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

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 𝖥: [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 𝖥(1/2) b, then there exists a 2-players, 2-values game without 𝖥-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 𝖥(1/2) = b, we show that: - Every normal, n-strategies game has an 𝖥-equilibrium when 𝖥( 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 𝖥(1/2) ≤ b, then every 2-players, 2-values, 3-strategies game has an 𝖥-equilibrium; if 𝖥(1/2) > b, then there exists a normal 2-players, 2-values, 3-strategies 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 concave function 𝖥, a counterexample game without 𝖥-equilibrium.