IA168 — Problem set 3
Problem 1 [5 points]
Consider the following two-player strategic-form game G:
X Y
A (4, 4) (−1, 5)
B (5, −1) (1, 1)
a) In Gavg
irep, find a subgame-perfect equilibrium whose outcome is (3.5, 3.2).
b) Calculate infs∈SP E(Gavg
irep) u1(s).
c) Calculate sups∈SP E(Gavg
irep) u1(s).
Justify your reasoning.
Problem 2 [4 points]
Consider the following two-player strategic-form game G, with real-valued parameters x, y:
A B
A (2, 1) (7, −1)
B (−2, 6) (x, y)
The players will play an infinite number of rounds, with a discount factor δ. Both will play the following
strategy: If only B’s have been played so far (i.e., the current history lies in (B, B)∗
), then the player plays B;
otherwise he plays A. Let s denote the corresponding strategy profile.
Find all pairs (x, y) ∈ R × R for which inf{δ ∈ R: 0 < δ < 1 ∧ s is a SPE in Gδ
irep} = 3/5.
Justify your reasoning.
Problem 3 [4 points]
Consider the incomplete-information game G = ({1, 2}, ({A, B, C}, {D, E, F}), ({P, Q}, {R, S}), (u1, u2)}),
where u1, u2 are given by the following matrices:
u1(−, −, P) D E F
A 6 5 4
B 1 2 5
C 1 2 3
u1(−, −, Q) D E F
A 6 5 4
B 1 2 3
C 1 5 3
u2(−, −, R) D E F
A 6 1 1
B 5 1 1
C 4 1 2
u2(−, −, S) D E F
A 1 5 1
B 2 4 2
C 3 3 3
For each X ∈ {A, B, C, D, E, F}, find all strictly, weakly, and very weakly dominant strategies in game G−X,
where G−X is created from G by deleting action X.
Problem 4 [7 points]
Consider the following Bayesian game: There are two players, they have two actions A, B, and they have
two types S, R. Type S means the player wants to play the same action as the other player, R means he wants
to play the other action. Specifically, the gain is +3 if this goal is achieved, plus there is bonus +1 for playing
action A.
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Formally: GP = ({1, 2}, ({A, B}, {A, B}), ({S, R}, {S, R}), (u1, u2), P), where u1, u2 are given by the following
matrices:
u1(−, −, S) A B
A 4 1
B 0 3
u1(−, −, R) A B
A 1 4
B 3 0
u2(−, −, S) A B
A 4 0
B 1 3
u2(−, −, R) A B
A 1 3
B 4 0
Let BNE(GP ) denote the set of Bayesian Nash equilibria in game GP . Moreover, let UV |XY denote the
strategy profile ({(S, U), (R, V )}, {(S, X), (R, Y )}) (i.e., player 1 plays U if he is S and he plays V if he is R;
similarly for player 2). Find a distribution P such that:
a) BNE(GP ) = ∅;
b) BNE(GP ) = {AA|AB, AB|AA};
c) BNE(GP ) = {AB|AB};
d) BNE(GP ) = {AB|AB, BA|BA};
e) BNE(GP ) = {AA|AB};
f) |BNE(GP )| = 5.
Furthermore, P is required to satisfy that for every player i ∈ {1, 2} and every type t ∈ {S, R}, the probability
that i is of type t is positive.
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