Exercise 1. Consider the following game. There are finitely many players, say n > 1. Each...

 

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Exercise 1. Consider the following game. There are finitely many players, say n > 1. Each player i's pure strategies are collected in the set S; = [0, 100]. Player i's payoff from a strategy profile s = ($₁,..., Sn) is given by u; (s) = ($1 + == - (₁₁-α a for some number a € (0, 1). (a) Compute player i's best response si to s. (b) Show that 100(n-1) na n 2 +5₂)) ² Sn) st (100,..., 100) = 100- (c) Show that any strategy s,> 100(n-1)/(n-a) is strongly dominated for every player i. (d) Let S = [0, 100(a(n-1)/(n − a))k]. Show that any strategy s() > 100(a(n-1)/(n-a))k is strongly dominated assuming that every player j is restricted to strategies in S. (d) Find the unique Nash equilibrium of this game and show that it is the unique rationalizable outcome, in the sense of being the only survivor of iterated elimination of strongly dominated strategies. Exercise 1. Consider the following game. There are finitely many players, say n > 1. Each player i's pure strategies are collected in the set S; = [0, 100]. Player i's payoff from a strategy profile s = ($₁,..., Sn) is given by u; (s) = ($1 + == - (₁₁-α a for some number a € (0, 1). (a) Compute player i's best response si to s. (b) Show that 100(n-1) na n 2 +5₂)) ² Sn) st (100,..., 100) = 100- (c) Show that any strategy s,> 100(n-1)/(n-a) is strongly dominated for every player i. (d) Let S = [0, 100(a(n-1)/(n − a))k]. Show that any strategy s() > 100(a(n-1)/(n-a))k is strongly dominated assuming that every player j is restricted to strategies in S. (d) Find the unique Nash equilibrium of this game and show that it is the unique rationalizable outcome, in the sense of being the only survivor of iterated elimination of strongly dominated strategies.