How do you value bet the river?

[Pokey]

Lately I've been adjusting my game to try and squeeze more EV out of my winning hands. I've heard the advice from other players that our opponents are nimrods and we can always value-bet for more than we think. The conclusion was that we should bet more than is "reasonable" to call, knowing that they'll call often enough to make it worth our while.

I question that wisdom.

I used to value bet the river for 2/3rds of the pot when I had a great hand. Sometimes my opponents would fold, yes, but when they called I made quite the bonanza. Everything seemed right with the world, but my winrate at showdown was lower than I preferred and my winrate in PTBB/100 wasn't where I wanted it to be.

I've changed my gears lately, to good success. These days, I'm betting about 1/2-pot or slightly less at the river when I've got a good hand. My results have been very strong for a few different reasons:

1. A smaller bet is easier to call. As a result, the strength of the average calling hand is lower, meaning that I win more often when called. A big bet that is ONLY called by a hand that beats you is not necessarily a better bet than a small bet that can be called by hands that you beat.

2. With a REALLY great hand, you often have the deck so crippled that your opponents can't have much. A big bet folds them out almost every time, whereas a small bet might get more "meh, maybe he's bluffing and my QQ holds up" crying calls.

3. Ignorant, unattentive, and aggressive opponents can misinterpret a modest bet as weak, and come over the top on you, even though they would have folded to a solid bet. This is an EXTREMELY +EV thing to have happen to you.

4. When your opponent has absolutely nothing on the river, he will often fold to ANY bet. Making smaller value bets means that your river bluffs can be smaller in size; that means that your bluffs don't have to succeed nearly as often for them to be profitable for you. Example: if your river bets and river bluffs are both 2/3-pot, your opponent must fold 40% of the time for the bluff to be profitable. If your river bets and river bluffs are both 1/3-pot, your opponent only needs to fold 25% of the time for the bluff to be profitable.

Obviously, I'll vary the size of my bet based on my opponent and the board -- if my opponent goes to town with garbage, I'm comfortable value-betting larger, but if he's relatively tight then those smaller river bets really pay off in the long run.

I'm very curious to hear other people's thoughts on the subject.

[BalugaWhale]

It depends.

The other day I flopped a K-high flush, and the turn was the A, giving me the nut flush. WHen I bet the turn and was called, I knew that villain had a decent flush. So, I overbet pushed the river and knew he wouldn't fold.

However... with a TPTK hand, I've felt like 1/2 pot is best on the river. 1/2 pot does get called a lot, though i'd be hesitant to go below 1/2 pot (a lot of missed value).

It really depends on what you have, what villain will call with, and how much he'll call.

[NinaWilliams]

IMO its very dependant on the opponent and the board. 1/2 is a good starting point, and ill bet more or less depending on how strong I think his hand is is, and how loose I think he is.

[cbloom]

I've been thinking about bet sizing and "The Price Is Right". Say you know you have the best hand on the river and he
has something mediocre. You want to bet as much as possible such that he'll call. The more you bet (that he calls)
the more profit you make. If you bet too much, he'll just fold and you make $0. This is like the "Price is Right" in that
you want to get as close as possible to his maximum call amount, but without going over.
Your profit as a function of bet
size is :
<P>
P(B) = (B < B_max) ? B : 0
<P>
The problem is you don't know B_max, but you can estimate it with some error. Let's say you know it's between L and H,
with an even probability of being anywhere in there. What should you bet to maximize profit? Clearly it's somewhere in
the range [L,H] , but where exactly?. Maybe I'll do the math and figure out where exactly.

<P>

Well, Sean beat me to the math and sent me this :

<P>
<PRE>
Is it clearly in the middle? Clearly we have the boundary cases,
let N be the bet amount:

N = L -> E = $N
N = H -> E = $0

So you might hope that it's a curve that grows as N increases,
and then starts decreasing again. But that might be overwon by
the person's tendency to fold. Now, if L is 0, and H is non-zero,
clearly there must be some > 0 value, though. So that sounds
good. But I think in practice it's not. (Also, this totally changes
for a gaussian instead of a uniform distributution).

For a bet of N, probability of folding is (N-L)/(H-L), so probabiliity
of calling is 1-(N-L)/(H-L).

E = N - N*(N-L)/(H-L)
E = N - (N*N-N*L)/(H-L)
H-L is constant, so computing the derivative ignores it:

dE/dN = 1 - (2N - L)/(H-L)

set that to 0:

1 = (2N-L)/(H-L)
(H-L) = (2N-L)
H = 2N
N = H/2

Weirdly, L canceled out, so I probably screwed up. But it could
be true, that that's always the peak of the quadratic, and therefore
the optimal result is:

L <= H/2: N = H/2
L >= H/2: N = L

This makes some intuitive sense. Let's ask whether it's a good
idea to bet (H+L)/2, versus betting L.

In other words, we have a sure thing of making $L, so let's factor
that out (I think this makes it more intuitive). Now we want to know
if we should increase that to $(H+L)/2.

If we increase, our increase versus $L is:
$(H-L)/2
and we have a 50/50 chance at it. If we increase and lose,
we lose the $L
-$L

so E = 0.5 * (H-L)/2 + 0.5 * -L
= 0.25*H - 0.25*L - 0.5*L
= 0.25*H - 0.75*L
So this says if H is less than _3 times_ L, it's a losing bet
to take the 50/50 gamble of raising to (H+L)/2. So it makes
sense; when L gets close to H, the rate of extra-money-beyond
L doesn't sufficiently offset the increased chance of losing L.
</PRE>

<P>

That all looks right to me, and is kind of interesting. If L is
significant at all, you want to bet the maximum that you're *sure*
they'll call, and trying to edge it a little higher is -EV because the
risk of losing them is too great. Say for example L is $20 and H is $40, you
must bet just $20 !! Even $21 is worse because it makes them fold 1/20th of
the time so your EV is just (19/20)*21 = $19.95. However if L is zero, eg. they have
something so weak they might not call any bet, the best bet is H/2.

<P>

In reality, they have more like a Gaussian distribution of call values,
though I doubt that changes the answer too much. Also you have to look
at multiple models, say there's a 75% they call based on a model like
this, and a 25% chance they call any bet (but your stacks are not huge
compared to the pot, so this isn't insane). In that case there's a bit
more reward for edging your bet up, the risk of losing them if they have
something weak is hedged a bit by the EV gain when they have something
they call any bet with.

[Ness]

lol

[Ness]

Pokey,

Holy [censored], you write long posts.

Anyway, I only read the first sentece or two.

In SSNL, the players are more likely to call smaller bets than big ones.

In most cases, I am more inclined to make a callabe bet, rather than a large bet with a big hand.

[Pokey]

[ QUOTE ]

Pokey,

Holy [censored], you write long posts.


[/ QUOTE ]

Oh, SURE, cbloom posts the Encylopedia Britannica on poker math and *I* get the tl;dr. [img]/images/graemlins/grin.gif[/img]

[math]
cbloom: Your post really screwed with me. I worked out the math six times and every time I got a max at N = H/2. The second derivative is -2/(H-L) which is negative by definition, so this is SUPPOSED to be a local max. Despite that, it's not. Mind you, it's been a decade since I took a calc class, but this isn't like graphing in polar coordinates or integrating arc-cosines or something -- it's straightforward maximizations. What's the scoop on this??

Oh, and as to the distributional problem: not to make the problem even more complicated than it already is, but I'd say you might consider something more like a Poisson distribution: highly unlikely to be a very low number, a "sweet spot" at around 1/2-pot, and then exponential decay all the way out to their stack size. If you wanted to make it even MORE accurate, you'd make it a combined discrete/continuous distribution, with a positive mass at $0. That takes into account that if your opponent missed a draw or is totally busted in his hand and knows it (which happens with positive and non-zero probability), then he'll fold to ANY bet; otherwise, he'll have an unknown distribution identifying his folding probability.
[/math]

[Tupacia]

Haha, oh man, math is hard. In response to Pokey's first post, I very rarely bet more than 1/2 pot for river value unless it's against someone on my buddy list (the type who call stacks off with bottom pair) in which case I'll pot it. Betting 2/3 of the pot for value is also bad as there is no way your blocking bets are that big, thus making your blocking bets easy to pick off. Ideally, your value bets should look exactly like your blocking bets making it impossible for a villain to tell the difference. Just my thoughts...

[Fallen Hero]

Taking meta-game into consideration is not reasonable in ssnl. Bet for value whatever you want to bet for value, bluff a different ammount if you think it's better, no one notices or cares.

I think we're being to general here, it really depends on the action I got during the rest of the hand, how obvious the hand is, etc.
Having said all that I really don't think betting 1/2 pot instead of 2/3 gets called by many more hands or by the same hands but more often.

[Jamougha]

Work out what they have, then work out what they'll call...