[Mathieu]

As promised, here is Aaron's analysis:

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Aaron,

I know this is off topic, but I have theory question regarding the river bet.

"Why do we (many of us do it) bet the river HU when our opponent has shown a lot of strength and we think that if we get raised it will be bad news."

I think if the raise will mean that we are more likely to be behind, then we should just check-call. This way we don't allow our opponent to get 2 bets in when we are behind.



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But then you also don't win a bet when villain has a hand which is worse than your and checks behind. In this case, I don't want him to check behind with two pair because he's afraid of the flush.

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My point is that it seems like we often bet hoping that our opponent will "just call" in situations where our opponent got the last raise on the turn and is obviously going to bet if we check.

So I think that for the bet/call to be better than the check/call we have to think that a raise by our opponent will imply that our winning chances are around 50%-65%. If there are any higher we should bet-reraise and any lower we should check-call.

Do you agree?


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The short answer is that the bet-call and the check-call lines are both close in value, and it goes either way unless you have extra information. HPFAP suggests that you want to believe you have the best hand 55% of the time when you are called to make a river bet correct. Since I'm always getting called here, that would imply I want to have the best hand 55% of the time to make betting correct (55% instead of 51% to cover the times I get re-raised and lose).

But that's only an estimate based on your assessment of your hand ONLY and (as I recall) is more about value betting than value calling (inducing a bluff... or in this case, getting villain to bet a strong hand that's worse than yours). I'm not going to worry about the bet-3-bet line because you've got to have a very very strong hand to get capped on the turn and still 3-bet the river.

Here are the four things that can happen (Hero is OOP):

1) check-check
2) check-bet-call
3) bet-call
4) bet-raise-call

We need to assign various probabilities to these. To do this, we need to give villain a hand range:

Hands that beat me
AA = 3 ways
JT = 16 ways
Flush = 3 ways (estimating that it's about a likely as AA based on the action)
Total = 22 ways

Hands that I beat
99 = 3 ways
88 = 3 ways
AQ = 3 ways
A9 = 9 ways

A8 = 9 ways
Q9 = 3 ways
98 = 9 ways
Total = 39 ways

This is about 65-35 for me to be winning. Since I'm not folding this, all we need to do is compute river EV on the extra bets we're putting in.

1) check-check -- This happens is more a function of how villain plays than what he holds. I suspect this gets checked through 20% of the time when villain gets cold feet because of the flush. Checking through is 0 EV because I don't win or lose any bets on the river, and it's just a showdown computation.

2) check-bet-call - So the 80% of the time villain bets the river, and I win 65% and lose 35% of these cases.

What is the EV of checking?

EV = 0 + (.80*.65)*(1) + (.80*.35)*(-1)
= .52 - .28
= .24

The betting line is much harder to compute because we just don't know what villain's playing style is. To make things concrete, we'll assume he raises A9 and better and calls everything else. You can tweak this around to see what other assumptions give.

3) bet-call - Villain here is on one of the 21 ways of holding two pair worse than A9. This is 21/61 of the possible hands. We win one bet from this situation.

4) bet-raise-call - Villain does this with the rest of his hands. We win two bets out of 18/61 of these and we lose two bets with 22/61 of these.

What is the EV of betting?

EV = (21/61)*(1) + (18/61)*(2) - (22/61)*2
= .34 + .59 - .72
= .21

Obviously, you don't do this in the game. And you can see how very close in value this runs (espcially considering that .03 BB is a tiny fraction of the 12 BB pot to start the river action.

What are the key factors?

1) How often does villain check behind? If he checks behind more, EV of checking goes down.
2) How aggressive is villain? More aggressive villains will call your river bet with fewer hands and re-raise more of them. This means you get two bets with the best hand more often than you get just one.
3) What range of hands does villain have? In this case, if you were to take away the flopped straight possibility, betting goes way up in value because now the number of hands which beat me is greatly diminished.

That was probably about ten times longer than you expected.