IA168 — Problem set 1
Throughout this problem set, we consider pure strategies only.
Problem 1 [2 points]
Consider the following table
Y1 Y2
X1 1, 0 2, 1
X2 2, 1 4, 0
X3 3, 2 6, 1
and a game defined as follows. Player 1 picks a row and Player 2 picks a column of the table independently of
each other. The payoff of Player 1 (Player 2) is the first (second) value in the corresponding cell.
Give a formal description of this game as a game in strategic form.
Problem 2 [4 points]
Consider a game G given by the following table
B1 B2 B3 B4
A1 5, 9 7, 7 1, 6 3, 4
A2 6, 3 8, 5 3, 4 5, 2
A3 7, 5 6, 4 6, 6 4, 6
A4 6, 2 3, 1 7, 3 1, 8
a) Find the games Gk
DS for k = 0, 1, . . . and determine the number of IESDS equilibria.
Is the game IESDS-solvable?
b) Find the games Gk
Rat for k = 0, 1, . . . and determine the number of rationalizable equilibria.
Is the game solvable by rationalizability?
Problem 3 [4 points]
Prove that rationalizability creates no new Nash equilibria in any finite two-player strategic-form game.
Definition. A strategy profile s ∈ S Pareto dominates a strategy profile s ∈ S if ui(s) ≥ ui(s ) for all i ∈ N,
and ui(s) > ui(s ) for at least one i ∈ N.
A strategy profile s ∈ S is Pareto-optimal if it is not Pareto dominated by any other strategy profile.
Problem 4 [4 points]
Find a game with exactly 2 Pareto-optimal strategy profiles and exactly 2 Nash equilibria such 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 5 [6 points]
Consider the following zero-sum game, defined by the payoff table for Player 1:
A2 B2
A1 1 x
B1 0 y
where x, y ∈ R and the payoffs of Player 2 are the opposite values of those of Player 1 in the table above
(e.g. u2(A1, A2) = −u1(A1, A2) = −1).
Player 1 and Player 2 will play this game infinitely many times. For i ∈ {1, 2} and j ∈ N, we denote by
si,j ∈ {Ai, Bi} the strategy chosen by Player i in the j-th iteration, and by sj we denote the strategy profile
(s1,j, s2,j).
For both i ∈ {1, 2}, let si,1 = Ai and for j ≥ 2 we have si,j = Ai iff Ai is a best response to s3−i,j−1 (i.e. a
best response to the strategy of the other player in the iteration before), si,j = Bi otherwise.
In dependence on the parameters x and y, determine the sequence of strategy profiles played by Player 1
and Player 2. Explain your reasoning.