Problem set 5
Advanced Microeconomics,
Ec 992, Spring 2024
Games of incomplete information I
1. Exercise 8.E.1 of MWG textbook.
2. Consider the following 2-player game of incomplete information. Player 1’s action set is {U, D} and Player 2’s action set is {L, R}. Player 1 can be of two possible types T1 or T2. Player 2 believes that Player 1 is of type T1 with probability 5/3. Table 1 and Table 2 provide the payoffs for each action profile when Player 1 is of type T1 and type T2 respectively. Find a Bayes Nash equilibrium of this game.
Table 1: Payoff matrix when Player 1 is type T1.
Table 2: Payoff matrix when Player 1 is type T2.
3. Exercise 8.E.3 of MWG textbook.
4. Consider the following duopoly market where firms compete by choosing prices simultaneously. The demand for firm 1’s good is
and for firm 2’s good is
where p1 ≥ 0 is price chosen by firm 1 and p2 ≥ 0 is price chosen by firm 2. Suppose ˜bn, n = 1, 2 are independently and identically distributed as follows.
where 0 < bL < bH. The value of ˜b1 is private information of firm 1 and that of ˜b2 is private information of firm 2. Both firms have zero marginal cost. Assume that bH −bL < 4 and solve for the symmetric pure strategy Bayes-Nash equilibrium of this game where both firms set strictly positive prices.
5. A public good is provided if at least one person pays the cost c > 0. There are N ≥ 2 individuals. The value of the good to individual n is vn, and is known only to individual n, n = 1, 2, . . . , N. The individuals believe that values v1, v2, . . . , vN are identically and independently distributed over the interval [0, ¯v] according to a distribution function G, where c < ¯v. Each individual can choose to pay c or to pay 0 and the individuals choose payments simultaneously. If the public good is not provided each individual’s payoff is 0.
(a) Under what conditions does there exist a pure strategy Bayes Nash equilib-rium of this game where individual 1 pays c if v1 ≥ c and pays 0 otherwise and individuals n = 2, . . . , N always pays 0? Are these conditions satisfied if G is the uniform. distribution over [0, ¯v]? Explain your answer.
(b) Answer both parts below.
i. Does there exist a (symmetric) pure strategy Bayes Nash equilibrium of this game where individual n pays c if and only if vn ≥ v
∗
for some v
∗ ∈ [0, ¯v], n = 1, 2, . . . , N? If yes, obtain the value of v
∗ and if not, explain why not. What is the value of v
∗
if G is the uniform. distribution? Explain your answer.
ii. What is the probability that the public good in provided in the sym-metric Bayes Nash equilibrium you obtained above? Under what con-dition(s) does this probability increase with N? Is this condition sat-isfied when G is the uniform. distribution? Explain your answer.
6. Answer both parts.
(a) Consider the complete information game given by the matrix in Table 3, where Player 1’s action set is {U, D} and Player 2’s action set is {L, R} and determine all the Nash equilibria (pure and mixed).
Table 3: Complete information game.
(b) Consider the incomplete information game where the payoffs for each ac-tion profile are as given in Table 4. Player 1’s action set is {U, D} and Player 2’s action set is {L, R}. The type of player i is denoted θi and is known only to player i, i ∈ {1, 2}. The players’ beliefs over the types are that θ1 and θ2 are identically and independently distributed according to a uniform. distribution over the interval [0, x], where x > 0 is a constant known to both players. Find a pure strategy Bayes-Nash equilibrium of
Table 4: Payoff matrix for incomplete information game.
the game where each player uses a threshold strategy. For this equilibrium, derive the limit of the probability that Player 1 plays D and the limit of the probability Player 2 plays R as x → 0.