IA168 — Problem set 1
Throughout this problem set, “game” means “two-player strategic-form game with pure strategies only”.
Problem 1 [6 points]
Find a game with exactly 2 Pareto optimal strategy profiles and exactly 2 Nash equilibria so that:
a) both of the Nash equilibria are Pareto optimal;
b) exactly one Nash equilibrium is Pareto optimal;
c) neither of the Nash equilibria is Pareto optimal.
Problem 2 [10 points]
Consider the following zero-sum game, defined by the payoff table for player 1:
A2 B2
A1 x y2
B1 y x2
where the payoffs of player 2 are the opposite values of these payoffs (e.g. u2(A1, A2) = −u1(A1, A2) = −x)
and x, y ∈ R.
Player 1 and player 2 will play this game infinitely many times and we will denote by s1,i ∈ {A1, B1} the
strategy that player 1 chooses in the i-th iteration, by s2,i ∈ {A2, B2} the strategy that player 2 chooses in the
i-th iteration and by si the strategy profile (s1,i, s2,i).
Suppose that s1,1 = A1, s2,1 = A2, s1,i = A1 iff A1 is a best response to s2,i−1 and s2,i = A2 iff A2 is a best
response to s1,i−1 for i > 1. Find the necessary and sufficient condition for x, y so that:
a) ∃i ∈ N: si = si+1 = (A1, A2)
b) ∃i ∈ N: si = si+1 = (B1, B2)
c) ∀i ∈ N: si = si+1, si = si+2 and si = si+3
d) ∃i ∈ N: si = si+1 and either si = si+2 or si = si+3
Explain your reasoning.
Problem 3 [4 points]
Prove that each Nash equilibrium is rationalizable and survives IESDS (Theorem 17.2).