Adversarial Search Game Theory

From the point of view of an intelligent agent, interaction with an unpredictable entity or environment introduces uncertainties that require the inclusion of contingencies in the intelligent agent’s problem-solving process. In a competitive environment, agents with conflicting goals interact, resulting in adversarial search problems commonly referred to as games. In mathematical game theory, any multiagent environment is considered to be a type of game. A type of game of particular interest to AI are turn-based, two-player, deterministic games with perfect knowledge with a winner and a loser. These are generally referred to as zero-sum games.

Two Person Turn-Based Games Perfect knowledge and deterministic moves suggest a search tree. Pruning allows us to ignore portions of the search tree that are unpromising. Heuristic evaluation functions allow us to approximate the value of a state without a complete search. S0 – the initial state of the game player(S) – indicates which player has the move in state S actions(S) - the set of legal moves from state S result(S,a) – performs the transition from state S by action a terminal(S) – a Boolean function that returns true if S is an end state (game over) utility(S,p) – gives the value of the state S for player p

Tic-Tac-Toe Partial Game Tree

A Two-Ply Game Tree The △ nodes are “MAX nodes,” in which it is MAX’s turn to move, and the ▽ nodes are “MIN nodes.” The terminal nodes show the utility values for MAX The other nodes are labeled with their minimax values. MAX’s best move at the root is a1, because it leads to the state with the highest minimax value MIN’s best reply is b1, because it leads to the state with the lowest minimax value.

Alpha-Beta Pruning

Alpha-Beta Pruning shallow pruning

Alpha-Beta Pruning shallow pruning

Alpha-Beta Pruning shallow pruning shallow pruning

Alpha-Beta Pruning shallow pruning shallow pruning

Alpha-Beta Pruning shallow pruning shallow pruning deep pruning

Reconsider Tic-Tac-Toe Adversarial search has a relatively high computational cost Tic-Tac-Toe is relatively easily solvable (i.e. we can draw the entire search tree) Symmetries greatly reduce search tree size How many rules would be needed to create a Rule-Based Tic-Tac-Toe Intelligent (i.e. Expert System)?

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TTT.exe Rule-Based Tic-Tac-Toe