Me and my friend walked into a poker room and there was a hand of no-limit hold'em going on. Two players were on the river. We couldn't see the board, but Alice had her chips pushed forward signifying she had gone all-in. Bob was in the process of moving his own chips forward for the call. Before the players turned over their cards, my friend suggested: "Let's bet on who wins this hand!"

Knowing nothing else about the players or the action on previous streets, which player do you pick?

My friend's reasoning:

My reasonging:

Is there any theory-based reason to pick one or the other?

Before I called, Alice had a big pot share and was more likely to win the hand. After I called, my share would have increased under the condition that I was playing rationally.

Really depends on the number of slowplays in my range, which would have a significant effect on my share.

In a perfectly polarized situation, the bettor would always win more often than the caller, unless he's overbluffing.

Slowplays (calls that beat some of the bettor's value range) can potentially change this though. It depends on the exact situation.

Can not be answered without bet size and pot size. All in could be two chips. If both players are GTO, then if the bet is below pot size, a correct call will win less than half the time. Conversely, a bet size above pot will have less frequent calls, but more winning calls. So, without any more info than this, if the bet is bigger than pot size, a GTO call wins more often than not. The reason for the different sizes in bets is the distribution of combos in the range of the caller.

Probably the one who is in position.

I think she wins more than half the time when called.

For IP at least then her betting range must win at least half the time when called because otherwise checking back everything would increase the EV of her whole range. I suspect she also wins more than half the time OOP but I'm less sure.

On the river in general the bettor should theoretically have more value than bluffs.

Consider also that pot odds make it so that Bob is priced in to call even when he is beat more than 50% of the time.

If Alice and Bob are playing rationally, the bettor should win more often.

It's a pot odds thing, as referenced in your spoiler. e.g. If Alice bets pot, she's risking 1x pot to win 1x pot. Bob only needs to risk 1x pot to potentially win 2x pot, so he doesn't have to win as often to break even.

If they played this game multiple times and Bob won more than 50% of the time when he called, it would be evidence that Alice was overbluffing, like ZKesic said. Either that, or Bob is a nit, gets bluffed off the best hand a lot, and only calls when he's certain of winning. Both cases would indicate someone wasn't playing optimally.

The bettor wins by having the best hand at showdown and making people fold but the caller can only win by having the best hand at showdown.

Yes, but in this case, we know that there was a call. Since the bettor may have relied partially on a fold to win the hand with his all-in bet, I think it reasonable to conclude that he probably has the lesser hand – essentially I’m stating the friend’s position.

Interesting question. I wonder if whosnext can come up with a simulation??

Hard to find fault in this logic. If ranges are perfectly polarized, it has to be the bettor who wins more often unless someone is playing a poor strategy.

Still not convinced this works for the general case, though. Yes it's true that Bob can call expecting to win less than 50% of the time, e.g. 33% in the case of a pot-sized bet... But surely it's the very bottom of his calling range that wins that percentage of the time, whereas the rest of that range falls anywhere between 33%-to-win and the stone nuts. It's not clear to me that all of those hands on average have to win less than 50% of the time.

And it's not like Alice is losing money by betting even if Bob wins 50% of the times when he calls... She still gets half the pot when called plus the whole pot every time Bob folds.

Hmm, surely Alice gets a bunch of EV from all the times Bob folds. Could that not make betting +EV even if equity-when-called alone was less than 50%?

Hmm, yeah. If Alice bets pot, she needs to win more than 50% of the time to make money with the bet. But again, Alice can also win by Bob folding...

There are no perfectly polarized ranges, just as there are no perfectly balanced uniform post flop ranges.

This question may be a sort of Rorschach test for poker players.

I was quite confident in my belief that the caller wins more often in real-life live poker. After reading the thread, I am only slightly less confident.

And, yes, I am hard at work at developing a simulation to answer this question once and for all.

Watch some hands online. How often does the caller snap-call (answer: hardly ever) and how often does he/she sigh-call or sigh-fold? (answer: very often).
In many cases, the player facing a river bet was really hoping for a river check, because if they had a strong range, they would often have bet (or raised) on the turn and then bet the river themselves.
I don't have my tracker on this PC, but from memory, the player facing aggression on the river calls about 50% of the time, and only wins about 35% of those pots.

Alice

If both players are playing correctly and Alice bets pot on the river she should have roughly 1/3 bluffs and 2/3 value bets and thus win 67% of the time. Bob's equilibrium is 33%- if he wins more than 33% when he calls then he is folding too much, and vice versa.

If using smaller bets, Alice wins more often (and vice versa).

In live poker depending on stake I think it's further skewed to the bettor winning more than equilibrium because the population average doesn't bluff enough rivers and calls too frequently al on streets including rivers. I didn't drive an hour here to fold damnit

Alice sizes her bets to attack Bob’s range on the river, induing her all in shoves. This is to make Bob indifferent to calling. This gives Bob a 50/50 *when he does call*. However, this does not ensure Bob will break even on the river with perfect defense, since Alice probably has left herself exactly one PSB left to shove with. This optimal stack size earns her more than the normal amount of the pot versus if both players were deep stacked.

This is how the river works.

I think a more in-depth thread specific to GTO rivers will be helpful. I will get around to this in the near future unless someone else wants to do it first.

Let's say a is the probability Alice is bluffing, and b is the probability Bob wins against Alice's value hands when calling. The total probability of Bob winning when he calls is a+b.

We know the bigger the opening size, the bigger the value of a. Also, a is always under 50% unless Alice is overbluffing.

The value of b is the piece that's missing from the puzzle. Intuitively it feels like b is closer to 0% when Alice bets big (closer to nuts/nothing) and higher when Alice bets small (thin value). Of course b is always under 50% unless Alice is making serious errors constructing her value range.

For a given bet size, we can calculate the value of a where Alice is betting a balanced range (meaning Bob is indifferent to calling with hands that beat all of Alice's bluffs but none of her value). This way we get the required value of b for Bob to win more than half the time.

Bet size	a	required b
0.5x pot	25%	25%
1x pot	33%	17%
2x pot	40%	10%
10x pot	48%	2%

So Bob would need to beat 25% of Alice's value bets when she bets half pot, 17% when she bets full pot and so on. On first glance it seems like a lot to ask from a calling range against a value-betting range, but can't figure out how to prove it one way or another...

This is the correct answer. Essentially everyone in here is pointing out factual things but only pointing at half the equation.

The people saying Alice are pointing to the fact that Bob doesn't have to win 50% of the time to call; the people saying Bob are pointing to the fact that Alice doesn't need to win 50% of the time to bet. Assuming a perfectly polarized scenario (Alice's value hands have 100% equity and her bluffs have 0% equity), Alice will win most of the time; but value hands only need to have <50% equity when called for the bet to be +EV, so assuming her value hands have barely more than 50% equity and her bluffs still have close to 0% equity, she'll have less than 50% equity when called.

So assuming she's bluffing at threshold and her bluffs have close to 0% equity, it's just a matter of how often Bob is ahead of her value hands.

In practice, even between two bots, I'm pretty confident asserting Alice-bot's equity with her value range is high enough on average that she's winning >50% when called. The river aggressor is generally the one who already had the uncapped range before betting* (so the top of the range has close to 100% equity), and Bob-bot's call-down range is generally made up of far more bluff-catchers than slowplays (so that the bottom of Alice's value range is generally well above 50% when called).

However, this would be less true in a scenario where the only bet size is all-in, since value bets are at their thinnest and in fact range merges are possible.

If applied to live poker between two humans, where thin value bets are rare, pots are generally underbluffed, and people call too light, it's especially safe to say Alice is going to be shipping the pot an enormous percentage of the time.


*For this reason, the fact that we're on the 4th betting street is important here. In an AIPF scenario, it is generally the case that both players took an aggressive action at each opportunity, thus there is no "slowplay" necessary and no tendency for one player to be more uncapped than the other. If it were an unopened pot in a blind vs blind scenario in a game with two big blinds that had checked down to the river, that would change things considerably. In almost all scenarios on the river, though, there is someone who was already at an advantage, and the fact that one player is betting implies it was more likely to be that person.