The M � Algorithm � Incorporating Opponent Models into Adversary Search

While human players adjust their playing strategy according to their opponent computer programs which are based on the minimax algorithm use tha same playing strategy against a novice as against an expert This is due to the assumption of minimax that the opponent uses the same strategy as the player This work studies the problem of opponent modelling in game playing We recursively de ne a player as a pair of a strategy and an opponent model which is also a player A strategy can be determined by the static evaluation function and the depth of search M an algorithm for searching game trees using an n level modelling player that uses such a strategy is described and analyzed We demonstrate experimentally the bene t of using an opponent model and show the potential harm caused by the use of an inaccurate model We then describe an algorithm M for using uncertain models when a bound on the model error is known Pruning in M is impossible in the general case We prove a su cient condition for pruning and present a pruning algorithm that returns the M value of a tree searching only necessary subtrees Finally we present a method for acquiring a model for an unknown player First we describe a learning algorithm that acquires a model of the opponent s depth of search by using its past moves as examples Then an algorithm for acquiring a model of the player s strategy both depth and function is described and evaluated Experiments with this algorithm show that when a superset of the set of features used by a xed opponent is available to the learner few examples are su cient for learning a model that agrees almost perfectly with the opponent