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The minimax algorithm and Monte-Carlo tree search (MCTS) can be used to implement agents which play deterministic (i.e., non-probabilistic) games, like chess or tic-tac-toe, that have complete information of the game.

Are there general methods that work for games with incomplete information and/or games with a probabilistic component (e.g., poker or bridge)?

nbro
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Ward Beullens
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1 Answers1

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Yes. You're asking several questions at once, though.

The simplest possibility is a game like backgammon, which includes probability, but full information. The extension to minimax is straightforward, and called expectiminimax.

Incomplete information is usually called "partial observability," and exists in games like kriegspiel, which is a variant of chess where you cannot see the opponent's pieces. Here, the extension to tree search is that your tree depends on sequences of percepts instead of individual board states. As you might imagine, this blows up the tree very quickly.

Card games are usually both at the same time, and need both techniques.

Note that these simple extensions only just scratch the surface. In the same way that chess and go require more than just simple search trees, partially observable random games require more than extensions. When the actions have probabilistic results (i.e., chance of failure) then you're into deep academic research territory.

Novak
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  • I see, i had already heard about the expectiminimax method, but i don't see how the tree with sequences would work. Can you explain or do you know a reference? Thanks! – Ward Beullens Jan 19 '15 at 20:20
  • Consider the dice as a third player who moves randomly, and apply probability theory as appropriate for expectiminimax. In general, see Russell and Norvig chapter 5. – Novak Jan 20 '15 at 02:03
  • Yeah i understand expectiminimax but I meant to ask about the method when facing partial observability. I didn't get what you said about a tree that depends on sequences. – Ward Beullens Jan 20 '15 at 16:18
  • It's rather beyond the scope of stack overflow. But the idea is that in an observable game tree, each node represents a board state, because that is all you need to know-- that is all there is to know-- about the state of the game. In a partially observable game, you need to carry around the sum of your knowledge which is captured in the history of actions and percepts. See Russell and Norvig. – Novak Jan 20 '15 at 20:01