HM Quickie 6. A Multiway EV Analysis
Multiway EV analysis is complicated. Typically, there will be a main pot and one or more side pots dependent on the number of players and how stack sizes are distributed. Yet, it can be done if one in his study is willing to do the math. We present here an example of a 3-way hand a poster on a poker forum asked about:
“If my stack is 10, Villain A stack is 10, and Villain B stack is 5, I can work out the EV for when I go all-In and just A calls, or the EV when I go all-in and just the B calls, but I don't know the proper way to do it for when they both call.”
Admittedly this is a fairly simple multi-way problem compared to some others but one that is useful for indicating the approach. We’ll make the following assumptions:
Pot = 1. Hero goes all in with a bet of 10.
Hero hand/stack: AKs /10. Villain A range/stack = 15% / 10. Villain B range/stack = 30% / 5.
First, we obtain Hero’s equity for the following cases using a hand evaluator such as Equilab.
H vs. A vs. B= 46%, 29%, 25%. (A wins, B wins, C wins)
For the relevant two-way contest, we have using the H and A equities,
H vs. A = 46 / (46+29) = 61%,
Note this result is very close to what you get if you use Equilab in a two-way confrontation (63%) but that would not account for card removal of the third player, B.
Based on the stack sizes, the 3-way or main pot is 1 + 10 + 5 = 16 and the 2-way side pot is 5. The side pot occurs if B wins the hand leaving H vs. A fighting for it.
The following table summarizes the situation:
Note: Sum of probabilities = 100%
The EV equation is as follows:
EV = Sum [Pr(Win)*Win$] – Sum [Pr(Lose)*Lose$]
EV_hero = 0.46 * 16 + 0.15 * 5 – (0.29 + 0.10) * 10 = 4.2
Including fold possibilities is not much more complex. However, more than three players as well as including raise possibilities could make such analysis not very practicable. However, 2 or 3 players closing the action is probably the most likely scenario and hopefully this post indicates how one might go about evaluating multi-way EV.