Formal analysis and algorithms for extracting coordinate systems of games

Abstract

A two-player game given in the normal form of payoff matrix may be alternatively viewed as a list of the outcomes of binary interactions between two sets of entities, solutions and tests. The internal structure of such interactions may be characterized by an appropriately constructed coordinate system, which spatially arranges the solutions with respect to coordinates identified with tests, while preserving their mutual relations as given by the matrix. Of particular interest are coordinate systems of minimal size that give rise to the notion of dimension of a game. Following [1], we investigate such coordinate systems and relate their features to properties of partially ordered sets (posets), mostly to poset width and poset dimension. We propose an exact algorithm for constructing a minimal correct coordinate system and prove its correctness. In the experimental part, we compare the exact algorithm to the heuristics proposed in [1] on a sample of random payoff matrices of different sizes to demonstrate that the heuristics heavily overestimates the size of the minimal coordinate system. Finally, we show how the game dimension relate to the a priori dimension of a game.