Game-theoretic modeling of players’ ambiguities on external factors

We propose a game-theoretic framework that incorporates both incomplete information and general ambiguity attitudes on factors external to all players. Our starting point is players’ preferences on return-distribution vectors, essentially mappings from states of the world to distributions of returns to be received by players. There are two ways in which equilibria for this preference game can be defined. Also, when the preferences possess ever more features, we can gradually add more structures to the game. These include real-valued functions over return-distribution vectors, sets of probabilistic priors over states of the world, and eventually the traditional expected-utility framework involving one single prior. We establish equilibrium existence results, show the upper hemi-continuity of equilibrium sets over changing ambiguity attitudes, and uncover relations between the two versions of equilibria. Some attention is paid to the enterprising game, in which players exhibit ambiguity-seeking attitudes while betting optimistically on the favorable resolution of ambiguities. The two solution concepts are unified at this game’s pure equilibria, whose existence is guaranteed when strategic complementarities are present. The current framework can be applied to settings like auctions involving ambiguity on competitors’ assessments of item worths.