Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs

This paper characterizes pure-strategy and dominant-strategy Nash equilibrium in noncooperative games that may have discontinuous and/or non-quasi-concave payoffs. Conditions called diagonal transfer quasi-concavity and uniform transfer quasi-concavity are shown to be necessary and, with conditions called diagonal transfer continuity and transfer upper semicontinuity, sufficient for the existence of pure-strategy and dominant-strategy Nash equilibrium, respectively. The results are used to examine the existence or nonexistence of equilibrium in some well-known economic games with discontinuous and/or non-quasi-concave payoffs.