Help understanding the Expected Value equation when applied to poker

Question

I don't play or follow poker, (just thought I'd declare that), but found it interesting from speaking to a few people that do that their way of calculating the Expected Value, $E(x)$, didnt account for the money they had paid up front to play. I asked why and was basically told it doesn't matter. I can't understand why.

Lets say person $A$ and $B$ are playing against each other and it is known player $B$ will fold $35$% of the time, meaning $65$% of the time they won't.

So a hypothetical game is played and each time Player $B$ doesn't fold Player $A$ has really bad luck and looses. So Player $A$ is loosing $65$% of the time and Player $B\ 45$% of the time

$35$% of the time Player $A$, will win $\$20,(\$40$ in the middle - $\$20$ dollars of their own they put in).

$65$% of the time Player $A$ will loose $\$20$ dollars

Therefore: I calculated $E(x)=(0.35*20)-(.65*20)=7-13=-6$.

However I have been told that is wrong

The people I ran this by said when calculating the loss value I should not consider the $\$10$ put in as the initial playing fee. Meaning the calculation should be

$E(x)=(0.35*20)-(.65*10)=7-6.5=0.5$.

To play each player puts in $\$10$ and then at the point where they raise they raise by $\$10$. So each player has put in $\$20$ when it comes time to show their hand (there is now $\$40$ in total sitting in the middle).

This makes no sense to me, despite it looking good by not counting the $10 they are not seeing that their funds are diminishing....

Or have I got this wrong? For the life of me I can't work out why they said not to count the $\$10$ that is put in at the start. One justified it by saying you need to pay the $\$10$ to be in to win the $\$20$ profit. Yes this is true they need to pay the $\$10$ to play and have a chance of making $\$20$, over time it appears they'd loose money??

Is anyone able to clear this up for please ..thanks (or should I be asking in a different forum ?)

Answer 1

I think the point you're failing to consider is really an economic one. It's the notion of having a sunk cost. Let me give you an example:

Suppose you have already bet \$10 on a particular hand (or it's part of the blind or whatever) and you are faced with the following choice:

• Fold and get no payoff
• Call a further bet of \$10 (i.e. put ten more dollars in) in which case you will win the pot of \$50 with probability 3/10.

If you call the bet, then your expected payoff is \$15 and you paid \$20 dollars total, so it seems like a bad idea because you're losing \$5 in expectation. However, if you folded, you would still have lost \$10. This is a sunk cost, because it's one that you have already incurred - that's why you only should consider the payoffs going forward. And, given that, you're actually better off calling the bet.

In terms of how their wealth is changing as they play, obviously the sunk cost still makes them less wealthy, but since it has already been committed to, it should not be considered in the choice of actions going forward in the game.