IA168 — Problem set 1
Problem 1 [4 points]
Give an example of a zero-sum two-player strategic-form game G with pure strategies only such that
G = G1
Rat = G2
Rat = G3
Rat.
Problem 2 [8 points]
Consider a two-player strategic-form game with mixed strategies, where each player has exactly two strategies,
called A1, B1, and A2, B2, respectively.
a) Define the utility functions of both players so that there are infinitely many mixed Nash equilibria, but
there are ¯σ1 ∈ Σ1, ¯σ2 ∈ Σ2 such that for all σ1 ∈ Σ1, σ2 ∈ Σ2 neither (¯σ1, σ2) nor (σ1, ¯σ2) is a mixed Nash
equilibrium.
b) Define the utility functions of both players so that there are as many mixed Nash equilibria as possible, but
only finitely many of them.
c) Prove that, provided there are only finitely many mixed Nash equilibria, there cannot be more of them than
in your example from b).
Problem 3 [8 points]
Prove or disprove:
a) IESDS creates no new Nash equilibria in any finite strategic-form game G with pure strategies.
b) IESDS creates no new Nash equilibria in any strategic-form game G with pure strategies.