One answers to an open question of Herings et al. (2008), by proving that their fixed point theorem for discontinuous functions works for mappings defined on convex compact subset of $\R^n$, and not only polytopes. This fixed point theorem can be applied to the problem of Nash equilibrium existence in discontinuous games.fixed point theorem; discontinuity; nash equilibrium

Philippe Bich

English

Research Papers in Economics

An answer to a question of Herings et al.
by Philippe Bich
Paris School of Economics, Centre d’Economie de la Sorbonne UMR 8174,
Universit´ e Paris I Panth´ eon Sorbonne, 106/112 Boulevard de l’Hˆ opital, 75013
Paris. email: bich@univ-paris1.fr
Abstract
One answers to an open question of Herings et al. (2008), by prov-
ing that their ﬁxed point theorem for discontinuous functions works
for mappings deﬁned on convex compact subset of Rn, and not only
polytopes. This rests on a ﬁxed point result of Toussaint (1984).
Keywords: Fixed point, discontinuity.
1 The result
In [2], Herings et al. prove the following new ﬁxed point theorem for possibly
discontinuous mappings:
Theorem 1.1 Let P a non empty polytope, i.e. the convex hull of a ﬁnite
subset of Rn; let f : P   P which is ”locally gross direction preserving”
in the following sense: for every x   P such that f(x)  = x, there exists
Vx, an open neighborhood of x in P such that for every u and v in Vx,
 f(u)   u,f(v)   v    0.
Then f admits a ﬁxed-point, i.e. there exists ¯ x   P such that f(¯ x) = ¯ x.
This theorem is a generalization of Brouwer ﬁxed point theorem (see [1])
which says that every continuous mapping from the unit closed ball of Rn
to itself admits a ﬁxed point. Yet, there is a restriction in Theorem 1.1:
the set P must be a polytope. But typical strategy sets in game theory are
rather compact and convex sets. Thus, an important question in practice
is to know if Theorem 1.1 holds true for such subset of Rn. In [3], one can
read: ”whether locally gross direction preserving is su cient to guarantee
the existence of a ﬁxed point on an arbitrary non empy convex and compact
set is still an open question”.
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8In this note, we prove that Herings et al. result holds true when P is
a nonempty convex and compact subset of Rn. This proof rests on a ﬁxed
point theorem of S. Toussaint ([4]) we describe at the end of this paper.
Here, and throughout this paper, for every x   Rn and y   Rn,  x,y 
denotes the eulidean scalar product of x and y. The set P is a nonempty
convex and compact subset of Rn.
Lemma 1.2 If f : P   P is locally gross direction preserving and has no
ﬁxed points, then for every x   P such that x  = f(x), there exists px   Rn
and Vx, an open neighborhood of x in P, such that for every x    Vx, one has
 px,f(x
 )   x
   > 0.
Proof. Suppose that f : P   P is locally gross direction preserving and
has no ﬁxed points. Let x   P, and let Vx be an open neighborhood of x in
P such that for every u and v in Vx,  f(u)   u,f(v)   v    0. Let {f(x1)  
x1,...,f(xk) xk} be a basis of the vector space F :=span{f(y) y,y   Vx},
where k   N , and x1, ...,xk are in Vx. Then deﬁne px =
 k
i=1(f(xi)   xi).
Let x    Vx. One clearly have
 px,f(x
 )   x
     0 (1)
from ”locally gross direction preserving” property and from the deﬁni-
tion of px. Besides, since one has  f(xi)   xi,f(x )   x     0 for every
i = 1,..,k, Inequation 1 is an equality if and only if for every i = 1,...,k, one
has  f(x )   x ,f(xi)   xi  = 0. This last property would imply f(x )   x   
F     F = {0}, a contradiction with the assumption that f(x )  = x . Thus,
Inequality 1 is strict, which ends the proof of the lemma.
Now, for every x   P, one introduces the (possibly empty) multivalued
mapping Tx deﬁned for every y   P by
Tx(y) = {z   P, z   y,px  > 0}.
Clearly, for every x   P, Tx has convex values. Besides, for every y   P,
one has y /   Tx(y). Lastly, for every z   P, the set T  1
x (z) = {y   P | z  
Tx(y)} is an open subset of P. A multivalued mapping Tx satisfying theses
assumptions is called KF in [4].
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8Besides, from the deﬁnition of px, there exists a neighborhood Vx of x
such that for every x    Vx, one has f(x )   Tx(x ): one says that Tx is a
KF-majorant of f at x (see [4]).
To summarize, we have proved that a locally gross direction preserving
mapping f : P   P without ﬁxed point is KF-majorized in the sense that
for every x   P, there exists a KF-majorant Tx of f at x.
Now, state Theorem 2.2. in [4]: it says that given any non empty convex
and compact subset X of a topological space, given any multivalued map-
ping F : X   X which is KF-majorized, there exists ¯ x   X such that
F(¯ x) =  . Apply this theorem to F = f and X = P. Thus, there should
exists ¯ x   P such that f(¯ x) =  , a contradiction. This proves that any lo-
cally gross direction preserving mapping f : P   P must admit a ﬁxed point.
Acknowledgements
I wish to thank an anonymous referee for valuable comments, and in
particular for suggesting me to simplify the main result of my paper by using
a ﬁxed point theorem by Sabine Toussaint.
References
[1] L. Brouwer, ¨ Uber Abbildung von Mannigfaltigkeiten, Math. Ann. 71,
1912, p. 97-115.
[2] J.J. Herings, G. van der Laan, D. Talman, Z. Yang, A ﬁxed point theo-
rem for discontinuous functions, Operations Research Letters, to appear,
2007.
[3] J.J. Herings, G. van der Laan, D. Talman, Z. Yang, A ﬁxed point the-
orem for discontinuous functions, Tinbergen Institute discussion paper,
2005-004/1.
[4] S. Toussaint, On the existence of equilibria in economies with inﬁnitely
many commodities and without ordered preferences, Journal of Eco-
nomic Theory 33 (issue 1), 1984, p. 98-115.
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8

A ﬁxed point theorem for discontinuous functions, Operations Research Letters, to appear,

A ﬁxed point theorem for discontinuous functions, Tinbergen Institute discussion paper,

On the existence of equilibria in economies with inﬁnitely many commodities and without ordered preferences,

An answer to a question of herings et al

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An answer to a question of herings et al

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