Test

Originally Posted by daviddem

At first glance this seems to be a violation of the fundamental theorem, but it's not.

Let's not only look at our EV but also the one of Ah8h:

1) if 9c5c calls (he makes a mistake)
- our EV 12.3421
- Ah8h EV -0.30758
- 9c5c EV -0.03632

Note that the total is $12 because that is the pot the players are fighting for to start with.

2) if 9c5c folds (he plays correctly):
- our EV 12.7111
- Ah8h EV -0.7111

And obv. the total is stilll $12.

So what happens when 9c5c folds is that he dumps his negative EV towards the other two players. So both players as a group loose from 9c5c's correct play, and the fundamental theorem of poker has not been invalidated.

The reason that our EV actually increases is that in the process, Ah8h's EV has decreased by more than the amount of total negative EV from 9c5c. This is because another effect of 9c5c folding is to rebalance the showdown equities between us and Ah8h, and our equity benefits much more from his fold than Ah8h's equity does.

Every time you play a hand differently from the way you would have played it if you could see all your opponents' cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.

The fundamental theorem of poker is not based on the total EV as a group, but only our EV. While your assessment that the total EV of all the players left in the hand stays the same (which is true because it's a zero-sum game), this doesn't hold the fundamental theorem intact.

In game theory we call this implicit collusion, and is a big part of why multiway games are so difficult to study.

Additionally, the fundamental theorem holds 100% of the time in heads-up situations, but it doesn't hold 100% of the time in multiway situations.

For further reading, see: Morton's theorem - Wikipedia, the free encyclopedia