Game tree

A game tree is a directed graph whose nodes represent positions in a game and whose edges represent moves between those positions. It is a fundamental concept in game theory and artificial intelligence, particularly in the context of creating game-playing algorithms.

The root node of the game tree represents the starting position of the game. Each subsequent level of the tree represents the possible moves available to players, alternating between the players involved. The leaf nodes of the tree typically represent terminal positions of the game, such as a win, loss, or draw. Each leaf node is associated with a value, which reflects the outcome of the game for a particular player, often expressed numerically.

Game trees are typically very large, even for relatively simple games. The size of the tree grows exponentially with the number of possible moves at each position and the length of the game. Because of this exponential growth, exploring an entire game tree is often computationally infeasible. Techniques such as pruning (e.g., alpha-beta pruning), heuristic evaluation functions, and Monte Carlo tree search are used to manage the complexity of game trees and enable effective decision-making in games.

The depth of a game tree corresponds to the number of moves from the starting position to a given position represented by a node. A complete game tree includes all possible moves from the starting position, leading to all possible terminal positions.

Game trees are used to analyze games, determine optimal strategies, and develop AI agents that can play games effectively. The concepts of minimax and negamax algorithms are often used in conjunction with game trees to determine the best moves for a player, assuming that the opponent is also playing optimally. They provide a theoretical framework for reasoning about games and making informed decisions.