Analysis of folding KJs

I attempted to reconstruct the chip counts before hand #117 from the cardplayer live update archive. I came up with the following chip counts. The blinds were 200K-400K with a 40K ante. Because the available information is neither entirely precise nor entirely consistent I had to occasionally perform minor rounding to keep the chip total at $22.6M.

Now Hasan posts the 400K BB, Maxfield the 200K SB, and each antes 40K.

There are three scenarios after Tuan moves all-in. The EV column is their expected win considering only the money in the top two spots; since they have each already locked up third place money, they are now playing a new tournament paying $0 for third.

If Hasan's probability of winning when calling with KJs is P(w) then the EV of calling is equal to that of folding when:

1.49M * P(w) + 1.09M * (1 - P(w)) = 1.25M

Solving for P(w) yields 40% exactly.

So Hasan should call if KJs is better than 40% to win against the hand range he gives Tuan. Here are a few example hand ranges and how KJs fares against them. Keep in mind as you estimate Tuan's range that he had fewer than five big blinds and was outchipped nearly 5-1 by both other players, so he had virtually no chance of sneaking up a spot. He had to play a hand, and this one was his last realistic shot at moving in without being called. If he folds, next hand he will have to post nearly 25% of his chips in the big blind.

So for Hasan to fold if his EV = EV(Hasan), he has to grant Tuan an extremely small range of strong hands.

Now look at Maxfield's EV in the three scenarios. If Hasan calls and wins he earns $140,000 compared to Hasan folding; if Hasan calls and loses he loses $40,000. That means Maxfield wants Hasan to call if he has even a 22% chance of winning.

If Hasan has more than a 22% chance of winning and folds, EV(Maxfield) suffers, and suffers more the higher the percentage goes. If Hasan has more than a 40% chance of winning and folds, EV(Hasan) suffers as well as EV(Maxfield).

SO WHAT'S THE POINT?

The point is that if everyone is self-interested, Hasan has to call when P(w) > 40% or he is losing money. But if he is getting an EV rebate thanks to owning part of Tuan, then he is not hurt nearly as badly as P(w) grows past 40%, while Maxfield IS hurt just as badly.

Correct play is altered even more dramatically if Hasan has less than 100% of himself while having 50% of Tuan. This is not at all unusual because of backing arrangements. If Hasan has only 35% of himself but 50% of Tuan, when should he call with KJs? Three guesses.

NOTES

I calculated tournament EV using a formula from this rgp thread which was simple enough not to discourage me from doing this while adequately reflecting my understanding of tournament EV.

Three players (a, b, and c) remain in a tournament, with chip counts of Ca, Cb, and Cc, where T = Ca+Cb+Cc. The probability of player a finishing in each spot are estimated as follows:

P(a1) = Ca / T
P(a2) = P(a2/b1) * P(b1) + P(a2/c1) * P(c1)
P(a3) = 1 - P(a2) - P(a1)
where P(a2/b1) = Ca / (T - Cb)
and P(a2/c1) = Ca / (T - Cc)

Similar calculations apply to the other two players. The value of finishing in each spot was as follows:

V(1) = $1,959,775
V(2) = $802,015
V(3) = $0

I used this openoffice spreadsheet to do the calculations. This is the very first spreadsheet I've made in my entire life so I'm sure there were more elegant ways to do it. If you manipulate the values in B2-B4 you will see their tournament EV recalculated on the fly.

With this many calculations it's always possible I've made an error somewhere so I'd be glad to hear about it if anyone finds one.

I'm going to police the comments on this one to keep it readable; there is too much noise lately and this isn't a good place for basic questions. If your comment disappears, that's why.