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TD2-7 second-round equities
I have just finished a tool that will calculate hot + cold heads-up equities for triple draw hands on the second draw. That is, how often will a given hand win the pot, when there is no more betting?
The decision to break or pat on the third draw is calculated by determining the game-theoretic optimal given the starting hands. (The alternatives are to use a strategy that works well against a range of opposing hands--- but what range?--- or to assume perfect drawing. This is sort of a compromise in that the pre-2nd-draw hands are known but later draws are not known.) Some sample hands. The 2nd column shows the 2nd-player equity on the 2nd round, and the 3rd column gives the equity for the same draw on the 3rd round (for comparison.) 7432 vs 7432: 0.505 0.500 7432 vs 8432: 0.448 0.462 7543 vs 7542: 0.627 0.608 7432 vs 9643: 0.391 0.433 9643 vs 7432: 0.617 0.567 7654 vs T732: 0.537 0.590 6542 vs 8732: 0.511 0.520 8732 vs 6542: 0.500 0.480 Here are the strategies for the last matchup. "break" means to pat if the other player is drawing, but draw otherwise. The 1's mean "do this 100% of the time", i.e., it is a pure strategy. Typically only T's and 9's will have a mixed strategy. First player: 8732 0.500015 2 draw 1 3 draw 1 4 pat 1 5 pat 1 6 pat 1 7 draw 1 8 draw 1 9 pat 1 T pat 1 J pat 1 Q draw 1 K draw 1 A draw 1 Second player: 6542 0.499985 2 draw 1 3 draw 1 4 draw 1 5 draw 1 6 draw 1 7 pat 1 8 pat 1 9 pat 1 T pat 1 J break 1 Q break 1 K draw 1 A draw 1 The calculator cannot handle situations in which it is correct to draw two, nor multiway pots. I find it interesting that hands typically get closer in value going to the last draw. The exception typically is draws with straight potential vs. a rougher draw--- the straight is less likely to make a good hand but the rough draw may be dead when it comes in. |
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