A group of n 2 firms uses a common resource to produce output. As more of...

 

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A group of n 2 firms uses a common resource to produce output. As more of the resource is used, any given firm can produce less output. Denote by x;>0 the amount of the resource used by firm i = 1,...,n. Then the total usage of the resource is X = x1 +...+ x = x; +[jai xj. Specifically, firm i's payoff is x; (1 - (x; +[jzi x;)) if X < 1, and zero if X > 1 (that is, the total supply of the resource is 1). Each firm i chooses x; to maximize its output. (a) Find a best response of firm i to actions of other firms xj, j + i. (Hint: to find a best response of player i, consider two cases: jix; 1 and jzi x; < 1). 2 (b) Find all pure strategy Nash equilibria. (Hint: There is a single equilibrium where X < 1. For this equilibrium you need to find equilibrium resource usages by each firm. There are infinitely many equilibria where X > 1. For these equilibria you only need to write a condition on xi, i = 1,...,n.). (c) Find an action profile (xi,...,x) at which each firm's payoff is higher than it is at any Nash equilibrium. Acti Go to A group of n 2 firms uses a common resource to produce output. As more of the resource is used, any given firm can produce less output. Denote by x;>0 the amount of the resource used by firm i = 1,...,n. Then the total usage of the resource is X = x1 +...+ x = x; +[jai xj. Specifically, firm i's payoff is x; (1 - (x; +[jzi x;)) if X < 1, and zero if X > 1 (that is, the total supply of the resource is 1). Each firm i chooses x; to maximize its output. (a) Find a best response of firm i to actions of other firms xj, j + i. (Hint: to find a best response of player i, consider two cases: jix; 1 and jzi x; < 1). 2 (b) Find all pure strategy Nash equilibria. (Hint: There is a single equilibrium where X < 1. For this equilibrium you need to find equilibrium resource usages by each firm. There are infinitely many equilibria where X > 1. For these equilibria you only need to write a condition on xi, i = 1,...,n.). (c) Find an action profile (xi,...,x) at which each firm's payoff is higher than it is at any Nash equilibrium. Acti Go to