Graduate micro II
Patrick Legros
Example of sequential equilibrium
Consider the following game
1
1− ε
ε
2
η
3
1 −η
b
a
1,1,1
δ
1− δ
δ
1− δ
0,0,0
3,2,2
0,0,1
4,4,0
Each player has two actions (left and right). Typical behavioral strategies
are represented in the figure; for instance ε is the probability with which player
1 plays L, etc.
There are many Nash equilibria; in particular (ε, η, δ) = (1, 0, 0) and (ε, η, δ) =
(0, 0, 1) are Nash equilibria. (1, 0, 0) is Nash because player 2 plays R with probability one - even if player 2’s action does not affect his payoff since player 1
plays L with probability one; however if player 2 plays L, then player 1 would
prefer to play R rather than L in order to get 4 instead of 3.
Consider a vector of strategies (ε, η, δ) . This vector is part of a sequential
equilibrium if we can find a belief structure at the information set of player
3, that is a probability p (a) assigned to node a such that (i) one can find a
sequence {(εk , η k , δ k )}∞
k=1 converging to (ε, η, δ) , and a probability p (b) such
that
k
(i) p (b) = limk→∞ εk η ε(1−ε
k)
k
(ii) (ε, η, δ) is sequentially rational.
We describe below the set of sequential equilibria. First, consider sequential
rationality.
Player 1 compares the expected utility 3 (1 − δ) of playing L to the expected
utility 1 − η + η4 (1 − δ) of playing R, since 3 (1 − δ) ≥ 1 − η + η4 (1 − δ) is
1
equivalent to 2 − 3δ ≥ (3 − 4δ) η, the best response correspondence for player 1
is
=0
<
=1
ε
as 2 − 3δ > (3 − 4δ) η
(SR1)
∈ [0, 1]
=
Similar computations show that the best response for player 2 is
η
=0
>
=1
as δ <
∈ [0, 1]
=
3
4
(SR2)
and for player 3 is
δ
=0
>
=1
as p (a) <
∈ [0, 1]
=
1
3.
(SR3)
• Case 1: η = 0. Then, (SR2) implies that δ ≥ 34 , and (SR3) implies that
p (a) ≤ 13 . Since η = 0 and δ ≥ 34 , 2 − 3δ < (3 − 4δ) η and (SR1) implies
that ε = 0. Now, since ε = η = 0, the information set of player 3 is not
reached. The belief p (a) ≤ 13 that makes δ ≥ 34 sequentially rational for
player 3 must therefore be obtained as the limit of posteriors obtained
from a sequence {(εk , η k , δ k )}∞
k=1 converging to (ε, η, δ) = (0, 0, δ) . For
k
each element of the sequence, pk (a) = εk +η ε(1−ε
; for instance, if εk = k12
k)
k
1
1
1
and η k = k , pk (a) = k2 → 0 < 3 .
Remark 1 It is straigthforward that any p (a) can be made fully consistent.
• Case 2: η > 0. Then, (SR2) implies that δ ≤ 34 and (SR3) implies in turn
that p (a) ≥ 13 . By (SR3), δ ≤ 34 requires p (a) ≥ 13 .Since η > 0, player 3’s
information set is reached with positive probability and beliefs are comε
ε
puted by Bayes law, p (a) = ε+η(1−ε)
and thererore we need ε+η(1−ε)
≥ 13 ,
or
η
ε≥
.
2+η
This inequality is impossible however. Since η > 0, ε ≥
(SR1) requires that
2 − 3δ ≥ (3 − 4δ) η.
η
2+η
> 0, and
— if η ∈ (0, 1) , then (SR2) requires δ = 34 , but then (*) becomes
which is absurd
(*)
−1
4
≥0
— if η = 1, then (*) is equivalent to δ ≥ 1, which contradicts δ ≤ 34 .
2
Therefore the set of sequential equilibria is the set
½
·
¸
·
¸¾
3
1
((0, 0, δ) , p (a)) : δ ∈
, 1 , p (a) ∈ 0,
.
4
3
In particular, Nash equilibria in which player 1 plays L cannot be sequential.
Remark 2 A sequential equilibrium must specify the strategy vector and the
belief structure.
3