When your opponent is correct to call (TOP)

[cmwck]

I was reading this example from Theory of Poker last night which got me thoroughly confused.

page 72:
"Let's say the odds are 5-to-1 against your opponent making a hand that beats yours. By betting $20 into a $150 pot, you are offering that player 8.5-to-1 odds ($170-to-$20), and so he is correct to call the $20."

It also goes on to say that it is correct for you to bet your better hand, even though you know he will call.

From this example I calculated each player's EV:

You bet $20. Your EV: (5*170 - 1*20) / 6 = +$138
He calls $20. His EV: (1*170 - 5*20) / 6 = +$12

Together both EVs sum to +$150, which happens to be the size of the pot. Now, we know that if someone has a positive EV, someone else must have a negative EV, and all EVs for all players must sum to 0. So, there is an EV of -$150 out there unaccounted for. But, we assume both players were heads up the whole time, so at any point in time their EVs must sum to 0. But their EVs sum to +$150. How can this be?

If it's +EV for you to bet with your better hand, and it's +EV for him to call with his drawing hand, it's an impossibility that both players will be making money in the same hand. By definition, heads up one player allways looses money to the other.

[GrekeHaus]

[ QUOTE ]
But, we assume both players were heads up the whole time, so at any point in time their EVs must sum to 0. But their EVs sum to +$150. How can this be?

[/ QUOTE ]

This is false. At any point during the hand, the total EV should sum of EVs is equal to the ammount of money in the pot. We're not talking about the entire hand here, just one point. Somebody likely made a -EV bet at some point during the hand, but once money is in the pot, it doesn't matter how it got there.

[Snoogins47]

[ QUOTE ]
[ QUOTE ]
But, we assume both players were heads up the whole time, so at any point in time their EVs must sum to 0. But their EVs sum to +$150. How can this be?

[/ QUOTE ]

This is false. At any point during the hand, the total EV should sum of EVs is equal to the ammount of money in the pot. We're not talking about the entire hand here, just one point. Somebody likely made a -EV bet at some point during the hand, but once money is in the pot, it doesn't matter how it got there.

[/ QUOTE ]

Bingo. Dead money in the pot.

To elaborate: you're mixing up the expectation of the ENTIRE hand, versus the expectation of a single betting round. Despite being heads up the entire hand, mistakes earlier in the hand (or what might not necessarily be mistakes, but would indeed be mistakes if you saw your opponent's cards) can easily, and in fact, often does, make you correct to call bets on later streets. These calls have a positive expectation, while in the context of the entire hand your expectation may be negative.

For instance, to use a ludicrous example to illustrate the point.

Suppose John has Ad 2d. Efraim has Js Jc.

The flop comes Jd 5d 9s.

The pot is $100, and both players have $1000.

John bets $950 with his flush draw. Efraim raises all in, for $50. John then (correctly) calls.

Looking at it from different perspectives: On the context of the entire hand from the flop on: (assuming a 25% chance to win for the flush draw, a fairly accurate approximation)

John (flush draw):

1100 x .25 = 275

-1000 x .75 = -750

= -475


Efraim (set)

.75 x 1100 = 825

.25 x -1000 = -250

= +575

Flush-Draw John's EV is -475, Set-'o-Jacks Efraim's is +575

Now, if we examine the clearly correct call John makes to close the action:

2050 x .25 = 512.5

-50 x .75 = -37.5

= +475

John's call on his flush draw there has a EV of +475. So when isolated, his play is clearly a hugely profitable call.

But he's still losing a shitload of money throughout the course of the entire hand.

[GrekeHaus]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
But, we assume both players were heads up the whole time, so at any point in time their EVs must sum to 0. But their EVs sum to +$150. How can this be?

[/ QUOTE ]

This is false. At any point during the hand, the total EV should sum of EVs is equal to the ammount of money in the pot. We're not talking about the entire hand here, just one point. Somebody likely made a -EV bet at some point during the hand, but once money is in the pot, it doesn't matter how it got there.

[/ QUOTE ]

Bingo. Dead money in the pot.

To elaborate: you're mixing up the expectation of the ENTIRE hand, versus the expectation of a single betting round. Despite being heads up the entire hand, mistakes earlier in the hand (or what might not necessarily be mistakes, but would indeed be mistakes if you saw your opponent's cards) can easily, and in fact, often does, make you correct to call bets on later streets. These calls have a positive expectation, while in the context of the entire hand your expectation may be negative.

[/ QUOTE ]

It's possible also that both people could be making +EV plays through the entire hand, and that all the money comes from the blinds/antes that were in the pot before the hand. For instance, consider an example from limit hold 'em.

Somebody in late position is dealt A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/club.gif[/img]. Everybody folds to him and he raises. The BB has Q[img]/images/graemlins/heart.gif[/img] J[img]/images/graemlins/heart.gif[/img]. Getting 3.5:1, he calls correctly.

Flop comes A[img]/images/graemlins/diamond.gif[/img] T[img]/images/graemlins/heart.gif[/img] 9[img]/images/graemlins/heart.gif[/img]

Now, the BB has a slight edge. He bets and LP raiser calls.

A blank hits on the turn, so LP now has the edge. BB checks and LP bets, BB calls. River is a blank BB checks, LP bets and BB folds.

The entire hand has gone by without either person making a -EV bet.