On Hurwicz–Nash Equilibria of Non–Bayesian Games under Incomplete Information ∗ Patrick Beißner † and M. Ali Khan ‡ December 5, 2018 Abstract We consider finite-player simultaneous-play games of private information in which a player has no prior belief concerning the information under which the other players take their decisions, and which he therefore cannot discern. This dissonance leads us to develop the notion of Hurwicz-Nash equilibria of non-Bayesian games, and to present a theorem on the existence of such an equilibrium in a finite-action setting. Our pure-strategy equilibrium is based on non-expected utility under ambiguity as developed in Gul and Pesendorfer (2015). We do not assume a linear structure on the individual action sets, but do assume private information to be “diffused” and “dispersed.” The proof involves a multi-valued extension of an individual’s prior to the join of the finest σ-algebra F of the information of the other players, and hinges on an absolute-continuity assumption on an individual’s belief with respect to the extended beliefs on F . (149 words) Key words and phrases : Non-Bayesian games, Hurwicz-Nash equilibria, Knightian uncertainty, private infor- mation, private beliefs, ambiguous beliefs. JEL Classification Numbers : C62, D50, D82, G13. ∗ The authors are grateful to Yongchao Zhang for ongoing collaboration, and to Achile Basile, Ying Chen, Amanda Friedenberg, Simon Grant, Faruk G¨ ul, Peter Hammond, Osama Khan, Jong Jae Lee, Alejandro Manelli, Ed Prescott, Frank Riedel, Eddie Schlee, Vernon Smith, Ron Stauber, Max Stinchcombe and Metin Uyanik for stimulating correspondence and conversation. Preliminary versions of this paper were presented at departmental seminars at The University of Queensland (November 7, 2018), Arizona State University (October 30, 2017) and Funda¸caoGetulioVargas (April 10, 2018); and at the 17th SAET Conference in Faro (June 30, 2017), the 6th Workshop in Equilibrium Analysis in Naples (January 27, 2018), the IMS-NUS Workshop in Game Theory, and the 27th EWGET Workshop in Paris (June 28, 2018). The authors warmly thank the Editor and her two anonymous referees for their careful reading: the second example, a fuller and more careful elaboration of the proof, and the connection to the literature on “imprecise probabilities” all presented herein, resulted from it. † Research School of Economics, The Australian National University, Canberra Australia. E-mail patrick.beissner@anu.edu.au ‡ Department of Economics, The Johns Hopkins University, Baltimore, MD 21218. E-mail akhan@jhu.edu 1