Part 19: Calling to Close the Action in Split Pot Games

Warning: This is math-intensive. Feel free to skip if you don't play split games.

Okay this is pretty cool and I'm going to pretty much just be posting this because I think it's neat. The general idea of split-pot games is that we have two equities, one for the high pot and one for the low pot, where the high pot and low pot are half of the total pot each. In games with a qualifier for the low, sometimes there won't be a low pot, which adds another layer of complexity to the EV calculation.

So let's suppose we're playing a game where there is a high/low split with no qualifier, someone shoves and we call. Let's call our high pot equity 'a', and call our low pot equity 'b'. Also, we're calling a bet of 1 with a pot of P when it's our turn to act. To find the EV of this call, first let's find the possible outcomes and our profit for each.

Outcome 1: We win both pots. The chance of this happening is ab. We will profit P.
Outcome 2: We win the high pot and lose the low pot. The chance of this happening is a(1-b). The profit here is a little tricky. Suppose the pot was 5 bb and we call our 1 bb, then we get 3 bb back. Our profit is (5+1)/2 - 1. So our profit is (P+1)/2 - 1.
Outcome 3: We win the low pot and lose the high pot. Same profit as outcome 2, but the chance of it happening is b(1-a).
Outcome 4: We lose both pots. The chance of this happening is (1-a)(1-b) and our profit is -1.

As always, we'll add all of these possible outcomes together to get our EV:

EV = abP + a(1-b)((P+1)/2 - 1) + (a-1)(b)((P+1)/2 - 1) + (1-a)(1-b)(-1)

This is a bit hairy because of that (P+1)/2 - 1 bit. Let's break it down to something easier to work with:

(P+1)/2 - 1
(P+1)/2 - 2/2
(P+1-2)/2
(P-1)/2

If we do the substitution we get:

EV = abP + a(1-b)((P-1)/2) + (1-a)(b)((P-1)/2) + (1-a)(1-b)(-1)

Which is still a bit hairy, but we can do it:

EV = abP + a(1-b)((P-1)/2) + (1-a)(b)((P-1)/2) + (1-a)(1-b)(-1)
EV = abP + a(1-b)(P-1)/2 + (b)(1-a)(P-1)/2 + (1-a)(1-b)(-1)
EV = abP + a(P - 1 - bP + b)/2 + b(P - 1 - aP + a)/2 + (-1)(1 - a - b + ab)
EV = abP + aP/2 - a/2 - abP/2 + ab/2 + bP/2 - b/2 - abP/2 + ab/2 - 1 + a + b - ab

Let's look at the break even point (i.e.: when EV=0).

0 = abP + aP/2 - a/2 - abP/2 + ab/2 + bP/2 - b/2 - abP/2 + ab/2 - 1 + a + b - ab
0 = abP + aP/2 - abP/2 + bP/2 - abP/2 - a/2 + ab/2 - b/2 + ab/2 - 1 + a + b - ab
0 = aP/2 + bP/2 - a/2 - b/2 - 1 + a + b
0 = aP/2 + bP/2 + a/2 + b/2 - 1
0 = aP + bP + a + b - 2
0 = P(a+b) + (a+b) - 2
0 = (P+1)(a+b) - 2
2 = (P+1)(a+b)
1/(P+1) = (a+b)/2

Remember that 1/(P+1) is just our standard bet/(bet+pot). Also notice that (a+b)/2 is just our average equity between the two. Therefore, when our average equity is greater than bet/(bet+pot), then we should call. (Notice also that this is a more general form of our earlier equation for finding the break even point for calling in a game without a split pot.)