IA168 — Problem set 4
Problem 1 [4 points]
Consider 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 2 [8 points]
Consider “3rd
price auction” as a game of incomplete information. The payoff of every player is 0 if their
bid was not (strictly) highest, and it is their type minus the 3rd
highest bid if they were the highest bidder.
The bid is a non-negative real number.
Formally, consider the following game of incomplete information
G = (N, (Ai)i∈N , (Ti)i∈N , (ui)i∈N ),
where N = {1, 2, 3, . . . , n}, n ≥ 3, (∀i ∈ N) Ai = Ti = R+
0 = {r ∈ R | r ≥ 0}, and
ui(a1, . . . , an; ti) =
0 (∃j ∈ N) aj ≥ ai,
ti − ai3 ai1 > ai2 ≥ ai3 ≥ · · · ≥ ain , i1 = i, {i1, . . . , in} = {1, . . . , n}.
a) Prove that there is no ex-post Nash equilibrium.
b) Prove or disprove the existence of an ex-post Nash equilibrium if the bids of each player are bounded by a
common bound, i.e.,
(∃vmax ∈ R+
0 ) (∀i ∈ N) Ai = [0, vmax].
c) Prove or disprove the existence of an ex-post Nash equilibrium if the types of each player are bounded by a
common bound, i.e.,
(∃vmax ∈ R+
0 ) (∀i ∈ N) Ti = [0, vmax].
d) Prove or disprove the existence of an ex-post Nash equilibrium if the bids of each player are bounded by
possibly different bounds, i.e.,
(∃v1, . . . , vn ∈ R+
0 ) (∀i ∈ N) Ai = [0, vi].
Problem 3 [8 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.
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.
We further require that P satisfies that for every player i ∈ {1, 2} and every type t ∈ {S, R}, the probability
that i is of type t is positive.