Zero-Sum Games in Social Choice and Game Theory

PhD thesis, Technische Universität München, 2018

Zero-sum games model the most extreme form of competition among players. When there are only two players, von Neumann’s minimax theorem shows that every zero-sum game admits a pair of maximin strategies that achieve unique optimal payoffs. After providing a new proof of the minimax theorem, we derive a set of epistemic conditions that necessitates maximin play. For the special case of symmetric zero-sum games, we determine the distribution over supports of maximin strategies in randomly chosen games. In decision theory, zero-sum games appear as representations of preferences over probabilistic outcomes through skew-symmetric bilinear utility functions. A subdomain of these preferences are preferences based on pairwise comparisons, for which one outcome is preferred to another outcome if and only if the former is more likely to yield a more preferred alternative. We show that three impossibility results of collective preference aggregation that obtain on the unrestricted domain cease to hold for preferences based on pairwise comparisons: Arrow’s dictatorship theorem, Moulin’s incompatibility of Condorcet consistency and resistance to the no-show paradox, and the conflict between consistency with respect to variable electorates and consistency with respect to components of similar alternatives.