Part 1-
1. Consider the following version of the Baliga - Sjostrom arms-race game:
B N
B -c1, -c2 2-c1, -10
N -10, 2 - c2 0, 0
Assume nontransferable utility. Suppose that each ci takes values in the set {1, 4}. That is, country i's cost of building arms can be low, ci = 1, or high, ci = 4.
a) Suppose that each country knows both its own cost and the other country's cost. For each of the four possible combinations (c1, c2), find the Pareto efficient outcomes.
b) Again suppose that each country knows both its own cost and the other country's cost. For each of the four possible combinations (c1, c2), find the pure-strategy Nash equilibria.
c) Now suppose that each country i knows ci, but knows only that cj = 1 with probability p, and cj = 4 with probability 1 - p. A pure-strategy for country i now consists of two actions, si(1) and si(4). For what values of p does there exist a pure-strategy Nash equilibrium with s1(4) = s2(4) = N. Show your derivation clearly.
Part II-
1. Let A = {a, b, c}. I = {Curly, Larry, Aloe}, and consider the social choice function g defined as follows: If a is the first choice of all players, then a is the social choice. Otherwise the social choice is the majority choice between b and c.
a) State the Gibbard-Satterthwaite Theorem.
b) Does the Gibbard-Satterthwaite Theorem imply that g is not straight forward? Answer YES or NO and explain your answer.
c) Give CM example showing that g is not straightforward. That is, state specific preferences for the three individuals and show how one individual can obtain a preferred outcome by reporting false preferences.
2. Suppose a public good level must be chosen in a society with three individuals, I = {1, 2, 3). The public good level x costs x units of the private good to produce, so the set of allocations is the set A = {(x, t1, t2, t3) : x > 0 and t1 + t2 + t3 = x}. Each Individual has a utility function ui(x, yi) =2αi√x + yi, with α1 = 1, α2 = 3 and α3 = 5. Each individual i has an endowment of the private good wi, so given on allocation (x, t1, t2, t3), each individual i's utility is ui(x, wi - ti) = 2αi√x + wi - ti.
a) Find the set of Pareto efficient allocations.
b) Suppose that each individual must pay one-third of the total taxes, t1 = x/3, for each i.
i) Find each individual's ideal point, xi* for the public good.
ii) Find the Condorcet winner. Is it Pareto efficient?
3. Consider the version of the Baliga - Sjostrom arms-race game in Part 1:
B N
B -c1, -c2 2-c1, -10
N -10, 2 - c2 0, 0
As in Part I, Question 2(c), suppose the ci's are random and independent, with ci = 1 with probability p and ci = 4 with probability 1 - p. Unlike Part I, suppose the game is sequential. First, country 1 observes c1 and choose B or N. Then country 2 observes c2 and country 1's action, and chooses B or N. Country 2 does not observe c1 but since she observes country 1's action, c1 is not relevant to her decision. A pure strategy for country 1 is a pair of actions depending on c1, (s1 (1), s1 (4)). A purest/ate& for country 2 is four actions, depending on c2 and player 1's action. (s2(1, B), s2(1, N ), s2(4, B), s2(4, N)).
For which values of p, If any, does there exist a pure-strategy subplot, perfect Nash equilibrium with s1(4) = N? Show your derivation clearly.
4. Consider a two-player Shubik auction in which each bid costs $2 but the players have differing values of the object, with V1 = 5 and V2 = 10.
a) Find a stationary mixed-strategy subgame perfect Nash equilibrium. Note that since the players have different values, they may have different bidding probabilities. Show your derivation clearly.
b) Give a brief explanation of why the difference in values causes the bidding probabilities to differ in the direction that they do.
5. Let I = (1, 2, 3), and for each of the following two power functions, find the core of the corresponding pillage game.
a) π(C, w) = 2#C+Σi∈Cwi.
b) Let α1 = 1, α2 = 2, and α3 = 2.
π (C, w) = Σi∈Cαiwi.