IA168 — Problem set 2
Throughout this problem set, “game” means “two-player strategic-form game with mixed strategies”.
Problem 1 [7 points]
Consider a game where each player has exactly five pure strategies, called Ai, Bi, Ci, Di, Ei for i ∈ {1, 2}.
The utility functions are defined by the following table:
A2 B2 C2 D2 E2
A1 (−4, 4) (−3, 2) (2, 2) (4, 1) (−2, −1)
B1 (0, 6) (−3, 3) (2, 3) (7, 3) (−3, 3)
C1 (−6, 0) (3, 6) (4, 1) (1, 2) (−6, 0)
D1 (−2, −1) (−2, 7) (2, 2) (5, 3) (−5, 3)
E1 (−6, 3) (−6, 3) (1, 3) (3, 2) (3, 6)
(a) Find a Nash equilibrium σ∗
= (σ∗
1, σ∗
2) such that | supp(σ∗
1)| + | supp(σ∗
2)| is maximal.
(b) Prove that σ∗
is a Nash equilibrium.
(c) Prove the maximality of | supp(σ∗
1)| + | supp(σ∗
2)|.
Problem 2 [5 points]
Give an example of a game where
(a) there is no weakly dominating pure strategy, but there exists a weakly dominating mixed strategy;
(b) there is no weakly dominating pure strategy, but there exists a very weakly dominating mixed strategy;
(c) there is no strictly dominated pure strategy, but there exists a strictly dominated mixed strategy;
(d) there is no very weakly dominated pure strategy, but there exists a strictly dominated mixed strategy
or prove that no such game exists.
Problem 3 [8 points]
Prove that for every k ∈ M there is a game with exactly k Nash equilibria, where
(a) M = {2n
− 1 | n ∈ N};
(b) M = N.