Simplified poker games

Question

Is there a list of simplified poker games - preferably ones for which there is either a complete solution (ie the optimal strategy for all players is known) or at least some extensive analysis, either theoretical or as a simulation.

Here are some examples of the kind of thing I'm talking about -

AK game

There are two players, and a deck containing a single A and a single K.

1. Both players ante $1.
2. Player 1 is dealt a card from the deck, and the other card is put aside.
3. Player 1 may either bet $1, or fold.
4. If player 1 bets, player 2 may either call or fold.
5. If player 2 calls, the game goes to a showdown - and player 1 wins if he holds A.

In this game it can be shown the the optimal strategies are for player 1 to bet if he has an A, and bluff 1/3 of the time if he holds K. Player 2 should call 2/3 of player 1's bets.

This game is simplified because (a) there are only two possible hands, (b) there is only one street, (c) only player 1 has any information, (d) there are no reraises, (e) the only options are bet $1 or fold (ie it is a limit game), (f) there are only two players.

AKQ game

There are two players, and the deck contains AKQ.

1. Both players ante $1.
2. Both players are dealt a card from the deck.
3. Player 1 may either bet $1 or fold.
4. If player 1 bets, player 2 may either call or fold.
5. If player 2 calls, the game goes to a showdown, with A > K > Q.

This seems more complicated since there are more cards and both players have information, but it's not that much more complicated. You can show that player 1 should bet all his aces, bet all his kings (it's a pure value bet - since p2 will always call with aces and fold with queens, your expectation from betting is -0.5 vs -1 from folding) and bluff at least some of his queens. Player 2 should always call with an ace, fold with a queen and call some of the time with a king.

This game illustrates the card removal effect - if player 2 sees he has a Q, he knows that player 1 must hold A or K, so he can always fold (there is no chance of winning a split pot in the case that player 1 also holds Q).

[0,1] games

In this situation the players aren't dealt a poker hand, but instead both receive a number uniformly from the range [0, 1]. The rules might allow reraises, they might not. There are various games of this kind discussed in The Mathematics of Poker by Chen and Ankenman, some of which can be solved exactly.

In some sense these games are slightly more realistic, as there is a range of possible hands. They have no card removal effects, as the player's hands are independent. However, they still only have two players and a single street.

Kuhn poker

This is essentially the same as the AKQ game above, except that player 1 can either check, fold or call, and player 2 is allowed to raise if player 1 checks. This is solved in Kuhn's 1950 paper "Simplified Two-Person Poker".

It demonstrates how the ability to reraise gives some of the advantage back to the second player.

I'm interested in games that are still simple enough to be solved, but are closer to "real" poker. The major thing that these examples leave out is the lack of further streets (more cards to come, which might improve or detract from a player's hand), the number of players and the effect of stack sizes. Can anyone point me in the right direction?

Answer 1

Games that are closer to "real" poker would not be simple enough to be solved. Recent research seems to be mostly focused on HU LHE and programming AI bots to play Game Theoretic Optimally. Most of this work seems to come out of the University of Alberta's Computer Poker Research Group. You can find their publications here.

One game that is closer to real poker, but not so complex as to be unsolvable is Rhode Island Hold'em. You can read more about it here. Even in this game, there are 3.1 billion nodes.

Here are some related articles that may be of interest:

• Rhode Island Hold'em
• Abstraction of Imperfect Information Games

By the way, there's nothing to stop you from creating your own simplified poker games that are closer to "real" poker, and solving them yourself. Obvious changes you can do are removing streets, removing cards, capping the raises at a lower number of bets. It should be an enlightening exercise.

Answer 2

Some more simple poker games (designed for simplicity and for game-theorists to analyse, rather than to persuade gamblers to gamble money by playing them):

von Neumann-Morgenstern poker. The protocol is similar to your AKQ game except that player 1's first action may be to bet or pass (not fold); if they pass, there is a showdown and high card wins the pot, which at that stage just contains the antes. It is thus similar to Kuhn poker, except that a single pass triggers a showdown, whereas in Kuhn poker, two passes are needed to do that.

One-card draw poker. In this blog entry, the author describes one-card versions of Guts (with the two players act simultaneously) and of Draw poker. In the latter game, there are two players, and each is dealt one card. Each in turn decides whether to stand pat, or to discard their card and get dealt a replacement card. The author does not give details of the betting.

The author also links to what he calls a stud form of the game, at a page by Geoffrey Gordon. This latter page contains an app where you can play the game, being Player 2 in every deal. If I have interpreted this app's action correctly, this game is Kuhn poker.

[von Neumann] discusses the above von Neumann-Morgenstern poker and a [0,1] variant.

[Reiley] discusses Borel's game (a [0,1] game whose protocol is like your AKQ game) and von Neumann's [0,1] game, though its focus is on a game like your AK poker.

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[Reiley] Reiley, David; Urbancic, Michael; Walker, Mark. "Stripped-down Poker: A Classroom Game with Signaling and Bluffing", Feb 2005 version

[von Neumann] John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior. Princeton, New Jersey: Princeton University Press, 1944.