IA168 — Problem set 2
Unless otherwise specified, “game” means “two-player extensive-form game with pure strategies only”
throughout this problem set.
Problem 1 [3 points]
We say that an imperfect information extensive-form game G is equivalent to a strategic form game G′
when
both have the same number of players and for every player i there is a bijection fi : Si → S′
i, such that for
every strategy profile s = (s1, . . . , sn) and every player i: ui(s) = u′
i(s′
), where s′
= (f1(s1), . . . , fn(sn)).
Find an imperfect information extensive-form game that is equivalent to this strategic form game:
A B C
X (0,1) (-1,5) (3,2)
Y (4,1) (-3,3) (2,6)
Problem 2 [5 points]
Consider this real-life situation: Adam is suspicious about the death of Alice, Bob’s wife. He confronts Bob,
but Bob tells him: ”If you keep sticking nose into my bussiness, I will kill you.” Adam decides whether to stay
quiet or try to force a confession out of Bob. If he escalates the situation, Bob may either confess that he
killed his wife and turn himself to police or attack Adam. If Bobs attacks Adam, then with probability 1 − p
he will win and Adam will be dead and with probability p Adam will be victorious and Bob will not survive
this confrontation.
We model this scenario as the perfect-information game depicted below.
..
A
.
(5, 10)
.
B
.
(15, −40)
.
(100p − 100, −100p)
.
q
.
f
.
c
.
a
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 3 [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 4 [5 points]
Let G be a perfect-information game. Prove or disprove:
a) For every Nash equilibrium with outcomes (x, y) in G, there exists a subgame-perfect equilibrium with
outcomes (x′
, y′
) where either x′
≥ x or y′
≥ y.
b) for every Nash equilibrium with outcomes (x, y) in G, there exists a subgame-perfect equilibrium with
outcomes (x′
, y′
) where x′
≥ x and y′
≥ y.
2