Entropy-Driven Evolutionary Approaches to the Mastermind Problem

Abstract

Mastermind is a well-known board game in which one player must discover a hidden color combination set up by an opponent, using the hints the latter provides (the number of places –or pegs– correctly guessed, and the number of colors rightly guessed but out of place in each move). This game has attracted much theoretical attention, since it constitutes a very interesting example of dynamically-constrained combinatorial problem, in which the set of feasible solutions changes with each combination played. We present an evolutionary approach to this problem whose main features are the seeded initialization of the population using feasible solutions discovered in the previous move, and the use of an entropy-based criterion to discern among feasible solutions. This criterion is aimed at maximizing the information that will be returned by the opponent upon playing a combination. Three variants of this approach, respectively based on the use of a single population and two cooperating or competing subpopulations are considered. It is shown that these variants achieve the playing level of previous state-of-the-art evolutionary approaches using much lower computational effort (as measured by the number of evaluations required).