## The Clairvoyant Game

Here is a well-known toy problem (the Clairvoyant Game) that doesn't converge: Suppose your hand is face-up. You have no hidden information. You don't know whether your opponent's hand is stronger. There is no drawing. The optimal strategy is for your opponent to bet all-in for value with all strong hands, and bluff with some weaker hands. If the chance for your opponent to have a stronger hand $p$ is too large, then you have to fold to all bets, even smaller bets. If $p$ is not too large, then you can call to neutralize bluffs while your opponent bluffs to neutralize your folds.

Calling risks $n$ to gain $n+1$ so your opponent should bluff $n$ times for every $n+1$ value bets, with absolute probability $\frac{n}{n+1}p$, if that is possible. If $\frac{2n+1}{n+1}p \gt 1$ then you have to give up. Bluffing risks $n$ to gain $1$ so if $p \lt \frac{n+1}{2n+1}$ then you should call with probability $\frac{1}{n}$.

As $n\to\infty$, the bet size does not converge. Further, your calling probability goes to $0$. However, the conditional probability that your opponent bluffs, given a bet, converges to $1/2$. The probability that your opponent bluffs converges to $p$.

If you have a strong hand less than $1/2$ of the time, betting $n$ into a pot of size $1$ is not optional. It's not that you make an uncallable overbet, but a smaller one would do. If you don't bet all-in, you lose equity because you can't protect as many bluffs, or if your opponent calls with the right frequency, you don't get paid off enough when you do get called.

If there are $r$ betting rounds, the optimal strategy is for your opponent to bet so the pot grows geometrically, $\frac{1}{2}(\sqrt[r]{2n+1}-1)$ times the pot. Your opponent sometimes bluffs $i$ times for each $i \in \{1,...,r\}$. As $n\to\infty$, the chance to bet on round $i$ goes to $2^{r+1-i}p$.

## Big Pots for Big Hands

Will Sawin mentioned a version of this result in his answer.

Suppose each player is dealt a hidden hand in $[0,1]$, with the lower number winning in case of a showdown.

Claim: For any SPR $n$, in the optimal strategy, the probability that you put more than $b$ into the pot against any strategy is $O(1/b)$.

Proof idea: If you put in more than $b$ more frequently, then you will often do so with hands that are not in the strongest $1/b$. So, your strategy will lose more than the pot to the suboptimal strategy of folding all hands except for the top $1/b$, and playing to provoke you to put at least $b$ into the pot, then reveal the hand and play the Clairvoyant Game above.

If this bound were $o(1/b)$ then it would imply convergence of the optimal strategy as $n\to \infty$ since we could play the optimal strategy for SPR $b$ and forfeit if the pot reached $b$, costing $o_b(1)$. I believe the solutions to some restricted games indicate that there is some power law with a power strictly less than $-1$.

As Will Sawin mentioned, this lets you prove some sorts of convergence of subsequences for $[0,1]$ games as long as you ensure that the ways for the pot to get up to $b$ are compact.

## Nut-Blockers

So, what about poker games people play? Should we see people use the full stack, so that we might get convergence of equities and probabilities of betting but not amounts, or does poker resemble a $[0,1]$ game where the pot can't get very big because players are afraid of paying off extremely strong hands?

The consensus among poker players may be that it almost never makes sense to bet more than the pot. However, the consensus of poker players is unreliable and it's wrong here. I've posted examples from my own play (in poker strategy forums) where I argued that particular overbets were much better than pot-sized bets. In some situations, this is to exploit suboptimal play, such as that people rarely want to fold a hand that has just improved, or to fold a full house or better, even if it is quite possible that someone has a stronger hand. However, there are also times when people have a limited range and the game resembles the Clairvoyant Game. You have to make large overbets to maximize the power of your range.

Will Sawin mentioned the example of a nut-blocker hand, a hand that might not be strong, but which blocks your opponent from having the strongest hand. For example, if you are playing Texas Hold'em and the community cards are KQQ, the nut hand is QQ for four-of-a-kind. The second-best hand is KK for a high full house. The third best hand is KQ for the low full house. It also blocks your opponent from having the nuts. So, if a significant portion of your opponent's range is KK, and you have just QQ and KQ in your range, then you should overbet with QQ, and for every time you do that, you can overbet almost as often with KQ, playing the Clairvoyant Game.