We study a persuasion game à la Kamenica and Gentzkow (2011) where players are ambiguity averse with maxmin expected utility (Gilboa and Schmeidler, 1989). With no prior ambiguity, a Sender may choose to use ambiguous communication devices. Our main result characterizes the value of optimal ambiguous persuasion, which is often higher than what is feasible under Bayesian persuasion. We characterize posteriors that are potentially plausible when they are generated by ambiguous devices. One way to construct an optimal ambiguous communication device is by using synonyms, messages that lead to the same posteriors, in which Sender can hedge himself against ambiguity while inducing actions from Receiver that would not be possible under standard Bayesian persuasion. We also show that the use of synonyms are a necessary * We are grateful to the Editor, Marciano Siniscalchi, and two anonymous referees for very constructive comments and suggestions that have significantly improved the paper. We are also grateful to Pierre

I consider games with ambiguous payoffs played by non-Expected Utility decision makers. Three equilibrium solutions are studied. Nash equilibrium in which equilibrium mixed strategies must be best responses, Crawford equilibrium in beliefs and pure equilibrium in beliefs in which equilibrium strategies are mixtures of best responses, with the latter restricting best responses to pure actions. I study the interactions between ambiguity preferences on one side and equilibrium properties on the other. I show how the equilibrium concepts differ, computing necessary and sufficient conditions for existence and equivalence. I also show how these solution concepts fare against two fundamental principles of Nash equilibrium in standard games: the principle of indifference and the reduction principle. Given both are central to the computation of Nash equilibrium in games with Expected Utility players, their failure indicates how relaxing the Expected Utility hypothesis may disrupt standard game theoretic results such as the characterization of equilibria in two-player games. KeywordsAmbiguity • Equilibrium in beliefs • Game theory • Nash equilibrium • Non-expected utility B Dorian Beauchêne

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