, your value is maximized. Not too much, not too little... Just right. However, as you approach this "optimal" point the benefits for increasing your bets have greater and greater risk at a smaller and smaller reward.
Consider risking half the kelly has 3/4ths the return with half the volatility. This means you recover from your all time highs in bankroll much faster and make new highs much faster on average, although the upswings will not be as large and the overall trend may be slightly less strong. Risking 1/4th the kelly still has 40% of the return as the full kelly with 25% of the risk. In other words, risking less is almost always better from a practical risk/reward perspective.
This is only what risk looks like given CERTAIN outcomes. With uncertain outcomes over an infinite amount of time there is risk of betting far too much. If you bet over 2 times the kelly the eventual result is losing everything.
As such, there is no reason to overbet the kelly at all.
Over FIXED time horizon, even as long as 10 years, everything changes. Betting the full kelly over a fixed time horizon results in a probability distribution to explain the results. Out of 1,000 players who play a kelly criterion strategy a very small number of them will have extraordinary results, while the majority will have below average results and the upper 25% may have above average results but it will be skewed by the top 1%. The bottom 25% will produce break even or even a loss, while the bottom 25% (larger or smaller depending on the number of profitable "bets") will have lost 90% of their bankroll.
An
infinite time horizon is a "long time" for the skew to "balance out". In
a fixed time horizon of even what seems like an enormous amount of
trials, like a billion, the results will still be quite skewed.
Since this essentially amounts to a lottery ticket chance of having enormous gains that are far beyond what you could ever spend, there isn't a lot of added value to taking on a "Kelly" worth of risk. 1/5th of the Kelly is acceptable to some, but probably not if you have to withdraw from your bankroll at regular intervals without some other means to live if you go through downswings.
However, risking less than 1/5th of the Kelly will produce both a high probability of very good results and good upside. There still will be a chance of pretty spectacular results, and still a small chance of poor results, but overall the outcome will be likely to be pretty good. There will still be a skew. Because the probable outcome is more money than less, THIS is a correct decision, and money is the probable side effect.
This amount of risk requires examination of your skill edge. That isn't something that is entirely certain so there is a slight "gamble" on your own skill. Or if you are playing an "equilibrium" strategy there is a gamble that others will be bad enough that you will beat the rake (which is a very good assumption) and that your mistakes will be small enough (playing without mistakes even with a fairly basic predetermined strategy is difficult and has mental and psychological challenges).
There is a reason that pros like Chris Ferguson and Phil Gordon and others have never gone broke and also live comfortable lives with poker playing at one time still being a reasonably significant part of it. Aside from having skill, they follow a set of bankroll management guidelines and very solid bankroll strategies. Since it is difficult to distinguish playing poorly with being unlucky and it is easy to convince yourself in defense of the ego that you were unlucky. Money doesn't care, more often than not it will flow towards those who make good decision, and some periods of time flow away from them, but overall if you do not make good decisions including bankroll management you will end up with less money. You can't afford to let ego get in the way so if your money says the right decision is to decrease the amount you risk, you should gear down.
Often times the way that seems like the fastest way to get rich quick ends up being the longest, as it will inhibit your ability to develop and grow when the chances fail, and when the chances succeed it will not bring you back to the same trajectory to overcome past and future failures as the result of trying to take this short cut.
In fact, the unprofessional player is less dedicated and therefore less prepared (even if they have natural talent) so they will have a smaller edge. While the non-professional may need experience to develop that edge, he needs to learn the fundamentals which requires sufficient play against enough bad players at stakes that are enough to matter, but not so much that emotion rules over logic. He also needs to not only survive long enough to develop that edge, but also maintain the ability to not have to move down in stakes or play less often.
However, if you have a separate income, and don't play as often there is SOME room for you to borrow from your future income and play in higher tournaments than your bankroll. (Please note: This is not done by going into debt, one should never practice debt as that will eliminate any bennefit having an edge has).
For example, if every weekend you like to play a little poker, and your excess cash after expenses is large proportionally to your bankroll, you may be willing to risk a bit more as long as:
1)You already have 3-6 months worth of expenses (some suggest more) on the side.
2)You don't have any outstanding debt.
3)You plan on playing for years
4)Playing at a particular stake this month won't prevent you from playing at the same stake next month without violating rules 1-3.
So if you earn $1000 a month after expenses and want to use all of it playing poker every month for 10 years you might say your future bankroll discounted for the future is maybe $50,000* (plus whatever you have as your bankroll now not including what's saved for other purposes).
(*Money now is worth more than money later, and money which requires risk must be discounted for volatility, and money used for this must be discounted each month for value that you must pass up later so rather than 10 years, you might only use 4 years worth after factoring this all in)
If playing at a particular stake prevents you from playing at that stake the next month, playing higher stake still inhibits your ability to save enough to move up, and earn something from poker (higher stakes are typically less lucrative in terms of profitability expressed in big blinds). If you want to play for say 10 years worth, take about 40% of the excess earnings over that period. Use the lesser number of either the average of what you made the last 3 years vs what you are making now.
There is a "bubble" bias of people having the most to spend right before a bubble. So it's possible you may overestimate your future earnings dramatically. Take monthly income times 12 to get yearly, then yearly excess times the number of years (in this case 4). This is your future bankroll in 10 years after being discounted. SO if it's $200*12*4=$9,600. Now 2% of that is $192.
So if you REALLY want to enter a particular $150 tournament you can borrow from your future bankroll. It comes at a pretty steep cost, but if you give yourself enough time to recoup the damages before you start again, you can give it a shot. Chris Ferguson went from 0 to $10,000 in about a year and most of it was spent at the 1cent 2cent table. That proves it is possible to be incredibly intelligent with your bankroll and still earn significant amounts, but he also took 8 of those 13 months grinding the very low stakes and freerolls and that doesn't mean it has to be done that way. He also played a few more hands than most will get to. He also earned his way to prove he is competent at higher stakes first after studying applied mathematics for nearly a decade and applying that work to poker so he knew he had a major edge.
It also certainly would help if you are FIRST able to prove your track record and profitability at a smaller stake for thousands of hands and then move up. Play is much worse live but you can compare the micros to the low stakes live and $2/$5 or $5/10 to $0.50/$1 online.
Daniel Negreanu on bankroll management and a foolproof plan to become a professional poker player.
Chris Ferguson 0 to 10k challenge.
Since the initial challenge you can see the huge downswing he went on which is totally possible. He had to go back down to the very low stakes again and he had the discipline to. He lost 97.5% of his bankroll when he dropped from 40k to about $1,000 again. His rules are not as conservative as you might think particularly for someone who does it professionally and depend on some of it for income, and especially as you rise in stakes and your edge declines. Many players just haven't really had a bad enough downswing to realize what's actually quite common over a long enough period of time and will happen to just about everyone at some point if they play enough hands.
This is why money is only a PROBABLE side effect of correct decision and it comes at the expense of a possible side effect of losing enough to want to throw up at even what many consider "conservative bankroll management". How well will you play when that happens? If you are focused on the result of money, you will tilt off and lose it all when you run badly. If you focus on making the correct decisions, and don't pay a lot of emotional attention to the money as best as possible, the money will eventually come over time. The slower path is ironically probably the more likely one to actually get to where you want to go. Study the game, EARN your advantage and go out and use that advantage over a large number of hands and continue to grow your advantage.
If you want to make money in poker it is possible, but it should never be the goal. Money is only the possible result. It is only a probable side effect of making good decisions. Having fun is a side effect of playing well and making money. Therefore the goal of poker is to make correct decisions. Money is the probable side effect of correct decisions. Let me repeat for emphasis. The goal of poker is to make the correct decisions. If you succeed in this goal, the probable side effect will be more money.
So if opponent is tight and plays JJ+,AQ+ in a spot, your Ace King or QQ is no good, fold. If opponent plays 88+,AQ in a certain spot your AQ suited is no good and your ace king suited is a coinflip and ace King offsuit is actually slightly behind. On the other hand, these hands can still be called if you are getting the right price so you win say twice the amount you risk. But each of these decisions require a lot of knowledge about what's good in certain spots or at least a good feel based upon doing the math. Experience at the table will never get you enough repetition to know with as much precision as those who do put in the work.
Expected Value
Good decisions are quantified as decisions that after examining all possible choices result in a positive expected value or a positive EV which is often denoted as +EV. But they also must include bankroll decisions.If you flip a coin and you lose $1 when you win, and lose $1 when you lose the expected value is ZERO. This can be calculated as probability times result plus probability times result.
(.50*1)+(.50*-1)=(.50)+(-.50)=0
Positive expected value calculations determine when the probable side effect over the long run is more money assuming good bankroll management. It can be calculated by the sum of the probability of each result multiplied by the corresponding result.
So let's say you have AQo and opponent only has one of a few cards. A pocket pair of 88 or better or AQ and he always will play any card in this range. You are 39.55% to win vs this range. Intuition should at least get you to estimate close to this number if you have done the work. say 40%. If your opponent has moved all in for $40 into a pot of $10 after you raised $4 and everyone else folds, should you call? You have to call $30 more. You win $50 when you are right, lose $30 when you lose. You are getting 5:3 or 1.667 to 1. This means you need to win at least 1/2.667 or 37.5% to win after rake in order to call.
Your expected value is about:
(.40*50)+(.60*-30)=$2. Assuming your raise attempt to risk $3 was a break even decision, the result of both raising and then calling profits $1 less than it would in situations when you get callers or opponents to fold but it profits $2 more than folding. In some cases, you make an unprofitable decision to raise in the first place, so the EV calculation of +$2 is actually only gaining back some of what you lose from raising. In other words if you always are raised and usually fold, you usually lose $3, but in this one instance, you are able to gain $2 of that $3 you usually lose back. So just because this is "+EV" or better than folding, doesn't mean the decision to get into the spot was necessarily profitable*.
This point of emphasis is made so you understand that the EV of any particular decision is often connected with the prior decisions. You can make a correct decision in one spot, but be incorrect to get into that spot in the first place.
*(To be clear, it is unlikely that raising Ace Queen is ever a mistake but if a hand like A2 against a reckless opponent yielded $2 to call their all in while the individual mistake after you raised may make money in most cases you usually will fold to everyone else's raise and usually you lose money so that $2 may not be enough to make up for where you lose a portion of that $3 you risk initially).
Every decision should be analyzed including what hands you play preflop.
Over the very long run with consistent bankroll management, your bankroll is the starting bankroll (and any additions or subtractions to it) plus or minus the sum of the EV of all decisions you make.
If 50 hands yield $1 each but 11 hands after that yields -$5, you may be capable of being a winning player, but overall you will not be because of tilt being the biggest leak in your game.
Unfortunately, break even decisions while in "expected value" are neutral, in terms of bankroll decisions, they are negative. Even positive decisions with poor bankroll management can lead to the probable outcome over a long period of time being losing money. The goal therefore cannot be solely based upon "expected value". It is vital to understand why.
The reason break even decisions lose money is because any bet has both the possibility of having a positive result and a negative result. However, several negative results strung together require a disproportional number of positive results to break even. To illustrate this start with $100 and imagine losing 50% then gaining 50%. Think you are back to even? Think again. A 50% loss from $100 takes you to $50. A 50% gain from $50 only takes you to $75. So a break even proposition has actually cost you 25% of your bankroll given this amount of volatility over this amount of time.
You may incorrectly deduce that you'll just increase the amount as you lose. But this doesn't solve the underlying problem is that you now have the same probability of an even greater loss, such that the gain once again is not enough to offset the loss. The only possible solution other than getting money from other sources (and even then sometimes it's not enough) is risking less to reduce volatility or to dramatically increase your edge so that type of drawdown doesn't happen and the gains are skewed to the upside large enough to overcome them. It's very uncommon to have that great of edge, and those with a great edge tend to be overconfident even more so and risk their bankroll more liberally than they should to their own detriment.
Money Management
Fortunately as we will illustrate below and explain later, risking 1/4th as much in the worst case scenario will still produce 40% of the results (in the long run of infinity). In the best case scenario you will go from a losing player to a winning player simply by having less to overcome on your downswings. You will also have a much greater probability of a positive outcome over a fixed period in time, and 1/4th of the volatility and emotional stress.
Greater volatility over longer periods of time will cost a fixed bankroll to go to zero, and a growing bankroll (such as a bankroll derived from excess income at a job) will continue to over time decline. Certainly it could go the other way. You COULD gain 50% first to $150 and then lose 50% to $75. The result is eventually the same over infinite time horizon. Timing it, or varying your bet size after you lose isn't going to compensate for a losing philosophy of playing a losing game.
This is what happens if you risk over 2 times the "Kelly"
However, respecting that you don't have an infinite time horizon is more reason to bet less if you are looking for the probable outcome of more money. Over a fixed time horizon the supposed "optimal" amount to bet is actually way too much unless you want a "lottery type" distribution where a very very very small minority ends up with a very large return and a large majority end up at a poor or negative return, while the upper 25% may have satisfactory returns.
Betting less instead would produce a satisfactory return maybe 40% of the time but without the absurdly unlikely outliers that will never actually occur to any one individual in the history of the next 100 generations to any one of billions attempting it. I don't think people realize that you can pretty much rule out the possibility of the "positive outlier event". The length and impact of the "infinite time horizon" cannot be overstated enough. Having the duration of time of the entire universe and all life forms ever in history of the universe would be less significant to infinity than the size of an atom to the size of the universe.
Even though it's a stock trading website, there is plenty of useful information in understanding modeling risk with a massive fixed, known edge at various risk levels produces distributions of results over a fixed time period. More about Risk Management here.
At the extreme, not even a single pixel can represent the difference between the number of people achieving outlier results and zero. At 10% most result in losing over 90% of their bankroll maybe 0.5% are outliers and 0.01% produce an average of an amazing return. At 20% risk 999.99 go effectively busto. Dream on.
You can see that with starting bankroll of $1000, larger amounts of risk produces a larger and larger percentage of people that actually lose money over 300 bets. It's difficult to really see what the actual percentage is, but the actual numbers wouldn't apply anyways. The concept that nearly ALL results effectively go bankrupt when they are betting closer to the supposed "optimal" amount to maximize long term growth goes to show you that even betting 1/3rd of this amount is risky.
In theory you might be able to eventually get your 10 cent bankroll up to a million when the 1/1000 outlier actually hits a series of several wins and eventually all those major outliers on either ide even out to the average. But we are talking about "quattroduodecatrillions" of hands until that happens and we are also talking about playing at stakes using fractions of a penny which isn't reality. Or if you have an income of $1,000, you are just going to reload and lose it all every single month or so and maybe over 1000 months or perhaps 4000 before you finally hit that one outlier month in which you pay for all the months before it and then some. The problem is a human life span is only 960 months, and only probably 480 of it if you lived and breathed poker could be spend playing poker. And how boring of life is that to be broke for most of your life?
What's worse, is you can never really retire with this strategy or play poker for a living with a high amount of risk if you don't have some income elsewhere. In order to really play poker for a living, you're going to have to play at absurdly low risk percentages such as 1/30th of the "optimal bet size" at risk at any one time (we're talking like 0.20% of your bankroll at risk) and have so much excess separate from your bankroll saved on the side that your probable result is centered right around break even anyways and most likely you are just withdrawing from 2 years worth of savings, but the overall "average" with 0.20% of your bankroll is theoretically enough to live off of if you made it each and every month with certainty.
So if you made $50 an hour at $2/5 live games buying in for $500, you'd need a bankroll of more than $250,000. Well, maybe earning $10 an hour is enough at $0.50/$1.00. You'd need $50,000. net worth. If you always had a way to make ends meet if you absolutely had to and were okay with having a risk of ruin and living life in fear of going broke, you could use the standard measurements of risk management. Maybe you are okay with knowing that you can always sell your house or take out a mortgage if you have your home paid off. Either way there are very few people in the world with
1)The skill to play poker
2)The understanding of bankroll management
3)The willingness to grind at less conservative stakes as a side job until capital has been built up
4)The discipline to stick to insanely conservative bankroll management strategies
5)The passion for the game to play for thousands of hours each year.
6)The discipline to not spend their earnings at at a higher rate than their "expected" earnings.
Realistically there are plenty of people that can earn professionally if they have sources to be staked and endorsement deals and/or business income, or book deals and poker coaching income, but the days of easy endorsements and free capital in the poker community is gone and in a bad economy your edge will likely go from positive to negative and the difficulty of games and easy money floating around will evaporate.
Furthermore, if you're an online player, you'd have a higher average and lower volatility multitabling at 2% risk with 5 tables than 10% risk at one assuming the edge was the same so risking more has no practical advantage in any normal form.
The principal of volatility hurting more than it helps is true even with an edge if you do not manage bankroll properly. While some people may chime in about compound interest and how several wins in a row will compound as well, that may be true to a very small degree, but the benefit is smaller than the drawback of risking more. It is not worth it and you will not benefit to a strong enough degree to compensate for the cost of volatility if you risk too much. The amount you can risk with a break even proposition to have positive results is zero. Not to mention it's not worth your time.
The casino will always have a larger bankroll than you, and if they don't, they will kick you out because they don't want that kind of risk. Phil Ivey was kicked out for playing craps, a game in which he had the worst of it from very small casinos because he was placing very large 100k bets. The casinos don't gamble, they play mathematical certainties over enough times so that they make money, and they didn't want to allow to put their entire operation at a risk of ruin. They will turn down your "business" if either they know you have the best of it at a small stakes or you are playing at such large stakes that their "probable outcome" is not more money over a very long term.
For similar reasons, decisions with positive EV can still have negative long term expected growth rates. That may sound like a contradiction, but it's not. The "Kelly criterion" is like "EV" but for your bankroll over an unlimited time horizon. It already compensates for the fact that a 20% drawdown in your account requires a 25% growth rate to get back to even and a 50% drawdown requires a 100% gain. But realize that over an infinite time horizon a 50% loss can eventually be recovered, even if for all practical purposes it cannot easily be in real life in a finite timeframe.
While an individual decision repeated for the same bet amount an infinite amount of times may be positive, repeated as a percentage of bankroll or within the realistic parameters of actually having a limited bankroll, the long term result is losing money.
When Winning With Bad Management Still Results In A Loss
Take for example a winning poker player who wins 10% of all tournaments. 10% of the time he wins an average of 20 times his buy in. Clearly this is tremendously profitable in terms of EV. Let's calculate how much in terms of buy in after 10 tournaments. (1win*20buy ins)+(9wins*-1buy ins)=11 buy ins profit every 10 or gain over cost*100 is 11/10=1.1*100 110% net ROI.However, the odds of going 6 tournaments in a row without a win are greater than 50% .9^6=53.14% If this player risks 100% of his bankroll he has a 90% chance of going bust. If he risks 50% of his bankroll, EVEN if he adjusts his bet every tournament so he only bets 50% of his remaining bankroll after each loss, he has an over 50% chance of having less than 2% of his bankroll before he wins. When he finally does win, he will be doing so at such small stakes that 20 buy in victory won't even take him remotely close to where he was. As such, he has a very good chance of not even being able to afford a low stakes buy in with this plan.
The theory of "adjusting downward" as you lose is reasonable, but only if you start low enough to begin with. And a fixed percentage at an amount between the "adjusting downward" amount and the starting amount probably has the same impact of adjusting downward with less volatility and more consistent gains. A "fixed percentage" actually does adjust upwards and downwards since it's a percent of your bankroll.
The theory of increasing your risk as you lose is unreasonable as this increases your chances of losing such a greater amount that you won't even have enough to continue even if you have Bill Gate's credit line, and if you did, you'd have to start at such a small bet that it wouldn't even be worth your time and you'd still put unreasonable risk of losing everything. Take a coinflip. Risk a $5 and double the risk 35 times or so and you'd lose all $85 billion of Bill Gate's bankroll. Gain $5 35 times and you've gained $175. Is $175 worth risking $85 billion? Is it worth the time? This strategy is guaranteed to lose eventually it is not nearly as logical as you'd like to wish it was.
You cannot change how you "time" the amount that you bet or to turn a profitable decision or unprofitable decision to be more profitable or less unprofitable. You can only influence the probability of a gain at the expense of the downside or probability of a huge loss. The best decision is to sit with a fixed percentage of your bankroll or lower than that fixed amount, and to determine which "fxied amount" suits your goals.
Fortunately there is a solution to increase the results if you are a winning player, and it involves understanding the nature of risk.
There is an amount to bet which over bets that approach an infinite amount, your value is maximized. Not too much, not too little... Just right. However, as you approach this "optimal" point the benefits for increasing your bets have greater and greater risk at a smaller and smaller reward.
Consider risking half the kelly has 3/4ths the return with half the volatility. This means you recover from your all time highs in bankroll much faster and make new highs much faster on average, although the upswings will not be as large and the overall trend may be slightly less strong. Risking 1/4th the kelly still has 40% of the return as the full kelly with 25% of the risk. In other words, risking less is almost always better from a practical risk/reward perspective.
This is only what risk looks like given CERTAIN outcomes. With uncertain outcomes over an infinite amount of time there is risk of betting far too much. If you bet over 2 times the kelly the eventual result is losing everything.
As such, there is no reason to overbet the kelly at all.
Over FIXED time horizon, even as long as 10 years, everything changes. Betting the full kelly over a fixed time horizon results in a probability distribution to explain the results. Out of 1,000 players who play a kelly criterion strategy a very small number of them will have extraordinary results, while the majority will have below average results and the upper 25% may have above average results but it will be skewed by the top 1%. The bottom 25% will produce break even or even a loss, while the bottom 25% (larger or smaller depending on the number of profitable "bets") will have lost 90% of their bankroll.
Since this essentially amounts to a lottery ticket chance of having enormous gains that are far beyond what you could ever spend, there isn't a lot of added value to taking on a "Kelly" worth of risk. 1/5th of the Kelly is acceptable to some, but probably not if you have to withdraw from your bankroll at regular intervals without some other means to live if you go through downswings.
However, risking less than 1/5th of the Kelly will produce both a high probability of very good results and good upside. There still will be a chance of pretty spectacular results, and still a small chance of poor results, but overall the outcome will be likely to be pretty good. There will still be a skew. Because the probable outcome is more money than less, THIS is a correct decision, and money is the probable side effect.
This amount of risk requires examination of your skill edge. That isn't something that is entirely certain so there is a slight "gamble" on your own skill. Or if you are playing an "equilibrium" strategy there is a gamble that others will be bad enough that you will beat the rake (which is a very good assumption) and that your mistakes will be small enough (playing without mistakes even with a fairly basic predetermined strategy is difficult and has mental and psychological challenges).
There is a reason that pros like Chris Ferguson and Phil Gordon and others have never gone broke and also live comfortable lives with poker playing at one time still being a reasonably significant part of it. Aside from having skill, they follow a set of bankroll management guidelines and very solid bankroll strategies. Since it is difficult to distinguish playing poorly with being unlucky and it is easy to convince yourself in defense of the ego that you were unlucky. Money doesn't care, more often than not it will flow towards those who make good decision, and some periods of time flow away from them, but overall if you do not make good decisions including bankroll management you will end up with less money. You can't afford to let ego get in the way so if your money says the right decision is to decrease the amount you risk, you should gear down.
The Non-Professsional Player
If you aren't a pro player and you don't want to make a living playing poker, but still want to give yourself a chance at getting rich, should you still care? Absolutely.Often times the way that seems like the fastest way to get rich quick ends up being the longest, as it will inhibit your ability to develop and grow when the chances fail, and when the chances succeed it will not bring you back to the same trajectory to overcome past and future failures as the result of trying to take this short cut.
In fact, the unprofessional player is less dedicated and therefore less prepared (even if they have natural talent) so they will have a smaller edge. While the non-professional may need experience to develop that edge, he needs to learn the fundamentals which requires sufficient play against enough bad players at stakes that are enough to matter, but not so much that emotion rules over logic. He also needs to not only survive long enough to develop that edge, but also maintain the ability to not have to move down in stakes or play less often.
However, if you have a separate income, and don't play as often there is SOME room for you to borrow from your future income and play in higher tournaments than your bankroll. (Please note: This is not done by going into debt, one should never practice debt as that will eliminate any bennefit having an edge has).
For example, if every weekend you like to play a little poker, and your excess cash after expenses is large proportionally to your bankroll, you may be willing to risk a bit more as long as:
1)You already have 3-6 months worth of expenses (some suggest more) on the side.
2)You don't have any outstanding debt.
3)You plan on playing for years
4)Playing at a particular stake this month won't prevent you from playing at the same stake next month without violating rules 1-3.
So if you earn $1000 a month after expenses and want to use all of it playing poker every month for 10 years you might say your future bankroll discounted for the future is maybe $50,000* (plus whatever you have as your bankroll now not including what's saved for other purposes).
(*Money now is worth more than money later, and money which requires risk must be discounted for volatility, and money used for this must be discounted each month for value that you must pass up later so rather than 10 years, you might only use 4 years worth after factoring this all in)
If playing at a particular stake prevents you from playing at that stake the next month, playing higher stake still inhibits your ability to save enough to move up, and earn something from poker (higher stakes are typically less lucrative in terms of profitability expressed in big blinds). If you want to play for say 10 years worth, take about 40% of the excess earnings over that period. Use the lesser number of either the average of what you made the last 3 years vs what you are making now.
There is a "bubble" bias of people having the most to spend right before a bubble. So it's possible you may overestimate your future earnings dramatically. Take monthly income times 12 to get yearly, then yearly excess times the number of years (in this case 4). This is your future bankroll in 10 years after being discounted. SO if it's $200*12*4=$9,600. Now 2% of that is $192.
So if you REALLY want to enter a particular $150 tournament you can borrow from your future bankroll. It comes at a pretty steep cost, but if you give yourself enough time to recoup the damages before you start again, you can give it a shot. Chris Ferguson went from 0 to $10,000 in about a year and most of it was spent at the 1cent 2cent table. That proves it is possible to be incredibly intelligent with your bankroll and still earn significant amounts, but he also took 8 of those 13 months grinding the very low stakes and freerolls and that doesn't mean it has to be done that way. He also played a few more hands than most will get to. He also earned his way to prove he is competent at higher stakes first after studying applied mathematics for nearly a decade and applying that work to poker so he knew he had a major edge.
It also certainly would help if you are FIRST able to prove your track record and profitability at a smaller stake for thousands of hands and then move up. Play is much worse live but you can compare the micros to the low stakes live and $2/$5 or $5/10 to $0.50/$1 online.
Daniel Negreanu on bankroll management and a foolproof plan to become a professional poker player.
Chris Ferguson 0 to 10k challenge.
This is why money is only a PROBABLE side effect of correct decision and it comes at the expense of a possible side effect of losing enough to want to throw up at even what many consider "conservative bankroll management". How well will you play when that happens? If you are focused on the result of money, you will tilt off and lose it all when you run badly. If you focus on making the correct decisions, and don't pay a lot of emotional attention to the money as best as possible, the money will eventually come over time. The slower path is ironically probably the more likely one to actually get to where you want to go. Study the game, EARN your advantage and go out and use that advantage over a large number of hands and continue to grow your advantage.
Monday, June 22, 2015
Optimal Preflop Poker 3: Optimal Preflop Poker With Antes
See Optimal Preflop Poker 1 and Optimal Poker 2 first.
Once the antes are involved, our raises tend to only be around the size of blinds plus antes (2.25x) rather than twice the blinds (3x). As a result, collectively opponents should raise us 50% of the time to prevent us from raising with any two and force our steal attempts to break even. As such, we need to have value over these hands.
How much value? Really, we only may need to call a small raise or 3bet a small amount proportional to the pot to take it down. Even if our opponent puts in 3 times our bet for 7.5 big blinds, we will only have to call 5 more big blinds into a pot that will be inflated to 17.5 after both our opponents and us call. So we only need 28.6% equity for a call to be better than a fold, once we are raised.
If we are in raise or fold mode, 3x our bet is about 1.5 times what's in the pot. To prevent our opponents from breaking even we must be able to raise 40% of the time. In other words we can multiply our "value" hands by 2.5 to form our initial steal range. Because of the antes and the equity both us and our opponent gain on the backs of those that fold, we actually can increase this more. However, we will not try to calculate this just yet.
If we are getting 1:1, our opponents need to raise us 50% of the time to force us to break even on our steal attempts.
Note:If they just call, they are forced to call even more often or make a raise at some point in the future more often or mix in a wider range of call with some mixture of raises this amount or tighter.
That translates to the following hand range given our raise started with X players remaining.
8 8.3% 88+, AJ+,ATs+,KTs+,QJs
7 9.43% 88+, AJ+, KQ, A9s+,KTs+,QJs,JTs
6 10.91% 66+, AJ+, KQ, A9s+,KTs+,QTs+,JTs,T9s
5 12.95% 66+, AJ+, KQ, A9s+,K9s+,Q9s+,J9s+,T8s+,98s,87s,76s,65s,54s
4 15.91% 66+, AJ+, KQ,QJ,JT, A8s+,KTs+,QTs+,J9s+,T8s+,97s+,87s,76s,65s,54s
3 20.63% 55+, AT+,KT+,QT+,JT,A7s+,KTs+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
2 29.29% 22+, A9+,KT+,QT+,JT,T9,98,87,76,65,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
1 50% 22+,A2+,K4+, Q8+, J9+,T8+,98,A2s+,K2s+,Q2s+,J5s+,T6s+,96s+,85s+,75s,64s+,54s.
So now what hands have equity over them? If you look at the pot odds after a reraise and you call, you are putting in 7.5 to win 17.5 and that means you need 43% equity assuming it's checked to the river. If you look at hands that have 33% equity over that range since you are getting 2:1 on your calls, then you only need 33% equity vs the calling range. Although once you put in the raise there are hands with more than 28% equity but less than 33% equity that would profit more than folding on calls, that is only attempting to salvage a portion of what already was risked with the steal attempt and it would be more ideal to have never raised at all with those hands. As such, if anything that should only allow you to widen your steal range because when raised you don't always lose the full 2.25 you risk even when your hand may not have value.
Since we don't know exactly what the optimal percentage of raising vs just calling is and how that manifests in equity postflop we cannot know the exact equity one should have since it's impossible to know just how often opponent should call or raise and how often we should call or raise when opponent raises. However, I am very confident the ideal range for when we have "value" vs a raiser is somewhere between 33% and 45% equity.
These numbers need to be much higher with "trouble hands" or "reverse implied odds" hands such as high cards with no drawing potential or weak kickers and drawing potential in an aggressive game, while they retain more of their value in passive games. Suited connectors and draw heavy hands can actually play much stronger when you are the aggressor and have implied odds. In other words, tend to play a lot more suited connector type hands and fewer weak kings and queens and weak aces type hands than the "optimal play" actually suggests.
We needed a way to be completely objective AND approximate optimal play, so we didn't account for actions beyond the flop and assumed all bets break even and are checked to the river. In reality, you can ignore that last bet unless it's break even or better when you are betting on a draw, where you can get paid off when you hit, and you tend to lose that last bet with the one pair, weak kicker hand and get in trouble with opponent has you outkicked or plays to beat one pair.
But I digress.
Opponents may call and defend blinds often enough to minimize the amount gained from our steal attempts in addition to raising occasionally. This probably requires us to have closer to 45% equity and as a result, a tighter strategy is better. However many times both the players in the pot win at the expense of the folders, which favors us forcing the action more often which somewhat counterbalances the need to be tighter to compensate for optimal blind defenders, with allowing looser action to still be profitable. Having positional advantage and postflop edge may allow for more hands, but it simultaneously doesn't require you to play as many to do well and in some cases in tournaments chip accumulation at lower or no risk is better than more chips at higher risk. Since I haven't calculated the optimal rate of defending the blinds, I can't factor everything in perfectly.
I do know that the optimal solution really will allow blinds to defend very liberally (especially for only a 2.25x big blind bet) by calling often. This means our steal attempts won't win the full pot but instead a fraction of a slightly larger pot which translates into slightly less than the full steal vs optimal opponents. However, the optimal solution also will require them to defend pretty liberally after the flop which may allow us to gain quite a bit of chips a high percentage of the time when they don't
So I'm not sure but since you never need to play "optimal" preflop and opponents will never be "equally skilled postflop" plus position does matter this may not be that far off. You also should in reality really consider the opponents and postflop actions and other variables. This is just a baseline for thinking about decision making and understanding how the antes effects play.
So let's provide two guidelines with the optimal play likely somewhere between:
These hands have 33% equity over initial hand range of opponent. (Every single hand in this range is individually profitable vs opponent's range if opponent just calls and checks to the river)
vs range of 3bet defender
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
These hands have 45% equity over 3bet range. (Will be profitable if opponent occasionally raises and we call and hands are checked to the river even if steal attempts from bluffs are break even).
45% equity:
8% range: TT+,AQ+,AJs (5% of hands)
9.5% range:TT+,AJ+,ATs,KQs (6.5% of hands)
11% range:88+,AJ+,ATs+,KQs (7.4% of hands)
13% range:77+,AT+,KQ,A7s+,KTs+ (11.2% of hands)
16% range:77+,A8+,KT+,A2s+,K9s+,QTs+ (16.3% of hands)
20% range:66+,A8+,KT+,A2s+,K9s+,QTs+ (16.75% of hands)
30% range:44+,A4+,K8+,Q9+,JT,A2s+,K5s+,Q7s+,J8s+,T9s (30% of hands)
50% range:33+,A2+,K7+,Q9+,JT,K4s+,Q8s+,J9s+,T9s (33% of hands)
In either case, we can widen our range considerably compared to non ante stages. That's not even including the "bluff raises" as our steal attempts should break even vs optimal opponent.
If all these hands have "value" vs a hand opponent plays, we should add in bluffs such that we have a "value" hand 40% of the time. That means multiply the hand range by 2.5.
For example, take the 33% equity.
Total hand range:
8:39.59%
7:69.76%
6:88.61%
5 or less players remaining: over 100% or Any Two Cards.
This is approximately:
8:22+,A2+,K8+,QT+,JT,T8+,98,K3s+,Q7s+,J7s+,T7s+,96s+,85s+,76s,64s+,54s
7:A2+,K2+,Q2+,J4+,T6+,96+,86+,J2s+,T2s+,93s+,84s+,74s+,64s+,54s+,43s
6:22+,A2+,K2+,Q2+,J2+,T4+,92+,84+,74+,64+,54,T2s+,92s+,82s+,72s+,62s+,52s+,42s+,32s
5 or less: any two.
OR
5:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
4:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
3:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
2:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
1:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
Button, cutoff, hijack, and even middle position raises any two.
I only multiplied the initial 45% equity range by 2 since I assumed this strategy was probably for a tighter strategy against larger reraises or opponent who bluffs more often and wanted the so called "conservative estimate" in which we want the true optimal strategy to fit somewhere between these two. Also, opponents who 3bet may 3bet with odds to keep you in, but if you choose to fold, you need a 4bet hand more often. So this leads to the following hands as the "tighter end" of correct play during the ante stages.
8: 66+,AJ+,ATs+,KTs+,QTs+,JTs (10%)
7: 44+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s (13%)
6: 33+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s (14.8%)
5: 22+,A8+,KJ+,QJ,JT,A2s+,K9s+,Q9s+,J9s+,T8s+,98s (22.3%)
4: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s (32.6%)
3: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s,65s (33.5%)
2: 22+,A2+,k2+,Q6+,J7+,T7+,97+,87,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,54s,43s (60.03%)
1: 22+,A2+,k2+,Q2+,J5+,T7+,97+,86+,Q2s+,J2s+,T2s+,94s+,84s+,74s+,64s+,54s,43s (65.8%)
If multiplied by 2.5 instead:
8 12.44%
7 16.21%
6 18.48%
5 27.90%
4 40.72%
3 41.86%
2 75.04%
1 82.20%
That translates into roughly
8 55+,AT+,KQ,A9s+,K9s,Q9s+,J9s+,T9s
7 33+,AT+,KJ+, A5s+,K9s,Q9s+,J9s+,T9s,98s
6 22+,A9+,KJ+, A4s+,K9s,Q9s+,J9s+,T9s,98s
5 22+,A4+,KT+,QJ,JT A2s+,K8s,Q9s+,J8s+,T8s+,97s+,87s
4 22+,A2+,K9+,QT+,JT,T8+,98,87,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
3 22+,A2+,K9+,QT+,JT,T8+,98,87,76,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
2 A2+,K2+,Q2+,J2+,T5+,96+,85+,75+,64+,54,T2s+,93s+,84s+,74s+,63s+,53s+,43s,
1 A2+,K2+,Q2+,J2+,T3+,94+,84+,74+,63+,53,43, any suited
Because the blinds are correct to defend more liberally and they will slightly eat into your profit, you can probably raise less often.
However, from an exploitative perspective, and because we don't know quite how often opponents should raise and they sometimes should call, if opponents just call, rather than raise, you can widen the ranges.
If you prefer not to 4bet light, and you don't want to be quite as much of a target, you probably can get away with raising much less often, particularly if opponents fold and you will still be able to accumulate chips. That will allow you to fold a lot more to aggressive 3bets and 4bets and avoid all ins. You may be able to win a tournament without ever having to show down a hand or ever being all in which supports cutting back on your profitability and loose play in exchange for lower variance and less risk, and also not being quite as much as a target by stealing less often.
In either case, we can dramatically widen our raise range but as a result our opponents should have a pretty wide 3bet range and we in turn will need a pretty wide 4bet range. One problem with this is it assumes stacks are infinitely deep meaning there is no all in move. Since there is, this strategy plays too many hands when a 3 or 5bet all in is perfect stack size for shoving and too few when a 4bet or 6bet or 2bet shove is perfect stack size. Eventually we no longer have equity over opponent and eventually we have to either call and gain whatever equity remains or fold. Also, because of post flop decisions and it being very unlikely the hand will play to the river, we need to start out much stronger if we want to continue to the river a high percentage of the time that we play.
The 33% equity "value" range also works for flat calling raises from the button. Since we may be squeezed this could be slightly tighter, particularly if a squeeze play is likely going to be all in, but we could potentially play very loose on the button vs raises because we only need 33% equity vs the initial raiser and we have position.
Another decision we may make is facing a 2x BB raise. Although game theory solution is to 3bet often enough to force opponent to break even, we can also look at when we have equity to call assuming the blinds fold.
33% equity to flat on button when facing X% raising range:
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
We can see that even against tight opponents, it's correct to have a very wide range when calling. However, this range should be tighter because blinds will not always fold, players will not check until the river, and we occasionally will get "squeezed" by the blinds and have to fold. Also, our opponent's hand ranges are not "known", and there are "reverse implied odds" for a lot of these weak kicker hands.
Nevertheless, the concept of occasionally calling on the button can never be that bad, and if we are likely to get squeezed, that favors just calling with premium hands occasionally too. Even though we may let our opponents outflop us or get money in and outdraw us sometimes giving up that risk for value and balance and for the potential to resqueeze shove with aces or getting opponent's entire stack when he hits top pair or decides to continue with a low pair or ace king on low card flops may be worth it.
Since it's mathematically such an advantage to play lots of hands and call, and since opponents do not always adjust, the ante stages is THE MOST profitable stage of the tournament. We want to both maximize our chances to get to the ante stages while also trying to get there with enough chips to be able to make a handful of failed raises and still have over 20 big blinds.
There may be some tradeoff between giving up some probability of getting to the ante stages, and ensuring that when you do get there you have enough chips. Every decision to play or not play a hand may cost us chips. Costing us chips may force us into future situations where we are more desperate which could cost us more chips. So any decision could hurt our chances of getting there. usuaully the decision of playing a hand decreases our chances of getting there more than not playing a hand, but usually not playing a hand decreases our chances of having well over 30 big blinds when we do actually get there. But having more than 30 big blinds is not really necessary since you can clearly get a lot more chips from 30 big blinds, and still be able to raise preflop, Cbet the flop and still have chips left over to do it again. I'm not convinced that having more than 30 big blinds really adds a lot of utility, beyond giving you a few more shots at a particularly level of utility and perhaps sustaining it for maybe one more level of play which you may be able to parlay into two more levels of play. However, you can parlay a tight image into additional value in future levels of play as well and tend to get there more often, so those effects are not really that significant if you know how to use a tight image.
Certainly there is a lot of missed opportunity by not having more than 20 big blinds in the ante stages. However, even with 15 big blinds it's probably only a matter of time before you double up. You can increase your stack significantly with a move in or two to maintain or even grow your chips. Once the double up happens you probably have a really tight image, a lot more chips, and the blinds may be much higher so it probably still is worth quite a lot and can nearly be made up for particularly AFTER a double up with a monster hand when opponents are now forced to recognize how tight we've been, then we can go to work with much less resistance. Even if you are still shortstacked after playing this patiently, you can shove a few times in a row after the double up then scale back for a rotation or two and then do it again. Then you can play ultra tight again and before you know it you pick up a big hand again and no longer have the tight image until you're called again. You win 2 of these spots where you are probably on average of 60-80% to win and now you have the ability even in a fast structure to win with only a few all ins all tournament and in a slow structure, you may be able to avoid ever being all in.
So being all in twice and called with 60-80% to win is not really all that different from being all in once with the 40-65% chance to win earlier on and parlaying it into a lot of chips in smaller pots early.
Either way, we have to recognize that it's profitable like no other to play a much crazier strategy than you'd ever play in a cash game. If there tends to be a lot of pots that make it to the turn and river with lots of bet, a lot of the preflop edge may be neutralized and the equity will have to be higher and a bit closer to the no ante stages. But when opponents start playing more hands, we then can profitably be aggressive to attempt the limp raise or squeze play which is another story to cover later.
Once the antes are involved, our raises tend to only be around the size of blinds plus antes (2.25x) rather than twice the blinds (3x). As a result, collectively opponents should raise us 50% of the time to prevent us from raising with any two and force our steal attempts to break even. As such, we need to have value over these hands.
How much value? Really, we only may need to call a small raise or 3bet a small amount proportional to the pot to take it down. Even if our opponent puts in 3 times our bet for 7.5 big blinds, we will only have to call 5 more big blinds into a pot that will be inflated to 17.5 after both our opponents and us call. So we only need 28.6% equity for a call to be better than a fold, once we are raised.
If we are in raise or fold mode, 3x our bet is about 1.5 times what's in the pot. To prevent our opponents from breaking even we must be able to raise 40% of the time. In other words we can multiply our "value" hands by 2.5 to form our initial steal range. Because of the antes and the equity both us and our opponent gain on the backs of those that fold, we actually can increase this more. However, we will not try to calculate this just yet.
If we are getting 1:1, our opponents need to raise us 50% of the time to force us to break even on our steal attempts.
Note:If they just call, they are forced to call even more often or make a raise at some point in the future more often or mix in a wider range of call with some mixture of raises this amount or tighter.
That translates to the following hand range given our raise started with X players remaining.
8 8.3% 88+, AJ+,ATs+,KTs+,QJs
7 9.43% 88+, AJ+, KQ, A9s+,KTs+,QJs,JTs
6 10.91% 66+, AJ+, KQ, A9s+,KTs+,QTs+,JTs,T9s
5 12.95% 66+, AJ+, KQ, A9s+,K9s+,Q9s+,J9s+,T8s+,98s,87s,76s,65s,54s
4 15.91% 66+, AJ+, KQ,QJ,JT, A8s+,KTs+,QTs+,J9s+,T8s+,97s+,87s,76s,65s,54s
3 20.63% 55+, AT+,KT+,QT+,JT,A7s+,KTs+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
2 29.29% 22+, A9+,KT+,QT+,JT,T9,98,87,76,65,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
1 50% 22+,A2+,K4+, Q8+, J9+,T8+,98,A2s+,K2s+,Q2s+,J5s+,T6s+,96s+,85s+,75s,64s+,54s.
So now what hands have equity over them? If you look at the pot odds after a reraise and you call, you are putting in 7.5 to win 17.5 and that means you need 43% equity assuming it's checked to the river. If you look at hands that have 33% equity over that range since you are getting 2:1 on your calls, then you only need 33% equity vs the calling range. Although once you put in the raise there are hands with more than 28% equity but less than 33% equity that would profit more than folding on calls, that is only attempting to salvage a portion of what already was risked with the steal attempt and it would be more ideal to have never raised at all with those hands. As such, if anything that should only allow you to widen your steal range because when raised you don't always lose the full 2.25 you risk even when your hand may not have value.
Since we don't know exactly what the optimal percentage of raising vs just calling is and how that manifests in equity postflop we cannot know the exact equity one should have since it's impossible to know just how often opponent should call or raise and how often we should call or raise when opponent raises. However, I am very confident the ideal range for when we have "value" vs a raiser is somewhere between 33% and 45% equity.
These numbers need to be much higher with "trouble hands" or "reverse implied odds" hands such as high cards with no drawing potential or weak kickers and drawing potential in an aggressive game, while they retain more of their value in passive games. Suited connectors and draw heavy hands can actually play much stronger when you are the aggressor and have implied odds. In other words, tend to play a lot more suited connector type hands and fewer weak kings and queens and weak aces type hands than the "optimal play" actually suggests.
We needed a way to be completely objective AND approximate optimal play, so we didn't account for actions beyond the flop and assumed all bets break even and are checked to the river. In reality, you can ignore that last bet unless it's break even or better when you are betting on a draw, where you can get paid off when you hit, and you tend to lose that last bet with the one pair, weak kicker hand and get in trouble with opponent has you outkicked or plays to beat one pair.
But I digress.
Opponents may call and defend blinds often enough to minimize the amount gained from our steal attempts in addition to raising occasionally. This probably requires us to have closer to 45% equity and as a result, a tighter strategy is better. However many times both the players in the pot win at the expense of the folders, which favors us forcing the action more often which somewhat counterbalances the need to be tighter to compensate for optimal blind defenders, with allowing looser action to still be profitable. Having positional advantage and postflop edge may allow for more hands, but it simultaneously doesn't require you to play as many to do well and in some cases in tournaments chip accumulation at lower or no risk is better than more chips at higher risk. Since I haven't calculated the optimal rate of defending the blinds, I can't factor everything in perfectly.
I do know that the optimal solution really will allow blinds to defend very liberally (especially for only a 2.25x big blind bet) by calling often. This means our steal attempts won't win the full pot but instead a fraction of a slightly larger pot which translates into slightly less than the full steal vs optimal opponents. However, the optimal solution also will require them to defend pretty liberally after the flop which may allow us to gain quite a bit of chips a high percentage of the time when they don't
So I'm not sure but since you never need to play "optimal" preflop and opponents will never be "equally skilled postflop" plus position does matter this may not be that far off. You also should in reality really consider the opponents and postflop actions and other variables. This is just a baseline for thinking about decision making and understanding how the antes effects play.
So let's provide two guidelines with the optimal play likely somewhere between:
These hands have 33% equity over initial hand range of opponent. (Every single hand in this range is individually profitable vs opponent's range if opponent just calls and checks to the river)
vs range of 3bet defender
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
These hands have 45% equity over 3bet range. (Will be profitable if opponent occasionally raises and we call and hands are checked to the river even if steal attempts from bluffs are break even).
45% equity:
8% range: TT+,AQ+,AJs (5% of hands)
9.5% range:TT+,AJ+,ATs,KQs (6.5% of hands)
11% range:88+,AJ+,ATs+,KQs (7.4% of hands)
13% range:77+,AT+,KQ,A7s+,KTs+ (11.2% of hands)
16% range:77+,A8+,KT+,A2s+,K9s+,QTs+ (16.3% of hands)
20% range:66+,A8+,KT+,A2s+,K9s+,QTs+ (16.75% of hands)
30% range:44+,A4+,K8+,Q9+,JT,A2s+,K5s+,Q7s+,J8s+,T9s (30% of hands)
50% range:33+,A2+,K7+,Q9+,JT,K4s+,Q8s+,J9s+,T9s (33% of hands)
In either case, we can widen our range considerably compared to non ante stages. That's not even including the "bluff raises" as our steal attempts should break even vs optimal opponent.
If all these hands have "value" vs a hand opponent plays, we should add in bluffs such that we have a "value" hand 40% of the time. That means multiply the hand range by 2.5.
For example, take the 33% equity.
Total hand range:
8:39.59%
7:69.76%
6:88.61%
5 or less players remaining: over 100% or Any Two Cards.
This is approximately:
8:22+,A2+,K8+,QT+,JT,T8+,98,K3s+,Q7s+,J7s+,T7s+,96s+,85s+,76s,64s+,54s
7:A2+,K2+,Q2+,J4+,T6+,96+,86+,J2s+,T2s+,93s+,84s+,74s+,64s+,54s+,43s
6:22+,A2+,K2+,Q2+,J2+,T4+,92+,84+,74+,64+,54,T2s+,92s+,82s+,72s+,62s+,52s+,42s+,32s
5 or less: any two.
OR
5:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
4:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
3:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
2:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
1:22+,A2+,K2+,Q2+,J2+,T2+,92+,82+,72+,62+,52+,42+,32
Button, cutoff, hijack, and even middle position raises any two.
I only multiplied the initial 45% equity range by 2 since I assumed this strategy was probably for a tighter strategy against larger reraises or opponent who bluffs more often and wanted the so called "conservative estimate" in which we want the true optimal strategy to fit somewhere between these two. Also, opponents who 3bet may 3bet with odds to keep you in, but if you choose to fold, you need a 4bet hand more often. So this leads to the following hands as the "tighter end" of correct play during the ante stages.
8: 66+,AJ+,ATs+,KTs+,QTs+,JTs (10%)
7: 44+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s (13%)
6: 33+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s (14.8%)
5: 22+,A8+,KJ+,QJ,JT,A2s+,K9s+,Q9s+,J9s+,T8s+,98s (22.3%)
4: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s (32.6%)
3: 22+,A2+,K9+,QT+,JT,a2S+,k5s+,Q8s+,J8s+,T7s+,97s+,86s+,76s,65s (33.5%)
2: 22+,A2+,k2+,Q6+,J7+,T7+,97+,87,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,54s,43s (60.03%)
1: 22+,A2+,k2+,Q2+,J5+,T7+,97+,86+,Q2s+,J2s+,T2s+,94s+,84s+,74s+,64s+,54s,43s (65.8%)
If multiplied by 2.5 instead:
8 12.44%
7 16.21%
6 18.48%
5 27.90%
4 40.72%
3 41.86%
2 75.04%
1 82.20%
That translates into roughly
8 55+,AT+,KQ,A9s+,K9s,Q9s+,J9s+,T9s
7 33+,AT+,KJ+, A5s+,K9s,Q9s+,J9s+,T9s,98s
6 22+,A9+,KJ+, A4s+,K9s,Q9s+,J9s+,T9s,98s
5 22+,A4+,KT+,QJ,JT A2s+,K8s,Q9s+,J8s+,T8s+,97s+,87s
4 22+,A2+,K9+,QT+,JT,T8+,98,87,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
3 22+,A2+,K9+,QT+,JT,T8+,98,87,76,K3s+Q6s+,J6s+,T6s+,97s+,85s,75s,64s+
2 A2+,K2+,Q2+,J2+,T5+,96+,85+,75+,64+,54,T2s+,93s+,84s+,74s+,63s+,53s+,43s,
1 A2+,K2+,Q2+,J2+,T3+,94+,84+,74+,63+,53,43, any suited
Because the blinds are correct to defend more liberally and they will slightly eat into your profit, you can probably raise less often.
However, from an exploitative perspective, and because we don't know quite how often opponents should raise and they sometimes should call, if opponents just call, rather than raise, you can widen the ranges.
If you prefer not to 4bet light, and you don't want to be quite as much of a target, you probably can get away with raising much less often, particularly if opponents fold and you will still be able to accumulate chips. That will allow you to fold a lot more to aggressive 3bets and 4bets and avoid all ins. You may be able to win a tournament without ever having to show down a hand or ever being all in which supports cutting back on your profitability and loose play in exchange for lower variance and less risk, and also not being quite as much as a target by stealing less often.
In either case, we can dramatically widen our raise range but as a result our opponents should have a pretty wide 3bet range and we in turn will need a pretty wide 4bet range. One problem with this is it assumes stacks are infinitely deep meaning there is no all in move. Since there is, this strategy plays too many hands when a 3 or 5bet all in is perfect stack size for shoving and too few when a 4bet or 6bet or 2bet shove is perfect stack size. Eventually we no longer have equity over opponent and eventually we have to either call and gain whatever equity remains or fold. Also, because of post flop decisions and it being very unlikely the hand will play to the river, we need to start out much stronger if we want to continue to the river a high percentage of the time that we play.
The 33% equity "value" range also works for flat calling raises from the button. Since we may be squeezed this could be slightly tighter, particularly if a squeeze play is likely going to be all in, but we could potentially play very loose on the button vs raises because we only need 33% equity vs the initial raiser and we have position.
Another decision we may make is facing a 2x BB raise. Although game theory solution is to 3bet often enough to force opponent to break even, we can also look at when we have equity to call assuming the blinds fold.
33% equity to flat on button when facing X% raising range:
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
We can see that even against tight opponents, it's correct to have a very wide range when calling. However, this range should be tighter because blinds will not always fold, players will not check until the river, and we occasionally will get "squeezed" by the blinds and have to fold. Also, our opponent's hand ranges are not "known", and there are "reverse implied odds" for a lot of these weak kicker hands.
Nevertheless, the concept of occasionally calling on the button can never be that bad, and if we are likely to get squeezed, that favors just calling with premium hands occasionally too. Even though we may let our opponents outflop us or get money in and outdraw us sometimes giving up that risk for value and balance and for the potential to resqueeze shove with aces or getting opponent's entire stack when he hits top pair or decides to continue with a low pair or ace king on low card flops may be worth it.
Since it's mathematically such an advantage to play lots of hands and call, and since opponents do not always adjust, the ante stages is THE MOST profitable stage of the tournament. We want to both maximize our chances to get to the ante stages while also trying to get there with enough chips to be able to make a handful of failed raises and still have over 20 big blinds.
There may be some tradeoff between giving up some probability of getting to the ante stages, and ensuring that when you do get there you have enough chips. Every decision to play or not play a hand may cost us chips. Costing us chips may force us into future situations where we are more desperate which could cost us more chips. So any decision could hurt our chances of getting there. usuaully the decision of playing a hand decreases our chances of getting there more than not playing a hand, but usually not playing a hand decreases our chances of having well over 30 big blinds when we do actually get there. But having more than 30 big blinds is not really necessary since you can clearly get a lot more chips from 30 big blinds, and still be able to raise preflop, Cbet the flop and still have chips left over to do it again. I'm not convinced that having more than 30 big blinds really adds a lot of utility, beyond giving you a few more shots at a particularly level of utility and perhaps sustaining it for maybe one more level of play which you may be able to parlay into two more levels of play. However, you can parlay a tight image into additional value in future levels of play as well and tend to get there more often, so those effects are not really that significant if you know how to use a tight image.
Certainly there is a lot of missed opportunity by not having more than 20 big blinds in the ante stages. However, even with 15 big blinds it's probably only a matter of time before you double up. You can increase your stack significantly with a move in or two to maintain or even grow your chips. Once the double up happens you probably have a really tight image, a lot more chips, and the blinds may be much higher so it probably still is worth quite a lot and can nearly be made up for particularly AFTER a double up with a monster hand when opponents are now forced to recognize how tight we've been, then we can go to work with much less resistance. Even if you are still shortstacked after playing this patiently, you can shove a few times in a row after the double up then scale back for a rotation or two and then do it again. Then you can play ultra tight again and before you know it you pick up a big hand again and no longer have the tight image until you're called again. You win 2 of these spots where you are probably on average of 60-80% to win and now you have the ability even in a fast structure to win with only a few all ins all tournament and in a slow structure, you may be able to avoid ever being all in.
So being all in twice and called with 60-80% to win is not really all that different from being all in once with the 40-65% chance to win earlier on and parlaying it into a lot of chips in smaller pots early.
Either way, we have to recognize that it's profitable like no other to play a much crazier strategy than you'd ever play in a cash game. If there tends to be a lot of pots that make it to the turn and river with lots of bet, a lot of the preflop edge may be neutralized and the equity will have to be higher and a bit closer to the no ante stages. But when opponents start playing more hands, we then can profitably be aggressive to attempt the limp raise or squeze play which is another story to cover later.
Game Theory Optimal Preflop Poker 2
In Optimal Preflop Poker, I took the approach of first seeing how our opponents should play us to force us to break even on bluffs, and then I showed how we would need to have approximately 45% equity due to the price we are getting because of the blinds. Those will be our value hands. We will then add enough bluff hands such that our opponent won't be able to exploit us by raising every time., and such that we have a "value hand" often enough to force him to break even on his bluffs.
Since our opponent will be betting 9 over our raise of 3 to pick up a total of 4.5 with the blinds, he needs to be able to win 2/3 pots to be able to bluff successfully. As such, we need a hand 1/3 times in which we can raise him. Some of these will be value hands, some will be bluffs.
So we can approximately multiply any hand with 45% equity by 3 and that gives us the range of hands in which we should open with initially.
So we need hands with 45% equity vs:
8 4.95% 99+,AQ
7 5.63% 99+,AQ,Ajs,KQs
6 6.53% 99+,AQ,Ats+,KJs,QJs
5 7.79% 88+,Ats+,KTs+,Qjs,JTs
4 9.64% 66+,AJ+,Ats+,KTs+,QTs+,JTs
3 12.64% 55+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s
2 18.35% 22+,A9+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s,98s
1 33.33% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,87s+
That turns out to be:
8: JJ+,AK
7: JJ+,AK,AQs
6: JJ+,AQ+
5: TT+,AQ+
4: 99+,AQ+,AJs,KQs
3: 88+,AJ+,ATs,KQs
2: 66+,AT+,KQ,A9s,KJs+
1: 44+,A7+,KT+,QJ,A2s+,K9s+,QTs+
Then determine percentage of all hands that makes up and multiply hand range by 3 to add in "bluff range" and have total range where 1/3 hands are value.
8 0.090497738
7 0.099547511
6 0.126696833
5 0.140271493
4 0.171945701
3 0.194570136
2 0.294117647
1 0.601809955
And that gives you total hand range given X players left to act. (This is approximately ideal if the solution didn't allow for a mixture of hands such that you might sometimes play 56s and mostly fold but mostly play JTs and rarely fold)
8 66+,AQ+,Ats+,Kts+,QTs+,JTs 9%
7 66+,AJ+,A9s+,Kts+,QTs+,JTs 9.9%
6 55+,AJ+,KQ,A9s+K9s+,Q9s+,J9s+,T9s 12.7%
5 44+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s 14%
4 22+,AT+,KJ+,A6s+,K9s+,Q9s+,J9s+,T9s 17.2%
3 22+,A9+,KJ+,A3s+,K9s+,Q9s+,J9s+,T8s 19.5%
2 22+,A3+,K9+,QT+K5s+,Q8s+,J7s+,T8s+,97s+,87s+ 29.4%
1 22+,A2+,k2+,Q6+,J7+,T7+,97+,87,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,54s,43s 60.2%
When we are reraised, we should 4bet a mixture of value bets and bluffs and also flat call a mixture of value hands and hands for future bluffing and to disguise our range.
That is how one can act with no antes and perfectly matched preflop and postflop play. Because the postflop play is evenly matched, the pot will be "chopped". In reality it will be very high volatility with a lot of actions on each street. A small difference in skill can be very dangerous for the person who starts with the worst hand, particularly one that is easily dominated. As such, it is almost always better to be tighter preflop and that allows you to be more aggressive and looser on the flop rather than fight for very small edges and break even hands to neutralize your valuable hands and force opponent to call you down lighter overall which allows value hands to make more at the expense of having to play a lot of "break even hands".
It also is "impossible" to instantly know opponent's strategy so if he raises you and you start out with a wide range of hands like this you can run into a lot of trouble.
As such, usually I recommend playing such that YOU have a greater than 50% chance of being best, rather than playing "bluff hands" to force opponent to play such that your bluffs break even and HE has 50% chance of being best.
However, it goes to show that in the right circumstances a "nutball strategy" not only is possible, but optimal against tough opponents and an even looser one may be best if you have a significant post flop skill advantage.
Up next we will get into Optimal Preflop Poker when antes are involved which hopefully will illustrate just how much you need to loosen up in tournaments when antes kick in.
Since our opponent will be betting 9 over our raise of 3 to pick up a total of 4.5 with the blinds, he needs to be able to win 2/3 pots to be able to bluff successfully. As such, we need a hand 1/3 times in which we can raise him. Some of these will be value hands, some will be bluffs.
So we can approximately multiply any hand with 45% equity by 3 and that gives us the range of hands in which we should open with initially.
So we need hands with 45% equity vs:
8 4.95% 99+,AQ
7 5.63% 99+,AQ,Ajs,KQs
6 6.53% 99+,AQ,Ats+,KJs,QJs
5 7.79% 88+,Ats+,KTs+,Qjs,JTs
4 9.64% 66+,AJ+,Ats+,KTs+,QTs+,JTs
3 12.64% 55+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s
2 18.35% 22+,A9+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s,98s
1 33.33% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,87s+
That turns out to be:
8: JJ+,AK
7: JJ+,AK,AQs
6: JJ+,AQ+
5: TT+,AQ+
4: 99+,AQ+,AJs,KQs
3: 88+,AJ+,ATs,KQs
2: 66+,AT+,KQ,A9s,KJs+
1: 44+,A7+,KT+,QJ,A2s+,K9s+,QTs+
Then determine percentage of all hands that makes up and multiply hand range by 3 to add in "bluff range" and have total range where 1/3 hands are value.
8 0.090497738
7 0.099547511
6 0.126696833
5 0.140271493
4 0.171945701
3 0.194570136
2 0.294117647
1 0.601809955
And that gives you total hand range given X players left to act. (This is approximately ideal if the solution didn't allow for a mixture of hands such that you might sometimes play 56s and mostly fold but mostly play JTs and rarely fold)
8 66+,AQ+,Ats+,Kts+,QTs+,JTs 9%
7 66+,AJ+,A9s+,Kts+,QTs+,JTs 9.9%
6 55+,AJ+,KQ,A9s+K9s+,Q9s+,J9s+,T9s 12.7%
5 44+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s 14%
4 22+,AT+,KJ+,A6s+,K9s+,Q9s+,J9s+,T9s 17.2%
3 22+,A9+,KJ+,A3s+,K9s+,Q9s+,J9s+,T8s 19.5%
2 22+,A3+,K9+,QT+K5s+,Q8s+,J7s+,T8s+,97s+,87s+ 29.4%
1 22+,A2+,k2+,Q6+,J7+,T7+,97+,87,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,54s,43s 60.2%
When we are reraised, we should 4bet a mixture of value bets and bluffs and also flat call a mixture of value hands and hands for future bluffing and to disguise our range.
That is how one can act with no antes and perfectly matched preflop and postflop play. Because the postflop play is evenly matched, the pot will be "chopped". In reality it will be very high volatility with a lot of actions on each street. A small difference in skill can be very dangerous for the person who starts with the worst hand, particularly one that is easily dominated. As such, it is almost always better to be tighter preflop and that allows you to be more aggressive and looser on the flop rather than fight for very small edges and break even hands to neutralize your valuable hands and force opponent to call you down lighter overall which allows value hands to make more at the expense of having to play a lot of "break even hands".
It also is "impossible" to instantly know opponent's strategy so if he raises you and you start out with a wide range of hands like this you can run into a lot of trouble.
As such, usually I recommend playing such that YOU have a greater than 50% chance of being best, rather than playing "bluff hands" to force opponent to play such that your bluffs break even and HE has 50% chance of being best.
However, it goes to show that in the right circumstances a "nutball strategy" not only is possible, but optimal against tough opponents and an even looser one may be best if you have a significant post flop skill advantage.
Up next we will get into Optimal Preflop Poker when antes are involved which hopefully will illustrate just how much you need to loosen up in tournaments when antes kick in.
Optimal Preflop Poker
To play true optimal preflop poker, we first must look at collectively how often our opponents would have to raise to keep us honest on our bluffs. From there we then want to determine which hands have value when faced with a raise and add in some bluffs at the optimal proportion which don't have value against that range but have value in stealing preflop.
If there were no antes a "standard raise" might be 3 times the big blind*. Our opponents ideally should force our bluffs to be break even so we couldn't do this with any two. Since we are winning 1.5 when we win but risking 3, we can win 1.5 twice and on the third if we are reraised and fold we break even on our bluffs.
Thus, collectively, our opponents can wait for a hand that triggers a raise 1/3rd of the time that WE raise.
What this means is if we raised with a hand that had a 66.667% chance of having the strongest ranked preflop hand out of all remaining opponents, and our opponents only raised with an equal hand or better than our opponents would have an equal hand or better 1/3rd of the time. In other words, our opponents can wait for a hand that is as strong as that range.
To solve for a hand that has a 66.67% chance of being best preflop we actually take the odds our opponents don't have a better hand to the Nth power, where N is the number of opponents left. We set this until it equals .667. In other words .667=(1-X)^N. We solve N for 1 through 8 players remaining.
8 4.95% 99+,AQ
7 5.63% 99+,AQ,Ajs,KQs
6 6.53% 99+,AQ,Ats+,KJs,QJs
5 7.79% 88+,Ats+,KTs+,Qjs,JTs
4 9.64% 66+,AJ+,Ats+,KTs+,QTs+,JTs
3 12.64% 55+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s
2 18.35% 22+,A9+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s,98s
1 33.33% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,87s+
So that gives us our opponent's range of hands.
The next step is to determine where we have value vs that range. Since we want to have a hand often enough to stand up to a reraise of about 3x our raise, we can multiply the hand range that has value by 3 and add in bluffs.
Against an opponent who is less aggressive and/or tighter than that, we can add in more bluffs. Against an opponent who is more aggressive and/or looser, we can subtract some bluffs from that baseline.
Additionally, if we are facing a 3bet, we also should set the baseline strategy such that our opponents break even on their bluffs. Thus with some of the bluff raise hands we may also 4bet to accomplish this.
Finally, the thing that further complicates this approximate solution is if our opponent would only raise or fold, we should tend to call more for value with the bottom half of the hands that beat an opponents 3bet range, but do not beat an opponent's 5bet range. We'd also replace the hands we call with by adding additional bluffs.
With this adjustment though, the opponent would adapt and have an informational advantage postflop that becomes even more refined by the turn, and even more so by the river. So a closer to optimal counter solution would be to mix in some calls with hands that don't have the pot odds to call, but in combination with the hands that have value may represent enough strength to allow for future bluffs. And then we would have to also occasionally raise with these hands and mix the strongest hands so there are some calls so the range is not as predictable.
At this point there becomes a tradeoff between randomizing it completely and giving up value now, or randomizing it incompletely giving information now. This is why the true optimal solution in poker is incredibly difficult, especially since this doesn't look at postflop actions and assumes it is checked down to the river or bets break even. That is probably not the case and more likely both position and stronger hands have a greater advantage as the weaker hands will have to fold and surrender equity.
--------
*You could solve this for larger or smaller bets preflop or with antes. Eventually we will make an adjustment for antes and smaller preflop bets but for now let's continue onto the next post where we will actually provide the optimal adjustment.
*The true optimal strategy would actually mix bet sizes as well such that value hands and stronger hands tend to be slightly larger, but are mixed up often enough so as to not give too much information about the strength of the hand. Varying bet sizes would only provide a very, very small edge over an near optimal opponent who's only weakness was that he didn't, however.
Continue To Part 2
See Part 3
If there were no antes a "standard raise" might be 3 times the big blind*. Our opponents ideally should force our bluffs to be break even so we couldn't do this with any two. Since we are winning 1.5 when we win but risking 3, we can win 1.5 twice and on the third if we are reraised and fold we break even on our bluffs.
Thus, collectively, our opponents can wait for a hand that triggers a raise 1/3rd of the time that WE raise.
What this means is if we raised with a hand that had a 66.667% chance of having the strongest ranked preflop hand out of all remaining opponents, and our opponents only raised with an equal hand or better than our opponents would have an equal hand or better 1/3rd of the time. In other words, our opponents can wait for a hand that is as strong as that range.
To solve for a hand that has a 66.67% chance of being best preflop we actually take the odds our opponents don't have a better hand to the Nth power, where N is the number of opponents left. We set this until it equals .667. In other words .667=(1-X)^N. We solve N for 1 through 8 players remaining.
8 4.95% 99+,AQ
7 5.63% 99+,AQ,Ajs,KQs
6 6.53% 99+,AQ,Ats+,KJs,QJs
5 7.79% 88+,Ats+,KTs+,Qjs,JTs
4 9.64% 66+,AJ+,Ats+,KTs+,QTs+,JTs
3 12.64% 55+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s
2 18.35% 22+,A9+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s,98s
1 33.33% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,87s+
So that gives us our opponent's range of hands.
The next step is to determine where we have value vs that range. Since we want to have a hand often enough to stand up to a reraise of about 3x our raise, we can multiply the hand range that has value by 3 and add in bluffs.
Against an opponent who is less aggressive and/or tighter than that, we can add in more bluffs. Against an opponent who is more aggressive and/or looser, we can subtract some bluffs from that baseline.
Additionally, if we are facing a 3bet, we also should set the baseline strategy such that our opponents break even on their bluffs. Thus with some of the bluff raise hands we may also 4bet to accomplish this.
Finally, the thing that further complicates this approximate solution is if our opponent would only raise or fold, we should tend to call more for value with the bottom half of the hands that beat an opponents 3bet range, but do not beat an opponent's 5bet range. We'd also replace the hands we call with by adding additional bluffs.
With this adjustment though, the opponent would adapt and have an informational advantage postflop that becomes even more refined by the turn, and even more so by the river. So a closer to optimal counter solution would be to mix in some calls with hands that don't have the pot odds to call, but in combination with the hands that have value may represent enough strength to allow for future bluffs. And then we would have to also occasionally raise with these hands and mix the strongest hands so there are some calls so the range is not as predictable.
At this point there becomes a tradeoff between randomizing it completely and giving up value now, or randomizing it incompletely giving information now. This is why the true optimal solution in poker is incredibly difficult, especially since this doesn't look at postflop actions and assumes it is checked down to the river or bets break even. That is probably not the case and more likely both position and stronger hands have a greater advantage as the weaker hands will have to fold and surrender equity.
--------
*You could solve this for larger or smaller bets preflop or with antes. Eventually we will make an adjustment for antes and smaller preflop bets but for now let's continue onto the next post where we will actually provide the optimal adjustment.
*The true optimal strategy would actually mix bet sizes as well such that value hands and stronger hands tend to be slightly larger, but are mixed up often enough so as to not give too much information about the strength of the hand. Varying bet sizes would only provide a very, very small edge over an near optimal opponent who's only weakness was that he didn't, however.
Continue To Part 2
See Part 3
Saturday, June 20, 2015
From Cash Games To Tournaments
There are many differences between cash games and tournaments, but the biggest difference I have found is the difference between just the blinds and when the antes become involved.
In a cash game if players were equally skilled after the flop, and it folded to you, you'd want to raise such that you have a 50% chance or more of having the best hand. (Plus adding bluff raises. The more opponents fold, the more bluff raises to add)
50% chance hand starts out as favorite opener strategy given X players left to act:
8 8.3% 88+, AJ+,ATs+,KTs+,QJs
7 9.43% 88+, AJ+, KQ, A9s+,KTs+,QJs,JTs
6 10.91% 66+, AJ+, KQ, A9s+,KTs+,QTs+,JTs,T9s
5 12.95% 66+, AJ+, KQ, A9s+,K9s+,Q9s+,J9s+,T8s+,98s,87s,76s,65s,54s
4 15.91% 66+, AJ+, KQ,QJ,JT, A8s+,KTs+,QTs+,J9s+,T8s+,97s+,87s,76s,65s,54s
3 20.63% 55+, AT+,KT+,QT+,JT,A7s+,KTs+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
2 29.29% 22+, A9+,KT+,QT+,JT,T9,98,87,76,65,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
1 50% 22+,A2+,K4+, Q8+, J9+,T8+,98,A2s+,K2s+,Q2s+,J5s+,T6s+,96s+,85s+,75s,64s+,54s
However, once the antes are involved you are often min raising or 2.5x the big blind raising. As such when you get called by a player not in the blinds, you have around 2.25 blinds dead in the pot plus your opponents 2.25 plus yours. You are getting 2 to 1 on your money and thus can lose that 2.25 big blinds twice and on the 3rd win it and you are break even.
If your opponents always call you only need to be 33% to win. IF opponents properly raise you more as a result of this strategy, you can still raise as long as you have 40% equity. Either way, the point is it is correct to loosen up once the antes are involved. The exception is if you have plans to use a tight image for high risk, high percentage moves in larger pots such as 3bets, 4bets, squeeze plays and "dead money grab" (raise over limpers) or if opponents are aggressive you might just flat call their raises a few times and be willing to resqueeze back over a squeeze play or flat call.
Although 33% chance of starting with the strongest hand is not the same as having 33% equity, it's a good first step and illustrates how your strategy should change when the antes are involved. Also, if opponent's correctly adapt, it becomes incorrect to raise quite this frequently because they more often than not will 3bet you, and you won't be able to have a 4bet hand often enough to make up for it if they correctly adjust. Nevertheless, it's an important starting point from which to establish the theoretical strategy and if your opponents are tighter or looser or aggressive, you can adjust from there.
8) 12.84% 44+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s+
7) 14.53% 33+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s+
6) 16.74% 33+,At+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s+,98s
5) 19.74% 22+,A9+,KJ+,QJ,A3s+,K9s+,Q9s+,J9s+,T8s+,98s
4) 24.02% 22+,A5+,KJ+,QJ,JT,A2s+,K9s+,Q9s+,J8s+,T8s+,98s
3) 30.66% 22+,A2+,KT+,QT+,JT,K8s+,Q8s+,J8s+,T8s+,97s+,87s,
2) 42.26% 22+,A2+,K8+,QT+,JT,T8+,98,87,K3s+Q6s+,J6s+,T6s+,96s+,85s,75s,64s+
1) 66.67% 22+,A2+,K2+,Q2+,J5+,T7+,97+,86+,J2s+,T3s+,95s+,84s+,74s+,64s+,54s,43s
3betting is something that also should change as a result of the antes. First, there is still more in the pot proportional to the bets since the bets and 3bets tend to be a bit smaller and the antes are still involved. Secondly, if your opponents correctly open up their hand ranges more to adjust for the antes, it becomes possible to raise them lighter.
Before I continue, please understand how you might deviate from the strategy provided for a greater edge. Always 3betting is not necessarily the most ideal solution in practice. In practice you should probably bluff 3bet a few hands that are normally outside of the playable range to a raise, and flat call with those that are on the bottom half of the 3bet range and mix in some calls and some 3bets with the strongest of hands to veil the strength of what each action means with a greater importance towards getting value over veiling information.
If you have an edge postflop, it's better to keep the pot smaller preflop more often so that you can exploit that edge to a greater extent. If opponent doesn't adapt to the information there's less reason to give information away. If opponent folds too much you can bluff raise more often and if he calls to much you can eliminate bluff raises and always 3bet the entire range.
3bet without antes
Since you are assuming opponents should range with a 50% chance of being best, to put in a 3bet you'll want a 75% chance hand is best preflop (This approximately means your hand has a greater than 50% it's best after an opponent raises)
8) 3.53% TT+,AK
7) 4.03% 99+,AK
6) 4.68% 99+,AQ+
5) 5.59% 99+,AQ+,AJs+,KQs
4) 6.94% 99+,AQ+, ATs+,KTs+,QJs
3) 9.14% 77+,AJ+ATs+,KTs+,QTs+,JTs
2) 13.40% 44+,AJ+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s
1) 25% 22+,A8+,KT+,QT+,JT+,A2s+,K7s+,Q9s+,J9s+,T8s+,98s,87s,76s
Ante stages
Aside from the obvious tournament pressure and players not wanting to play a pot that may risk their tournament life with marginal holdings, 3 betting light also works because of the antes. The added price of the antes does make a difference, and the opponent's hand range in theory SHOULD be larger as well.
Ante stages 3 bet for opponent's likely to call or fold and unlikely to 4bet exploit, or likely to fold too much.
57% of opponent's range when opponent opens with handrange with 40% chance of being best.
40% because at an aggressive table opponents should not actually raise with 33% chance it's best or 50% due to the antes, but around 40%. We 3bet with 57% of opponent's range because the pot odds allows for us to have the pot odds when we're called so we don't have to be a favorite vs opponent's range we just need to be best around 43% of the time.
8) 6.17% 99+,AQ+,ATs+,KQs
7) 6.99% 99+,AQ+,ATs+,KTs+,QJs
6) 8.07% 88+,AQ+,ATs+,KTs+,QTs,JTs
5) 9.55% 66+,AJ+,KTs+,QTs+,JTs
4) 11.67% 55+,AJ+,KQ+,A9s+,K9s+,QTs+,JTs
3) 15.00% 33+,ATo+,A7s+,K9s+,Q9s+,J9s+,T9s
2) 20.95% 22+,A8+KJ+,QJ,A2s+,K9s+,Q9s+,J9s+,T8s+,98s
1) 34.20% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,86s+,76s
The real ideal solution would look for equity rather than odds hand is favorite but that requires some deep calculations vs various hand ranges. This is close enough to show how we might widen our range.
Flat calling on the button.
As I said, a 33% chance of having the best hand is not the same as having a 33% chance to win. While it's difficult to know what hand range we will be called with if we raised one thing we can do is look at flat calling an opponent's hand range from the button.
This time we will say the opponent doesn't adjust to antes so we are looking at flat calling an opponent with the SAME opening hand range. We are now looking at EQUITY vs that range. Without antes we need about 45% equity. With antes we need about 33%.
Flat from button without antes (45% equity vs range)
8% range: TT+,AQ+,AJs,
9.5% range:TT+,AJ+,ATs,KQs,
11% range:88+,AJ+,ATs+,KQs
13% range:77+,AT+,KQ,A7s+,KTs+
16% range:77+,A8+,KT+,A2s+,K9s+,QTs+
20% range:66+,A8+,KT+,A2s+,K9s+,QTs+
30% range:44+,A4+,K8+,Q9+,JT,A2s+,K5s+,Q7s+,J8s+,T9s
50% range:33+,A2+,K7+,Q9+,JT,K4s+,Q8s+,J9s+,T9s
flat from button when antes are in play (33% equity).
33% equity
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
Note:33% equity assumes after the flop hands are checked down to the river this is not the case and a player should actually adjust for the fact that it is not. This means "trouble hands" with high card and kicker problems become less playable while "implied odds" hands like suited connectors become more playable.
I would rather have a hand with slightly worse equity and use post flop position and skill and ability to have better implied odds and less likely to get trapped for a big pot. Also, because I'm picking specific hands in opener's range some hands may be overvalued against an unknown range and others may be undervalued. This is just a guideline to determine an approximate calling range such that you have equity in the ante stages.
As you get shorter stacked, suited connectors lose a lot of that value and the actual hands should be closer to the solution. As they get even shorter still there are more semibluff opportunities with suited connectors so having a larger probability of being able to make a big move with outs may have enough value to add some of the higher suited connectors again.
Mostly, this is to illustrate how much you should widen your calling range preflop and encourage you to play more hands.
After the flop, not a ton really changes in terms of decisions, other than the fact that the remaining stack size proportional to the pot should always change decisions and what hand strength is "good enough". The antes tends to create far more situations where the stacks are short postflop relative to the blinds.
You may also have to fight a little harder with the worse preflop hand for the 33% equity. So you need to fight to see a showdown or get your money's worth enough to make up for playing more hands. You should work a little harder at getting a cheap showdown as it is of benefit to both you AND your opponent at the expense of all the players not in the hand that have contributed.
One thing that does change slightly is the all in decision. The pot becomes large, much faster proportional to the "big blinds",
Let's consider a 3bet all in. If there are no antes, 2.5x big blinds raise plus blinds equals 4bbs in the pot. If there are antes in the pot 2.5x raise and antes is probably about 5bbs. An all in for 3.5 times the pot without antes would take place if your stack was 14 big blinds. An all in with antes would take place if you had 17.5 big blinds. With antes you can make your move a little sooner or loosen up your range of push hands at a given level of big blinds.
All in preflop as the first one in also changes with antes. A preflop move all in with 15 big blinds increases stack by 10% without antes. Preflop with antes that same 15% big blind all in increases stack by 15%. You could increase your stack by 10% if you open shoved with 22-25 big blinds. This means your moves can take place sooner or with a wider range of hands if you'd like.
The other key difference is a tournament player could accumulate every single chip and win and only get 25% of the prize money. Every all in that is risked actually benefits the players not in the pot slightly because they move closer to the money. While this can be taken to an extreme and entire models can be made off of this to determine decisions, I'd hesitate to go that route completely.
The ICM model is basically if everyone flipped a coin all in for there chips what would be the probability that each remaining player finished in each place, and as a result, what would the overall payout be. That carries with it the assumption of equal skill. Even those who try to quantify a skill would say that a skilled player has a 60% chance of winning those coinflips.
In reality you are taking low variance edges over hundreds of hands until ultimately the blinds and antes force an all in decision. A good player may have doubled or tripled his stack without high varience. In theory if the blinds went up extremely slowly, a player could turn down any time he risked too much of his bankroll so that he never went broke and even the smallest skill advantage would mean a first or second place finish everytime if he were the only one to play that way.
Well the blinds and antes don't go up that slow, but even so, considerable value can be added.
ICM will teach you that only decisions should change close to the money and close to the final table but for the most part giving up an edge in exchange for avoiding all in confrontation isn't advantageous. But that's based on equal skill. When you introduce skill, the opposite is true.
Losing your chance to play during the ante stages is giving up a huge amount if you have a skill edge. The opportunity cost is not worth the chips you gain by taking risks. If you can win a net of 3 times what's in the middle per rotation in the ante stages, you never have to be all in to win a tournament. If you can win 2 times what's in the middle you probably will never have to be all in more than once. I think winning 2 times what's in the middle per rotation is fairly reasonable with the asterisk of doing so may cost you enough volatility that you actually do have to be all in more than once, and your skill edge tends to decrease the later in a tournament you get. However, you still should be able to finish tournaments being all in less than 4 times before the final table even in very large tournaments if you do this well. This probably translates into an over 10% chance of making the final table. If this drops to 6-8% in exchange for a few more wins it is probably worth it. If the field is 500 players in theory you should make the final table just less than 2% of the time. In practice that is not the case for many good players.
Either knowing the ICM solutions for the late stages of the tournament and bubble stages OR taking the approach of high probability of survival, high aggression on the bubble, and patience afterwards until you are then going to take the higher risk aggressive strategy and get a big stack or go home. Then once you have a large enough stack to go on autopilot and basically automatically finish 5th without ever being all in then you can play hyper tight and raise a large amount with JJ+,AK and fold JJ and AK to large shoves and sometimes QQ. You can perhaps widen range a bit in late position and under the gun because under the gun garners more respect and late position when it folds to you doesn't require as strong of a hand to raise or call an all in. You are virtually guaranteed a fighting chance at the final table by this strategy and regardless of how many chips you have the final table is always going to be high varience.
Final table you survive to around 5th place, pick up the aggression dramatically. You also will try to manipulate who you steal from such that all the players end up fighting for second place. You want to keep the shortest stack alive while stealing from the others fairly evenly but trying to prevent anyone from becoming a huge threat. So you might stay away from the second place guy in chips for awhile so as to not let him catch up with a move or two over you and wait to come after him later so he thinks you are staying out of his way and playing tighter against him until the blinds raise and you are maybe 3 handed.
In a cash game if players were equally skilled after the flop, and it folded to you, you'd want to raise such that you have a 50% chance or more of having the best hand. (Plus adding bluff raises. The more opponents fold, the more bluff raises to add)
50% chance hand starts out as favorite opener strategy given X players left to act:
8 8.3% 88+, AJ+,ATs+,KTs+,QJs
7 9.43% 88+, AJ+, KQ, A9s+,KTs+,QJs,JTs
6 10.91% 66+, AJ+, KQ, A9s+,KTs+,QTs+,JTs,T9s
5 12.95% 66+, AJ+, KQ, A9s+,K9s+,Q9s+,J9s+,T8s+,98s,87s,76s,65s,54s
4 15.91% 66+, AJ+, KQ,QJ,JT, A8s+,KTs+,QTs+,J9s+,T8s+,97s+,87s,76s,65s,54s
3 20.63% 55+, AT+,KT+,QT+,JT,A7s+,KTs+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
2 29.29% 22+, A9+,KT+,QT+,JT,T9,98,87,76,65,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,54s
1 50% 22+,A2+,K4+, Q8+, J9+,T8+,98,A2s+,K2s+,Q2s+,J5s+,T6s+,96s+,85s+,75s,64s+,54s
However, once the antes are involved you are often min raising or 2.5x the big blind raising. As such when you get called by a player not in the blinds, you have around 2.25 blinds dead in the pot plus your opponents 2.25 plus yours. You are getting 2 to 1 on your money and thus can lose that 2.25 big blinds twice and on the 3rd win it and you are break even.
If your opponents always call you only need to be 33% to win. IF opponents properly raise you more as a result of this strategy, you can still raise as long as you have 40% equity. Either way, the point is it is correct to loosen up once the antes are involved. The exception is if you have plans to use a tight image for high risk, high percentage moves in larger pots such as 3bets, 4bets, squeeze plays and "dead money grab" (raise over limpers) or if opponents are aggressive you might just flat call their raises a few times and be willing to resqueeze back over a squeeze play or flat call.
Although 33% chance of starting with the strongest hand is not the same as having 33% equity, it's a good first step and illustrates how your strategy should change when the antes are involved. Also, if opponent's correctly adapt, it becomes incorrect to raise quite this frequently because they more often than not will 3bet you, and you won't be able to have a 4bet hand often enough to make up for it if they correctly adjust. Nevertheless, it's an important starting point from which to establish the theoretical strategy and if your opponents are tighter or looser or aggressive, you can adjust from there.
8) 12.84% 44+,AJ+,KQ,A9s+,K9s+,Q9s+,J9s+,T9s+
7) 14.53% 33+,AT+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s+
6) 16.74% 33+,At+,KJ+,A5s+,K9s+,Q9s+,J9s+,T9s+,98s
5) 19.74% 22+,A9+,KJ+,QJ,A3s+,K9s+,Q9s+,J9s+,T8s+,98s
4) 24.02% 22+,A5+,KJ+,QJ,JT,A2s+,K9s+,Q9s+,J8s+,T8s+,98s
3) 30.66% 22+,A2+,KT+,QT+,JT,K8s+,Q8s+,J8s+,T8s+,97s+,87s,
2) 42.26% 22+,A2+,K8+,QT+,JT,T8+,98,87,K3s+Q6s+,J6s+,T6s+,96s+,85s,75s,64s+
1) 66.67% 22+,A2+,K2+,Q2+,J5+,T7+,97+,86+,J2s+,T3s+,95s+,84s+,74s+,64s+,54s,43s
3betting is something that also should change as a result of the antes. First, there is still more in the pot proportional to the bets since the bets and 3bets tend to be a bit smaller and the antes are still involved. Secondly, if your opponents correctly open up their hand ranges more to adjust for the antes, it becomes possible to raise them lighter.
Before I continue, please understand how you might deviate from the strategy provided for a greater edge. Always 3betting is not necessarily the most ideal solution in practice. In practice you should probably bluff 3bet a few hands that are normally outside of the playable range to a raise, and flat call with those that are on the bottom half of the 3bet range and mix in some calls and some 3bets with the strongest of hands to veil the strength of what each action means with a greater importance towards getting value over veiling information.
If you have an edge postflop, it's better to keep the pot smaller preflop more often so that you can exploit that edge to a greater extent. If opponent doesn't adapt to the information there's less reason to give information away. If opponent folds too much you can bluff raise more often and if he calls to much you can eliminate bluff raises and always 3bet the entire range.
3bet without antes
Since you are assuming opponents should range with a 50% chance of being best, to put in a 3bet you'll want a 75% chance hand is best preflop (This approximately means your hand has a greater than 50% it's best after an opponent raises)
8) 3.53% TT+,AK
7) 4.03% 99+,AK
6) 4.68% 99+,AQ+
5) 5.59% 99+,AQ+,AJs+,KQs
4) 6.94% 99+,AQ+, ATs+,KTs+,QJs
3) 9.14% 77+,AJ+ATs+,KTs+,QTs+,JTs
2) 13.40% 44+,AJ+,KQ,A8s+,K9s+,Q9s+,J9s+,T9s
1) 25% 22+,A8+,KT+,QT+,JT+,A2s+,K7s+,Q9s+,J9s+,T8s+,98s,87s,76s
Ante stages
Aside from the obvious tournament pressure and players not wanting to play a pot that may risk their tournament life with marginal holdings, 3 betting light also works because of the antes. The added price of the antes does make a difference, and the opponent's hand range in theory SHOULD be larger as well.
Ante stages 3 bet for opponent's likely to call or fold and unlikely to 4bet exploit, or likely to fold too much.
57% of opponent's range when opponent opens with handrange with 40% chance of being best.
40% because at an aggressive table opponents should not actually raise with 33% chance it's best or 50% due to the antes, but around 40%. We 3bet with 57% of opponent's range because the pot odds allows for us to have the pot odds when we're called so we don't have to be a favorite vs opponent's range we just need to be best around 43% of the time.
8) 6.17% 99+,AQ+,ATs+,KQs
7) 6.99% 99+,AQ+,ATs+,KTs+,QJs
6) 8.07% 88+,AQ+,ATs+,KTs+,QTs,JTs
5) 9.55% 66+,AJ+,KTs+,QTs+,JTs
4) 11.67% 55+,AJ+,KQ+,A9s+,K9s+,QTs+,JTs
3) 15.00% 33+,ATo+,A7s+,K9s+,Q9s+,J9s+,T9s
2) 20.95% 22+,A8+KJ+,QJ,A2s+,K9s+,Q9s+,J9s+,T8s+,98s
1) 34.20% 22+,A2+,K9+,QT+K5s+,Q8s+,J7s+,T7s+,97s+,86s+,76s
The real ideal solution would look for equity rather than odds hand is favorite but that requires some deep calculations vs various hand ranges. This is close enough to show how we might widen our range.
Flat calling on the button.
As I said, a 33% chance of having the best hand is not the same as having a 33% chance to win. While it's difficult to know what hand range we will be called with if we raised one thing we can do is look at flat calling an opponent's hand range from the button.
This time we will say the opponent doesn't adjust to antes so we are looking at flat calling an opponent with the SAME opening hand range. We are now looking at EQUITY vs that range. Without antes we need about 45% equity. With antes we need about 33%.
Flat from button without antes (45% equity vs range)
8% range: TT+,AQ+,AJs,
9.5% range:TT+,AJ+,ATs,KQs,
11% range:88+,AJ+,ATs+,KQs
13% range:77+,AT+,KQ,A7s+,KTs+
16% range:77+,A8+,KT+,A2s+,K9s+,QTs+
20% range:66+,A8+,KT+,A2s+,K9s+,QTs+
30% range:44+,A4+,K8+,Q9+,JT,A2s+,K5s+,Q7s+,J8s+,T9s
50% range:33+,A2+,K7+,Q9+,JT,K4s+,Q8s+,J9s+,T9s
flat from button when antes are in play (33% equity).
33% equity
8.3% range: 44+,A9+,KT+,K9s,Q9s+
9.5% range: 22+,A2+,KT+,QJ,K4s,Q9s+,JTs
10.91% range:22+,A2+,K9+,Q9+,JT,T9,K2s+,Q7s+,J8s+,T8s+,98s
12.95% range:22+,A2+,K2,Q6+,J8+,T8+,98,Q2s+,J2s+,T4s+,96s+,86s+,76s,
15.91% range:22+,A2+,K2+,Q2+,J8+,T8+,97+,J2s+,T5s,95s+,85s+,75s+,65s,54s
20.63% range:22+,A2+,K2+,Q2+,J7,T8,97+,87,J2s+,T6s+,95s+,84s+,74s+,64s+,53s+
29.29% range:22+,A2+,K2+,Q2+,J2+,T2+,95+,85+,75+,65,54,92s+,82s+,73s+,63s+,52s+,43s
50% range:22+,A2+,K2+,Q2+,J2s,T3+,95+,85+,75+,64+,54,92s+,83s+,73s+,62s+,52s+,43s
Note:33% equity assumes after the flop hands are checked down to the river this is not the case and a player should actually adjust for the fact that it is not. This means "trouble hands" with high card and kicker problems become less playable while "implied odds" hands like suited connectors become more playable.
I would rather have a hand with slightly worse equity and use post flop position and skill and ability to have better implied odds and less likely to get trapped for a big pot. Also, because I'm picking specific hands in opener's range some hands may be overvalued against an unknown range and others may be undervalued. This is just a guideline to determine an approximate calling range such that you have equity in the ante stages.
As you get shorter stacked, suited connectors lose a lot of that value and the actual hands should be closer to the solution. As they get even shorter still there are more semibluff opportunities with suited connectors so having a larger probability of being able to make a big move with outs may have enough value to add some of the higher suited connectors again.
Mostly, this is to illustrate how much you should widen your calling range preflop and encourage you to play more hands.
After the flop, not a ton really changes in terms of decisions, other than the fact that the remaining stack size proportional to the pot should always change decisions and what hand strength is "good enough". The antes tends to create far more situations where the stacks are short postflop relative to the blinds.
You may also have to fight a little harder with the worse preflop hand for the 33% equity. So you need to fight to see a showdown or get your money's worth enough to make up for playing more hands. You should work a little harder at getting a cheap showdown as it is of benefit to both you AND your opponent at the expense of all the players not in the hand that have contributed.
One thing that does change slightly is the all in decision. The pot becomes large, much faster proportional to the "big blinds",
Let's consider a 3bet all in. If there are no antes, 2.5x big blinds raise plus blinds equals 4bbs in the pot. If there are antes in the pot 2.5x raise and antes is probably about 5bbs. An all in for 3.5 times the pot without antes would take place if your stack was 14 big blinds. An all in with antes would take place if you had 17.5 big blinds. With antes you can make your move a little sooner or loosen up your range of push hands at a given level of big blinds.
All in preflop as the first one in also changes with antes. A preflop move all in with 15 big blinds increases stack by 10% without antes. Preflop with antes that same 15% big blind all in increases stack by 15%. You could increase your stack by 10% if you open shoved with 22-25 big blinds. This means your moves can take place sooner or with a wider range of hands if you'd like.
The other key difference is a tournament player could accumulate every single chip and win and only get 25% of the prize money. Every all in that is risked actually benefits the players not in the pot slightly because they move closer to the money. While this can be taken to an extreme and entire models can be made off of this to determine decisions, I'd hesitate to go that route completely.
The ICM model is basically if everyone flipped a coin all in for there chips what would be the probability that each remaining player finished in each place, and as a result, what would the overall payout be. That carries with it the assumption of equal skill. Even those who try to quantify a skill would say that a skilled player has a 60% chance of winning those coinflips.
In reality you are taking low variance edges over hundreds of hands until ultimately the blinds and antes force an all in decision. A good player may have doubled or tripled his stack without high varience. In theory if the blinds went up extremely slowly, a player could turn down any time he risked too much of his bankroll so that he never went broke and even the smallest skill advantage would mean a first or second place finish everytime if he were the only one to play that way.
Well the blinds and antes don't go up that slow, but even so, considerable value can be added.
ICM will teach you that only decisions should change close to the money and close to the final table but for the most part giving up an edge in exchange for avoiding all in confrontation isn't advantageous. But that's based on equal skill. When you introduce skill, the opposite is true.
Losing your chance to play during the ante stages is giving up a huge amount if you have a skill edge. The opportunity cost is not worth the chips you gain by taking risks. If you can win a net of 3 times what's in the middle per rotation in the ante stages, you never have to be all in to win a tournament. If you can win 2 times what's in the middle you probably will never have to be all in more than once. I think winning 2 times what's in the middle per rotation is fairly reasonable with the asterisk of doing so may cost you enough volatility that you actually do have to be all in more than once, and your skill edge tends to decrease the later in a tournament you get. However, you still should be able to finish tournaments being all in less than 4 times before the final table even in very large tournaments if you do this well. This probably translates into an over 10% chance of making the final table. If this drops to 6-8% in exchange for a few more wins it is probably worth it. If the field is 500 players in theory you should make the final table just less than 2% of the time. In practice that is not the case for many good players.
Either knowing the ICM solutions for the late stages of the tournament and bubble stages OR taking the approach of high probability of survival, high aggression on the bubble, and patience afterwards until you are then going to take the higher risk aggressive strategy and get a big stack or go home. Then once you have a large enough stack to go on autopilot and basically automatically finish 5th without ever being all in then you can play hyper tight and raise a large amount with JJ+,AK and fold JJ and AK to large shoves and sometimes QQ. You can perhaps widen range a bit in late position and under the gun because under the gun garners more respect and late position when it folds to you doesn't require as strong of a hand to raise or call an all in. You are virtually guaranteed a fighting chance at the final table by this strategy and regardless of how many chips you have the final table is always going to be high varience.
Final table you survive to around 5th place, pick up the aggression dramatically. You also will try to manipulate who you steal from such that all the players end up fighting for second place. You want to keep the shortest stack alive while stealing from the others fairly evenly but trying to prevent anyone from becoming a huge threat. So you might stay away from the second place guy in chips for awhile so as to not let him catch up with a move or two over you and wait to come after him later so he thinks you are staying out of his way and playing tighter against him until the blinds raise and you are maybe 3 handed.
Raise Over Limpers Steal - "dead money grab"
One good move is to raise over several limpers when you are on the button or in one of the blinds. You can do it without necessarily having a good hand, particularly if players tend to limp in weak. Limps typically signify weakness from players who aren't particularly tricky and so you have a pretty good chance of picking up the pot provided you make a larger than usual raise.
Even if an opponent did limp with a wide range that included strength, maybe only about 5% of all hands is going to stand up to a large raise or less. SO if they limp in with 40% of all hands and only play 5% that represents 12.5% of their holdings. So I estimate an 87.5% chance of opponents to fold plus the SB and BB playing maybe 10% and 8% of all hands or less to a large raise..
# of limps % chance all opponents fold given X limpers when you act from button
1 72.45%
2 63.39%
3 55.47%
4 48.54%
5 42.47%
6 37.16%
7 32.52%
8 28.45%
Assuming break even postflop or hand checked to river:
2x the pot requires 40% equity vs opponent's calling range if you get 1 caller 100% of the time.
2x the pot requires 35% equity vs opponent's calling range if you get 1 caller 75% of the time.
2x the pot requires 30% equity vs opponent's calling range if you get 1 caller 50% of the time.
1.5x pot requires 37.5% equity if 1 caller 100% of the time.
1.5x pot requires 31.25% equity if 1 caller 75% of the time.
1.5x pot requires 25% equity if 1 caller 50% of the time.
Should adjust the above based upon number of limpers so you can develop isolation raise/blind steal range.
Use equity against the top 5% hands to determine which hands have the equity to make the move given the number of limpers
Now let's look at given the fold percentages of the reraise over the limpers how often we need to win when called to profit (assuming opponents never raise and we check down to the river)
Assuming limp in range of 40% of hands and opponents playing only top 5% of hands to a raise with 1.5 times pot bet.
You can just about play any two cards if these assumptions are correct. However, for these assumptions to be correct and opponents to fold, it may take a slightly larger raise. So what about 2x pot with same hand range?
Now you can't use any two cards, particularly as there are more limpers but you can still be pretty liberal about your hand range.
But just in case these assumptions are incorrect, what if opponents limp in with 25% of hands and call with 8% with big and small blind just as tight?
Now you may need to be more selective about your hand range. Having 33% equity over a 8% range is a little easier to find a raising hand than needing 33% equity over a 5% range of hands, but even still it's a bit more difficult to find a spot to raise. Either way, after representing such strength preflop and having the lead, I would be comfortable with my chances with less equity since a continuation bet may be able to take it down often enough to show a slight profit. While I'm probably not going to be able to put multiple bets in, a single bet on the flop or turn should win the flop often enough.
At the most conservative generalization 35% equity when called over say a 5% range of hand is strong enough. Once the first few limpers are in the hand range after that tends to be wider.
Let's actually go for 5.7% range of hands and define it as
99+,AQ+,AJs, KQs.
So what hand has a 35% chance of winning vs these hands?
99+,AJ+,ATs, KTs+
There are a lot of hands that are relatively close, particularly if our opponent is limping more hands and particularly if we can say he doesn't limp with AA or KK. Since we are probably making too conservative of assumptions intentionally, what about a more authentic situation?
What if opponent has 22-JJ,AJ,AQ, and any two broadway cards in his range? Sometimes he won't call with all of these hands so it wouldn't be surprising if he folds a lot of these sometimes and often times won't limp in with many of these hands either.
Now we can play A2+, K2+ and a whole lot of hands. Even Q2 gets 32.8% equity which is pretty close and if he limps in wider and folds more often that probably would be strong enough to raise. The problem with assuming 33% equity is enough is that often times opponents will limp raise, rather than just call and we may not have any equity and we may just have to fold. We probably won't have a strong enough hand to stand up to an all in. As such, I wouldn't steal with hands this weak, those types of hands may even be profitable due to positional advantage and it may get me close enough that if a few other variables align or if I have maybe moderate strength hands like Q9 and J9 I might make the play.
So if either the equity is only 30% or the hand range is more realistic once we're called we can profitably attempt a steal with a lot more hands.
When we don't have to worry about AA and KK, A2+ and K2+ always have at least one overcard to a pair which makes them 33% to win in most situations. Many other hands that were a few percentage points off improve substantially and become profitable. Small differences in our assumptions make very large differences which makes this very tricky.
Of course, it's really hard to conclusively say that opponent always limps weak and the earlier position limps you have to be careful with. But if the first two positions have not limped I would be a lot more comfortable raising a wide range of hands. Maybe not quite as wide as A2+ and K2+ but close. Also, unless the game is incredibly passive, I'd prefer having a bit less equity in exchange for more clarity in the decision. So 56s for this reason is better than K2 by far, but perhaps neither are playable.
This is speculative but in general I might raise with A9+,any two broadway cards,A2s+,K9s+,Q8s+,J8s+,T8s+ and possibly suited connectors over limpers. I would speculate a lot less if there is a tricky or extremely tight player who never plays a hand in early position limping or early position limps in general. I would be more willing to play perhaps even a wider range if the limpers come from middle position and late position with straight forward players that don't seem to try to trap.
Even though it's possible some of the individual hands at the bottom of the range may not have equity against opponent's calling range, it's not the worst thing in the world to have some slightly unprofitable hands in your range. It certainly doesn't add a ton of value either to have them but by making you less predictable and seemingly more loose it may give you enough additional action in marginal situations to make up the difference, but the cost is that your bluffs won't be belived as much and opponents will play back at you. In any rate, it certainly is unlikely that a few extra hands will make that big of difference and typically you can find far more spots that allow higher confidence. But if you are feeling confident about your edge over your opponents (and do not have a tendency to be over confident), it's not terrible to speculate, and it may even make money, particularly if you have a lot of extra chips or very few and can make all in semibluffs on the flop without it being a massive overbet and prevent opponent chance to see showdowns
Negreanu's Take On This Play
Daniel Negreanu's has a slightly different approach with this play. As you probably would guess his approach is a bit less mathematical and more based upon table image, perception and using this move infrequently as well as the psychology behind opponents based upon their stacks, but the math is still a consideration. He also would prefer to raise with a polarized range. Either he is making a play or he has a very strong hand. That allows him to actually just limp behind or complete from SB or check from big blind when he has the hands with some value and get value out of it that way. That's perfectly fine at an aggressive table with no knowledge of you, but at a passive one it's probably better to seek out the maximum equity in case you get called.
His strategy is more about taking one single stab at the pot and giving up where as the one detailed above is more about taking one stab and then playing some poker after the flop which probably means representing that the flop hit you when it checks to you by betting fairly frequently. If you want a balanced strategy, checking behind on dry flops and high card flops just as you would with strength to induce a bluff and weaker calls and also be more believable and consistent with your style. Checking of the flops makes it more likely to get to showdown as well when hand has value but worse hands will not continue if you bet like on an ace high flop. Since the pot will probably be quite bloated it makes sense to want to check at least one street at some point.
Floating the turn or bluff raising becomes possible if you check the flop against an aggressive opponent. Floats will allow you to represent draws on a draw heavy board and slow playing a monster on no draw boards. Raises on a draw heavy board will represent a strong hand and denying free cards while raising on a dry board will represent either that you are really strong and you hope opponent has top pair or that you have top pair, weak kicker or perhaps less and are "betting for information" to avoid putting more money in by the river if you are raised while also potentially making a slightly stronger kicker or higher pair than you fold.
The minraise to set up a bluff is a good play on the turn depending on the opponent's bet size. Most drawing hands on the turn will not have the odds to call. This also looks strong on dry boards as well. It will often set up a larger bluff while raising for information on the turn. It looks like you checked to induce a bet, and raised to induce a call. On the turn it doesn't look like you are trying to buy a free card like it does on the flop. It particularly does well on boards with all low cards or a pair and low cards even though typically you will not check these boards (but on occasion you might).
Anyways, here's Daniel on the "dead money grab"
http://youtu.be/eQT8udBfhdE
Even if an opponent did limp with a wide range that included strength, maybe only about 5% of all hands is going to stand up to a large raise or less. SO if they limp in with 40% of all hands and only play 5% that represents 12.5% of their holdings. So I estimate an 87.5% chance of opponents to fold plus the SB and BB playing maybe 10% and 8% of all hands or less to a large raise..
# of limps % chance all opponents fold given X limpers when you act from button
1 72.45%
2 63.39%
3 55.47%
4 48.54%
5 42.47%
6 37.16%
7 32.52%
8 28.45%
Assuming break even postflop or hand checked to river:
2x the pot requires 40% equity vs opponent's calling range if you get 1 caller 100% of the time.
2x the pot requires 35% equity vs opponent's calling range if you get 1 caller 75% of the time.
2x the pot requires 30% equity vs opponent's calling range if you get 1 caller 50% of the time.
1.5x pot requires 37.5% equity if 1 caller 100% of the time.
1.5x pot requires 31.25% equity if 1 caller 75% of the time.
1.5x pot requires 25% equity if 1 caller 50% of the time.
Should adjust the above based upon number of limpers so you can develop isolation raise/blind steal range.
Use equity against the top 5% hands to determine which hands have the equity to make the move given the number of limpers
Now let's look at given the fold percentages of the reraise over the limpers how often we need to win when called to profit (assuming opponents never raise and we check down to the river)
Assuming limp in range of 40% of hands and opponents playing only top 5% of hands to a raise with 1.5 times pot bet.
| # of Limpers | equity needed to steal |
| 1 | 19.39% |
| 2 | 21.65% |
| 3 | 23.63% |
| 4 | 25.37% |
| 5 | 26.88% |
| 6 | 28.21% |
| 7 | 29.37% |
| 8 | 30.39% |
| # of limpers | Equity needed to steal |
| 1 | 25.51% |
| 2 | 27.32% |
| 3 | 28.91% |
| 4 | 30.29% |
| 5 | 31.51% |
| 6 | 32.57% |
| 7 | 33.48% |
| 8 | 34.31% |
But just in case these assumptions are incorrect, what if opponents limp in with 25% of hands and call with 8% with big and small blind just as tight?
| 1.5x pot | 2x pot | ||
| # of limpers | equity needed | equity needed | |
| 1 | 23.42% | 28.74% | |
| 2 | 27.93% | 32.32% | |
| 3 | 30.99% | 34.79% | |
| 4 | 33.08% | 36.46% | |
| 5 | 34.49% | 37.59% | |
| 6 | 35.45% | 38.38% | |
| 7 | 36.11% | 38.89% | |
| 8 | 36.56% | 39.24% |
Now you may need to be more selective about your hand range. Having 33% equity over a 8% range is a little easier to find a raising hand than needing 33% equity over a 5% range of hands, but even still it's a bit more difficult to find a spot to raise. Either way, after representing such strength preflop and having the lead, I would be comfortable with my chances with less equity since a continuation bet may be able to take it down often enough to show a slight profit. While I'm probably not going to be able to put multiple bets in, a single bet on the flop or turn should win the flop often enough.
At the most conservative generalization 35% equity when called over say a 5% range of hand is strong enough. Once the first few limpers are in the hand range after that tends to be wider.
Let's actually go for 5.7% range of hands and define it as
99+,AQ+,AJs, KQs.
So what hand has a 35% chance of winning vs these hands?
99+,AJ+,ATs, KTs+
There are a lot of hands that are relatively close, particularly if our opponent is limping more hands and particularly if we can say he doesn't limp with AA or KK. Since we are probably making too conservative of assumptions intentionally, what about a more authentic situation?
What if opponent has 22-JJ,AJ,AQ, and any two broadway cards in his range? Sometimes he won't call with all of these hands so it wouldn't be surprising if he folds a lot of these sometimes and often times won't limp in with many of these hands either.
Now we can play A2+, K2+ and a whole lot of hands. Even Q2 gets 32.8% equity which is pretty close and if he limps in wider and folds more often that probably would be strong enough to raise. The problem with assuming 33% equity is enough is that often times opponents will limp raise, rather than just call and we may not have any equity and we may just have to fold. We probably won't have a strong enough hand to stand up to an all in. As such, I wouldn't steal with hands this weak, those types of hands may even be profitable due to positional advantage and it may get me close enough that if a few other variables align or if I have maybe moderate strength hands like Q9 and J9 I might make the play.
So if either the equity is only 30% or the hand range is more realistic once we're called we can profitably attempt a steal with a lot more hands.
When we don't have to worry about AA and KK, A2+ and K2+ always have at least one overcard to a pair which makes them 33% to win in most situations. Many other hands that were a few percentage points off improve substantially and become profitable. Small differences in our assumptions make very large differences which makes this very tricky.
Of course, it's really hard to conclusively say that opponent always limps weak and the earlier position limps you have to be careful with. But if the first two positions have not limped I would be a lot more comfortable raising a wide range of hands. Maybe not quite as wide as A2+ and K2+ but close. Also, unless the game is incredibly passive, I'd prefer having a bit less equity in exchange for more clarity in the decision. So 56s for this reason is better than K2 by far, but perhaps neither are playable.
This is speculative but in general I might raise with A9+,any two broadway cards,A2s+,K9s+,Q8s+,J8s+,T8s+ and possibly suited connectors over limpers. I would speculate a lot less if there is a tricky or extremely tight player who never plays a hand in early position limping or early position limps in general. I would be more willing to play perhaps even a wider range if the limpers come from middle position and late position with straight forward players that don't seem to try to trap.
Even though it's possible some of the individual hands at the bottom of the range may not have equity against opponent's calling range, it's not the worst thing in the world to have some slightly unprofitable hands in your range. It certainly doesn't add a ton of value either to have them but by making you less predictable and seemingly more loose it may give you enough additional action in marginal situations to make up the difference, but the cost is that your bluffs won't be belived as much and opponents will play back at you. In any rate, it certainly is unlikely that a few extra hands will make that big of difference and typically you can find far more spots that allow higher confidence. But if you are feeling confident about your edge over your opponents (and do not have a tendency to be over confident), it's not terrible to speculate, and it may even make money, particularly if you have a lot of extra chips or very few and can make all in semibluffs on the flop without it being a massive overbet and prevent opponent chance to see showdowns
Negreanu's Take On This Play
Daniel Negreanu's has a slightly different approach with this play. As you probably would guess his approach is a bit less mathematical and more based upon table image, perception and using this move infrequently as well as the psychology behind opponents based upon their stacks, but the math is still a consideration. He also would prefer to raise with a polarized range. Either he is making a play or he has a very strong hand. That allows him to actually just limp behind or complete from SB or check from big blind when he has the hands with some value and get value out of it that way. That's perfectly fine at an aggressive table with no knowledge of you, but at a passive one it's probably better to seek out the maximum equity in case you get called.
His strategy is more about taking one single stab at the pot and giving up where as the one detailed above is more about taking one stab and then playing some poker after the flop which probably means representing that the flop hit you when it checks to you by betting fairly frequently. If you want a balanced strategy, checking behind on dry flops and high card flops just as you would with strength to induce a bluff and weaker calls and also be more believable and consistent with your style. Checking of the flops makes it more likely to get to showdown as well when hand has value but worse hands will not continue if you bet like on an ace high flop. Since the pot will probably be quite bloated it makes sense to want to check at least one street at some point.
Floating the turn or bluff raising becomes possible if you check the flop against an aggressive opponent. Floats will allow you to represent draws on a draw heavy board and slow playing a monster on no draw boards. Raises on a draw heavy board will represent a strong hand and denying free cards while raising on a dry board will represent either that you are really strong and you hope opponent has top pair or that you have top pair, weak kicker or perhaps less and are "betting for information" to avoid putting more money in by the river if you are raised while also potentially making a slightly stronger kicker or higher pair than you fold.
The minraise to set up a bluff is a good play on the turn depending on the opponent's bet size. Most drawing hands on the turn will not have the odds to call. This also looks strong on dry boards as well. It will often set up a larger bluff while raising for information on the turn. It looks like you checked to induce a bet, and raised to induce a call. On the turn it doesn't look like you are trying to buy a free card like it does on the flop. It particularly does well on boards with all low cards or a pair and low cards even though typically you will not check these boards (but on occasion you might).
Anyways, here's Daniel on the "dead money grab"
http://youtu.be/eQT8udBfhdE
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