Savage Games: A Theory of Strategic Interaction with Purely Subjective Uncertainty

Abstract. We define and discuss Savage games, which are ordinal games that are set in L. J. Savage’s framework of purely subjective uncertainty. Every Bayesian game is ordinally equivalent to a Savage game. However, Savage games are free of priors, prob- abilities and payoffs. Players’ information and subjective attitudes toward uncertainty are encoded in the state-dependent preferences over state contingent action profiles. In the games we study player preferences satisfy versions of Savage’s sure thing principle and small event continuity postulate. An axiomatic innovation is a strategic analog of Savage’s null events. We prove the existence of equilibrium in Savage games. This result eschews any notion of objective randomization, convexity, and monotonicity. Applying it to games with payoffs we show that our assumptions are satisfied by a wide range of decision-theoretic models. In this regard, Savage games afford a tractable framework to study attitudes towards uncertainty in a strategic setting. We illustrate our results on the existence of equilibrium by means of examples of games in which players have expected and non-expected utility.