Game Theory with Applications to Finance and Marketing

A similar reasoning applies to σ 2 (L). Hence (U,L) is the unique proper
equilibrium of this strategic game.
5. Myerson also proves that any finite strategic game has a proper equilibrium,
and hence any finite game has a trembling-hand perfect equilibrium
and an NE. Let us sketch Myerson’s proof. Note that it suffices to
show that for any ɛ k ∈ (0, 1), an ɛ k -proper equilibrium σ k exists, since
by the compactness of Σ, a convergent subsequence of {σ k ; k ∈ Z + }
exists. Thus fix any ɛ ∈ (0, 1). Define
m ≡ max{#(S i ); i = 1, 2, · · · , I},
where recall that #(A) is the cardinality of set A (the number of elements
of A). Define d ≡ ɛm . For all i = 1, 2, · · · , I, define
m
Σ d i ≡ {σ i ∈ Σ i : σ i (s i ) ≥ d, ∀s i ∈ S i }.
Note that Σ d i is a non-empty compact subset of Σ 0 i . Define
Σ d ≡ Π I i=1 Σd i .
For all i = 1, 2, · · · , I, consider the correspondence F i : Σ d → Σ d i defined
by
F i (σ) = {σ i ∈ Σ d i : u i (s i , σ −i ) < u i (s ′ i, σ −i ) ⇒ σ i (s i ) ≤ ɛσ i (s ′ i), ∀s i , s ′ i ∈ S i }.
Note that given each σ ∈ Σ d , F i (σ) is convex and closed. We claim
that F i (σ) is also non-empty. To see this, for each s i ∈ S i , define ρ(s i )
to be the number of pure strategies s ′ i ∈ S i with
Define
σ ′ i(s i ) ≡
u i (s i , σ −i ) < u i (s ′ i , σ −i).
ɛ ρ(s i)
∑
s ′′
i ∈S i ɛρ(s′′
i ) , ∀s i ∈ S i .
By construction, we have σ ′ i (s i) ≥ d, and ∑ s i ∈S i
σ ′ i (s i) = 1. Moreover,
it can be verified that σ ′ i ∈ F i (σ), showing that F i (σ) is indeed nonempty.
Finally, one can verify that F i is upper hemi-continuous. Define
F ≡ Π I i=1F i : Σ d → Σ d .
5