I've Never Felt So Sexy...

I see weak players check top pair on the river all the time. I have heard many 'tight' players erronously tell me to "never bet one pair on the river for value" (I of course always told them how right they were!). However, it is true he could easily have folded a worse hand here.

[W. Deranged]

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I see weak players check top pair on the river all the time. I have heard many 'tight' players erronously tell me to "never bet one pair on the river for value" (I of course always told them how right they were!). However, it is true he could easily have folded a worse hand here.

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I agree. It's amazing how often, particularly in live games, that players will check good top pair hands and even better on the river.

The question of how often villain is folding a better hand and how often he's folding a worse hand is really not that important. The important questions are:

1. How often is our raise correct for value against "the Nut"?

2. Will "the Unknown" fold a better hand often enough compared to the times he calls with a better hand to make the play worthwhile?

Think of it this way:

If we have a hand that beats the Unknown, the primary consideration for whether we would raise or fold is how our hand stacks up against the Nut. Hand-reading suggests that he probably doesn't have a K or better, and so, unless he's on a big slowplay or hit a weird river, we probably have him beat. So those times that villain has a worse hand, raising is a viable option and in fact probably a bit better than calling (and certainly not significantly worse).

So, one way to simplify the model here is simply to say that whatever percentage of the time that villain has a worse hand, raising and calling are essentially equal EV. (They obviously aren't, but they're probably close or at least raising is better so we can be "conservative" and say they're equal).

So we only need to consider the situation when villain has a better hand.

Let's say that if we call, villain will always call with a better hand (a pretty reasonably assumption here; rarely might villain check AQ here with the intent of not overcalling or something like that).

Let's say that if we raise he'll call with proportion c . (In other words, he'll call 100c% of the time).

With the pot ~12 BB when we are facing a call, we need to compare the EV of calling and raising:

If we call, we lose 1 BB every time he has a better hand.

If we raise and villain has a better hand, we lose 2 BB the c proportion of the time he calls, and win 12 BB the (1-c) proportion of the time he folds. So, we simply need to satisfy:

-2c + 12(1-c) > -1

Which reduces to c < 13/14; namely, villain just has to fold 1 out of 14 times with a better hand to make this worthwhile.

Unless he c-three bets your ass with a hand that you beat and you fold the winner :P