We formulate and prove a modification of Eilenberg-Montgomery fixed-point theorem, which is a generalization of Kakutani’s theorem. It enables us to provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. We construct a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by the modified version of Eilenberg-Montgomery theorem.Perfect equilibrium, best response correspondence, unit simplex, absolute neighborhood retract, deformation retract, fixed point

I. Topolyan

English

Research Papers in Economics

KRANNERT SCHOOL OF 
MANAGEMENT 
 
    Purdue University 
    West Lafayette, Indiana 
 
 
Existence of Perfect Equilibria: A Direct Proof 
 
By 
 
I. Topolyan 
 
 
 
Paper No. 1226 
Date:  October, 2009 
Institute for Research in the 
Behavioral, Economic, and 
Management Sciences EXISTENCE OF PERFECT EQUILIBRIA: A DIRECT PROOF★
I. TOPOLYAN 1
1 Department of Economics, Krannert School of Management, Purdue University, 100 S. Grant
Street,
W. Lafayette, IN 47907–2076, USA; itopolya@purdue.edu
Abstract. We formulate and prove a modiﬁcation of Eilenberg-Montgomery
ﬁxed-point theorem, which is a generalization of Kakutani’s theorem. It
enables us to provide a direct proof of the existence of perfect equilibria in
ﬁnite normal form games and extensive games with perfect recall.
We construct a correspondence whose ﬁxed points are precisely the per-
fect equilibria of a given ﬁnite game. Existence of a ﬁxed point is secured
by the modiﬁed version of Eilenberg-Montgomery theorem.
JEL classiﬁcation: C7
Keywords: Perfect equilibrium, best response correspondence, unit sim-
plex, absolute neighborhood retract, deformation retract, ﬁxed point.
1. Introduction
Perfect equilibria were introduced in 1975 by Selten [?] as a reﬁnement of subgame
perfect and Nash equilibria. Perfect equilibria play important role in contemporary
game theory due to its stability with respect to slight imperfections of rationality,
or “trembling-hand perfection”.
Date: October 7, 2009.
I acknowledge a huge intellectual debt to my advisor, Charalambos D. Aliprantis, who passed
away February 27, 2009. This work carries many of his prominent ideas and is a result of years
of our work together. I express gratitude to my co-advisors Jack Barron and Bill Novshek, Subir
Chakrabarti, Gabriele Camera, Dan Kovenock, James Moore and many other people for their
guidance, support, helpful conversations, encouragement and advice.2
Perfect equilibria exist for every normal form game, and for every extensive form
game with perfect recall there exists a perfect equilibrium in behavior strategies.
However, the proof of existence, given by Selten [?], is indirect and relies on the
existence of a Nash equilibrium in every perturbed game. This makes the perfect
equilibria not as “tangible” and complicates their treatment.
We are the ﬁrst to provide a direct proof of the existence of perfect equilibria in
normal form games and in extensive form games with perfect recall. We construct
a correspondence whose ﬁxed points are precisely the perfect equilibria of a given
game. This correspondence coincides with the best response correspondence on the
interior of the strategy space, however the constructed correspondence possesses
much more general boundary behavior. In order to prove existence of a ﬁxed point,
we formulate and prove a version of Eilenberg-Montgomery ﬁxed point theorem. Its
proof is inspired by and borrows the idea of Selten’s proof of existence of perfect
equilibria [?].
The paper is organized as follows. Chapter 2 provides deﬁnitions of the game-
theoretic concepts invoked in this paper. In Chapter 3 we introduce the necessary
tools of algebraic topology, formulate and prove a version of Eilenberg-Montgomery
ﬁxed point theorem. In Chapter 4 we construct a correspondence whose ﬁxed points
are the perfect equilibria of a given game, and apply the version of Eilenberg-
Montgomery theorem introduced in the previous chapter to proof the existence of a
ﬁxed point.
2. Definitions and Methodology
Deﬁnition 2.1. A normal form Γ of a ﬁnite n-player game is a tuple ( 1,⋅⋅⋅ ,  ,ˆ ℎ),
where    is a ﬁnite set of pure strategies of player  ,   =
Q 
 =1   , and ˆ ℎ :  → →ℝ 
is the payoﬀ function that assigns to every   ∈   the vector of payoﬀs ˆ ℎ( ) =
(ˆ ℎ1( ),⋅⋅⋅ ,ˆ ℎ ( )).
Deﬁnition 2.2. A mixed strategy    for player   is a probability distribution over
  . The set of all such probability distributions is denoted by    ≡ Δ i, where    is
the cardinality of   . The set of mixed strategies for the game Γ is   =
Q 
 =1   .
Mixed strategy    ∈    for player   is completely mixed if    is in the interior of
  , i.e.,    ∈  ∘
 . Mixed strategy   ∈   is completely mixed if for every player  ,   
is completely mixed.3
We can now deﬁne an expected payoﬀ function ℎ, which is an extension of the
payoﬀ function ˆ ℎ to all of  .
Deﬁnition 2.3. An expected payoﬀ function is a function ℎ :  → →ℝ  such that
ℎ( ) =
Y
 ∈ 
 1( ) 2( )⋅⋅⋅  ( )ˆ ℎ( ),
where   ( ) is the probability that   assigns to the   ℎ component of  , i.e., the
probability with which player   chooses   .
Deﬁnition 2.4. A best response correspondence of player   is the correspon-
dence    :  −  =
Q
 ∕=    → →   deﬁned for each  −  ∈  −  as
  ( − ) = {˜    ∈    : ℎ (˜   , − ) ≥ ℎ (  , − ) ∀   ∈   }.
Deﬁnition 2.5. A best response correspondence for an  -person normal form
game is the correspondence   :  → →  deﬁned for each   ∈   as an  -tuple
( 1( −1), 2( −2),⋅⋅⋅ ,  ( − )), where for each  ,   ( − ) is player  ’s best response
correspondence deﬁned above.
We will deﬁne perfect equilibrium in terms of substitute sequences (as appeared
in Selten (1975) as an alternative characterization of perfection).
Deﬁnition 2.6. A substitute sequence for a strategy proﬁle ¯   ∈   is a sequence
of completely mixed strategy proﬁles approaching ¯  , i.e, a sequence
￿
  ￿
⊆  ∘ such
that    → ¯   as   → ∞.
Deﬁnition 2.7. A mixed strategy  ∗ ∈   is called a perfect equilibrium point
of a normal form game Γ if  ∗ is a best response to at least one substitute sequence
for  ∗.
3. A New Version of the Eilenberg-Montgomery Fixed Point
Theorem
We ﬁrst provide some deﬁnitions invoked in Eilenberg-Montgomery ﬁxed-point
theorem and the formulation of the theorem itself, and then formulate and prove a
slight modiﬁcation of the theorem.4
Deﬁnition 3.1. Given a topological space  , a nonempty-valued correspondence
  :  → →  is closed if it has a closed graph.
Deﬁnition 3.2. Let   be a topological space, then a nonempty-valued correspon-
dence   :  → →  is closed if it has a closed graph.
Deﬁnition 3.3. Let   be topological space and   a subspace of  . Then a contin-
uous map   :   →   is a retraction if the restriction of   to   is the identity map
on  . Equivalently, if we denote the inclusion map on   by   :   ֒→  , a retraction
is a continuous map   :   →   such that   ∘   is homotopic to the identity map on
 .
Deﬁnition 3.4. A subspace   of a topological space   is called a retract if such a
retraction exists.
Deﬁnition 3.5. Let   be topological space and   a subspace of  . Then   is called
a neighborhood retract of   if there exists an open set   ⊆   such that   ⊆  
and   is a retract of  .
The notion of a neighborhood retract is a weakening of a retract: let   be a
subspace of a topological space   such that   is a retract of  . Then, take   to
be the neighborhood of   in Deﬁnition to deduce that   is a neighborhood retract
of  .
Deﬁnition 3.6. A topological space   is an absolute neighborhood retract (or
ANR) if for every embedding of   as a closed subset of a normal space   the image
of   is a neighborhood retract of  .
Deﬁnition 3.7. A topological space   is a deformation retract of   if there
exists a continuous function   :   →   which is homotopic to the identity map of
  and ﬁxes  . Equivalently, there exists a continuous map   :   ×   →   (where
  = [0,1]) such that  ( ,0) =   and  ( ,1) ∈   for all   ∈  , and  ( , ) =  
for all   ∈  . The homotopy   is called a deformation retraction of   onto  .
Some authors do not require a deformation retract to ﬁx  , and call the space de-
scribed above a strong deformation retract. An important property of a deformation
retract as stated in Deﬁnition is that it is homotopy invariant and preserves homol-
ogy groups, which is crucial in proving the new version of Eilenberg-Montgomery
theorem.5
Deﬁnition 3.8. A nonempty compact metric space   is said to be acyclic provided:
(1) the homology groups   ( ) vanish for all   > 0,   ∈  , and
(2) the reduced 0 ℎ homology group ˆ  0( ) vanishes for   ∈  .
Theorem 3.9. (Eilenberg-Montgomery ﬁxed point theorem) Let   be an
acyclic absolute neighborhood retract and   :  → →  be a closed correspondence
such that for every   ∈   the set  ( ) is acyclic. Then   has a ﬁxed point.
The proof can be found in [?]. Notice also that according to Deﬁnition 3.7 if  
is acyclic, then it is understood that   is a nonempty compact metric space.
We are now ready to formulate and prove a version of Eilenberg-Montgomery
ﬁxed point theorem. It allows much more general boundary behavior of the corre-
spondence (as long as the closedness of the graph is preserved), however it requires
existence of a nicely behaved deformation retract. Notice, however, that the latter
assumption is always fulﬁlled in certain cases (for instance, when   is a convex
set; see Example 3.11), which produces a stronger version of Eilenberg-Montgomery
theorem for those cases.
Theorem 3.10. Let   be an acyclic absolute neighborhood retract,   be a dense
subset of  , and   ⊂   be a deformation retract of   such that  ( , ) ∈   for
every   ∈   and   > 0. Let   :  → →  be a closed correspondence such that for
every   ∈   the set  ( ) is acyclic. Then   has a ﬁxed point.
Proof. Fix   ∈  ,   > 0, and consider  ( , ). Clearly,  ( , ) itself is a deformation
retract of  , hence it is ANR and acyclic (since deformation retraction preserves
homology groups). Observe that  ( , ) ⊆  , and apply Theorem ?? to  ( , )
in place of  , and   restricted to  ( , ). Denote the corresponding ﬁxed point of
  on  ( , ) by   .
Consider the net {  } ∈(0,1] in  . Since ¯   =   is compact, the net {  } has a
limit point in   (see Theorem 2.31 in Aliprantis and Border (2006) ). So passing to
a subsequence without loss of generality, we get    →  ∗ as   → ∞. Denoting by   
the graph of the correspondence  , we see that (  ,  ) ∈    since    is a ﬁxed point
of   over the set  ( , ). Therefore by the closedness of    we have ( ∗, ∗) ∈   ,
so that  ∗ is a ﬁxed point of   over  , which completes the proof.
Example 3.11. (Existence of a nicely behaved deformation retract) Let   be a
convex subset of some topological vector space. If   is a singleton, then the only6
dense subset of it is   itself. Assume   is not a singleton, then convexity implies
 ∘ ∕= ∕⃝. Let   =  ∘, ﬁx  ∗ ∈  ∘ and   ∈ (0,1). Clearly   is dense in  .
Consider a straight-line homotopy   :   ×  → →  deﬁned for each   ∈   as
 ( , ) =   ∗ + (1 −  ) . Clearly,   is continuous,  ( ,0) =   and  ( ,1) =  ∗
for all   ∈  . Notice also that  ( ∗, ) =  ∗ for all   ∈  . Therefore the homotopy
  is a deformation retraction, and the singleton  ∗ is a deformation retract of  .
Notice also that  ( , ) ⊆   for all 0 <   ≤ 1. Therefore  ,   =  ∘,   =  ∗ and
  satisfy the hypotheses of Theorem 3.10.
4. A Direct Proof of the Existence of Perfect Equilibria
Recall that given an  -player normal or extensive form game Γ with a strategy
space    for each player   (as usual let   =
Q 
 =1    ), the graph of the best response
correspondence
   = {( , ) ∈   ×   :   ∈  ,  ∈  ( )},
where   :  → →  is the best response correspondence as stated in Deﬁnition 2.5.
Now consider the graph of the best response correspondence restricted to  ∘× ,
i.e.,
  = {( , ) ∈  
∘ ×   :   ∈  
∘,  ∈  ( )}.
Deﬁne    to be the closure of the set   in  × , and identify with    a correspon-
dence from   to  , call it  , having    as its graph. Clearly, such correspondence is
well-deﬁned as long as    as a subset of   ×   is well deﬁned, but it is so because
a closure of the set   in   ×   is well-deﬁned.
From the above it follows immediately that   is a closed correspondence (i.e., has
a closed graph). Indeed,    = ¯   in   ×   implies that    is closed in   ×  .
Lemma 4.1. The correspondence   is nonempty-valued and closed.
Proof. It remains only to show that   is nonempty-valued. So, ﬁx any   ∈  . If
  ∈  ∘, then  ( ) ∕= ∕⃝ since in the interior of  ,   coincides with the best response
correspondence, which is nonempty-valued.7
Let   ∈ ∂ . Take a sequence {  } of completely mixed strategies (i.e.,    ∈  ∘
for each   ∈ ℕ) such that lim →∞    =  . Since each    is in the interior of  , then
 (  ) ∕= ∕⃝. For each   ∈ ℕ pick some    ∈  (  ). Thus we obtained a sequence of
points {  } in a compact set  , hence this sequence has a convergent subsequence,
without loss of generality   m →   as   → ∞ for some   ∈  .
Notice also that (  ,  ) ∈    for each   and by relabelling (  ,  ) → ( , ) as
  → ∞. However, since    is closed, then ( , ) ∈   , i.e.,   ∈  ( ). This establishes
that   is nonempty-valued.
Notice that the Closed Graph Theorem (see Theorem 17.11, p. 561 of Aliprantis
and Border(2006)) implies that   is upper hemicontinuous and closed-valued.
The following theorem shows that for any normal form game Γ, the set of its
perfect equilibria coincides with the set of ﬁxed points of the constructed correspon-
dence  .
Theorem 4.2. A strategy proﬁle  ∗ ∈   is a ﬁxed point of   if and only if  ∗ is a
perfect equilibrium point of Γ.
Proof. (⇒) Let the strategy proﬁle  ∗ ∈   be such that  ∗ ∈  ( ∗), that is, ( ∗, ∗) ∈
  . Since    is the closure of   in  × , there exists a sequence (  ,  ) ⊆   such
that (  ,  ) → ( ∗, ∗) as   → ∞.
Observe that for each  ,  (  ) is a Cartesian product of some faces of the unit
simplices   , hence there exists   ∈ ℕ such that
 
∗ ∈  ( 
 )
for all   ≥  , and    →  ∗.
This shows that  ∗ is a perfect equilibrium point of Γ.
(⇐) Assume  ∗ ∈   is a perfect equilibrium point of Γ, then there exists a
substitute sequence for  ∗, say
￿
  ￿
such that  ∗ ∈  (  ) for all   ∈ ℕ.
Notice that (  , ∗) ∈   for all   and (  , ∗) → ( ∗, ∗) as   → ∞. Therefore
( ∗, ∗) ∈ ¯   =   , q.e.d.
The diﬃculty with proving that the correspondence   has a ﬁxed point arises
from the fact that   need not be convex-valued or even acyclic-valued. However, it
is nicely behaved in the interior of the strategy space.8
Notice that the correspondence   satisﬁes the hypotheses of Theorem ??. Indeed,
  is a nonempty, convex and compact metric space, hence it is acyclic ANR, and
a deformation retract satisfying the hypothesis of Theorem ?? exists (see Example
3.11). Since the correspondences   and   coincide on  , then  ( ) is acyclic for
every   ∈  . Therefore by Theorem ?? correspondence   has a ﬁxed point over  .
This establishes existence of a perfect equilibrium for a ﬁnite normal form game.
Our result extends to prove existence of a perfect equilibrium in behavior strate-
gies in extensive games with perfect recall. To show this, we employ a fundamental
result of Selten [?]. It establishes a bijection (one-to-one and onto map) between the
set of perfect equilibria (in behavior strategies) of an extensive game with perfect
recall and the set of perfect equilibria of the corresponding agent normal form. Then
existence of perfect equilibria for an extensive game with perfect recall is implied
by perfect equilibrium existence for the corresponding agent normal form game,
established earlier.
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[8] J. Nash, Equilibrium goints in N-person games, Proceedings of the National Academy
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Continuity and Equilibrium Stability, Purdue Working Paper.

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Fixed-point theorems for multivalued transformations,

Game Theory,

Game Theory: Analysis of Conﬂict,

Games and Decision Making,

Inﬁnite Dimensional Analysis: A Hitchhikers Guide, 3rd Edition,

Non-cooperative games,

Re-examination of the perfectness concept for the equilibrium points in extensive games,

Spieltheoretische Behandlung eines Oligopolmodells mit Nachfragetragheit, Zeitschrift fur die gesamte

Theory of Correspondences,

Existence of Perfect Equilibria: A Direct Proof

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Existence of Perfect Equilibria: A Direct Proof

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