Using minimax search for card games with imperfect information

Question

I want to use minimax search (with alpha-beta pruning), or rather negamax search, to make a computer program play a card game.

The card game actually consists of 4 playe

Answer 1

I want to clarify details that the accepted answer doesn't really go into.

In many card games you can sample the unknown cards that your opponent could have instead of generating all of them. You can take into account information like short suits and the probability of holding certain cards given play so far when doing this sampling to weight the likelihood of each possible hand (each hand is a possible world that we'll solve independently). Then, you solve each hand using perfect information search. The best move over all of these worlds is often the best move overall - with some caveat.

In games like Poker this won't work very well -- the game is all about the hidden information. You have to precisely balance your actions to keep the information about your hand hidden.

But, in games like trick-based card games, this works pretty well - particularly since new information is being revealed all the time. Really good players have a good idea what everyone holds anyway. So, reasonably strong Skat and Bridge programs have been based on these ideas.

If you can completely solve the underlying world, that is best, but if you can't, you can use minimax or UCT to choose the best move in each world. There are also hybrid algorithms (ISMCTS) that try to mix this process together. Be careful about the claims here. Simple sampling approaches are easier to code -- you should try the simpler approach before a more complex one.

Here are some research papers that will give some more information on when the sampling approach to imperfect information has worked well:

Understanding the Success of Perfect Information Monte Carlo Sampling in Game Tree Search (This paper analyzes when the sampling approach is likely to work.)

Improving State Evaluation, Inference, and Search in Trick-Based Card Games (This paper describes the use of sampling in Skat)

Imperfect information in a computationally challenging game (This paper describes sampling in Bridge)

Information Set Monte Carlo Tree Search (This paper merges sampling and UCT/Monte Carlo Tree Search to avoid the issues in the first reference.)

The problem with rule-based approaches in the accepted answer is that they can't take advantage of computational resources beyond that required to create the initial rules. Furthermore, rule-based approaches will be limited by the power of the rules that you can write. Search-based approaches can use the power of combinatorial search to produce much stronger play than the author of the program.