You may be familiar with the no-limit clairvoyance game from Mathematics of Poker. There are two players, an initial pot size of 1, and stacks of n. Player 1 is dealt 50/50 a winning and losing hand, and player 2 always has a bluffcatcher. Player 1 is allowed to bet any size up to n (or to check), then player 2 can call or fold (but not raise). They show that the NE is for P1 to always bet n (all-in) with a winning hand, and bet n with probability n/(1+n) with a losing hand (and check otherwise). And vs. any bet size x in (0,n] P2 calls with probability 1/(1+x).

It turns out that the analysis is actually a bit more nuanced. For suboptimal bet sizes x < n, player 2 can actually call with a range of frequencies while still being in equilibrium. If P2 calls with any probability in the range
[1/(1+x), min{n/(x(1+n)), 1}] then it is a Nash equilibrium. (The lower bound makes P1 indifferent between checking and betting x with losing hands, and the upper bound makes player 1 indifferent between betting n and betting x with a winning hand.)

So the question arises, if you are player 2 and facing a suboptimal bet size of x from player 1, how often should you call? Since there are infinitely many values in the equilibrium range, standard Nash equilibrium analysis doesn't give a conclusive answer. Obviously the answer doesn't matter from the perspective of a worst-case EV guarantee, since all strategies are NE. But against opponents who actually make suboptimal bet sizes it could make a big difference.

I've been thinking about the question for a long time, and I haven't been very satisfied with the solutions of several standard Nash equilibrium "refinement" concepts, which are game-theory solutions that select certain Nash equilibria over others.

To make this concrete, suppose that n = 2 (both players have 2*pot behind), and let x = 1 (player 1 makes a suboptimal pot-sized bet). Then player 2 can call any probability in the range [1/2,2/3] and be in equilibrium. But in practice is any one of these strategies better than the others?

Obviously to properly answer the question you'd need to think about reads on the particular opponent, population tendencies, maybe do experiments and/or look at historical data, etc. But I am also interested in a theoretically-motivated approach that is not totally reliant on this information (so that it can be also used for other similar non-poker situations).

So my question for you is what probability do you choose to call with in the game for n = 2 facing a bet of x = 1 and why?

If you want to see a more detailed discussion and analysis, I have a recent paper here (Section 3 is about the poker example specifically)
https://arxiv.org/pdf/2210.16506.pdf
To summarize, the prior main equilibrium refinement concepts say to call with prob 2/3, while a new one I came up with that I think is better motivated theoretically says to call with prob. 5/9. Though my intuition as a poker player says to call with prob 1/2.

I dont see why x being different to n would make any difference to player 2's response??

And if all answers give player 2 the same EV, I dont see any reason to use one over another?? The thing is, you are asking for a practical solution but the solutions you're offering are not practical. Player 2 cant actually call 2/3 of the time, as she only has 2 options. She can either call or fold. If instead we are saying she can call 2/3 of the time then we are saying that both players are able to make multiple choices in the future which would allow them to change their strategies over time. And if we are saying they are able to change their strategy over time it becomes impossible to judge the best strategy, as it would include an understanding of the reaction of the opponent. Eg. If we decide to fold at a lower frequency this would encourage the opponent to bluff at a higher frequency, but we dont know how long it would take them to adjust, nor the strategy they will use when adjusting.

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I have always found this interesting about mixed strategies... In reality it is impossible to fold 1/3 of the time in a certain spot. The truth is that your options are to either fold or not fold. If you only see that spot once in your life you are certain not to use the GTO ratios even if you use an RnG. I think this is fundamentally misunderstood factor in all similar games. I mean, consider this... If it is GTO in a certain spot to bet 1/3 of my range with a 1bb bet, and 1/2 of my range with a 2bb bet, the first time I play in this spot I could use the 1bb sizing and bet that much of my range to hit GTO, but then I would have to stick to that sizing to remain at GTO. If I suddenly changed my sizing to 2bbs my strategy now includes two different sizing's which would give a different ratio of hands for each. As their are almost infinite sizings to choose from, I could constantly change my strategy within some parameters and would constantly hit GTO. I could bet 1bb with, basically, any range that I wanted, provided I planned to add the appropriate sizings in future situations, which I need not actually do when the time comes!

I think the confusion exists because there are actually 2 different GTO's. One of which is a theoretical GTO, and the other is a realistic GTO. The theoretical GTO is the one that everyone knows and LOVES, with mixed ratios, which offers the opponents no EV in a zero sum game. The other GTO does not take you to equilibrium, even though it is the exact same strategy. The realistic GTO is always changing, but, provided the opponent uses a rational exploit based on our past actions it will lead to the same ratios that the theoretical GTO dictates. EG, instead of using a mixed bet/check ratio, if I were to bet every time in those spots, after the opponent adjusted to this, it would lead me to use the check option instead. The opponent and I would bounce back and forth forever and after an infinite amount of time we would end up with the ratios that the theoretical GTO dictates. I have long understood that there are two different perspectives from which you can consider your situation, you can either look at the present moment, at the hand you actually hold, or you can consider all that hands that you could possibly hold in a particular situation (our actual range). When you look at GTO from these two different perspectives, you are given these two different versions of GTO. One of which is a unrealistic GTO, as it is impossible to actually use a mixed ratio for one individual spot, and, it is irrelevant to consider the hands you could have held (your actual range) in a particular situation. The other is the realistic GTO because it relates to the hand you do actually hold, and gives you a strategy that is possible, a strategy that, although it is constantly adapting, makes perfect sense to use at any given time.

I think what we are really seeing here is the difference between the two different sets of laws that govern all existence. If the only laws that governed existence were those that governed information from the past, the theoretical GTO would be the only GTO. However, in reality we also live in a world where a different set of laws govern the future. These laws dictate that the GTO solution will be a fluid strategy that always changes.

Can you give us page number from MoP

If you pick any freq. between 1/2 and 2/3 P1 loses ev with both value hands and bluffs if he picks suboptimal sizing. OTOH if you call 1/2 he can still keep his NE ev by only bluffing with smaller sizing and if you call 2/3 he can retain his NE EV by only value betting with small sizing. To me both 1/2 and 2/3 seem like a bad choice since there are some P1 strats that wont lose EV.

The game is the "half-street no-limit clairvoyance game" pages 148-150 1st edition.

I also linked my paper in the OP that has more detailed analysis and gives a solution of 5/9.

Interesting point that any value except 1/2 or 2/3 guarantees an improvement in EV regardless of what P1 does (given that he sometimes bets 1), while 1/2 and 2/3 don't necessarily.

It's just a single-bet polar vs bluff-catcher toy game.

This is a cool question. It's one of the rare cases where MDF isn't a single number, but rather a range of numbers.


The general solution when Facing a non all-in bet:

• You need to call enough to make their bluffs worth 0 or -EV
• You need to fold enough to make their value bets indifferent or lower EV than if they would have shoved instead.

In the example you gave, the value bet gains more EV by shoving 2x pot, until you defend more than 2/3rds of the time facing 1x pot. At that point, betting smaller is better exploitatively.
The bluff is worth 0EV in the 2x pot line, and <=0 EV in the 1x pot line, so long as you defend at least 1/2 the time.

Note - we have to assume sufficient bluffs. If the aggressor could reach an equilibrium where they can bet all their bluffs, then the defender should always fold regardless of bet size. This is because the aggressor claims 100% pot share in equilibrium, so calling any bet allows the aggressor to increase their pot share by value betting those lines.

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Plugging this into Piosolver gives a defence frequency of 52.54%, which is just enough to make the polarized player never choose the 1x pot size. But this is just one solution.

Nodelocking the defence frequency between 50% and 66.6% yields no change to the aggressor's strategy or expected value, as betting smaller is never chosen.





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In summary, any solution where you call more than 1/2 and less than 2/3 of the time makes the 1x pot bet strictly worse than the 2x pot bet. Calling exactly 1/2 or 2/3 is technically the same EV, but slightly worse strategically as it's harder for your opponent to make a mistake.

Ok, but why is Sam Ganzfried claiming 5/9 is best?

If the aggressor is playing perfectly then they'll never choose the suboptimal 1x pot size, so long as you defend between 1/2-2/3 of the time. So against perfect strategy there is no change to your EV.

But what if the aggressor sometimes chooses the suboptimal 1x pot size?


If I understand your argument, you're saying that 5/9 lower's the aggressor's EV the most if they are forced to use the smaller bet size at some frequency?

The paper has the full technical details: https://arxiv.org/pdf/2210.16506.pdf.

Basically, there have been several concepts that are refinements of Nash equilibrium in the literature, and I wasn't convinced by them in this game, and created a new one I think is more compelling theoretically that leads to 5/9.

You can google trembling-hand perfect equilibrium, quasiperfect equilibrium, and proper equilibrium. These are popular refinements and they all lead to 2/3. Basically they assume that players "tremble" and make mistakes with arbitrarily small probability epsilon on all actions, and they look for a Nash equilibrium that is a limit of a sequence of Nash equilibria of these "perturbed games" where the players are forced to make mistakes with probability epsilon.

My new concept, which I call observable perfect equilibrium, assumes only that the opponent makes a mistake consistent with our observation that they bet 1 sometimes. If you work out the analysis, the only solution ends up being 5/9, though I don't have any simple intuition behind it. But I think it is better than the 2/3 result that the other refinements give.

in case of 5/9 both value and bluffs lose same ev compered to NE ev. They lose 1/9, thats probably why you get that as some kind of special value.

Good to see some academic game theorists on this forum, Welcome Sam!

• Defending 1/2 makes their value bets as bad as possible without becoming exploitable.
• Defending 2/3 makes their bluffs as bad as possible without becoming exploitable.
• Defending 5/9 makes their value bets and bluffs equally bad compared to NE.

Good intuition, you beat me to it!

If the aggressor is using a random unknown value:bluff ratio for the pot-sized bet, then defending 2/3 will punish them the most without making yourself exploitable, as they're more likely to be overbluffing than underbluffing. This is (probably?) why other pseudo equilibria defend 2/3.

I still think you're all mad lol. The effective ranges are the size of the bet. This bet being bigger or smaller wont make a difference to the solutions described in the very first paragraph of this thread. Seems to me like the authors of the legendary MoP explained all the relevant logic.

I can reach out to Bill and Jerrod and see if they can comment here.

Their original solution is correct, but it is just one of infinitely many solutions.

Awesome!!

Then I can give them some jip for naming an equilibrium strategy set "optimal". They caused me soooo much trouble over the years! Bloody Negreanu once said to me on Twitter that I am actually mistaking the word "optimal" for "best". Like, wtf, in the dictionary if you look up optimal it only gives a one word answer: Best!!... fml...Perhaps those two will understand that this strategy is never optimal, thanks to God giving us free will, the option to do nothing.