(In)Existence of Equilibria for 2-Player, 2-Value Games with Semistrictly Quasiconcave Cost Functions

We consider 2-player , 2-value cost minimization games where the players’ costs take on two values, a , b , with a < b . The players play mixed strategies and their costs are evaluated by semistrictly quasiconcave cost functions representable by strictly quasiconcave, one-parameter functions 𝖥: [0, 1] →ℝ . Our main result is an impossibility result stating that: If the maximum of F is obtained in (0,1) and 𝖥 (1/2 ) b , then there exists a 2-player, 2-value game without F-equilibrium. The counterexample to the existence of equilibria game used for the impossibility result belongs to a new class of very sparse 2-player, 2-value bimatrix games which we call simple games . In an attempt to investigate the remaining case 𝖥 (1/2 ) = b , we show that: Every simple, n -strategy game has an F-equilibrium when 𝖥 (1/2 ) = b . We present a linear time algorithm for computing such an equilibrium. For 2-player, 2-value, 3-strategy games, we have that if 𝖥 (1/2 ) ≤ b , then every 2-player, 2-value, 3-strategy game has an F-equilibrium; if 𝖥 (1/2 ) > b , then there exists a simple 2-player, 2-value, 3-strategy game without F-equilibrium. To the best of our knowledge, this work is the first to provide an (almost complete) answer on whether there is, for a given function F, a counterexample game without F-equilibrium.