Formulate the situation as strategic game - nash equilibrium, Game Theory

Two individuals use a common resource (a river or a forest, for example) to produce output. The more the resource is used, the less output any given individual can produce. Denote by xi the amount of the resource used by individual i (where i = 1, 2). Assume specifically that individual i's output is xi(1 (x1 + x2)) if x1 + x2 < 1 and zero otherwise. Each individual i chooses xi ? [0, 1] to maximize her output.

(a) Formulate this situation as a strategic game.

(b) Find the best response correspondences of the players.

(c) Find its Nash equilibria.

(d) Does the Nash equilibrium value of x1; x2 maximize the total output? (Is there any other output pro le that results in a higher total output than the Nash equilibrium?)

(e) Suppose now there are n individuals and hence the payoff function of individual i (where i = 1; 2;...; n) is given by xi(1 (x1 +x2 + ... +xn)) if x1 +x2 +...+xn 6 1 and zero otherwise. Find the Nash equilibria of this game.

 

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