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(In)Existence of Equilibria for 2-Players, 2-Values Games with Concave Valuations

by   Chryssis Georgiou, et al.

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]→ℝ 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) ≀ 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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