CHARACTERIZATIONS OF THE EXISTENCE OF EQUILIBRIA IN GAMES WITH DISCONTINUOUS AND NON-QUASICONCAVE PAYOFFS

This paper characterizes pure-strategy and dominant-strategy Nash equi librium in noncooperative games which may have discontinuous and/or no n-quasiconcave payoffs. Conditions called diagonal transfer quasiconca vity and uniform transfer quasiconcavity are shown to be necessary and , with conditions called diagonal transfer continuity and transfer upp er semicontinuity, sufficient for the existence of pure-strategy and d ominant-strategy Nash equilibrium, respectively. The results are used to examine the existence or non-existence of equilibrium in some well- known economic games with discontinuous and/or non-quasiconcave payoff s. For example, we show that the failure of the existence of a pure-st rategy Nash equilibrium in the Hotelling model is due to the failure o f an aggregator function to be diagonal transfer quasiconcave-not the failure of payoffs to be quasiconcave, as has been elsewhere conjectur ed.