A DIRECT PROOF OF THE EXISTENCE OF
SEQUENTIAL EQUILIBRIUM AND A BACKWARD
INDUCTION CHARACTERIZATION
S. K. CHAKRABARTI1 AND I. TOPOLYAN 2
1
Department of Economics, IUPUI, 425 University Blvd, Indianapolis, IN 46202,
USA; [email protected]
2
Department of Economics, University of Cincinnati, Cincinnati, OH 45221,
USA; [email protected]
Abstract. We give an example that shows that the set of
perfect and hence sequential equilibrium points is a strict
subset of the set of equilibrium points of the Agent normal
form. We then provide a direct proof of the existence of a
sequential equilibrium for finite games that relies solely on
the structure of the sequential game. We also show that in
the case of sequential games with perfect recall that have
simple information structures, the sequential equilibria can
be found by using the backward induction method. Most
sequential games in economics have this simple information
structure and thus can be solved using this backward induction method.
JEL classification: C7
Keywords: Sequential equilibrium, Perfect Bayesian equilibrium, sequential games, imperfect information, backward induction, information sets, consistent assessment, beliefs.
Date: January 18, 2011.
An early draft of the paper was presented at the Midwest Theory Conference,
Northwestern University, May 2010. We acknowledge very useful conversations
with the late C. D. Aliprantis on this topic and would also like to thank Will Geller
for some very useful comments. Some parts of this paper have been drawn from
I. Topolyan’s Ph. D. dissertation, Purdue University, 2009. The usual disclaimer
applies.
2
1. Introduction
Sequential equilibrium is one of the most widely accepted equilibrium
concepts for sequential games with imperfect and/or incomplete information. As such it plays a central role in the analysis of many games in
economics, finance and other disciplines in which players have imperfect
or incomplete information at the time they make their choices. Thus
an understanding of the regularity conditions under which sequential
equilibrium exists and finding efficient methods for computing sequential equilibria are extremely important. In [8], in which the concept
of sequential equilibrium is first developed and discussed, the question
of existence of sequential equilibrium for finite sequential games with
perfect recall is addressed by using the observation that every trembling hand perfect equilibrium is a sequential equilibrium, and that a
trembling hand perfect equilibrium always exists for finite sequential
games with perfect recall as argued in [13]. The proof of the existence of trembling hand perfect equilibrium in [13], however, relies on
the observation that a perfect equilibrium of the agent normal form is
a trembling hand perfect equilibrium of the extensive form, and that
a perfect equilibrium of the agent normal form always exists. This
approach to the question of existence of a sequential equilibrium thus
relies on the use of the agent normal form to analyze the extensive form
game. But the use of the agent normal to find a sequential equilibrium
of a game opens up the possibility that many details of the structure
of the game tree and the sequential nature of the game are not fully
utilized. Indeed, it is not hard to find examples of sequential games in
which the equilibrium of the agent normal form is not an equilibrium of
the sequential game and therefore not a perfect equilibrium and hence
not a sequential equilibrium of the game. The set of sequential equilibrium is thus a strict subset of the set of equilibrium points of the agent
normal form of a sequential game and the issue then is whether this
strict subset is nonempty. An argument that uses the Agent normal
form to establish the existence of a perfect equilibrium or a sequential equilibrium therefore would need to independently verify that one
of the equilibrium points of the Agent normal form is a perfect or a
sequential equilibrium. Further, the use of the agent normal form to
analyze the sequential game does not provide any immediate intuition
as to how one should go about computing a sequential equilibrium.
Therefore, a direct proof of the existence of a sequential equilibrium
that relies solely on the structure of the sequential game would be both
more useful and insightful. We show here that the issue of existence
of a sequential equilibrium in finite games with perfect recall can be
3
addressed more transparently by using a much more direct approach.
This consists of first finding the best response strategy of a player in
the entire game by using a dynamic programming approach, and then
finding an equilibrium in these best response strategies. This approach
of using the entire game tree to analyze the best response strategies and
the resulting equilibrium points provides some direct intuition as how
one can proceed to actually compute a sequential equilibrium. This
direct approach to the question of the existence of a sequential equilibrium also indicates that frequently the best way to find and compute
the sequential equilibria of a sequential game is to use the backward
induction process.
As is well-known, the backward induction procedure works extremely
well for sequential games with perfect information. For sequential or
extensive form games with imperfect information, the issue is more
subtle as one has to deal with beliefs at the information sets of the
players. The optimal choice at a given information set in these cases
depends on the beliefs at the information set, and these beliefs in turn
depend on the strategies used by the players who choose at nodes and
information sets that precede that given information set. This is unlike
the case for sequential games with perfect information in which the
players know exactly which node they are at when making choices.
What we show here is that in many cases of interest it is still possible
to use a general backward induction process to find the set of sequential
equilibria of games with incomplete or imperfect information. Thus one
can look at the optimal responses of a player for each possible belief at
the information set and then use these responses to find the optimal
choices of the players at the information sets in the preceding stage
of the game, again for each possible belief at these information sets.
Proceeding in this manner one can work back to the initial information
set. If one uses such a method one should be able to find not just
one sequential equilibrium of the game but possibly the entire set of
sequential equilibrium. We illustrate this idea here by using backward
induction to find the sequential equilibria of the game of Figure 1.
The optimal behavior strategy of player 3 at the information set
{C, D} is given by b3
⎧
if μ(C) > 14
⎨ b3 (l) = 1
b (l) ∈ [0, 1] if μ(C) = 14
⎩ 3
b3 (l) = 0
if μ(C) < 14 .
where b3 (l) denotes the probability with which player 3 chooses l when
the behavior strategy of player 3 is given by b3 .
4
(0, 0, 3)
*
l
C X
XXX
*
r XX
z (3, 2, 2)
L 3
A
: (0, 0, 0)
γ
l 1 @
@
3H
R@
L DHHH
r HH
@
R
@
j (4, 4, 1)
2 Z
BZ
Z
RZZ
~ (1, 1, 1)
Z
Figure 1
Now given the optimal behavior strategy b3 of player 3, and the behavior strategy b1 of player 1 at node A, the optimal behavior strategy
b2 of player 2 is given by
⎧
b (R ) = 1
if b1 (L) > 14 so that μ(C) > 14
⎪
⎪
⎨ 2 b2 (R ) = 1
if b1 (L) = 14 so that μ(C) = 14 , and if b3 (l) > 14
⎪ b2 (R ) ∈ [0, 1] if b1 (L) = 14 so that μ(C) = 14 , and b3 (r) = 14
⎪
⎩ b (L ) = 1
otherwise.
2
Given these optimal behavior strategies of players 2 and 3, the optimal choice of player 1 at the node A is then b1 (R) = 1 as in that case
μ(C) = 0 and b2 (L ) = 1 and b3 (r) = 1 and the payoff of player 1 is 4.
In the backward induction procedure used to find the sequential equilibrium, the information sets of a player in the game play the same
role as the decision nodes in the standard backward induction method
used to solve sequential games with perfect information. Thus, the
first step is to find the optimal choices of player 3 at his information set {C, D} and then work back to the optimal choices of player
2, and finally, find the optimal choices of player 1. An interesting
feature of this procedure is that it also clearly indicates that in the
game of Figure 1, the only sequential equilibrium of the game is the
one given by the consistent assessment (μ , b ) where μ (C) = 0 and
(b1 (R) = 1, b2 (L ) = 1, b3 (r) = 1). This shows that this procedure can
often be used to find not just a sequential equilibrium of a sequential
game but the entire set of sequential equilibria of the game.
This procedure can be used quite effectively to find the sequential
equilibria of finite sequential games with perfect recall that have what
we refer to here as a simple information structure. Roughly speaking
5
this means that the nodes in the information sets do not interlock, that
is, the information set of a player either unambiguously follows or succeeds that of another player or the information sets are independent of
each other. We show here that if this condition, that the game have a
simple information structure, holds, then the backward induction process yields very good results. The condition that a sequential game
have a simple information structure is quite broad and easily verifiable. Indeed, it seems that the majority of the sequential games used
in applications in Economics, Finance and other disciplines have a simple information structure. These include various models of sequential
bargaining (see for instance [2, 9]), Entry Deterrence Game (see for
example [9]), various signaling games (for instance, Beer-Quiche Game
in [4]), Joint Venture Entry Game [9], limit-pricing games [11], as well
as various models of cheap talk games. One may also refer to [12] and
[5] for other examples.
The paper is organized as follows. In Section 2 we describe the
notation and give all the definitions regarding sequential games and
sequential equilibrium. In Section 3 we present an example that shows
that the sequential equilibrium points of a sequential game may be a
strict subset of the set of equilibrium points of the agent normal form
of the game. In Section 4 we provide a direct proof of the existence of
a sequential equilibrium that relies solely on the structure of the game
tree and the information sets. In Section 5 we define sequential games
with simple information structures and games with complex information structures. In Section 6 we show that the backward induction
process will identify all the sequential equilibria of a finite sequential
game with perfect recall that has a simple information structure. In
Section 7 we conclude.
2. Definitions and Methodology
As usual, a reflexive, antisymmetric and transitive binary relation
on a set X is called a partial order.
Definition 2.1. A pograph is a pair (X, ), where X is a finite set
of nodes, and is a partial order on X. The arbitrary node of X will
be denoted by t.
Intuitively, the binary relation designates precedence, and the notation t1 t2 informs us that t2 is among the successors of t1 .
Definition 2.2. If y x, then y is a predecessor of x, and x is a
successor of y.
6
Definition 2.3. A node t = x is an immediate predecessor (or a
parent) of x if t x and there is no other node s such that t s x.
In this case x is called an immediate successor, or a child of t.
A node with nonempty set of children would be referred to as a
decision node, while a node having no children would be called a
terminal node. A node with no parent would be called a root. Denote the set of all decision nodes by Y , the set of all terminal nodes by
Z, and the set of roots by W . Clearly, W ⊆ Y and X = Y ∪ Z.
Definition 2.4. If x y, then a path from x to y is a chain
x = x1 x2 . . . xn = y
such that xi−1 is an immediate predecessor of xi for each i = 1, . . . , n.
Definition 2.5. A frame T is a pograph such that for every non-root
node x there exists a unique root w such that:
(1) there exists a unique path from w to x, and
(2) for any y ∈ W different from w, there is no path from y to x.
Note that this definition implies that every non-root node has exactly
one parent. A frame is called finite if the set of nodes X is finite. In
this paper we consider only finite frames.
Definition 2.6. An n-player extensive form (or sequential)
game G is a finite game in extensive form, which is a tuple (T, P, U, C, H),
whose elements are interpreted as follows.
A finite frame T is as described in Definition 2.5. We work with
frames instead of trees to account for the possibility of multiple roots.
Denote by P a player partition, P = {P0 , P1 , . . . Pn }. Each Pi is
called player i’s set and is a set of all nodes at which player i is decisive.
Player 0 incorporates a random mechanism that may determine the
game path, and may be inactive,
An information partition is denoted by U, which is a refinement
of the player partition P. It partitions each set Pi into information
sets u. Denote the set of all information sets of player i by Ui . Each
u ∈ Ui has the property that:
(1) if a, b ∈ u, then a b, and
(2) for every a ∈ u the set of choices available at a is the same.
Given a decision node x ∈ Y , we will denote the information set
containing x by U(x).
An information partition has a property that there is an information
set IW , called the root information set, which consists of nodes that
7
have no predecessors, i.e., it is the set {w ∈ W }. A probability measure
ρ on IW is specified.
Denote by C a choice partition that partitions the (finite) set of all
choices available throughout the game, M into the subsets Cx , x ∈ X,
where Cx represents the set of all choices (actions) available at decision
node x. It satisfies the following condition: for every u ∈ U and x, y ∈
u, we have Cx = Cy . Thus we can introduce another partition C ,
which partitions the set of all choices for the game into the subsets Cu ,
each of those containing all choices available at the information set u.
To each choice c available at a decision node x there corresponds a
unique edge originating from x, and vice versa.
A payoff function H is a vector-valued function that assigns to
every terminal node z ∈ Z a vector H(z) = (H1 (z), H2 (z), · · · , Hn (z)),
whose components are the payoffs of players 1, · · · , n at the terminal
node z.
Definition 2.7. Given a sequential game G = (T, P, U, C, H), the
quadruple Ξ = (T, P, U, C) is called an extensive form of the game
G.
We consider extensive games with perfect recall only. Let us introduce the following notation. Let D be a subset of X × X consisting
of all (x, y) ∈ X × X such that x y. Let α : D →
→ M be a function
defined as α(x, y) = c, where c ∈ Cx is the choice at x that lies on the
path from x to y.
Definition 2.8. An extensive game G is called an extensive game with
perfect recall if for every player i the following condition is satisfied:
for every u, v ∈ Ui , z, t ∈ u and x, y ∈ v, if z x and t y, then
α(z, x) = α(t, y).
Definition 2.9. Given an information set u ∈ Ui of player i, define a
local strategy biu to be a probability distribution over Cu . Denote the
set of all local strategies of player i at u by Biu = Δdu , where du is a
cardinality of Cu (the number of choices available at u), and Δdu is a
unit (d(u) − 1)-simplex.
◦
, i.e., if
A local strategy biu is called completely mixed if biu ∈ Biu
every choice at Cu is played with some positive probability.
Definition 2.10. A behavior strategy bi of player i is a tuple (biu )u∈Ui ,
i.e., an assignment of some local strategy biu to every u ∈ U
i . The set
of all behavior strategies of player i is denoted by Bi , B = u∈Ui Biu .
8
Definition 2.11. A behavior strategy combination b = (b1 , · · · , bn )
is an n-tuple whose ith component is a behavior strategy of player i.
We will call a behavior strategy bi completely mixed if for each u ∈
Ui , biu is completely mixed. A behavior strategy combination b is
completely mixed if each bi is completely mixed.
Fix a behavior strategy combination b ∈ B. It induces a probability
measure P on Z as follows. Fix a terminal node z ∈ Z, without
loss of generality let w ∈ W be the unique root predecessor of z, and
w x1 · · · xr−1 xr = z be the path from w to z. Given a
non-root node x, denote by b(x) the probability assigned by b to the
edge connecting x with its parent (recall that in a frame every nonroot node has exactly one parent). Then the realization probability of
z given b ∈ B is:
P (z|b) = ρ(w) ·
r
b(xj ).
j=1
Now we can define the expected payoff of player i given a behavior
strategy combination b. Assume without loss of generality the set of
terminal nodes is Z = {z1 , . . . , zm }. Then the expected payoff of player
i can be calculated as follows:
Ei =
m
Hi (zj )P (zj ).
j=1
Definition 2.12. A system of beliefs μ is a function that prescribes
to every non-singleton information set u ∈ U a probability measure
over the nodes in u.
Given a sequential game G with perfect recall, the set of all beliefs
systems will be denoted by M.
Definition 2.13. An assessment is a pair (μ, b), where μ ∈ M is a
system of beliefs and b ∈ B is a behavior strategy combination.
Fix a completely mixed behavior strategy profile b ∈ B ◦ . Then, every
node of the extensive form is reached with some positive probability. As
Kreps and Wilson [8] argue, given b ∈ B ◦ reasonable beliefs are those
computed from b via Bayes’ rule. That is, given a non-root decision
node x,
μ(x) =
P (x|b)
P (x|b)
=
.
P (u(x)|b)
y∈u(x) P (y|b)
(2.1)
9
Denote by Ψ◦ ⊆ B ◦ × M the set of all assessments (μ, b) such that
b is a completely mixed behavior strategy profile and μ is computed
from b via the above formula. Let Ψ be the closure of Ψ◦ in B × M.
Definition 2.14. An assessment (μ, b) is consistent if (μ, b) ∈ Ψ
(i.e., (μ, b) is a limit point of some sequence (μk , bk ) ⊆ Ψ0 ).
Definition 2.15. A belief correspondence φ : B →
→ M is defined
for each b ∈ B as φ(b) = {μ ∈ M : (μ, b) ∈ Ψ}.
Clearly, φ is nonempty-valued and closed (i.e., has a closed graph).
Note that it is a function on B ◦ (because for any completely mixed
behavior strategy profile, every information set is reached with some
positive probability, so that the ratio in Equation 2.1 is well-defined
for each non-root decision node).
For each information set u ∈ U, denote by Z(u) ⊆ Z the set of all
terminal successors of u. A node z ∈ Z belongs to Z(u) if and only if
some node x ∈ u is among the predecessors of z.
Given an assessment (μ, b), for every terminal node z and every information set u ∈ U we can calculate the conditional probability of
reaching z given that the information set u is reached, as follows:
Pu (z|μ, b) =
μ(pm (z)) ·
m−1
l=0
b(λ(pl (z)))
if
z ∈ Z(u)
0 otherwise
Then, we can define the expected payoff of player i starting from an
information set u ∈ U, given an assessment (μ, b) as follows:
Ei (u, b, μ) =
Hi (z)Pu (z|μ, b).
z∈Z(u)
Definition 2.16. An assessment (μ, b) is sequentially rational if
for each player i and each u ∈ Ui ,
Ei (u, b, μ) ≥ Ei (u, (bi , b−i ), μ) for every bi ∈ Bi .
Definition 2.17. An assessment (μ, b) is called a sequential equilibrium if it is both consistent and sequentially rational.
3. An Example: The sequential equilibrium points are a
strict subset of the set of equilibrium points of the
agent normal form
The example here shows that an equilibrium of the agent normal
form of a sequential game may fail to be an equilibrium of the sequential
game.
10
1
2
a
O
1
b
X
E
T
F L
R
B
Y T
2 B
L
R
1
1
G
L
R
H L
1
R
(1, 1)
(0, 0)
(0, 0)
(1, 1)
(1, 2)
(3, 0)
(3, 1)
(2, 1)
Figure 2.
A sequential game
In the sequential game in Figure 2, player 1 has three information
sets, namely, I11 = {O}, I12 = {E, F } and I13 = {G, H}, while player 2
has one, namely, I21 = {X, Y }.
If player 1 is represented by an agent at each of the three information
sets, where each agent has the same payoff as player 1, then the sequential game reduces to a strategic form game with four players, namely
agents 1, 2 and 3 of player 1 and player 2. The strategy set of agent 1
is {a, b}, the strategy set of agent 2 is {L, R} and the strategy set of
agent 3 is {L , R }. We observe that the strategy profile (a, L, L , T )
is a Nash equilibrium of the four player strategic form game with the
payoffs (1, 1, 1, 1). This can be verified by observing that if agent 1
deviates to b, then the payoff is still 1. If agent 2 deviates to R, then
the payoff falls from 1 to 0. If agent 3 deviates to R the payoff remains
1 and if player 2 deviates to B then the payoff falls to 0.
We claim that the strategy profile {(a, L, L ), T } is not an equilibrium of the sequential game. Consider the deviation by player 1 to
the strategy (b, L, R ), so that player 1 coordinates a change of choices
at two information sets. Player 1 gets a payoff of 3 rather than 1.
This shows that the strategy profile {(a, L, L ), T } is not an equilibrium of the sequential game and, hence, not a sequential equilibrium of
the game of figure 2. What goes wrong here is the fact that a player’s
strategy involves choices at each information set and that a player could
gain by coordinating choices at two or more information sets as happens in this case. While the example suggests that the agent normal
form approach to finding a sequential equilibrium may be inadequate,
it does not imply that the game does not have a sequential equilibrium. Indeed, a sequential equilibrium can be found by looking at the
optimal choices of the players at the information sets, starting with the
information sets {E, F } and {G, H}, and then working backwards.
11
Let μ(E) denote player 1 s belief that node E has been reached.
Then, the expected payoff of player 1 is Eu1 (L) = μ(E) if he chooses
L and Eu1 (R) = 1 − μ(E) if he chooses R. Thus, player 1 s optimal
choice at {E, F } is:
⎧
⎨ L if μ(E) ≥ 12
⎩
R if μ(E) ≤ 12 .
At the information set {G, H}, if μ(G) denotes player 1 s belief that
node G has been reached, then the expected payoff of player 1 is
Eu1 (L ) = μ(G) + 3(1 − μ(G)) if he chooses L and Eu1 (R ) = 3μ(G) +
2(1 − μ(G)) if he chooses R . Therefore, player 1 s optimal choices are
given by:
⎧
⎨ L if μ(G) ≤ 13
⎩
R if μ(G) ≥ 13 .
Player 2 now observes that if he chooses T , then his expected payoff
is:
1 if player 1 chooses a at {O},
0 if player 1 chooses b at {O}.
And if he chooses B at {X, Y }, then the expected payoff of player 2 is
1 if player 1 chooses a at {O},
0 if player 1 chooses b at {O}.
Player 2 also knows that if he chooses so that prob.(T ) = 13 and
prob.(B) = 23 then his expected payoff is 43 , if player 1 chooses b at
{O} and L at {G, H}. Note here that the choice L by player 1 is optimal if player 2 s choice is given by prob.(T ) = 13 . It can now be verified
that the strategy profile {(b, L, L ), (prob.(T ) = 13 , prob.(B) = 23 )} with
the belief system {(μ1 (E) = 13 , μ1 (G) = 13 ), (μ2 (Y ) = 1)} is a sequential
equilibrium of the game.
The example highlights two important facts. First, using the agent
normal form to look for sequential equilibrium may often be unsatisfactory and inadequate and a more direct approach to the question of
existence is necessary. In fact, the example shows that finding a perfect
equilibrium point of the agent normal form may not be enough as one
has to then check independently whether any of these is a sequential
equilibrium point. Indeed, it seems from the example that a direct
proof of the existence of a perfect equilibrium of the sequential game
and, hence, a sequential equilibrium is needed. Second, in actually trying to compute the sequential equilibrium, an approach that relies on
analyzing the optimal choices of the players at information sets, and
12
then using some form of backward induction, seems to yield good results. In what follows we provide a proof of the existence of sequential
equilibrium using a direct approach that relies only on the game tree.
We also provide a characterization of sequential equilibrium for a class
of games that validates the backward induction approach to finding the
sequential equilibria.
4. Existence of Sequential Equilibrium
Here we present the result that shows that every finite sequential
game with perfect recall has a sequential equilibrium. The existence
proof that we present here is a proof that does not rely on the agent
normal form of the game but works directly with the structure of the
sequential game itself. We start with a definition.
Definition 4.1. A behavior strategy profile b = (b1 , · · · , bn ) is said to
be -bounded if, for > 0, the behavior strategy bi of every player i
satisfies the condition that
bi (c) ≥ for every choice (edge) c at each information set of player i.
It should be noted that an -bounded behavior strategy profile is a
completely mixed behavior strategy profile as it imposes the constraint
that every choice in the sequential game is made with a positive probability. Because of this, every information set in the game is reached
with positive probability. Hence, a unique belief system consistent with
the behavior strategy profile is induced on every information set by an
-bounded behavior strategy profile. We now move onto the next result
which is on the existence of a sequential equilibrium. In order to prove
the theorem we need to recall some notation. Recall from Section 2
that
Ei (Ii , b, μ)
is the expected payoff of player i at the information set Ii when the
strategy profile b is played and μ is the belief at Ii that is induced
by the strategy profile π. We further recall that Δ(C(Ij )) is the set
of probability distributions over the choices C(Ij ) of player j at the
information set Ij . For an -bounded behavior strategy profile b , the
probability distributions over the choices at any C(Ij ) is bounded below
by > 0 so that the choices are given by a probability distribution in
a closed subset Δ (C(Ij )) of Δ(C(Ij )).
13
Now consider any strategy profile b−i of players other than i. It is
not hard to check that this is gives us an element of
K
j
Πj=i (Πk=1
Δ (C(Ij )).
Similarly, the strategy bi of player i gives us an element of
i
ΠK
k=1 Δ (C(Ii ).
Therefore,
Ei (b−i , bi ) = Ei (p−i , pi )
where Ei (p−i , pi ) is the expected payoff of player i when players make
choices at the information sets according to the probability distributions given by (p−i , pi ), which are induced on the choices at the information sets by the strategy profile (b−i , bi ).
In a sequential game with imperfect information a player may have
multiple information sets with some information sets preceding other
information sets. Thus, some information sets of a player will be
reached at earlier stages of the game. Let Li ≥ 1 denote the number of stages at which player i has an information set. Then given any
bi and, hence, probability distributions pi , one can split pi as
pi = (pi,s< , pi,s≥)
where pi,s< are the probability distributions at information sets in
stages that precede stage , where 1 ≤ ≤ Li . Note also that since
any -bounded strategy profile is in the interior of the strategy space,
it induces a unique belief system at each information set. For every
strategy profile b , we will indicate this unique belief system by μ(b ).
As this belief system is induced by the probability distributions p we
will often write μ(p ). We will write
Ei (Ii , (p−i , pi ), μ(p−i , pi ))
for the expected payoff of player i at the information set Ii when the
probability distributions induced by the strategy profile is given by
(p−i , pi ), and the resulting unique belief system by μ(p−i , pi ).
Given p−i , the probability distributions of players other than i on
the choices at their information sets, the optimal response of player i
involves the choice of a behavior strategy, and hence, probability distributions pi on the choices at the information sets of player i, that
maximizes
Ei (p−i , pi ).
In general, this cannot be done by choosing a probability distribution
pi (Iiki ) independently at each information set Iiki of player i so as to
14
maximize
Ei (Iiki , (p−i , pi ), μ(p−i , pi ))
given the distributions pi (Ii ) at Ii = Iiki . This is because a probability
distribution pi (Iiki ) at an information set Iiki determines the beliefs at
all the information sets that succeed Iiki , and a distribution over choices
that are optimal for one set of beliefs will not, in general, be optimal
for another set of beliefs. Further, a player can conceivably increase
his expected payoff in a sequential game by changing the distribution
on choices simultaneously at two or more information sets.
Therefore, the optimal behavior strategy of player i, given p−i , is
best determined by a backward induction process in which the optimal
probability distributions are first determined at information sets of
player i at the last stage Li as a function of the distributions chosen
at stages 1 through Li − 1. Next, the optimal distributions at stages
Li − 1 and Li are then determined as functions of the distributions at
stages 1 through Li − 2, and so on.1 In the next result we use this
approach to find the optimal behavior strategy of player i given the
behavior strategy of the players other than i.
The following result shows there is an -bounded strategy profile
such that if each player uses only -bounded strategies then each player
maximizes his expected payoff given the -bounded strategy profile of
the other players. In other words, the result shows that there is an
equilibrium in -bounded strategies.
Lemma 4.2. For every > 0, there is an -bounded behavior strategy
profile b, and a system of beliefs μ, consistent with b, such that for
ever player i
,
, ,
Ei (I, (b,
−i , bi ), μ ) ≥ Ei (I, (b−i , bi ), μ )
at each information set and for every -bounded behavior strategy of
player i and belief system μ induced by the strategy profile (b,
−i , bi ).
Proof: Given b−i and, hence, distributions p−i on choices C(I) at
information sets I of players other than i, and distributions (pi,s )s<Li
at all information sets in stages 1 through Li − 1, there is a unique
belief system induced at each information set in stage Li of player i.
Then, because the expected payoffs Ei (I, p , μ (p )) are continuous on
kL
the compact sets Δ (C(I)), there is a p̂i (Ii i ) that maximizes
kL
Ei (Ii i , (p−i , (pi,s )s≤Li ), μ (p−i , (pi,s )s≤Li ))
1Readers would correctly observe here that player i has to solve a finite-horizon
dynamic programming problem in order to determine the optimal behavior strategy.
15
kL
at each information set Ii i at stage Li of player i.
Define the correspondence
kLi
Φi
K
k
: Πj=i Πkjj=1 Δ (C(Ij j )) × Π<Li Δ (C(Iik ))
as
kL
→
→
kL
Δ (C(Ii i ))
kL
Φi i ((p−i , (pi,s )s<Li ) = {(p̂i (Ii i ))kLi }.
The correspondence thus gives the optimal probability distributions on
the choices at the information sets at stage Li of player i, given the distributions at the other information sets of player i and the information
sets of the players other than i. It is not hard to verify that the graph
of this correspondence denoted by G kLi is closed because of Berge’s
Φi
maximum theorem (see [1, Theorem 17.31, p. 570]) and, therefore,
GΦLi = ∪kLi G
i
kL
i
Φi
kL
is also closed, where ΦLi i (p−i , (pi,s )s<Li ) = ∪kLi Φi i ((p−i , (pi,s )s<Li ) defines the correspondence ΦLi i .
kL −1
Further, the expected payoff function at each information set Ii i
at stage Li of player i
kLi −1
Eimax Li (Ii
kLi −1
= Ei (Ii
, (p−i , (pi,s )s<Li ), μ (p−i , (pi,s )s<Li ))
kL
, (p−i , (pi,s )s<Li , (p̂i (Ii i ))kLi , μ (p−i , (pi,s )s<Li ))
K
k
defined on Πj=i Πkjj=1 Δ (C(Ij j ))×Π<Li Δ (C(Iik )) is continuous. This
observation also follows from Berge’s maximum theorem ( [1, Theorem 17.31, p. 570]).
kL −1
Now consider the choices at the information sets (Ii i )kLi −1 at stage
Li − 1 of player i. Define the correspondence
kLi −1
Φi
:
K
k
Πj=i Πkjj=1 Δ (C(Ij j )) × Π<Li −1 Δ (C(Iik ))
→
→
kLi −1
Δ (C(Ii
kL
)) × Δ (C(Ii i ))
as
kLi −1
Φi
kLi −1
((p−i , (pi,s )s<Li −1 ) = {(p̂i (Ii
kLi −1
where p̂i (Ii
kL
))kLi −1 , p̂i (Ii i ))kLi }
) maximizes
kL
Eimax Li (Ii i , (p−i , (pi,s )s<Li ), μ (p−i , (pi,s )s<Li ))
kL
and each p̂i (Ii i ) is in
kL
kLi −1
Φi i ((p−i , (pi,s )s<Li −1 , p̂i (Ii
))
16
kL
for each information set Ii i at stage Li of player i. By [1, Theorem 17.31, p. 570] it follows that the graph G kLi −1 is closed and the
Φi
function
kLi −2
Eimax Li −1 (Ii
kLi −2
= Ei (Ii
, (p−i , (pi,s )s<Li −1 ), μ (p−i , (pi,s )s<Li −1 ))
kLi −1
, (p−i , (pi,s )s<Li −2 , (p̂i (Ii
K
))kLi −1 , μ (p−i , (pi,s )s<Li−2 ))
k
defined on Πj=i Πkjj=1 Δ (C(Ij j )) × Π<Li −1 Δ (C(Iik )) is continuous.
Therefore, the graph of the correspondence
GΦLi −1 = ∪kLi −1 G
i
kL −1
i
Φi
is also closed, where the correspondence ΦiLi −1 is defined as
kLi −1
ΦiLi −1 (p−i , (pi,s )s<Li −1 ) = ∪kLi −1 Φi
((p−i , (pi,s )s<Li −1 ).
It should be clear from the definitions of these correspondences that
GΦLi −1 ⊂ GΦLi .
i
i
Proceeding recursively in this manner, at stage 1 we have the correspondence
K
k
Φi : Πj=i Πkjj=1 Δ (C(Ij j ))
→
→
ki
i
ΠK
ki =1 Δ (C(Ii ))
defined as
i
Φi (pi ) = {(p̂i (Iiki ))K
ki =1 }
i
where (p̂i (Iiki ))K
ki =1 maximizes the expected payoff
i
Ei (p−i , (pi (Iiki ))K
ki =1 )
of player i given the behavior strategy profile b−i of the players other
than i, and therefore, the corresponding probability distributions p−i
at the information sets of the players other than i. This correspondence
is nonempty and has a closed graph. Therefore, the correspondence
K
k
Φ : Πnj=1 Πkjj=1 Δ (C(Ij j ))
K
K
→
→
k
Πnj=1 Πkjj=1 Δ (C(Ij j ))
k
on the compact subset Πnj=1 Πkjj=1 Δ (C(Ij j )) of a Euclidean space given
by
Φ (p ) = (Φ1 (p−1 ), · · · , Φn (p−n ))
is nonempty and has a closed graph. It also can be checked that the
correspondence is convex-valued as are all the correspondences Φki for
1 ≤ ≤ Li of each player i = 1, · · · , n.
17
Hence, by Kakutani’s fixed point theorem, there is a p, ∈ Φ(p,)
and, hence, a behavior strategy2 profile b, that satisfies the required
.
conditions.
The next result shows that if the are small enough then the equilibrium in -bounded strategy profile is an approximate equilibrium. It
is a result that is used in the proof of the final result of this section.
Lemma 4.3. Given a finite sequential game, and an equilibrium in
1
-bounded strategies, there is a number K > 0, that depends on the
s
game, such that
K
0 ≤ max[Ei (bs , μs , Iiki ) − Ei ((bs−i , bi ), μn (bs−i , bi ), Iiki )] ≤
bi
s
for any behavior strategy bi of player i.
We omit the proof as it can be clearly seen to depend on the observation that if a behavior strategy of a player maximizes expected
payoff over the set of 1s -bounded behavior strategies, then the behavior
strategy that maximizes the expected payoff would assign zero probability to at most a finite number of choices, and hence, the expected
payoff can be increased by increasing the probabilities assigned to the
choices associated with the highest payoffs by a factor of (1 + Ms ), for
some M > 0. As the game is finite this M will have an upper bound.
Theorem 4.4. Every finite sequential game with perfect recall has a
sequential equilibrium.
Proof: Choose = n1 where n is large and consider the sequence of
behavior strategies and consistent beliefs {bn , μn }n such that for each
(bn , μn ), for every player i and each information set Iiki of player i
Ei (bn , μn , Iiki ) ≥ Ei ((bn−i , bi ), μn (bn−i , bi ), Iiki )
(4.1)
for any behavior strategy bi with induced probability distributions in
1
ki
n
n
i
n
ΠK
ki =1 Δ (C(Ii )). We know that such a pair {b , μ }n exists for each
n from Lemma 4.3.
Now consider the pair of behavior strategies and beliefs (b̂, μ̂) given
by
(b̂, μ̂) = lim (bn , μn ).
n→∞
1
K
k
Such a limit exists because Πnj=1Πkjj=1 Δ n (C(Ij j )) is a compact subset
of a Euclidean space so that every sequence {π n , μn }n has a limit point.
2This is the behavior strategy profile associated with the vector of probability
distributions p, that gives a probability distributions at each of the information
sets.
18
We claim that for each player i and each information set Iiki of
player i
Ei (b̂, μ̂, Iiki ) ≥ Ei ((b̂−i , bi ), μ(b̂−i , bi ), Iiki )
for any behavior strategy bi of player i.
Suppose not. Then for some player i and some behavior strategy bi
of player i, we have
Ei (b̂, μ̂, Iiki ) < Ei ((b̂−i , bi ), μ(b̂−i , bi ), Iiki ).
Let
Ei ((b̂−i , bi ), μ(b̂−i , bi ), Iiki ) − Ei (b̂, μ̂, Iiki ) = a > 0.
(4.2)
Now as the expected payoffs are continuous in the probability distributions at the information sets and the beliefs, it follows that there is
an n0 sufficiently large such that for all n ≥ n0 ,
a
|Ei (bn , μn , Iiki ) − Ei (b̂, μ̂, Iiki )| ≤
(4.3)
4
and
a
|Ei ((bn−i , bi ), μ(bn−i , bi ), Iiki ) − Ei ((b̂−i , bi ), μ(b̂−i , bi ), Iiki )| ≤ . (4.4)
4
From (4.2), (4.3) and (4.4) for all n ≥ n0 , we have
a
Ei ((bn−i , bi ), μ(bn−i , bi ), Iiki ) ≥ Ei ((b̂−i , bi ), μ(b̂−i , bi ), Iiki ) −
4
3a
ki
= Ei (b̂, μ̂, Ii ) +
4
a
ki
n
n
(4.5)
≥ Ei (b , μ , Ii ) + .
2
Now observe that by Lemma 4.3, for a given sequential game, there is
a K > 0 ( that depends on the sequential game) such that
|Ei ((bs−i , bi ), μ(bs−i , bi ), Iiki ) − Ei (bs , μs , Iiki )| ≤
K
s
where bi = lims→∞ bsi of a sequence {bsi }s of 1s - bounded behavior
strategies of player i. For the sequence {bn , μn } we now choose an n1
sufficiently large such that Kn < a4 . From this and from (4.5), for any
behavior strategy bi of player i, we have
K
Ei ((bn−i , bni ), μ(bn−i , bni ), Iiki ) ≥ Ei ((bn−i , bi ), μ(bn−i , bi ), Iiki ) −
n
a
ki
n
n
≥ Ei (b , μ , Ii ) + .
4
But this contradicts (4.1). Hence the claim must hold.
19
5. Games of Simple and Complex Information Structure
Given a sequential game, an information set can precede or succeed
another information set in the game or occur simultaneously, the latter intuitively meaning that simultaneous information sets are tangled
together, or there is no way to tell that one information set strongly
precedes the other.
Definition 5.1. Given an information set u of the extensive game G,
define the set of first-order successors (or immediate successors) of u, Succ1 (u), to be the set of all nodes x ∈ X such that x is
in the same information set with an immediate successor of a node y
for some y ∈ u. That is, x is in the set of immediate successors of the
information set u if there is an information set I and a node w such
that x, w ∈ I and the node w is an immediate successor of a node y of
u.
Definition 5.2. Given an information set u, for every natural number
n define recursively the set of k th -order successors of u, Succk (u),
to be the set Succ1 (Succk−1(u)). We let Succ0 (u) = {x ∈ X : x ∈ u}.
Notice that for every finite extensive form game there exists a maximum m such that Succk (u) = ∅ for every u ∈ U and every k ≥ m.
One can now describe the concept of an information set that precedes
or succeeds another information set.
Definition 5.3. Given an information set u, the set of all successors
of u, Succ(u), is the set of nodes
∞
Succk (u).
k=0
We can now define the precedence of information sets.
Definition 5.4. Let I1 and I2 be two information sets of the extensive
game G. We say that I1 precedes I2 if there is a node y ∈ I2 such
that y ∈ Succ(I1), and write I1 p I2 .
Thus an information set precedes another information set if its nodes
precedes the nodes of the successor information set.
As can be seen, the relation p is reflexive (since x ∈ Succ(x) for
every x ∈ X) and transitive, but is not antisymmetric (and in fact is
not complete).
Definition 5.5. Two distinct information sets I1 and I2 are called
simultaneous if both I1 p I2 and I2 p I1 .
20
Define an equivalence relation ∼p on the set of information sets of
the extensive form game as follows: I1 ∼p I2 if I1 p I2 and I2 p I1 .
An interesting feature is that the relation p modulo ∼p gives us a
partial order on the information sets of a sequential game.
Definition 5.6. An equivalence class of the relation ∼p is called a
knot.
Let K denote the set of all knots for the extensive form game G.
Notice that since G is a finite game, the set K is finite, without loss
of generality K = (K1 , K2 , · · · , Km ). Define a binary relation e on K
as follows: K1 e K2 if u v for some information sets u ∈ K1 and
v ∈ K2 . Then, K1 e K2 can be interpreted as knot K1 precedes
knot K2 .
Definition 5.7. An extensive form game G of perfect recall is said to
be a game of simple information structure if all its knots are singletons, otherwise it is a game of complex information structure.
1
1
a
a
L
/
b C 2
C
C
L
CR
C
CW
B d
l
BB r
BBN
S
S
S
R
S
S
w c
S
A
3
A
l AAr
A
AU
e J
L JJR
JJ
^
(a)
L
S
S
/
b C
C
C
L
CR
C
d
CWB
l
BB r
BBN
S R
S
S
w c
S
A
2
A
L AAR
A
3
e
U
A
J
l JJr
JJ
^
(b)
Figure 3
As an example, consider two sequential games depicted in Figure 3.
The game of Figure 3(a) is a game of complex information structure:
Here I1 = {a}, I2 = {b, e} and I3 = {c, d} are the information sets
of players 1, 2 and 3, respectively. Since b d and c e, we have
both I2 p I3 and I3 p I2 , so that I2 ∼p I3 . On the other hand,
I1 p I2 and I1 p I3 . Therefore this game has two knots: K1 = I1
and K2 = {I2 , I3 }.
On the other hand, the game of Figure 3(b) is a game of simple
information structure: here all three knots K1 = I1 = {a}, K2 = I2 =
{b, c} and K3 = I3 = {d, e} are singletons.
21
Notice that for the games of simple information structure the relation
p modulo ∼p is the relation p itself. Intuitively, in a game of simple
information structure information sets are aligned one after another,
so that relation p gives us a nice partial order of the information sets
of the game. We shall see that for these class of sequential games a
backward induction process can be used to find the sequential equilibria
of the game.
6. Sequential Equilibria and Backward Induction
Let G be an extensive form game of simple information structure.
Then, the binary relation p on the set of information sets U is a partial
order. Let F0 be the set of all minimal elements of U with respect to
the partial order p , that is, the set of all information sets u such that
there is no information set v with the property u p v.
We claim that F0 is nonempty. Indeed, notice that since G is a
finite game, the set U is finite and partially ordered by p . Also U has
finitely many chains, call them C1 , · · · , Cq , and every chain has finitely
many elements. Hence every chain Ci has a (unique) minimal element
ui . Then F0 = {u1, · · · , uq } = ∅,
Call children of F0 the set of children of all nodes from any information set of F0 , and denote the set of children of F0 by N0 . The following
result is crucial for defining the generalized backward induction, which
fails for games with a complex information structure.
Lemma 6.1. Let G be a sequential game of simple information structure. Then all children of F0 are terminal nodes.
Proof. Assume G is a sequential game of simple information structure.
It was shown above that F0 = ∅. Suppose by contradiction there exists
an information set u ∈ F0 , node x ∈ u and a child y of x such that y is
a non-terminal node (this implies y is a decision node). Without loss
of generality, let v be the information set containing y. Notice that u
is different from v by definition of an information set (since we have
x y).
Also, since x y, we have u p v. We claim that u ∼p v. Indeed, if
u ∼p v, then the order p is not antisymmetric, and hence not a partial
order, contradiction. Then we have both u p v and u ∼p v, which
implies u p v. But then u is not a minimal element with respect to
p , contradiction. This completes the proof.
Fix a system of beliefs μ ∈ M and proceed recursively as follows.
Denote by μF0 the restriction of μ to the set of information sets F0
(note that F0 = ∅). Given μF0 , the set of best responses, or sequentially
22
rational behavior strategy profiles b∗F0 (call it ΛF0 (μ)) is nonempty,
convex and compact subset of BF0 .
Fix b∗F0 ∈ ΛF0 (μ). Truncate the game tree Γ of the sequential game
G by deleting all the nodes in N0 and assigning to each node in F0 the
expected payoff vector generated by b∗F0 (we are using here Lemma 6.1,
which guarantees that all nodes in N0 are terminal nodes). This step
generates a new sequential game, call it G1 , with the corresponding
extensive form Γ1 .
It can be easily verified that p restricted to Γ1 is a partial order.
Denote by F1 the set of minimal elements among the information sets
of G1 with respect to the partial order p restricted to Γ1 . By the
earlier argument, the set F1 is nonempty. Repeat the above steps,
applied to the game G1 .
Since Γ has finitely many nodes, at some point the process will stop.
Thus, for a given system of beliefs μ, we can recursively construct
a finite sequence of games G1 , · · · , Gr using this backward recursion
process and a strategy profile b∗ .
Definition 6.2. We will call b∗ a generalized backward induction
strategy profile for the system of beliefs μ. If μ is consistent with
b∗ , that is μ is in φ(b∗ ), we call the assessment (μ, b∗ ) a backward
induction assessment.
Notice that since p is a partial order, the generalized backward
induction process will stop at the root information set IW . This is due
to the fact that IW is the unique maximal element of U with respect
to p for the game G.
Lemma 6.3. Let G be an extensive form game of simple information
structure. Then IW , the root information set, is the unique maximal
element of U with respect to the partial order p .
Proof. First, let us show that IW is a maximal element. Suppose by
contradiction there exists u ∈ U such that u p IW . This implies there
exist nodes x ∈ u, w ∈ IW such that x w, that is, x is a predecessor
of w. But this is impossible since w ∈ IW is a root.
Next, let us show that there is no maximal element other than IW .
Suppose by contradiction there exists an information set v different
from IW , such that v is a maximal element of U with respect to p .
Fix an element y ∈ v. By assumption y is a non-root node, hence there
exists a root node w ∈ IW such that there is a unique path from w
to y. But this implies w y, hence IW p v. This contradicts our
assumption that v is a maximal element. This establishes that IW is
the unique maximal element.
23
It is important to emphasize that for a given μ ∈ M, the generalized
backward induction process may yield infinitely, in fact uncountably
many behavior strategy profiles. This is because the set of best responses at any step may be uncountable. Also, in order to find all
backward induction assessments, one needs to perform the generalized
backward induction for every system of beliefs μ ∈ M.
Next, we need to show that the backward induction assessments are
precisely the sequential equilibria of a given sequential game G, and the
set of backward induction assessments is nonempty. The first result
(Theorem 6.5) can be established using mathematical induction. Its
proof also relies on the following lemma, which is an easy consequence
of the continuity of the dot product.
Lemma 6.4. Let G be an extensive form game, and information sets
u, v ∈ U be such that every node in v has a predecessor among nodes
of u. Let (μ, b) be a consistent assessment. If Pu (y|μ, b) = 0 for some
y ∈ v, then for every x ∈ v
μ(x) = Pu (x|μ, b)
.
y∈v Pu (y|μ, b)
Theorem 6.5. Let G be an extensive form game of simple information structure. Then an assessment (μ∗ , b∗ ) is a backward induction
assessment if and only if it is a sequential equilibrium of G.
Proof. Assume (μ∗ , b∗ ) is a backward induction assessment. Then the
system of beliefs μ∗ is consistent with b∗ by definition. It remains to
show that (μ∗ , b∗ ) is sequentially rational. This can be established using
the method of mathematical induction. WIthout loss of generality,
suppose the backward induction process stops at Fr for some r ∈ N,
i.e., the root information set IW is in Fr .
Fix a natural number 0 ≤ k < r. By induction hypothesis, assume
∗ ∗
(μ , b ) is sequentially rational starting from any information set in
Fj , for all 0 ≤ j ≤ k. We need to show that (μ∗ , b∗ ) is sequentially
rational starting from any information set in Fk+1. Let u ∈ Fk+1 be
an information set owned by say player i.
If player i does not own any information set among Succ(u), we are
done, since b∗iu is a best response of player i at u at the (k + 2)th step
of the generalized backward induction process.
Assume player i owns some information set in Succ(u). Suppose by
contradiction player i wants to deviate, starting from u, at a nonempty
set Vi of information sets. Consider two possible cases:
(1) Vi ∩ Fk+1 = ∅. Notice that the set Vi is finite and partially
ordered by p . Fix a chain C in Vi . Since it is finite and totally
24
ordered, it has a unique maximal element, call it v. But this
implies that player I wants to deviate at v, which contradicts
our induction hypothesis, since v ∈ Fj for some 0 ≤ j ≤ k.
(2) Vi ∩ Fk+1 = ∅, which means Vi ∩ Fk+1 = u. Then:
(a) If Vi \u = ∅, we arrive at a contradiction, since b∗iu is a
best response of player i at u at the (k + 2)th step of the
generalized backward induction process.
(b) Suppose Vi \u = ∅. Without loss of generality let b̃ ∈ B
be the behavior strategy profile such that b̃m = bj for all
m = i, and Ei (u, b̃, μ) > Ei (u, b, μ). The perfect recall
assumption together with Lemma 6.4 implies that for every
choice sj ∈ Cu of player i at u,
Ei (u, (sj , b̃\sj ), μ) > hi (u, (sj , b\sj ), μ).
But then, deviating to b̃i from bi by player i is equivalent to
deviating at the information set u only. As before, this is
a contradiction, since b∗iu is a best response of player i at u
at the (k + 2)th step of the generalized backward induction
process.
This shows (μ∗ , b∗ ) is sequentially rational starting from any information set in Fk+1. Hence by mathematical induction (μ∗ , b∗ ) is sequentially rational.
Let (μ∗ , b∗ ) is a sequential equilibrium. We want to show that it is a
backward induction assessment. Since (μ∗ , b∗ ) is a sequential equilibrium, b∗F0 is optimal starting from any information set in F0 given μ∗F0 .
Therefore b∗F0 is selected in the first step of the generalized backward
induction.
Assume by induction hypothesis that b∗Fk is selected on the (k + 1)th
step of backward induction, after truncating the extensive form according to (b∗Fl )l=0,··· ,k−1. Suppose b∗Fk+1 is not optimal starting from some
information set u ∈ Fk+1 , after truncating the extensive form according to (b∗Fl )l=0,··· ,k . But this implies the owner of u has an incentive to
deviate from b∗ , starting at u, hence (μ∗ , b∗ ) is not sequentially rational,
which is a contradiction. This completes the proof.
The next result shows that the set of backward induction assessments
is nonempty. This together with theorem 6.5 then shows that every
sequential equilibrium of a sequential game with a simple information
structure can be found by the method of backward induction.
25
Define a correspondence τ : M × B →
→ M × B as follows: given
(μ, b) ∈ M × B, let
τ (μ, b) = M̃ × B̃,
where M̃ is the set of all beliefs systems consistent with b, and B̃ =
B̃Fr × · · · × B̃F0 , such that for each j = 0, · · · , r, B̃Fj is the set of best
replies at Fj given ρ (the probability at the root information set IW ),
μFj and the truncation according to (bFj−1 , · · · , bF0 ).
It follows immediately from the definition of a backward induction
assessment that (μ, b) ∈ M × B is a backward induction assessment if
and only if (μ, b) is a fixed point of τ .
It remains to show that τ has a fixed point. Note that τ has a
problematic boundary behavior: if b ∈ ∂B, then the set of beliefs
consistent with b is not necessarily convex or even acyclic. However, τ
is nicely behaved on the interior of M × B. We use a generalization of
Kakutani’s fixed point theorem (see the Appendix) that allows us to
conclude that τ has a fixed point in M × B even though it does not
satisfy all the hypotheses of Kakutani’s theorem on the boundary of
B.
Lemma 6.6. The correspondence τ is convex-valued on the interior of
M × B.
Proof. Fix (μ, b) ∈ M ◦ ×B ◦ , without loss of generality τ (μ, b) = M̃ × B̃.
Then M̃ is a singleton, hence is a convex subset of M.
For each j = 0, · · · , r, B̃Fj is the set of best replies at Fj given ρ,
μFj and the truncation according to (bFj−1 , · · · , bF0 ). Therefore B̃Fj
is nonempty and convex for each j = 0, · · · , r. This implies B̃ =
B̃Fr ×· · ·× B̃F0 is a convex subset of B. Consequently, τ (μ, b) = M̃ × B̃
is convex.
Lemma 6.7. The correspondence τ is nonempty-valued and closed.
Proof. (i) First let us show that τ is closed. Fix a sequence (μk , bk ) ⊆
M × B such that (μk , bk ) → (μ∗ , b∗ ) ∈ M × B as k → ∞. Let (μ̃k , b̃k ) ∈
τ (μk , bk ) for each k ∈ N such that (μ̃k , b̃k ) → (μ̃∗ , b̃∗ ). We need to show
that (μ̃∗ , b̃∗ ) ∈ τ (μ∗ , b∗ ).
Without loss of generality τ (μk , bk ) = M k × B k for each k ∈ N and
τ (μ∗ , b∗ ) = M ∗ × B ∗ . By the closedness of the belief correspondence φ,
μ̃∗ ∈ M ∗ .
Fix j ∈ {0, · · · , r}, and notice that the payoff of the player decisive at
each information set u ∈ Fj , given ρ, μFj and the truncation according
to (bFj−1 , · · · , bF0 ), is jointly continuous in μ and b. Therefore b∗ ∈ B ∗ .
This shows (μ̃∗ , b̃∗ ) ∈ τ (μ∗ , b∗ ), hence τ has a closed graph.
26
(ii) By the closedness of τ and compactness of M × B it suffices
to show that τ is nonempty-valued on the interior of M × B. Fix
(μ, b) ∈ M ◦ ×B ◦ , and follow exactly the lines of the proof of Lemma 6.6
to establish that τ (μ, b) = ∅.
Lemmas 6.6 and 6.7 imply that τ satisfies the hypotheses of Theorem
7.1 in the Appendix. Also, M × B satisfies the hypotheses of Theorem
7.1, since M and B are nonempty, convex and compact subsets of some
Euclidean space. Therefore, we get the following existence result.
Theorem 6.8. τ has a fixed point over M × B, and consequently, the
set of backward induction assessments for an extensive game of simple
information structure is nonempty.
7. Conclusion
The fact that finite sequential games with perfect recall have sequential equilibrium is an observation that goes back to [8]. What
we do here is show that one needs to rely on the sequential nature of
the game in order to make a full and comprehensive analysis of this
question. In the process of providing a direct proof of the existence of
sequential equilibrium we also observe that backward induction plays
an important role in the analysis. This is certainly the case for finding the optimal strategy of any player. It is also the case that for an
important (and broad) class of sequential games, namely, those with
simple information structures all the sequential equilibrium points of
a sequential game can be obtained by using the process of backward
induction. The two results put together then show that a useful and
efficient method of computing the sequential equilibria of a game is to
proceed by computing the optimal choices of the players given their
beliefs at their information sets, and working back recursively from
information sets at the latest stages of the game, to optimal choices
at information sets at progressively earlier stages of the game. In the
case of games with simple information structure this method not only
allows us to find a sequential equilibrium, but it enables us to find the
entire set of sequential equilibria of the game.
As most sequential games that arise in applications, particularly in
economics and finance, are sequential games with a simple information
structure, the results presented here are especially useful for these sequential games. Indeed, in many applications, the equilibria of these
sequential games are mostly found by using variations of the generalized backward induction method outlined here.
While we have presented the results for sequential equilibrium, these
results also extend to the closely related concept of perfect Bayesian
27
equilibrium, see [6] for a full discussion of the concept of Perfect Bayesian
equilibrium. As this concept is widely used in many applications,
see for example [2], [5] and [9], it is important to have a well understood method of finding these equilibria. As a sequential equilibrium is also a perfect Bayesian equilibrium, Theorem 4.4 shows that a
perfect Bayesian equilibrium always exists for finite sequential games
with perfect recall. Further, the results for games with simple information structures also apply to perfect Bayesian equilibrium points,
and so does the generalized backward induction method for finding
these equilibrium points. The results presented here are, therefore, not
only useful for analyzing the sequential equilibrium points but also the
perfect Bayesian equilibrium points of a sequential game.
While we have focused here on analyzing only finite sequential games,
it should be clear that the techniques can be used to find the equilibrium points of sequential games with information sets that are not
necessarily finite or do not have a finite horizon. However, it would not
be clear whether these equilibrium points are sequential equilibrium
points as the consistency property of a sequential equilibrium for these
more general classes of sequential games is not that well understood
at this point. Perhaps these might be Perfect Bayesian equilibrium
points. We hope to pursue these and some other related issues in some
of our future research.
28
Appendix
Theorem 7.1. A Generalization of Kakutani’s Fixed Point
Theorem3 Let X be a nonempty, convex and compact subset of Rn ,
and τ : X →
→ X be a closed correspondence such that the set τ (x) is
convex for each x ∈ X ◦ . Then τ has a fixed point in X.
Proof. If X is a singleton, then the result is trivial. Assume X is not a
singleton, and pick some a∗ ∈ X ◦ . Consider a straight-line homotopy
H : X × I → X defined for each x ∈ X as H(x, t) = ta∗ + (1 − t)x,
where t ∈ I = [0, 1]. Clearly, H is continuous. For any fixed t, denote
by Ht the restriction of H to X given t.
Define a sequence of “approximating correspondences” as follows.
For each t ∈ (0, 1] let Xt = Ht (X). Notice that each Xt is a nonempty,
convex and compact subset of Rn since H is a straight-line homotopy.
→ Xt as follows:
Define τt : Xt →
τt (x) = Ht (τ (x)) for each x ∈ Xt .
Fix t ∈ (0, 1]. We claim that the correspondence τt is closed. Note
that τt is upper hemicontinuous since it is a composition of an upper
hemicontinuous correspondence τ and a continuous function Ht (see [1,
Theorem 17.23, p. 566]). Therefore, by the Closed Graph Theorem ([1,
Theorem 17.11, p. 561]) τt is closed. Also, τt (x) is compact since τ (x)
is compact and Ht is continuous. But since Rn is a Hausdorff space,
τt (x) is also closed for each x ∈ Xt .
Finally, note that for t = 0, Xt ⊂ X ◦ so that τt (x) is a convex set
for each x ∈ Xt . Therefore, for t = 0, Ht (τ (x)) is a convex set for each
x ∈ Xt as the straight line homotopy preserves convexity.
Hence, by Kakutani’s fixed point theorem, τt attains a fixed point in
Xt for each t ∈ (0, 1], say zt . Since zt ∈ X for each t and X is compact,
the net {zt }(t∈(0,1]) has a limit point in X, say z, as t → 0.
We claim that z is a fixed point of τ . Introduce a correspondence
ψ:I→
→ X × X defined for each t ∈ I as
ψ(t) = Gτt ,
where Gτt is a graph of the correspondence τt defined as
Gτt = {(x, y) ∈ Xt × Xt : x ∈ Xt , y ∈ τt (x)} .
Notice that (zt , zt ) ∈ Gτt for each t = 0, and hence (t, zt , zt ) ∈ Gψt . To
show z ∈ τ (z), it suffices to show that ψ has a closed graph.
First, notice that ψ is nonempty- and closed-valued since Gτt is
nonempty and closed for each t. It remains to show that ψ is upper
3This
result is also presented in 14.
29
hemicontinuous. Indeed, notice that ψ can be written as a composition
of correspondences
ψ = j ◦ (H × idI ) ◦ g ◦ h,
where j : I → X × I is the inclusion map,
H × idI is a product of H and an identity map on I, idI ,
g :X ×I→
→ X × X × I is defined for each (x, t) ∈ X × I as
g(x, t) = (x, τ (x), t),
h : X ×X ×I →
→ X × X is defined for each (x, y, t) (where y ∈ τ (x))
as
h(x, y, t) = (x, Ht (y)).
Notice that the correspondence H × idI is upper hemicontinuous
by [1, Theorem 17.28, p. 568] since both H and idI are continuous
functions (and so can be viewed as upper hemicontinuous correspondences with compact values). It is easy to check that j, g and h are
also upper hemicontinuous, hence ψ is upper hemicontinous by [1, Theorem 17.23, p. 566].
But since it was shown that ψ is nonempty- and compact-valued, it
follows by the Closed Graph Theorem (see [1, Theorem 17.11, p. 561])
that ψ is closed.
So, since (t, zt , zt ) ∈ Gψ for each t ∈ (0, 1) and zt → z as t → 0, then
(0, z, z) ∈ Gψ , so that (z, z) ∈ Gτ0 . As τ0 (x) = τ (x) for all x ∈ X, this
shows that z ∈ τ (z), that is, z is a fixed point of τ . This completes the
proof.
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