IA168 — Problem set 3
For an extensive-form game G, let SPE(G) denote the set of subgame-perfect equilibria of G.
Problem 1 [10 points]
Consider the following two-player strategic-form game G:
X Y
A (4, 4) (−1, 5)
B (5, −1) (1, 1)
a) In dependence on the parameter t ∈ N+
, calculate the number of strategy profiles in Gt−rep. Try to
express the result as explicitly as possible.
b) In Gavg
irep, find a subgame-perfect equilibrium whose outcome is (3.2, 3.5).
c) Calculate sups∈SP E(Gavg
irep) u1(s).
Justify your reasoning.
Problem 2 [10 points]
Let G be a two-player strategic-form game. Prove or disprove the following three propositions:
a) if 0 < δ < δ < 1, then SPE(Gδ
irep) ⊇ SPE(Gδ
irep);
b) if 0 < δ < δ < 1, then SPE(Gδ
irep) ⊆ SPE(Gδ
irep);
c) sups∈SP E(Gδ
irep) u1(s) + u2(s) is a continuous function of δ ∈ (0; 1) (here u1 and u2 correspond to the
given δ);