IA168 — Problem set 2
Throughout this problem set, “game” means “two-player extensive-form game with pure strategies only”.
Problem 1 [5 points]
Consider this real-life situation: Bob tells his wife Alice: “If you are ever unfaithful to me, I will kill you.”
Then, Alice decides to be either faithful or unfaithful. If she is unfaithful, Bob eventually finds it out and either
keeps quiet or really kills her and then, with probability p, is proved guilty and put in prison.
We model this scenario as the perfect-information game depicted below.
A
(10, 10) B
(15, −10) (−100, −40p)
f u
q k
In dependence on the parameter p, 0 ≤ p ≤ 1, answer the following questions:
How many strategies does each player have?
Which of them are never-best-response?
Which of them are maxmin?
How many strategy profiles are there?
Which of them are Nash equilibria?
Which of them are subgame-perfect equilibria?
Is Bob’s threat actually credible?
Problem 2 [7 points]
Find a perfect-information game 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 for player 1 is better than that of any subgame-perfect equi-
librium;
• there are exactly two subgame-perfect equilibria s, s , and the outcome of s is for both players better than
that of s .
Problem 3 [8 points]
For a strategy profile s of an imperfect-information game G, 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.