FIXED POINTS AS NASH EQUILIBRIAJUAN PABLO TORRES-MART´INEZ Received 27 March 2006; Revised 19 September 2006; Accepted 1 October 2006 The existence of fixed points for single or multival
Trang 1FIXED POINTS AS NASH EQUILIBRIA
JUAN PABLO TORRES-MART´INEZ
Received 27 March 2006; Revised 19 September 2006; Accepted 1 October 2006
The existence of fixed points for single or multivalued mappings is obtained as a corollary
of Nash equilibrium existence in finitely many players games
Copyright © 2006 Juan Pablo Torres-Mart´ınez This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
In game theory, the existence of equilibrium was uniformly obtained by the application
of a fixed point theorem In fact, Nash [3,4] shows the existence of equilibria for nonco-operative static games as a direct consequence of Brouwer [1] or Kakutani [2] theorems More precisely, under some regularity conditions, given a game, there always exists a cor-respondence whose fixed points coincide with the equilibrium points of the game However, it is natural to ask whether fixed points arguments are in fact necessary tools to guarantee the Nash equilibrium existence (In this direction, Zhao [5] shows the equivalence between Nash equilibrium existence theorem and Kakutani (or Brouwer)
fixed point theorem in an indirect way However, as he points out, a constructive proof is
preferable In fact, any pair of logical sentencesA and B that are true will be equivalent
(in an indirect way) For instance, to show thatA implies B it is sufficient to repeat the
proof ofB.) For this reason, we study conditions to assure that fixed points of a
continu-ous function, or of a closed-graph correspondence, can be attained as Nash equilibria of
a noncooperative game
2 Definitions
LetY ⊂ R nbe a convex set A functionv : Y → R is quasiconcave if, for each λ ∈(0, 1),
we havev(λy1+ (1− λ)y2)≥min{ v(y1),v(y2)}, for all (y1,y2)∈ Y × Y Moreover, if for
each pair (y1,y2)∈ Y × Y such that y1= y2the inequality above is strict, independently
of the value ofλ ∈(0, 1), we say thatv is strictly quasiconcave.
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2006, Article ID 36135, Pages 1 4
DOI 10.1155/FPTA/2006/36135
Trang 22 Fixed points as Nash equilibria
A mapping f : X ⊂ R m → X has a fixed point if there is x ∈ X such that f (x) = x A
vectorx ∈ X is a fixed point of a correspondence Φ : XX if x ∈ Φ(x).
Given a gameᏳ= { I,S i,v i }, in which each playeri ∈ I = {1, 2, ,n }is characterized
by a set of strategiesS i ⊂ R ni, and by an objective functionv i:n
j =1S j → R , a Nash equi-librium is a vector s =(s1,s2, ,s n)∈Πn
i =1S i, such thatv i(s) ≥ v i(s i,s − i), for alls i ∈ S i, for alli ∈ I, where s − i =(s1, ,s i −1,s i+1, ,s n)
Finally, letᏴ= { S : ∃ n ∈ N,S ⊂ R nis nonempty, convex, and compact}
3 Main Results
Consider the following statements
[Nash-1] GivenᏳ= { I,S i,v i }, suppose that each setS i ∈Ᏼ and that objective functions
are continuous in its domains and strictly quasiconcave in its own strategy Then there is
a Nash equilibrium forᏳ
[Nash-2] GivenᏳ= { I,S i,v i }, suppose that each setS i ∈Ᏼ and that objective functions
are continuous in its domains and quasiconcave in its own strategy Then there is a Nash
equilibrium forᏳ
[Brouwer] Given X ∈ Ᏼ, every continuous function f : X → X has a fixed point [Kakutani ∗ ] Given X ∈ Ᏼ, every closed-graph correspondence Φ : XX, with Φ(x) ∈
Ᏼ for all x ∈ X, has a fixed point, provided that Φ(x) =m
j =1π m
j(Φ(x)) for each x∈ X ⊂
Rm (For each j ∈ {1, ,m }, the projectionsπ m j :Rm → R are defined byπ m j (x) = x j, wherex =(x1, ,x m)∈ R m.) (The last property,Φ(x) =m
j =1π m j (Φ(x)), is not necessary
to assure the existence of a fixed point, provided that the other assumptions hold How-ever, when objective functions are quasiconcave, [Kakutani∗] is sufficient to assure the existence of a Nash equilibrium.)
Our results are [Nash-1]→[Brouwer]
Proof Given a nonempty, convex, and compact set X ⊂ R m and a continuous function
f : X → X, consider a game Ᏻ with two players I = { A,B }, which are characterized by the sets of strategiesS A = S B = X and by the objective functions: v A(x A,x B)= − x A − x B 2
andv B(x A,x B)= − f (x A)− x B 2, where · denotes the Euclidean norm inRm
As objective functions are continuous inS A × S B and strictly quasiconcave in own strategy, there exists a Nash equilibrium (x A,x B) Moreover, x A = x B andx B = f (x A), that assure the existence of a fixed point of f : X → X.
In fact, ifx A = x B, thenv A(x A,x B)< 0 Thus, player A can improve his gains choosing
a responsex A = x B ∈ X, as v A(x B,x B)=0, a contradiction Analogous arguments prove thatx B = f (x A) becausef (x A)∈ X.
Proof Fix a set X ⊂ Ᏼ and a correspondence Φ : XX that satisfies the assumptions
of [Kakutani∗] Define, for each 1≤ i ≤ m, the functions κ m
i :Rm × R m → R m × Rby
κ i m(x, y) =(x, y i), wherey =(y1, , y m)∈ R m
Trang 3Juan Pablo Torres-Mart´ınez 3 Consider a gameᏳ with m + 1 players, {0, 1, ,m }, characterized by the sets of strate-gies (S0, (S i; i > 0)) : =(X,(π i m(X); i > 0)) and by the objective functions: v0(x0,x −0)=
− x0− x −0 2,v i(x0,x1, ,x m)= −min(r,si)∈ κ m
i(Gr[Φ]) (x0,x i)−(r,s i) max, whereGr[Φ]
denotes the graph ofΦ, · maxis the max-norm, andx −0:=(x1, ,x m)
As hypotheses of [Nash-2] hold (see the appendix), there is an equilibrium (x0; (x1, ,x m)) for Ᏻ It follows that xi ∈ π m
i (Φ(x0))(If x i ∈ / π m
i (Φ(x0)) then, by defini-tion, (x0,x i)∈ / κ m i (Gr[Φ]) Thus, player i’s utility, v i(x0,x1, ,x m)< 0 However,
choos-ing anyx i ∈ π i m(Φ(x0))= ∅, the playeri can improve his gains, as (x0,x i)∈ κ m i (Gr[Φ])
and, therefore, his utility will be equal to zero, a contradiction.) Therefore, by Assump-tion [Kakutani∗], (x1, ,x m)∈ Φ(x0) Finally,x0=(x1, ,x m)∈ Φ(x0)(Ifx0= x −0:=
(x1, ,x n), we have thatv0(x0,x −0) := − x0− x −0 2< 0 Thus, player 0 can improve his
position choosingx0= x −0∈ X.) That concludes the proof.
Appendix
It follows from definitions above that the sets of strategies satisfy the assumptions of [Nash-2], objective functions are continuous andv0is quasiconcave in it own strategy Thus, rest to assure that functionsv i, with i ≥1, are quasiconcave in its own strategy This will be a direct consequence of the following lemma, takingZ = κ m i (Gr[Φ]).
Givenx ∈ R m+1 and a nonempty setZ ⊂ R m+1, the distance fromx to Z is d(x,Z) =
infz ∈ Z x − z max Note that the functionx → d(x,Z) is continuous, because | d(x1,Z) − d(x2,Z) | ≤ x1− x2 max For convenience of notations, letπ :Rm × R → R mbe the pro-jection defined byπ(x, y) = x.
Lemma A.1 Suppose that Z ⊂ R m+1 is a nonempty and compact set such that, for each
x ∈ R m , both π(Z) and Z ∩ π −1(x) are convex sets Then the function f :Rm × R → R defined by f (x,t) = d((x,t),Z) is quasiconvex in t.
Proof Fix a vector x0∈ R m We have to show thatL c = { t ∈ R; d((x0,t),Z) ≤ c }is a convex set for everyc ≥0 Assume by contradiction that, givenc ≥0, there are scalars
t1< t ∗ < t2 such thatt1,t2∈ L c andt ∗ ∈ L c LetA : = { x ∈ π(Z); x − x0 max≤ c }and consider the following sets:
A1= { x ∈ A; ∃ t ∈ Rs.t (x,t) ∈ Z, t ≤ t ∗ − c };
A2= { x ∈ A; ∃ t ∈ Rs.t (x,t) ∈ Z, t ≥ t ∗+c } (A.1)
Sinced((x0,t ∗),Z) > c, we have A = A1∪ A2 Moreover,A1∩ A2= ∅ (If there exists a vectorx ∈ A1∩ A2, then x − x0 max≤ c and, by the convexity of Z ∩ π −1(x), (x,t ∗)∈ Z,
contradictingd((x0,t ∗),Z) > c.) Since Z is compact, A1andA2are compact as well And sinceπ(Z) is convex, so is A In particular, A is connected Therefore, A1= ∅orA2= ∅
On the other hand, d((x0,t1),Z) ≤ c, so there exists a point (x ,t )∈ Z c-close to
(x0,t1) Then x − x0 max≤ c and t ≤ t1+c < t ∗+c, therefore x ∈ A \ A2, proving that
A1= ∅ Analogously, it follows fromd((x0,t2),Z) ≤ c that A2= ∅ We have obtained a
Trang 44 Fixed points as Nash equilibria
Acknowledgments
I am indebted with Jairo Bochi for useful suggestions and comments I also would like to thank the suggestions of Carlos Herv´es-Beloso, Alexandre Belloni, and Eduardo Loyo
References
[1] L E J Brouwer, ¨ Uber Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71 (1912),
no 4, 598.
[2] S Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8
(1941), no 3, 457–459.
[3] J F Nash, Equilibrium points in n-person games, Proceedings of the National Academy of
Sci-ences of the United States of America 36 (1950), no 1, 48–49.
[4] , Non-cooperative games, Annals of Mathematics Second Series 54 (1951), 286–295.
[5] J Zhao, The equivalence between four economic theorems and Brouwer’s fixed point theorem,
Working Paper, Departament of Economics, Iowa State University, Iowa, 2002.
Juan Pablo Torres-Mart´ınez: Department of Economics, Pontif´ıcia Universidade Cat ´olica do Rio de Janeiro (PUC-Rio), Marquˆes de S˜ao Vicente 225, Rio de Janeiro 22453-900, Brazil
E-mail address:jptorres martinez@econ.puc-rio.br
... ∅ We have obtained a Trang 44 Fixed points as Nash equilibria
Acknowledgments
I... generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8
(1941), no 3, 457–459.
[3] J F Nash, Equilibrium points in n-person... ∗),Z) > c.) Since Z is compact, A1andA2are compact as well And sinceπ(Z) is convex, so is A In particular, A is connected Therefore, A1=