Game tree

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In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. The complete game tree for a game is the game tree starting at the initial position and containing all possible moves from each position.

The first two ply of the game tree for tic-tac-toe.

The diagram shows the first two levels, or ply, in the game tree for tic-tac-toe. We consider all the rotations and reflections of positions as being equivalent, so the first player has three choices of move: in the centre, at the edge, or in the corner. The second player has two choices for the reply if the first player played in the centre, otherwise five choices. And so on.

The number of leaf nodes in the complete game tree is called the game tree complexity. It is the number of possible different ways the game can be played. The game tree complexity for tic-tac-toe is 26,830.

Game trees are important in artificial intelligence because one way to pick the best move in a game is to search the game tree using the minimax algorithm or its variants. The game tree for tic-tac-toe is easily searchable, but the complete game trees for larger games like chess are much too large to search. Instead, a chess-playing program searches a partial game tree: typically as many ply from the current position as it can search in the time available.

[edit] Solving Game Trees

With a complete game tree, it is possible to "solve" the game – that is to say, find a sequence of moves that either the first or second player can follow that will guarantee either a win or tie. The algorithm can be described recursively as follows.

An arbitrary game tree that has been fully colored

Color the final ply of the game tree so that all wins for player 1 are colored one way, all wins for player 2 are colored another way, and all ties are colored a third way.
Look at the next ply up. If there exists a node colored opposite as the current player, color this node for that player as well. If all immediately lower nodes are colored for the same player, color this node for the same player as well. Otherwise, color this node a tie.
Repeat for each ply, moving upwards, until all nodes are colored. The color of the root node will determine the nature of the game.

The following diagram shows a game tree for an arbitrary game, colored using the above algorithm.