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
No comments:
Post a Comment