3/01/2006

03-01-06 [poker] - 1

03-01-06 [poker]

On the river the pot has $60 in it, he bets $40 into you. You have $60 left. Should you fold, call or push? Well, any of them is reasonable. Many people make the mistake of thinking that calling & pushing are the same decision here, they are not. In order to call, your hand must be good >= 29% of the time. You won't have much left, but there's no need to put in any more chips unless you're good >= 50% of the time.

Another common case is when you have something like two pair on the river, but the flush card has hit. He checks. You think he either has one pair or a flush. Should you try to get in a value bet here? The crucial factor here is actually whether he will raise with anything but the flush. If he will only raise with the flush, you can fold to a raise and you should value bet. Let's look at that. Let's say 75% of the time he has a pair and 25% of the time he has a flush. The pot is currently $100. You bet $20, which is the most he'll call with a pair. With a pair he'll call, with a flush he'll raise and you fold.

EV(check) = 0.75 * 100 = $75

EV(bet) = -20 + 0.75 * 140 = $85

Obviously. But now, what if he'll raise without the flush? What if P of the time he has just a pair he pushes for $100 more and he also does that with the flush? First of all, should you call that? The pot is $140 and you have to call $100 to win $340. You win (0.75*P)/(0.25 + 0.75*P), so

EV = -100 + (0.75*P)/(0.25 + 0.75*P) * 340

Let's say he chooses P so that it's neutral whether you call or fold (which is the correct game theory thing to do).

100/340 * (0.25 + 0.75*P) = (0.75*P)
100/340 * (1/3 + P) = P
100/3 = P * 240
10/(3*24) = P
5/36 = P = 13.889%

So, in the case that he pushes you just fold and it's zero ev. Assuming he still calls in the other cases you get :

EV(bet) = -20 + 0.75*(1-P)*140 = $70.4

That's worse than the EV of just checking, and that's still assuming that he calls your bet. If he just folded his one pairs it would be disastrous. Say for example he folds his one pair 50% of the time, then your EV is :

EV(bet) = -20 + 0.75*(1-P)*(0.5*140 + 0.5*120) = $64

Note that this is even worse than if they just always folded their one pair and always pushed their flush. Of course we could get some value back by betting more bluffs in addition to two pairs which would make it neutral for them to fold their one pairs, etc. etc.

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