Hand Evaluation

To assess the quality of a hand after the flop, the Hand Evaluator combines together the strength and the potential of the hand in a value called effective hand strength (EHS). The EHS is an estimate which gives the probability that the given hand is currently the strongest one, or that it will become the strongest one by the showdown with the next community cards (potential).

$$EHS = hand\_strength + (1 - hand\_strength) \times hand\_potential$$

To calculate a hand's strength (HS) against a single opponent, the Hand Evaluator enumerates all the possible opponent hands and sums the weights of the hands that would win, lose or tie the given hand. Recall that the weight of a hand is the probability that an opponent would still be active with that particular hand.

$$HS = \frac{ahead + tied / 2}{ahead + tied + behind = total\_number\_of\_hands}$$

For instance, assume that Loki's hole cards are A $\heartsuit$-8 $\heartsuit$ (the same as in the above preflop example), the community cards on the flop are 9 $\heartsuit$-8$\clubsuit$-2$\clubsuit$, and all weights are equal to 1 (uniform distribution). From the $47\choose2$= 1081 possible opponent's two-card hands on the flop: 903 hands lose against Loki's hand, six hands tie and 172 hands defeat Loki's^3.1. Therefore, Loki's HS is

$$HS = \frac{903 + 6 / 2}{903 + 6 + 172 = 1081} = 0.84$$

To extrapolate the hand strength value to multiple opponents, the Hand Evaluator raises it to the power of the number of opponents still active in the game (HS_n). For the above example, if there are four players active in the game (including Loki), the Hand Evaluator calculates HS _n = (0.84)³ = 0.59.

The potential of a hand can be either positive or negative. Positive potential (PPOT) is the probability of a hand becoming the strongest one when it is behind. Negative potential (NPOT) is the probability of a hand falling behind when it is ahead. Both potentials are calculated by enumerating all possible opponents' hands and community cards to come in the next rounds. Potential calculations on the flop can be done by looking ahead either one round (considering the 45 possible cards on the turn) or two rounds (considering the $45\choose2$=990possible two-card combinations on the river). PPOT is calculated by adding the weights of the cases where Loki's hand improves.

$$PPOT = \frac{be\_behind\_end\_ahead + be\_behind\_end\_tied /... ..._tied\_end\_ahead /2} {total\_be\_behind + total\_be\_tied /2}$$

NPOT is given by:

$$NPOT = \frac{be\_ahead\_end\_behind + be\_ahead\_end\_tied / ... ..._tied\_end\_behind /2} {total\_be\_ahead + total\_be\_tied /2}$$

Consider the same above example. Table 3.1 shows the number of cases where Loki's hand situation at the flop changes (or remains the same) by the time the other two community cards are dealt. Assuming uniform weights, this table shows the sum of the weights of all the cases. $Total\_be\_ahead$ is equal to the number of cases where Loki's hand on the flop is the strongest one multiplied by the number of possible next two-card combinations ( 903 * 990 = 893,970). $Total\_be\_tied$ is 5,940 (6 * 990) and $Total\_be\_behind$ is 170,280 (172 * 990). The total number of cases enumerated in the potential calculations is 1081 * 990 = 1,070,190. Thus, PPOT for Loki's hand (A $\heartsuit$-8 $\heartsuit$) is

$$PPOT = \frac{37,659 + 51 / 2 + 270 /2} {170,280 + 5,940 /2} = 0.22$$

and NPOT is :

$$NPOT = \frac{170,249 + 1,258 / 2 + 90 /2} {893,970 + 5,940/2} = 0.19$$

 

Table 3.1: Number of cases where Loki's hand situation changes after two community cards are dealt

Flop's situation	River's situation	Number of cases
	Ahead	722,463
Ahead	Behind	170,249
	Tied	1,258
	Ahead	37,659
Behind	Behind	132,570
	Tied	51
	Ahead	270
Tied	Behind	90
	Tied	5,580