IA168 — Problem set 3
Problem 1 [7 points]
Consider this strategic-form game G:
A2 B2 C2
A1 (x, x) (26, −1) (−20, −1)
B1 (−1, 26) (13, 13) (−20, 26)
C1 (−1, −20) (26, −20) (2x − 10, 2x − 10)
a) For x = 10, state (as explicitly as possible) the number of SPEs in the 10-stage game G10-rep.
b) Find all x ∈ R, 0 ≤ x ≤ 10, for which the maximal outcome for player 1 in an SPE is as large as possible.
Formally: find all x ∈ [0, 10], for which sup{u1(s) | s ∈ SPE(G10-rep)} is maximal. Explain your solution.
Problem 2 [5 points]
Consider this strategic-form game G:
A2 B2
A1 (2, 1) (7, −1)
B1 (−2, 6) (x, y)
Consider also strategy profile s :
si(h) =
Bi if h ∈ {(B1, B2)}∗
Ai otherwise
Find all the pairs (x, y) ∈ R2
for which the minimal discount required for s to be an SPE is equal to 3
5 .
Formally: find all the pairs (x, y) ∈ R2
such that inf{δ ∈ (0, 1) | s is SPE in Gδ} = 3
5 .
Problem 3 [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 their type minus the 3rd
highest bid, if they were the highest bidder. The
bid is a nonnegative real number.
a) Prove or disprove the existence of a weakly dominant strategy for player 1.
b) Prove or disprove the existence of ex-post Nash equilibrium.
Now consider the ”3rd
price auction” as a Bayesian game, where the type of every player is uniformly
distributed on interval [0, vmax].
c) Prove that strategy profile s given by si(ti) = n−1
n−2 ti is a Bayesian Nash equilibrium.