Week 2: Refinements
September 7, 2022
In this reader we introduce and discuss a few refinements of Nash equilirium for extensive
form games. We consider the notions of subgame perfect equilibrium (SPE), perfect Bayesian
equilbrium (PBE), and sequential equilibrium (SE). In the specific context of signaling games
we also discuss the intuitive criterion (IC).
We will not give formal definitions, but rather try to develop an intuition for these refinements
in specific examples, and explain how to compute them.
1 SPE and PBE
Consider the following two person game in extensive form.
1
I
A
1
8
1
1
B
C
2
[α]
l
0
0
II
L
[1 − α]
R
2
r
l
r
a
8
1
0
0
8
1
0
0
2
b
6
2
c
2
6
d
0
0
The dotted line indicates a non-singleton information set for player 2. Thus, a (pure) strategy
profile for this game is a vector ((X, Y, Z), (u, v, w)), where X ∈ {I, II}, Y ∈ {A, B, C}, Z ∈
{L, R}, u ∈ {l, r} v ∈ {a, b}, and w ∈ {c, d}.
1
Refinements
2
NASH EQUILIBIUM.
The profile ((X, Y, Z), (u, v, w)) is a Nash equilibrium of this game when
(X, Y, Z) is a best response for player 1 against (u, v, w), and conversely (u, v, w) is a best
response for player 2 against (X, Y, Z).
Claim 1.
Exercise 1.
The profile ((I, A, L), (l, a, d)) is a Nash equilibrium.
Prove claim 1.
SUBGAME PERFECTION.
A subgame of an extensive form game is a part of the game tree
that starts in a singleton information set, and does not have an information set that contains
decision nodes both within and outside the subgame. For example, the above game tree has
5 subgames, namely the game itself (starting at the root), the subgame starting at the node
following action I, the subgame starting at the node following action II, the subgame starting
at the node following action L, and the subgame starting at the node following action R.
Note that for example the subtree starting at the node following action B is not a subgame,
because that node is in the same information set as the node following action C, which is a
node outside the subtree under discussion.
The profile ((X, Y, Z), (u, v, w)) is a subgame perfect equilibrium (an SPE) of the above game
when, for each subgame, the restriction of the profile to that subgame is a Nash equilibrium
of that subgame.
Exercise 2.
Claim 2.
Exercise 3.
1
Show that the Nash equilibrium ((I, A, L), (l, a, d)) is not subgame perfect.
The profile ((II, A, L), (l, b, c)) is an SPE.
Prove claim 2.
PERFECT BAYESIAN EQUILIBRIUM.
Given a general extensive form game, consider a pair
(σ, β), where σ is a profile of mixed behavioral strategies in an extensive form game, and β is
a profile of probability distributions over information sets. Such a pair (σ, β) is often called an
assessment, and the β component of an assessment is usually called a belief.
An assessment (σ, β) is a perfect Bayesian equilibrium (a PBE) if
[1] Given the profile σ, the belief β is consistent with Bayesian updating, and
[2] given the belief β, the profile σ maximizes expected payoffs at each information set.
A few remarks are in order.
1 Just to clarify, in this example, the strategy profile (Z, (v, w)) is the restriction of the profile
((X, Y, Z), (u, v, w)) to the subgame that starts after action II.
Refinements
3
[a] The first requirement is called consistency (CONS), while the second requirement is called
sequential rationality (SR).
[b] Note that requirement [1] (Bayesian updating) can only be applied to those information
sets that are reached with positive probability (in some subgame). In those information
sets in which [1] cannot be applied, the choice for β is free, as long as it is in line with
requirement [2]. We say that, in those cases, CONS is automatically satisfied.
[c] Formally, a PBE is an assessment (σ, β), consisting of a strategy profile and a belief.
However, often only the strategy component σ of the assessment is called a PBE as well
in that case, and the supporting belief is omitted.
We argue that the SPE ((II, A, L), (l, b, c)) in the above extensive form game is not a PBE.
Note that the non-singleton information set for player 2 is reached with probability zero, even
in the subgame following action I (since player 1 chooses A at the root of that subgame). So,
CONS is automatically satisfied.
So, take any belief [α, 1−α] at that information set. We check SR. The best response of player 2
given this belief is to play r, with an expected payoff of 1 (versus an expected payoff of 0 for the
action l). Thus, the choice of l in the strategy (l, b, c) of player 2 does not maximize expected
payoff given the belief [α, 1 − α], and SR does not hold. Hence, the SPE ((II, A, L), (l, b, c)) is
not a PBE.
Claim 3.
Exercise 4.
For any λ ∈ [0, 1], the profile ((I, λB + (1 − λC, L), (r, b, c)) is a PBE.
Prove claim 3.
Refinements
Exercise 5.
4
(SPE) Consider the following game.
1
a
b
2
c
5
4
d
1
e
f
g
e
C
1
2
5
4
f
g
C
1
2
7
2
1
9
7
2
1
2
5
4
1
2
3
6
8
3
3
6
(a) Find the unique SPE of this game.
(b) Now revise the game by eliminating player 1’s move “e” at his second information set. Find
the unique SPE of the revised game.
Refinements
5
2 PBE and SE
Consider the following two-person game in extensive form. The C indicates a chance move.
C
2
3
2
2
1−p
1
3
I
I
p
1−q
2
8
q
II
[α]
[1 − α]
z
1−z
z
1−z
3
1
0
2
3
2
0
1
We first compute all perfect Bayesian equilibria in behavioral strategies
((p, 1 − p), (q, 1 − q), (z, 1 − z))
for this game. We also compute the corresponding beliefs [α, 1 − α] for player II for each PBE.
Notice that for player 2, with belief [α, 1 − α],
l⪰r
A.
⇔
2−α≥1+α
⇔
α ≤ 12 .
If α < 21 . Then z = 1 by SR. Then, the unique best response of player 1 is p = q = 1.
Then however by CONS,
·p
2
1
= > .
1
3
2
·p+ 3 ·q
So, in this case we do not find any PBE.
α=
B.
2
3
2
3
If α > 21 . Then z = 0 by SR. Then p = q = 0. So, the info set of player 2 is reached with
zero probability, so CONS is automatic.
C.
If α = 12 . If p = q = 0. Then the info set is not reached, and CONS is automatic. Also,
any z ∈ [0, 1] satisfies SR. For p = q = 0 to be a best response, we need that 2 ≥ 3z, which
yields z ≤ 23 . All these profiles are PBE.
If however the info set is reached with strictly positive probability. Then by CONS,
1
2
=α=
2
3 ·p
2
1
3 ·p+ 3 ·q
,
Refinements
6
which can be rewritten to 2p = q. So, 0 < p < 1. Then player 1 mixes between playing In and
Out in the left decision node. Then necessarily 2 = 3z, so that z = 32 .
In total we find 3 line segments of PBE.
p
0
0
λ
q
0
0
2λ
z
0
z
2
3
Range parameters
1
2 ≤α ≤1
0 ≤ z ≤ 23
0 ≤ λ ≤ 12
α
α
1
2
1
2
Next, we compute a refinement of PBE, namely sequential equilibrium (SE) for this game. An
assessment (σ, β) is a sequential equilibrium (an SE) if
[1] there exists a completely mixed sequence (σ k )k∈N of strategy profiles with σ k → σ as
k → ∞, and β k → β as k → ∞ for the beliefs β k induced by σ k via Bayesian updating.
[2] given the belief β, the profile σ maximizes expected payoffs at each information set.
Again, a few remarks are in order.
[1] Since each σ k is assumed to be a completely mixed strategy profile, the belief β k is
uniquely defined since all information sets are reached with strictly positive probability.
[2] Note that [1] in fact is a strong version of CONS (confusingly usually also called CONS).
Thus, SE is indeed a refinement of PBE.
For
1
2
≤ α ≤ 1, we check that the PBE (p, q, z, α) = (0, 0, 0, α) is an SE. First we consider the
case where α < 1. Take a completely mixed sequence (pk , q k , z k ) that converges to (p, q, z).
We try to achieve that αk = α for all k. Then, by CONS, necessarily
α=
k
2
3 ·p
2
1 k
k
3 ·p + 3 ·q
,
so that α · q k = (2 − 2α) · pk . Choose
(pk , q k , z k ) =
(α
k,
2−2α 1
k , k
)
.
The indeed this sequence is completely mixed. Moreover,
2
3
·
2
3
pk
· pk
2pk
α
=
=
= α,
1
k
k
k
2p
+
q
2α
+
2 − 2α
+ 3 ·q
so that
(pk , q k , z k , αk ) → (p, q, z, α)
Refinements
7
as k → ∞. So, indeed, the strong form of CONS is satisfied. Finally, SR is immediate, because
we started with a PBE.
For the case α = 1, we need to be a bit more careful, to guarantee that the sequence is
completely mixed. Take for example αk =
k
k+1 .
we get that
(pk , q k , z k ) =
(
Then, using the same computations as above,
1
k
2
k(k+1) , k , k+1
)
.
We can now check that all requirements are indeed satisfied. In the same way, we can check
that all PBE for this game are in fact SE.
The following example shows that a perfect Bayesian equilibrium does not have to be sequential.
2
a
0
I
0
b
c
0
2
0
[1 − α]
d
II
[α]
d
II
e
0
3
0
e
[β]
f
0
2
0
[1 − β]
III
III
g
g
0
0
1
f
1
0
1
0
0
0
The strategy profile (a, e, f ) with beliefs α = 1 and β = 0 is a PBE. However, this assessment
is not an SE, because in every SE we need to have that α = β.
Exercise 6.
(PBE and SE) Consider the following two-person game in extensive form. The
C indicates a chance move.
Refinements
8
C
2
3
2
1
1−p
1
3
I
I
p
1−q
2
1
q
II
[α]
[1 − α]
z
1−z
z
1−z
5
3
0
2
4
2
0
3
(a) Compute all perfect Bayesian equilibria in behavioral strategies
((p, 1 − p), (q, 1 − q), (z, 1 − z))
for this game. Don’t forget to report the corresponding beliefs [α, 1 − α] for player II for
each PBE.
(b) Compute all sequential equilibria of this game. Don’t forget to report the completely mixed
strategy profiles for each equilibrium.
Refinements
9
Exercise 7.
(SE) Consider the following two-person game in extensive form. The C indicates
a chance move.
C
2
3
5
6
1−p
1
3
I
I
p
1−q
2
6
q
II
[α]
[1 − α]
z
1−z
z
1−z
9
5
0
3
9
0
0
3
(a) Compute all sequential equilibria in behavioral strategies
((p, 1 − p), (q, 1 − q), (z, 1 − z))
for this game.
(b) Compute all sequential equilibria of the game that results when the payoffs (2, 6) when
player I opts out in the right hand branch are replaced by (7, 6).
Refinements
10
3 PBE in signaling games
Consider the following signaling game we analysed last week.
u−1, 0
0, 1 u
r
1−r
0, 0 u
u
[α]
u
L
t
u
p
d
r
1−r
u
0, 1
1−p
uChance
u
d
u
[1 − α]
q
L
u
t′
s
1−s
u 1, 1
II
u 1, 1
.5
u
d
[β]
.5
II
2, 0 u
R
u
u
1−q
u
R [1 − β]
d
s
1−s
u 1, 0
We found that the pure strategy profiles (p, q, r, s) = (1, 1, 1, 1) and (p, q, r, s) = (0, 0, 0, 0) are
Nash equilibria in behavioral strategies. We check whether these profiles are PBE.
A.
For (p, q, r, s) = (1, 1, 1, 1). Consider the left information set. Since player 1 moves to the
left in both the upper and the lower node, each decision node for player 2 in the left info set is
reached with probability 12 . So, by CONS, α = 12 . Then, since the left info set is reached with
strictly positive probability (and hence the belief in the left information set is identical to the
probabilities generated by the strategy of player 1), SR is automatically satisfied.
Now consider the right info set. Since this info set is reached with zero probability, Bayesian
updating does not apply, and CONS is automatically satisfied. We check SR. Given belief
[β, 1 − β], the expected payoff of u is 1 − β, and the expected payoff of d is β. Thus,
u⪰d
⇔
1−β ≥β
⇔
β ≤ 12 .
It follows that any
(p, q, r, s, α, β) = (1, 1, 1, 1, 12 , β) with 0 ≤ β ≤
1
2
is a PBE.
B.
For (p, q, r, s) = (0, 0, 0, 0). A similar computation yields that any
(p, q, r, s, α, β) = (0, 0, 0, 0, α, 12 ) with 0 ≤ α ≤
is a PBE.
1
2
Refinements
Exercise 8.
Exercise 9.
11
Verify the claim in part B above.
(PBE) For the signaling games in Exercises 1 and 2 of the previous week,
compute all pure strategy perfect Bayesian equilibria of those games.