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 x_{i}(1 (x_{1} + x_{2})) if x_{1} + x_{2} < 1 and zero otherwise. Each individual i chooses x_{i} ? [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 x_{i}(1 (x_{1} +x_{2} + ... +x_{n})) if x_{1} +x_{2} +...+x_{n} 6 1 and zero otherwise. Find the Nash equilibria of this game.