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## Hand Playability

17/05/2018 by Michael Acevedo
Hand Analysis
D&B MAGAZINE

If there was no betting in poker and players were forced to go all-in every hand, the expected value for each player would be their hand equity times the pot.

Ev=Eq*Pot

Example: In a gambling HU game each player has to ante \$100, and there is no future betting, Player1 gets dealt 44, and player gets 98 What is each player’s expected payoff?

44 Equity = 47.26% Player1 Ev=47.26%*\$200=\$94.52

98 Equity = 52.74% Player2 Ev=52.74%*\$200=\$105.48

In this toy game both players get to realize 100% of their equity, but in real poker games where betting can happen across multiple streets, if a player folds his hand before showdown he is forbidding his equity in the pot and that equity will go to the remaining players. This dynamic is known as Equity realization.

Equity Realization (EqR) EqR refers to the fraction of Equity that is materialized in Ev. Hands that capture a bigger percentage of the pot than their equity share are said to over-realize their equity, and hands that capture a smaller piece of the pot than their equity share are said to under-realize their equity.

Example: Game: \$55 9-max online MTT Stacks: UTG= 50bb, BB=40bb Players: 9 (12.5% ante) Preflop: (2.625bb) UTG is a regular who raises 2bb, folds to Hero who is on the BB with 95 and has a decision.

UTG is a regular, so we assume he is opening a standard 16% Range that can be easily defended vs 3bets so, 3bet bluffing with 95 is out of the question and the decision is between calling or folding.

If we were calling for all our chips, the solution would be simply to compare the pot odds to our hand’s equity and call if our hand’s equity is greater than the pot odds:

95 Equity vs UTG Range = 29.5%

Pot Odds= Risk/(Risk+Reward)

Pot Odds= 1bb/(1bb+(2.625bb+2bb))= 1/5.625= 18%

Clearly our hand equity is a lot bigger than the pot odds, but since we are not all-in, if we make the call we have to play post flop and various unpleasant things can happen. Villain can bet aggressively and sometimes force us to fold the best hand, he can make a stronger hand than ours and we lose a big pot, or we could flop a monster but get no action, or even occasionally stack the villain, but unfortunately Raw Equity does not account for any of these possibilities.

How can we know if our call is +Ev if we have no idea of the type of situation we will end up facing?

The Equity Realization Factor tells us how much Equity we can expect to realize on average across all possible scenarios so, if we know both: a Hand Equity (Eq) and Equity Realization Factor (EqR) we can calculate the hand expectation for more complex scenarios including post flop play. Ev Call=EqR(Eq*Pot)

Conversely if we know the hand Equity and Ev we can solve the Ev equation for EqR:

EqR=Ev/((Eq*Pot))

Rescaling Ev to %Ev:

Ev/Pot=%Ev

Equity Realization simplified formula:

EqR=(%Ev)/Eq

Calculating each poker hand EqR for every single spot is almost impossible as it would require having the solution of the entire game of poker, but there are a couple ways we can get very good approximations.

1. We could use empirical data by taking a sample of millions of hands played in online games and compare their equity to the fraction of the pot they capture in every spot.
2. Use modern solvers to get each hand EqR on various spots and average the results.

Single Raised Pot Average EqR Factors (Generated with Pio Solver)

Going back to our Example, using 95o EqR factor from heatmap in the Ev equation:

Ev Call=EqR(Eq*Pot)-Risk

Ev Call=0.58(0.295*4.625)-1

Ev Call=-0.21bb

Calling with 95 has negative expectation, therefore it should be folded.

Another way to tackle this problem is to start by finding out how much equity our hand 95 needs to realize for our call to break even:

Min EqR= (Pot Odds)/(Hand Equity)

Min EqR= 18/29.5

Min EqR= 61%

95 EqR Factor 58% is 3% lower than the minimum 61%, which means that even if we have a lot of equity we just can’t realize enough of it post flop to make this call worth it.