IA168 — Problem set 3
For an extensive-form game G, let SPE(G) denote the set of subgame-perfect equilibria of G and NE(G)
denote the set of Nash equilibria of G.
Problem 1 [6 points]
Consider the following two-player strategic-form game G:
C S
C (−5, −5) (0, −20)
S (−20, 0) (−1, −1)
a) Calculate the number of strategies of Player 1 and Player 2 in G2−rep;
b) calculate the number of strategy profiles in G2−rep;
c) calculate the number of Nash equilibria in G2−rep.
Moreover, in dependence on parameter t ∈ Z+
d) calculate the number of strategies of Player 1 and Player 2 in Gt−rep.
e) calculate the number of strategy profiles in Gt−rep.
f) find all subgame perfect equilibria in Gt−rep.
Use the definition, not the example from the lecture. Justify your reasoning.
Problem 2 [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 (3.2, 3.5).
b) Calculate infs∈SPE(Gavg
irep) u1(s).
c) Calculate sups∈SPE(Gavg
irep) u1(s).
Justify your reasoning.
Problem 3 [4 points] Give an example of a two-player strategic-form game G = ({1, 2}, (S1, S2), (u1, u2))
such that all of the following conditions are satisfied
a) |S1| + |S2| = 5;
b) maxs∈SPE(Gavg
irep) u1(s) = 0;
c) maxs∈NE(Gavg
irep) u1(s) = 5.
Find the SPE s such that u1(s) = 0 and NE s such that u1(s ) = 5. Explain your reasoning.
Problem 4 [5 points] Consider the following strategic-form game G
A2 B2
A1 (2, 1) (7, −1)
B1 (−2, 6) (x, y)
Consider also strategy profile s = (s1, s2)
si(h) =
Bi if h ∈ (B1, B2)
∗
Ai otherwise
Find all 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
irep} = 3
5 .
Justify your reasoning.