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.