(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations
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We consider 2-players, 2-values minimization games where the players' costs take on two values, a,b, a<b. The players play mixed strategies and their costs are evaluated by unimodal valuations. This broad class of valuations includes all concave, one-parameter functions π₯: [0,1]ββ with a unique maximum point. Our main result is an impossibility result stating that: If the maximum is obtained in (0,1) and π₯(1/2) b, then there exists a 2-players, 2-values game without π₯-equilibrium. The counterexample game used for the impossibility result belongs to a new class of very sparse 2-players, 2-values bimatrix games which we call normal games. In an attempt to investigate the remaining case π₯(1/2) = b, we show that: - Every normal, n-strategies game has an π₯-equilibrium when π₯( 1/2) = b. We present a linear time algorithm for computing such an equilibrium. - For 2-players, 2-values games with 3 strategies we have that if π₯(1/2) β€ b, then every 2-players, 2-values, 3-strategies game has an π₯-equilibrium; if π₯(1/2) > b, then there exists a normal 2-players, 2-values, 3-strategies game without π₯-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 concave function π₯, a counterexample game without π₯-equilibrium.
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