In a game between two or more players, a Nash equilibrium is a set of strategies for all players so that given the strategies of all other players, no individual player has an incentive to deviate from their strategy. In the context of poker, such a set of Nash equilibrium strategies are often referred to as “Game Theory Optimal” or GTO. Finding a Nash equilibrium for poker is very difficult due to its complexity. So-called “solvers” can approximate the Nash equilibrium strategies (subject to some assumptions on bet sizes and perhaps other factors if I understand it correctly).

Out of curiosity, I was trying to figure out how to construct Nash strategies in a very simple toy version of poker. This version was inspired by the toy game described by @Yorunoame here:

Here is my toy version:

2 Players: Player 1 in the Small Blind/Button, Player 2 in the Big Blind.

Blinds: $1/$2.

Initial stacks: $100 for both (although I think it’s not important for this particular game).

3 cards in the deck: 1 Ace, 1 King, 1 Queen.

Each player is dealt 1 card from the deck at random.

No community cards, the hand ends at showdown (the highest card wins of course) or if one of the players folds.

Normal no-limit betting with normal minimum raising amounts.

Only one betting round, but multiple raises and re-raises are allowed.

Here is the problem:

Find Nash strategies for both players.

What’s the expected value (EV) of the optimal strategies in the SB and BB?

Note:

The strategies can be randomized; e.g., “If …, Player 1 raises to $6 50% of the time.”).

The Nash equilibrium concept assumes that each player knows the opponent’s strategy perfectly. This is a fairly reasonable assumption as players can infer their opponents’ strategies by observing frequencies of different betting patterns over sufficiently many hands.

After playing around with different strategies, basically a combination of guessing, trying to make the opponent indifferent to bluffs, and computing the EV of different options, I found a candidate Nash equilibrium, but I’m still not 100% sure if it’s correct.

I can post the strategy that I came up with later if anyone is interested. But first, I’d like to ask you to propose your strategies for this game. We might even converge to the equilibrium by iteratively constructing maximally exploitative strategies to each others proposals.

Nash Equilibrium in poker is when player’s ranges are in equilibrium with each other, or in other words, when opponents are playing an “optimal” game against one another whereby they both cannot gain anything by deviating from equilibrium strategy.

This leads to an eventual stalemate (long term break-even proposition) and it makes no sense for either opponent to deviate from Nash if the other player is strictly following Nash – this means they will be deviating from “optimal” strategy and the other player may benefit. If one player deviates from Nash then the other player can do better by re-adjusting their ranges accordingly.

I think the massive over bets are not viable here, as you don’t have a polarization advantage in this construction. Stated another way, I can’t make massive bluffs with the Q if my opponent is just as likely to have A as K.

Not being able to make massively oversized bluffs with bottom of range then takes out the benefit of making massively oversized bets with top of range, as the opponent’s incentive to call is removed. So I’ll need to ponder this a bit more. Curious to see others ideas.

I really like the concept of the three card deck but I quickly came to the realisation that in a shove / fold game the player with a King is completely exploited because Q always folds and A always calls against them.

Would a deck with two of each card be a more viable game?

If there was a push/fold constraint, then the SB would have to fold anything but an Ace and the game would become trivial. However, note that there is no push/fold constraint in my version of the game. Such constraints are sometimes enforced to simplify the analysis, but bet sizing should really be an output of the equilibrium, not an input.

My objective for the rules for this game was to find the simplest possible game that allows normal no-limit betting and has a nontrivial Nash equilibrium. The 2 card deck is obviously not interesting, but I believe the 3 card deck already has a nontrivial solution.

From there, one can make the game more complex in at least three dimensions:

Add multiple cards of the same value as you suggested.

I do not have much of a background on stats and equilibrium(besides chemistry LOL) despite having a lot of interest in it, cool thread, its funny how just one extra variable can turn a trivial nash equilibrium into a complex strategy and a hard-to-find equilibrium strategy

Just a few thoughts I had while I was out walking:

optimal bet size seems likely to be pot sized or smaller (I’d guess well under pot)

it seems hard to bet kings ever, as your opponent will always know if they are ahead or behind; about the only interesting case would be if they have Q, when they don’t know if you might possibly be betting with K, and thus could be bluffed

you need to find bluffs with Q if you ever want to get value with A

it still feels like you’ll mostly only be making smaller bets with A of a size small enough to force calls from K some amount of the time, with bluffs from Q at a frequency dictated by your bet size (smaller bets generating fewer bluffs)

These are all good observations, I think you are on the right track. Here are some more important observations:

Raising a K from the SB can never be optimal. Even if you only min raise with the K, you risk 3 to win 3. But even in the best-case scenario where you only get called by the A and any Q folds, this play has EV 0. But you could sometimes get bluffed by the Q, which makes it -EV.

For the same reasons, raising a Q from the SB is even worse.

If you can’t raise K’s or Q’s, you also can’t raise A’s (well you can, but this will generate 100% folds and thus only generate EV 1/3 * 3 = 1 from the SB).

It thus remains to analyze a limping strategy from the SB…

It should be obvious that there’s no pure strategy that would be a Nash equilibrium, so the equilibrium strategy has to be a mixed strategy.

This game is almost identical to one proposed by Harold Kuhn in a paper published in the early 1950s. If memory serves, he included a complete game theory Nash solution. The paper was named “Simplified Two-Person Poker,” and is probably floating around somewhere. Maybe search for it by name on Google’s specialized search engine for peer-reviewed papers at: scholar.google.com

Alright, since nobody else seems to dare to propose a solution, I’ll post my candidate:

Decision tree

P1 = Player 1 (in the SB/Button), P2 = Player 2 (in the BB)

P1 action: complete (call) with all A and all K.

P2 action: raise to 4 with all A and 1/3 of Q.

P1 action: call with all A and 1/3 of K
Note: P1 could also backraise with A, but then P2 would fold their Q, which effectively leads to the same result.
Result:

If P1 has A: P1 wins 8.

If P1 has K: P2 wins 8 75% of the time, P1 wins 8 25% of the time.

P1 action: fold with 2/3 of K.
Result: P2 wins 6.

P2 action: check back all K and 2/3 of Q.
Result: P1 wins 4 (if P2 has K, P1 has A and if P2 has Q, P1 has A or K).

P1 action: fold with all Q.
Result: P2 wins 3.

EV calculations perhaps later if anyone is interested.

It’s interesting that even in this simplest possible example, hands can be split into three categories:

A is the nuts, which tries to get max value.

K is the marginal made hand, which tries to get to showdown cheaply and is used as a bluff catcher.

Q is the trash hand that can only win by bluffing.

I can’t see how folding all queens from P1 can be optimal. If P2 will mostly fold kings and raise aces when faced with a raise, I would raise queens most of the time.

If P1 has a Queen, then P2 has an Ace 50% of the time and a King 50% of the time. By min raising, you risk 3 to win 3 (the initial pot). For this to be profitable, it needs to work at least 50% of the time. But you will get called by all Aces and some Kings, so it’s a -EV play. Choosing a larger raise makes the situation worse for P1.

I avoid equilibrium at all cost, LOL. If I suspect that my opponent is playing optimally or nearly so, my optimal strategy is to go find a softer game.

Anyway, I just searched for “Kuhn Poker” and see that Wikipedia has a diagram of the Nash solution to Kuhn’s original game. Note, however, that the game he proposed had each player posting an ante, not a SB/BB.

If you always fold a queen when first to act, your opponent will always know exactly what you have when you don’t fold.

When he has a king, he will know you have the ace every time and fold. When he has the ace, he knows you have the king. Maybe I am missing something, but this doesn’t seem optimal to me.