GAMES AND ECONOMIC BEHAVIOR
ARTICLE NO.
18, 277]285 Ž1997.
GA970523
An Axiomatization of Nash Equilibria in
Economic SituationsU
Bezalel Peleg †
Center for Rationality and Interacti¨ e Decision Theory,
The Hebrew Uni¨ ersity of Jerusalem, Feldman Building, Gi¨ at-Ram, 91904
Jerusalem, Israel
and
Peter Sudholter
¨ ‡
Institute of Mathematical Economics, Uni¨ ersity of Bielefeld, Postfach 100131,
D-33501 Bielefeld, Germany
Received November 15, 1995
We consider the class of all abstract economies with convex and compact
strategy sets, continuous and quasi-concave payoff functions, and continuous and
convex-valued feasibility correspondences. We prove that the Nash correspondence
is the unique solution on the foregoing class of abstract economies that satisfies
nonemptiness, rationality in one-person economies, and consistency. Journal of
Economic Literature Classification Numbers: C72, D50. Q 1997 Academic Press
1. INTRODUCTION
Abstract economies were introduced by Debreu Ž1952. in order to help
in the proof of existence of Walras equilibrium for competitive economies
ŽArrow and Debreu Ž1954... They have since been used extensively in
general equilibrium theory Žsee, e.g., Shafer and Sonnenschein Ž1975.,
Ichiishi Ž1983., and Border Ž1985... Abstract economies are essentially
strategic games in which the feasible set of strategies of a player may
depend on the strategies chosen by the other players. Thus, an abstract
* We are indebted to M. Majumdar, D. Pallaschke, and J. Rosenmuller
for several helpful
¨
conversations. We are also very grateful to two anonymous referees for insightful remarks.
†
The work of the first author was supported by the WARSHOW endowment at Cornell
University. Fax: Ž972.-2-6513681.
‡
Fax: qq49-521-106-2997. E-mail: [email protected].
277
0899-8256r97 $25.00
Copyright Q 1997 by Academic Press
All rights of reproduction in any form reserved.
278
PELEG AND SUDHOLTER
¨
economy is a strategic game combined with a finite sequence of feasibility
correspondences, one for each player. The main solution concept for
abstract economies is the social equilibrium Žsee Debreu Ž1952.., which is
a generalization of the Nash equilibrium. We provide an axiomatization of
the Nash correspondence of abstract economies.
We now briefly describe our result. Consider the class of all abstract
economies with the following properties: Ži. The strategy sets are convex
and compact. Žii. The payoff functions are continuous and quasi-concave.
Žiii. The feasibility correspondences are continuous and have nonempty
convex values. Then, the Nash correspondence is the unique solution on
the foregoing class of abstract economies that satisfies nonemptiness,
rationality in one-person games, and consistency.
The Nash correspondence has been axiomatized recently for various
classes of games Žsee Peleg and Tijs Ž1996., Peleg et al. Ž1996., and Norde
et al. Ž1996... In particular, Norde et al. Ž1996. contains a complete
characterization for the class of mixed extensions of finite games and for
the class of games with continuous concave payoff functions. This paper is
devoted to the characterization of the Nash correspondence of generalized
games with quasi-concave payoff functions. Although we use some results
of Peleg et al. Ž1996. and Norde et al. Ž1996., our result does not directly
follow from the foregoing papers. The main difficulty is that the class of
quasi-concave functions, unlike the class of concave functions, is not closed
under addition Ži.e., the sum of two quasi-concave functions may not be
quasi-concave..
2. THE MODEL
Let V be an infinite set Žthe set of ‘‘potential players’’.. An abstract
economy is a list
E s ² N, Ž A i . igN , Ž u i . igN , Ž Fi . igN : ,
where N s N Ž E . ; V is a finite nonempty set Žthe set of players.; A i is
the Žnonempty. set of strategies of player i g N; u i : AŽ E . ª R is the
payoff function for i g N Žhere AŽ E . s = i g NŽ E . A i and R is the real line.;
and Fi , i g N, is a correspondence from Ayi s = j/ i A j to A i . Fi is the
feasibility correspondence of i g N. Thus, if x g AŽ E . then Fi Ž xyi . ; A i is
the set of feasible strategies of i Žwhen the rest of the players choose
xyi s Ž x j . j g N _i4 .. ˆ
x g AŽ E . is a Nash equilibrium ŽNE., or a social
equilibrium ŽSE., if Ži. ˆ
x i g Fi Ž ˆ
xyi . for all i g N and Žii. u i Ž ˆ
x . G uiŽ x i, ˆ
xyi .
for all x i g Fi Ž ˆ
xyi . and i g N.
AN AXIOMATIZATION OF NASH EQUILIBRIA
279
Abstract economies were first studied in Debreu Ž1952. in connection
with the Arrow]Debreu existence theorem of Walras equilibrium. They
also later continued to play an important role in general equilibrium
theory Žsee, e.g., Shafer and Sonnenschein Ž1975. and Border Ž1985,
Chapters 19 and 20...
We shall consider the class j q Žthe subscript q indicates that we deal
with quasi-concave utility functions. of abstract economies that have the
following properties:
E s ² N, Ž A i . igN , Ž u i . igN , Ž Fi . igN : g j q
if
A i is a nonempty, compact, and convex subset of some
Ž finite-dimensional . Euclidean space Ei for every i g N.
Ž 2.1.
u i is continuous on the graph of Fi , gr Fi , for every i g N.
Ž We recall that gr Fi s x g AŽ E . ¬ x i g Fi Ž xyi . 4 ..
Ž 2.2.
For every i g N and xyi g Ayi , u i Ž ?, xyi . is a quasi-concave
function on Fi Ž xyi . .
Ž 2.3.
For every i g N, Fi is continuous Ž i.e., both upper
hemi-continuous and lower hemi-continuous . on Ayi .
Ž 2.4.
For every i g N and xyi g Ayi , Fi Ž xyi . is nonempty,
closed, and convex.
Ž 2.5.
Without loss of generality we may assume
u i Ž x . - u i Ž y . if x g A Ž E . _gr Fi and y g gr Fi for all i g N. Ž 2.6.
Indeed, the values of u i on AŽ E ._gr Fi are completely arbitrary.
A solution on j q is a function w that assigns to each E g j q a subset
w Ž E . of AŽ E .. For example, the function SE, which assigns to each
E g j q its set of social equilibria SEŽ E ., is a solution. A solution w on j q
satisfies nonemptiness ŽNEM. if w Ž E . / B for every E g j q . ŽWe remark
that, by Theorem 4.3.1 in Ichiishi Ž1983., SEŽ?. satisfies NEM.. w satisfies
one-person rationality ŽOPR. if for every one-person abstract economy
E s ² i4 , A i , u i , Fi : in j q ,
w Ž E . ; x i g Fi ¬ u i Ž x i . G u i Ž yi . for all yi g Fi 4 .
Ž Fi is constant for one-person abstract economies.. Obviously SEŽ?. satisfies OPR.
PELEG AND SUDHOLTER
¨
280
Let E g j q , x g AŽ E ., and S ; N Ž E ., S / B. The reduced abstract
economy of E with respect to S and x, E S, x is given by
E S , x s² S, Ž A i . igS , Ž u Si , x . igS , Ž FiS , x . igS: ,
xŽ
where u S,
aS . s u i Ž aS , x N _ S . and FiS, x Ž aS _i4 . s Fi Ž aS _i4 , x N _ S . for evi
ery aS g A S s = j g S A j and i g S. ŽHere the notation x T s Ž x j . j g T is
used whenever x g AŽ E . and T ; N.. Under the foregoing assumptions
E S, x g j q , that is, j q is closed in the sense of Peleg and Tijs Ž1996.. A
solution w on j q is consistent ŽCONS. if for every E g j q , S ; N Ž E .,
S / B and x g w Ž E ., x S g w Ž E S, x .. The interpretation of the consistency
property for abstract economies is actually the same as the interpretation
for strategic games Žsee Peleg and Tijs Ž1996... Indeed, w is consistent if
for every E g j q the following condition is satisfied: If x is an ‘‘equilibrium,’’ that is, x g w Ž E ., S ; N Ž E ., S / B, and all the members of
N _ S announce their strategies x i , i g N _ S, and leave the economy E,
then the members of S do not have to revise their strategies x j , j g S. As
the reader may easily verify, SEŽ?. satisfies CONS on j q .
3. A CHARACTERIZATION OF THE NASH
CORRESPONDENCE ON j q
Our main result is the following theorem.
THEOREM 3.1. If a solution w on j q satisfies NEM, OPR, and CONS,
then w Ž E . s SEŽ E . for e¨ ery E g j q .
The following simple result is needed for the proof of Theorem 3.1.
LEMMA 3.2. Let E g j q and x g AŽ E .. If x i g SEŽ Ei4, x . for e¨ ery
i g N Ž E ., then x g SEŽ E ..
Proof. Let i g N Ž E .. Because x i g SEŽ Ei4, x . it follows that
x i g Fii4, x
and
x
x
ui4,
Ž x i . G ui4,
Ž yi .
i
i
for all yi g Fii4, x . Ž 3.1.
By the definition of reduced games, Ž3.1. is equivalent to
x i g Fi Ž xyi .
and
u i Ž x . G u i Ž yi , xyi .
for every yi g Fi Ž xyi .. Thus, x g SEŽ E ..
Q.E.D.
Now we shall prove that SEŽ?. is the maximum solution on j q which
satisfies OPR and CONS.
LEMMA 3.3. If a solution w on j q satisfies OPR and CONS, then
w Ž E . ; SEŽ E . for e¨ ery E g j q .
AN AXIOMATIZATION OF NASH EQUILIBRIA
281
Proof. Let E g j q and x g w Ž E .. If < N Ž E .< s 1 then w Ž E . ; SEŽ E .
by OPR. ŽIf S is a finite set, then < S < is the cardinality of S.. Assume now
< N Ž E .< G 2. By CONS, x i g w Ž Ei4, x . for all i g N Ž E .. Hence, by the first
part of the proof, x i g SEŽ Ei4, x . for all i g N Ž E .. Therefore, by Lemma
3.2, x g SEŽ E ..
Q.E.D
In order to prove the converse inclusion we shall show that SEŽ?. has the
ancestors property ŽAP. on j q Žsee Peleg et al. Ž1996. and Proposition 3 of
Norde et al. Ž1996...
x g SEŽ E ., then there exists H g j q
LEMMA 3.4 ŽAP.. If E g j q and ˆ
such that the following conditions hold:
NŽ H . > NŽ E. ;
Ž 3.2.
H has exactly one social equilibrium y ;
Ž 3.3.
yNŽ E . s ˆ
x;
Ž 3.4.
H NŽ E ., y s E.
Ž 3.5.
We postpone the proof of Lemma 3.4 and shall now prove Theorem 3.1.
Proof of Theorem 3.1. Let E g j q and ˆ
x g SEŽ E .. By Lemma 3.4
there exists H g j q such that Ž3.2. ] Ž3.5. are satisfied. By NEM of w and
Lemma 3.3, w Ž H . s y4 . By CONS of w , Ž3.4., and Ž3.5. we obtain
ˆx s yNŽ E . g w Ž H NŽ E ., y . s w Ž E . .
Thus, w Ž E . > SEŽ E .. Because of Lemma 3.3, w Ž E . s SEŽ E ..
Q.E.D.
Proof of Lemma 3.4. Let
E s ² N Ž E . , Ž A i . igN Ž E . , Ž u i . igN Ž E . , Ž Fi . igN Ž E . :
be a member of j q and ˆ
x g SEŽ E .. We define
H s² N Ž H . , Ž AUi . igN Ž H . , Ž uUi . igN Ž H . , Ž FiU . igN Ž H .:
in the following way:
N Ž H . s N j Nq
where N l Nqs B, < N < s < Nq< , and N s N Ž E . .
Ž 3.6.
The existence of N Ž H . ; V is guaranteed by the fact that V is infinite.
We denote by t a bijection of N onto Nq and we choose
AUi s AUtŽ i. s A i
for all i g N.
Ž 3.7.
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PELEG AND SUDHOLTER
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In order to define uUi and FiU , i g N Ž H ., we need the following concepts
and notations. A continuous function p i : Ei ª A i , i g N Ž E . Žsee Ž2.1.., is
a retraction if p i Ž x i . s x i for all x i g A i . Because A i is a convex and
compact subset of Ei there exits a retraction p i of Ei on A i . Thus we may
define
FiU Ž x N _i4 , x N q . s Fi Ž p j Ž x j q ˆ
x j y x tŽ j. . . jgN _ i4
ž
/
Ž 3.8.
for every i g N and x g AŽ H . and
FtŽUi. Ž xytŽ i. . s A i
Ž 3.9.
for every i g N and every xyt Ž i. g AUytŽ i. , where
AUyt Ž i. s
= A j = = AUtŽ j. .
jgN
j/i
As the reader may verify, FiU satisfies Ž2.4. and Ž2.5. for all i g N Ž H ..
In order to define uUi , i g N Ž E ., we first introduce auxiliary functions
¨U
x j y x tŽ j. .
i Ž x . s ui xi , p j Ž x j q ˆ
ž Ž
. jgN _ i4 /
Ž 3.10.
for all x g AŽ H . and i g N. ¨ Ui Ž?, xyi . is quasi-concave on
FiU Ž xyi . s Fi Ž p j Ž x j q ˆ
x j y x tŽ j. . . jgN _ i4
ž
/
for every xyi g AUyi , and ¨ Ui continuous on gr FiU . Using Ž3.10. we further
define
bi :
= Aj = =
j/i
hgN q
AUh ª R
by
bi Ž x N _i4 , x N q . s bi Ž xyi . s max ¨ iU Ž x i , xyi . ¬ x i g FiU Ž xyi . 4
for i g N. By the Maximum Theorem Žsee Section 2.3 of Ichiishi Ž1983..,
bi Ž?. is continuous. Furthermore
bi Ž Ž x j . jgN _ i4 , Ž ˆ
x h . hgN q . s max u i Ž x i , x N _i4 . ¬ x i g Fi Ž x N _i4 . 4 . Ž 3.11.
ŽHere ˆ
x g AŽ H . is defined by ˆ
x tŽ i. s ˆ
x i for i g N.. Now we introduce our
second family of auxiliary functions. For every i g N let wUi : AŽ H . ª R
be given by
wiU Ž x . s wiU Ž x i , xyi . s bi Ž xyi . y 5 x tŽ i. y ˆ
xi 5 5 xi y ˆ
xi 5.
ŽHere 5 ? 5 denotes the Euclidean norm on Ei ..
Ž 3.12.
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AN AXIOMATIZATION OF NASH EQUILIBRIA
It is obvious that wUi satisfies Ž2.2. and Ž2.3.. Hence
uUi Ž x . s min ¨ iU Ž x . , wiU Ž x . 4
is an admissible payoff function for i g N Ž E .. Finally, we define
uUtŽ i. Ž x . s y5 x i y x tŽ i. 5
for all i g N Ž E . .
Ž 3.13.
This completes the definition of H. Clearly, H g j q . Now we claim
If y g SE Ž H .
then yi s ytŽ i. s ˆ
xi ,
for all i g N Ž E . . Ž 3.14.
Indeed, if y g SEŽ H . then yi s ytŽ i. for all i g N Ž E . by Ž3.13.. Hence
¨ iU Ž y . s u i Ž yi , ˆ
x N _i4 . and wUi Ž y . s u i Ž ˆ
x N . y 5 yi y ˆ
x i 5 . Thus, for i g
2
N Ž E ., yi s ˆ
x i , because yi is a best response to yyi . Therefore, our claim is
proved. So far we have proved Ž3.2. ] Ž3.4.. In order to prove Ž3.5. we
observe that for all i g N Ž E .:
x h . hgN q . , wiU Ž Ž x j . jgN , Ž ˆ
x h . hgN q .
min ¨ iU Ž Ž x j . jgN , Ž ˆ
½
s min u i Ž x N . , bi Ž Ž x j . jgN _ i4 , Ž ˆ
x h . hgN q .
½
/5
5
s min u i Ž x N . , max u i Ž z i , x N _i4 . ¬ z i g Fi Ž x N _i4 . 4 4
s u i Ž x N . Ž see Ž 3.11. and Ž 2.6. . .
Also,
FiU Ž Ž x j . jgN _ i4 , Ž Ž ˆ
x h . hgN q . s Fi Ž Ž x j . jgN _ i4 .
for each i g N Ž E ..
Q.E.D.
4. CONCLUDING REMARKS
We considered the set of abstract economies with convex and compact
strategy sets, continuous and quasi-concave payoff functions, and continuous and convex-valued feasibility correspondences. The Nash correspondence is completely characterized on the foregoing class of economies by
the following three axioms: nonemptiness, rationality for one-person games,
and consistency.
Let E s ² N, Ž A i . i g N , Ž u i . i g N , Ž Fi . i g N : be an abstract economy. E is a
game if for each i g N and xyi g Ayi , Fi Ž xyi . s A i . Thus, E is a game if
there are no feasibility constraints. If E is a game, then we shall also write
PELEG AND SUDHOLTER
¨
284
E s ² N, Ž A i . i g N , Ž u i . i g N :. Denote by Gq the set of all games in j q . By
modifying the proof of Theorem 3.1 we can show the following result.
THEOREM 4.1. If a solution w on Gq satisfies NEM, OPR, and CONS,
then w Ž G . s NEŽ G . for e¨ ery game G g Gq Ž for G g Gq we denote NEŽ G .
s SEŽ G ...
The axiomatization of the Nash correspondence on Gq is not covered by
the results of Norde et al. Ž1996.. Also, our results may be generalized to
infinite-dimensional strategy spaces. Indeed, one may replace Ž2.1. by a
weaker assumption. First, we recall that a normed linear space X Žwith
norm 5 ? 5., is strictly con¨ ex if for every pair of linearly independent
vectors x, y g X it holds that 5 x q y 5 - 5 x 5 q 5 y 5. The following result is
useful.
LEMMA 4.2. If A is a nonempty, con¨ ex, and compact subset of a con¨ ex
compact subset B of a normed space X, then there is a retraction of B on A.
Proof. It is well known that a compact set is separable, and hence the
affine hull of B Žthe set
aff B s
½
r
r
Ý g i bi ¬ r g N, Ý g i s 1, g i g R, bi g B
is1
is1
for all i with 1 F i F r
5/
is separable. Indeed, if bi ¬ i g N4 is a countable dense subset of B, then
½
r
r
Ý g i bi ¬ r g N,
Ý g i s 1, g i g Q
is1
is1
5
is a countable dense subset of aff B.
If there is a retraction of B y b 0 4 on A y b 0 4 for some b 0 g B, then
there is a retraction of B on A, because translations are continuous.
Therefore we assume w.l.o.g. that 0 is a member of B, hence aff B is a
linear subspace of X. Moreover, it can be assumed that aff B is a Banach
space. Otherwise take a ‘‘smallest’’ completion of aff B. This new space
inherits separability and convex or compact subsets of the original space
Ž1966... Thus aff B has a
retain these properties Žsee p. 130 of Kothe
¨
strictly convex isomorphic norm Žsee p. 160 of Day Ž1973... The proof of
existence of a retraction of a strictly convex normed space on a nonvoid
convex compact subset is the same as the proof when the normed space is
Euclidean.
Q.E.D.
AN AXIOMATIZATION OF NASH EQUILIBRIA
285
Now, using the generalizations of existence of social equilibria and the
Maximum Theorem for infinite-dimensional spaces Žsee Section 2.3 and
Theorem 4.7.2 of Ichiishi Ž1983.., we obtain
COROLLARY 4.3. Theorem 3.1 remains true if Ž2.1. is replaced by:
For every i g N, A i is a nonempty, compact, and
convex subset of a normed linear space.
Ž 4.1.
Indeed, in view of Ž3.10. it is sufficient to verify the existence of a
retraction of A i q A i y A i to A i for i g N Ž E .. This existence is guaranteed by Lemma 4.2 applied to B s A i q A i y A i and A s A i .
Finally, it should be remarked that obvious examples show the logical
independence of NEM, OPR, and CONS in the results ŽTheorems 3.1 and
4.1 and Corollary 4.3..
REFERENCES
Arrow, K. J., and Debreu, G. Ž1954.. ‘‘Existence of an Equilibrium for a Competitive
Economy,’’ Econometrica 22, 265]290.
Border, K. C. Ž1985.. Fixed Point Theorems with Applications to Economics and Game Theory.
Cambridge, UK: Cambridge Univ. Press.
Day, M. M. Ž1973.. Normed Linear Spaces. Berlin: Springer-Verlag.
Debreu, G. Ž1952.. ‘‘A Social Equilibrium Existence Theorem,’’ Proc. Nat. Acad. Sci. 38,
886]893.
Ichiishi, T. Ž1983.. Game Theory for Economic Analysis, New York: Academic Press.
Kothe,
G. Ž1966.. Topologische lineare Raume.
Berlin: Springer-Verlag.
¨
¨
Norde, H., Potters, J., Reijnierse, H., and Vermeulen, D. Ž1996.. ‘‘Equilibrium Selection and
Consistency,’’ Games Econ. Beha¨ . 12, 219]225.
Peleg, B., Potters, J., and Tijs, S. Ž1996.. ‘‘Minimality of Consistent Solutions for Strategic
Games, in Particular for Potential Games,’’ Econ. Theory 7, 81]93.
Peleg, B., and Tijs, S. Ž1996.. ‘‘The Consistency Principle for Games in Strategic Form,’’ Int.
J. Game Theory 25, 13]34.
Shafer, W. J., and Sonnenschein, H. Ž1975.. ‘‘Equilibrium in Abstract Economies without
Ordered Preferences,’’ J. Math. Econ. 2, 345]348.