Freeroll revisited.

There was a comment to my last post, about whether or not villain should call my push knowing that he was splitting the pot, or losing it to a possible flush. It seemed wrong to me to fold the nuts with these stack sizes, but it is a situation where it is easy to calculate the correct action.

Let's put ourselves in his position. We know we both have the ace high straight, and that we are losing to a made flush. Our equity in the pot is 39.8%. Pot size is 1220. Bet size is 1514. The expected value of calling the shove can simply be calculated as:

EV(call) = equity * (pot size + 2 * bet size) - bet size = $177.

So if we fold here, that is what we are costing ourselves.

I messed with the bet size to figure out how large a bet we can profitably call. It turns out the sweet spot is about $2380. So if the effective stacks had been $900 larger, our villain should have folded, unless he thought there was a chance I was bluffing, or that I didn't in fact have a flush draw. If these probabilities are estimated to be significant, the bet would have had to be much larger to correctly fold. I suppose it is somewhat reassuring that the theory tells us not to go about folding the nuts too often, even with redraws on the board.

Situations were you can actually precisely calculate the correct action come up rarely in poker, but this was one of those cases.