Game Theory with Applications to Finance and Marketing

and given any ɛ ∈ R ++ , σ ∈ Σ 0 is called an ɛ-perfect equilibrium if
∀i ∈ I, ∀s i , s ′ i ∈ S i,
u i (s i , σ −i ) < u i (s ′ i , σ −i) ⇒ σ i (s i ) ≤ ɛ.
A trembling-hand perfect equilibrium is then a profile σ ∈ Σ (which
need not be totally mixed!) such that there exists a sequence {ɛ k ; k ∈
Z + } in R ++ and a sequence {σ k ; k ∈ Z + } in Σ 0 with (i) lim k→∞ ɛ k = 0;
(ii) σ k is an ɛ k -perfect equilibrium for all k ∈ Z + ; and (iii) lim k→∞ σ i,k (s i ) =
σ i (s i ), ∀i ∈ I, ∀s i ∈ S i . It can be shown that a trembling-hand perfect
equilibrium must exist for a finite game, and the trembling-hand perfect
equilibrium is itself an NE, but the reverse is not true. 1 In particular,
the above profile (D,D) is not a trembling-hand perfect equilibrium.
3. Consider the extensive game with two players where player 1 first
chooses between L and R, and the game ends with payoff profile (2, 2)
if R is chosen, but if instead L is chosen, then player 2 can choose
between l and r, with the game ending with payoff profile (1, 0) if r is
chosen, and if instead player 2 chooses l, then player 1 can choose between
A and B, with the game ending with respectively payoff profiles
(3, 1) and (0, −5). This game has a unique SPNE, (L,l,A), but (R,r,B)
1 Let us prove that a trembling-hand perfect equilibrium σ is an NE. Recall the following
definition of NE: a profile σ ∈ Σ is an NE if and only if for all i ∈ I, for all s i , s ′ i ∈ S i,
u i (s i , σ −i ) < u i (s ′ i, σ −i ) ⇒ σ i (s i ) = 0.
Note that for all i ∈ I, for all s i , s ′ i ∈ S i such that
there exists K ∈ Z + such that
u i (s i , σ −i ) < u i (s ′ i , σ −i)
k ≥ K ⇒ u i (s i , σ k −i) < u i (s ′ i, σ k −i),
by the fact that σ k → σ, and hence for any such k, we have
implying that
σ k i (s i) ≤ ɛ k ,
0 ≤ σ i (s i ) = lim
k→∞ σk i (s i ) ≤ lim
k→∞ ɛ k = 0.
This shows that a trembling-hand perfect equilibrium is an NE.
2