Two people are engaged in a joint project if each person i

Question

Problem 1: Two people are engaged in a joint project. If each person I puts in the effort x_i, a nonnegative number equal to at most 1, which costs her c(x_i), the outcome of the project is worth f(x₁, x₂). The worth of the project is split equally between the two people, regardless of their effort levels. Formulate this situation as a strategic game. Find the Nash equilibria of the game when

a. f(x₁, x₂) = 3x₁x₂ and c(x_i) = x_i²  for i = 1,2

b. f(x₁, x₂) = 4x₁x₂ and c(x_i) = x_i for i = 1, 2

In each case, is there a pair of effort levels that yields higher payoffs for both players than do the Nash equilibrium effort levels?

Problem 2: Consider the public good game when u_i(c₁, c₂) is the sum of three parts: the amount c₁ + c₂ of the public good provided, the amount w_i - c_i person i spends on private goods, and a term (w_i - c_i)(c₁ + c₂) that reflects an intersection between the amount of the public good and her private consumption-the greater the amount of the public good game, the more she values her private consumption. In summary, suppose that person i's payoff is c₁+c₂+w_i-c_i+(w_i-c_i)(c₁+c₂), or w_i+c_j+(w_i-c_i)(c₁+c₂), where j is the other person. Assume that w₁ = w₂ = w, and that each player i's contribution c_i may be any number (positive or negative, possibly larger than w). Find the Nash equilibrium of the game that models this situation. (You can calculate the best responses explicitly. Imposing the sensible restriction that c_i lie between 0 and w complicates the analysis but does not change the answer.) Show that in the Nash equilibrium both players are worse off than they are when both contribute half of their wealth to the public good.

Problem 3: For the game in Table 1, determine, for each player, whether any action is strictly dominated or weakly dominated. Find the Nash equilibria of the game; determine whether any equilibrium is strict.

Problem 4: Find the Nash equilibrium of Cournot's game when there are two firms; the inverse demand function is given by

and the cost function of each firm i is C_i(q_i) = q_i².

Problem 5: Two firms use a common resource (a river or a forest, for example) to produce output. As the total amount of the resource used by the two firms increases, each firm can produce less output. Denote by x_i the amount of the resource used by firm i. Assume that firm i's output is x_i(1 - x₁ - x₂) if x₁ + x₂ ≤ 1, and zero otherwise. Each firm i chooses x_i to maximize its output. Formulate this situation as a strategic game. Find the Nash equilibria of the game. Find an action profile (x₁, x₂) at which each firm's output is higher than it is at any Nash equilibrium.

Problem 6: Consider Bertrand's duopoly game under a variant of the assumptions that the firms' unit costs are different, equal to c₁ and c₂, where c₁ < c₂. Denote by p₁^m the price that maximizes (p - c₁)(α - p), and assume that c₂ < p₁^m and that the function (p - c₁)(α - p) is increasing in p up to p₁^m.

a. Suppose that the rule for splitting up consumers when the prices are equal assigns all consumers to firm 1 when both firms charge the price c₂. Show that (p₁, p₂) = (c₂, c₂) is a Nash equilibrium and that no other pair of prices is a Nash equilibrium.

b. Show that no Nash equilibrium exists if the rule for splitting up consumers when the prices are equal assigns some consumers to firm 2 when both firms charge c₂.