The Semantics of Expectation

[Rick Nebiolo]

A two-week old thread on the holdem forum regarding a Roy Cooke hand has just resurfaced. The following question regarding expectation is obviously simple mathematically; however, we are having some problems with the semantics of expectation. All help is appreciated.

The pot is $100. Your lone opponent bets $40 on the river. He either has a big hand that won’t fold to a raise or a total bluff. You have a hand that can beat a total bluff (let’s say ace high). A personal omnipotent poker genie that is always right and never lies (but won’t do math for you) tells you that there is a 50% chance your opponent is bluffing.

If you call, what is your expectation?

If you fold, what is your expectation?

What is the difference in expectation between calling and folding?

Regards,

Rick

[BruceZ]

The pot is $100. Your lone opponent bets $40 on the river. He either has a big hand that won’t fold to a raise or a total bluff. You have a hand that can beat a total bluff (let’s say ace high). A personal omnipotent poker genie that is always right and never lies (but won’t do math for you) tells you that there is a 50% chance your opponent is bluffing.

If you call, what is your expectation?

.5($140) + .5(-$40) = $50

If you fold, what is your expectation?

$0. The expectation of a fold is always 0. You are dead certain to win exactly nothing so 1($0) = $0.

What is the difference in expectation between calling and folding?

[b]$50.

To compute expectation, simply compute the sum of each dollar amount that can be won or lost multiplied by the probability of winning or losing that dollar amount for every possible outcome.

[Rick Nebiolo]

Bruce,

I'm on my way out for the day but let me ask another question. I'll check in late tonight.

Let's say a player is faced with a choice of folding or calling and calling has a positive expectation of $50. The player folds (obviously a mistake). What terms would you use to quanttify the magnitude of his mistake?

Regards,

Rick

[BruceZ]

Let's say a player is faced with a choice of folding or calling and calling has a positive expectation of $50. The player folds (obviously a mistake). What terms would you use to quanttify the magnitude of his mistake?

Folding is an error that costs $50 in ev compared to calling. That is because folding has an ev of $0 while calling has a positive ev of $50.

[Jay]

Here may be a dumb question. The ev is $50 and now suppose you had to bet $60 (maybe $20 has to go as a toke or something - as the full $60 going into the pot would change the ev), I'm assuming the correct play is a fold (since betting $60 to win ev of $50 is a losing play???)?

[Graham]

I think I was responsible for returning to haunt Rick with this one...

For your question, I think that in the common understanding of EV that, if a certain play has an EV of +$50, then that means that when you put your $60 into the pot, then you expect - on average - to return $110 to your stack. The EV is over and above the actual amount you put at risk.

G

[Graham]

This is really a response for Rick, but you mentioned it, Bruce. The problem has cropped up 'cos rick is looking at it as a relative error, whereas EV is really an absolute quantity.

[MrRothstein]

I hope I don't come across as clueless, but...
Why wouldn't you calculate the expectation on a fold like this:

.5(-140) + .5(40) = -50

Thanks

[Graham]

Because expectation is what you win or lose beyond your betting stake.

A fold is betting $0 to win/lose exactly $0. Always.
You don't need to do a calculation for the EV of a fold; folding always has EV of $0.

Your calculation is simply the inverse of the true calculation for the EV of calling. You're thinking the same way as Rick and it makes my head hurt trying to avoid the confusion... /forums/images/icons/crazy.gif


G

[MrRothstein]

I think I get it now.
The problem for me was that $0 EV for a fold did not demonstrate how bad it was in this situation. I guess you would just look at what you are giving up by folding (which is $50).
Is that right?

Thanks

[tewall]

The original equation was EV = .5($140) - .5($40) = $50. If you had an additional expense of $20 that would affect the negative half of the equation: EV = .5($140) - .5($60) = $40, so the EV would still be positive. The EV calculation takes into account how much you're investing. In this situation your expected value is .5($140) less one half the cost of your bet, so as long as you don't have to bet $140 or more your EV would be positive.

[Rick Nebiolo]

Graham,

You wrote: "You're thinking the same way as Rick and it makes my head hurt trying to avoid the confusion."

I haven't had a chance to check back this forum for a day or so but this line made me laugh so hard now my head hurts! /forums/images/icons/grin.gif

Regards,

Rick

[Graham]

Yep,

Knowing the EV doesn't tell you whether you should take a particular action. The comparison of EV's tells you which actions you should take.

First you look at the EV of all your possible options (folding, calling, raising, screaming & tearing your hair out). Then you compare those EV's to get your best course of action.

All could be positive EV - pick the best one.
All could be negative EV - pick the least bad.

Everything's relative.

G

[Graham]

Ah, it wasn't so bad really, Rick. Glad you got a laugh though.

I'm no expert on anything, so it's good for me to chew through these things. Besides, it got my post count up - pretty miserable tally so far.

All the best,
G

Ok, next topic...

[Rick Nebiolo]

Graham,

You wrote: "All could be negative EV - pick the least bad.

But not if one choice is fold and you can always chose to fold. Then the EV is always zero. Didn't you tell me that? /forums/images/icons/grin.gif

Anyway, good explanation. Now I really get it.

Regards,

Rick

[Graham]

"you can always chose to fold. Then the EV is always zero. Didn't you tell me that?:


Doh!! Can I claim I was just testing you..?