IA168 — Problem set 3
Problem 1 [8 points]
Find a perfect-information extensive-form game with pure strategies only where all of the following conditions
are satisfied:
• there is a strategy profile whose outcome is for both players better than that of any Nash equilibrium;
• there is a Nash equilibrium whose outcome is for both players better than that of any subgame-perfect
equilibrium;
• there are exactly two subgame-perfect equilibria s, s , and the outcome of s is for both players better
than that of s .
Problem 2 [8 points]
For a strategy profile s of an imperfect-information extensive-form game G with pure strategies only, consider
the following property (∗):
For every information set I, there exists a node h ∈ I
such that sh
is a Nash equilibrium in Gh
.
Prove or disprove the following two propositions: In every imperfect-information game where no path leads
twice through the same information set, it holds that:
a) every subgame-perfect equilibrium satisfies (∗);
b) every strategy profile which satisfies (∗) is a subgame-perfect equilibrium.
Problem 3 [4 points]
Consider this strategic-form game G:
A2 B2 C2
A1 (x, x) (0, 0) (10y, 0)
B1 (0, 0) (3x, 3x) (0, 0)
C1 (0, 10y) (0, 0) (y, y)
Consider game Gt-rep. Find the necessary and sufficient condition for x, y, t so that there is an SPE τ such
that u1(τ) > 3xt. Shortly explain why your answer is correct.