Behavioural strategies in weighted Boolean games

Abstract

This paper studies the computation of mixed Nash equilibria in weighted Boolean games. In weighted Boolean games, players aim to maximise the total expected weight of a set of formulas by selecting behavioural strategies, that is, randomisations over the truth assignments for each propositional variable under their unique control. Behavioural strategies thus present a compact representation of mixed strategies. Two results are algorithmically significant: (a) behavioural equilibria satisfy a specific independence property; and (b) they allow for exponentially fewer supports than mixed equilibria. These findings suggest two ways in which one can leverage existing algorithms and heuristics for computing mixed equilibria: a naive approach where we check mixed equilibria for the aforesaid independence property, and a more sophisticated approach based on support enumeration. In addition, we explore a direct numerical approach inspired by finding correlated equilibria using linear programming. In an extensive experimental study, we compare the performance of these three approaches.