The introduction of an additional player to serve as coordinator in an N-person abstract economy leads in a natural way to an N+1-person noncooperative game. Sufficient conditions on the abstract economy are considered which lead to the existence of equilibrium in the resulting game and hence for the abstract economy

Border, Kim C.

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125
NONCOOPERATIVE GAMES, ABSTRACT ECONOMIES AND
WALRASI.AN EQUILIBRIA
Kim C. Border
"""c;1\lUTf Of: ''"" 1'1; �'to � 
� � � � 
� 0 
� Q 
� :ii 
�� � 
'Pt,.� .... � " ..L� SHALL tAP,." 
SOCIAL SCIENCE WORKING PAPER 396 
July 1981
ABSTRACT 
The introduction of an additional player to serve as 
coordinator in an N-person abstract economy leads in a natural way to 
an N+l-person noncooperative game. Sufficient conditions on the 
abstract economy are considered which lead to the existence of 
equilibrium in the resulting game and hence for the abstract economy. 
NONCOOPER.ATIVE GAMES, ABSTRACT ECONOMIES AND WALRASIAN EQUILIBRIA 
Kim C. Border 
O. Introduction. 
Dcbreu [1952] introduced the concept of an abstract economy as 
a generalization of a noncooperative game (Nash [1951]). In a 
noncooperative game each player has a set of strategies available to 
him regardless of what strategies the other players select. The 
choice of strategies by all of the players determine an outcome and 
the players are all assumed to have preferences over outcomes. These 
preferences over outcomes then generate preferences over strategy 
vec�ors. An equilibrium is a strategy vector such that no player can 
change his strategy so as to yield a more preferred outcome. In an 
abstract economy the set of strategies available to a player may be a 
proper subset of strategies which depends on the choices of the other 
players. An equilibrium for an abstract economy is a strategy vector 
which is feasible for all players given everyone's choices and no 
player can alter his strategy so as to effect a more preferred 
outcome. Axrow and Debreu [1954] used Debreu's result on the 
existence of equilibrium for an abstract economy to find sufficient 
conditions for the existence of Walrasian equilibrium for a market 
economy. The reduction of a market economy to an abstract economy is 
accomplished by introducing an "auctioneer" or "market participant", a 
new player whose strategy set is the set of prices. The auctioneer's 
choice of price restricts each consumer' s choice of consumption 
through the budget correspondence. Prof its of producers also depend 
on the auctioneer's choice of price, but their set of available 
productions does not depend on the price. 
2 
The preferences of consumers were assumed to be representable 
by real valued utility functions by Axrow and Debreu. Sonnenschein 
[1971] showed that transitivity of preferences could be dispensed with 
for proving the existence of Walrasian equilibrium. Mas-Colell [1974] 
was further able to dispense with completeness of preferences. Gale 
and Mas-Colell [1975] present a proof of existence of equilibrium for 
noncooperative games without complete or transitive preferences and 
use it to establish existence of Walrasian equilibrium. Shafer and 
Sonnenschein [1975] present a proof of existence of equilibrium for 
abstract economies without complete or transitive preferences and 
Shafer [1976] uses this to prove the existence of Walrasian 
equilibrium under fairly general conditions. 
The theorem of Gale and Mas-Colell on noncooperative games 
follows from Shafer and Sonnenschein' s theorem on abstract economies, 
but not conversely. This suggests that the Shafer-Sonnenschein 
theorem is stronger than needed to prove the existence of Walrasian 
equilibrium. By imposing additional hypotheses on the feasible 
strategy correspondences, while somewhat weakening assumptions on 
preferences, we can convert an N-person abstract economy to an N+l­
player noncooperative game. These extra assumptions were essentially 
introduced by Borglin and Keiding [1976], and are automatically 
satisfied by Walrasian budget correspondences. This is why Gale and 
Mas-Colell were able to prove the existence of Walrasian equilibrium 
with their theorem. Borglin and Keiding reduced an abstract economy 
to a 1-person game. Our technique of expanding the player set has a 
neat interpretation in terms of stationary points of a tatonnement 
procedure. 
1. Definitions. 
An N-person noncooperative game is a tuple G N (Xi, Pi) i=l' 
where for each i, 
P. 
1 
x -7-7 x. 1 
N 
is a possibly empty-valued correspondence, 
3 
where X = lT X.. An equilibrium for G is a strategy vector x e X such 
j=l J 
that for each i 
Pi(;) " or x. eP.(;). 1 1 
Here Xi is the set of strategies of player i and Pi(x) is interpreted 
in one of two ways. The set Pi (x) may be the set of strategies which 
player i strictly prefers to xi given x1
, • • •  ,xi-l'xi+l•••••xN. Then 
Pi(x) = rJ means that there are no preferred strategies to xi given the 
others' choices. Under this interpretation it never happens that 
x
i£ Pi(x). The second way we may interpret Pi(x) is as the set of 
"best replies" to x1, ••• ,xi-l'xi+l•···•xw Then Pi (x) is never empty 
and xi E Pi(x) is the appropriate equilibrium notion. Our definition 
of equilibrium allows for Pi to have one interpretation for some 
players and the other interpretation for the rest. 
An N-person abstract economy is a tuple E = (X.,S.,P.)� 1, 1 1 1 1= 
where for each i, 
and 
P. 
1 
s .  
1 
N 
lT x
J
. -7-7 X. is a possibly empty-valued correspondence 
j=l 
1 
lTx. -7-7 X. is a nonempty-valued correspondence. 
j=l J 
1 
j#i 
N 
Put X lT X.. We will write S.
j=l J 1 
x -7-7 xi. with the understanding 
that Si(x) does not depend on xi
. 
and 
An equilibrium for E is a vector x e X such that for each i, 
x. £ s .«;> 1 1 
P
i(x
) n S
i(x
) ". 
Again X
i is player i' s strategy set and Si(x) is the set of 
feasible strategies for i given the choices x1, • • •  ,xi-l'xi+l•·
···�· 
4 
The set Pi(x) is the set of strategies preferred to xi given everyone 
else' s choice. The equilibrium condition is that the vector be 
feasible for everyone and no one has a strategy which is both 
preferred and feasible. 
5 
2. Theorems. 
The following theorem is a slight variation of a theorem of 
Gale and Mas-Colell [1975] . Denote by co Pi, the correspondence whose 
value at x is the convex hull of Pi(x). 
Jheorem 1. Let G (Xi,Pi)i=l be a noncooperative game satisfying for 
each i, 
( i) Xi is a nonempty compact convex subset of a finite dimensional 
euclidean space. 
(ii) Either 
(a) co Pi has an open graph in X x Xi' and for each x e X, 
xi I co Pi(x). 
or 
(b ) Pi is a continuous singleton-valued correspondence, 
i.e., a continuous function, 
Then G has an equilibrium. 
Proof, The proof is virtually identical to that of Gale and Mas-
Colell [1975, p.10] and so we only sketch the proof, If Pi satisfies 
(ii.a) let Ui = {x : co Pi(x) F d} and let f i be a continuous 
selection from co Pi lu.· Define yi : X ��Xi via 
1 
yi(x) 
{ {fi(x)} 
X. 
1 
x e ui 
x I u. 
1 
6 
If Pi satisfies (ii.b) set Yi = Pi. 
Then y = TTyi satisfies the hypotheses of Kakutani's theorem and has a 
fixed point x. By construction, if Pi satisfies (ii.a)· then 
co P.(;) = d so P.(;) = d and if P. satisfies (ii.b) then x. e P.(x}. 1 1 1 1 1 
Q,E,D, 
Theorem 2 below is a modification of a theorem of Shafer and 
Sonnenschein [1975], The assumptions on preferences have been 
slightly weakened and those on the feasibility correspondences 
strengthened, but in a way so as to remain useful for proving the 
nonemptiness of Walrasian equilibrium. The proof proceeds by 
converting the abstract economy to an N+l person noncooperative game, 
Theorem 2, Let E (Xi,Si,Pi)�=l be an abstract economy satisfying, 
for each i, 
( i) Xi is a nonempty compact convex subset of a finite dimensional 
Euclidean space, 
(ii) Si has a closed graph in X x Xi, the correspondence 
x �� int Si(x) has an open graph in X x Xi' and for each 
x e X, Si(x) is compact, convex, and has nonempty interior 
(relative to Xi).
(iii) co pi has an open graph in x x  xi. 
(iv) For each x e X, xi J co Pi(x). 
Then E has an equilibrium. 
Remark. Shafer and Sonnenschein assume that Si is lower hemi-
continuous as well as closed, We have strengthened this in (ii.a) to 
assuming int Si is a nonempty-valued correspondence with open graph. 
Note that since Si(x) is compact and convex with nonempty interior 
that Si(x) = cl (int Si(x)), i.e., Si(x) is topologically regular. 
Note that interior here is relative to Xi, not the underlying 
Buel ide an space, 
7 
Shafer and Sonnenschein assume that Pi has an open graph. We 
have weakened this to assuming co Pi has an open graph. Assuming that 
Pi has open sections (see Bergstrom, Parks and Rader [1976]), then 
co Pi will have open graph. ( In this case co Pi will be convex-valued 
with open sections. See Bergstrom, Parks and Rader [1976] and Shafer 
[1974],) The assumption of open sections for Pi is strictly weaker 
than the assumption of open graph as long as Pi is not convex-valued. 
Proof of Theorem 2. We define an N+l person game as follows. 
N N 
Put Zo = ITx . •  and for i=l, • • •  ,N put z. = xi. Set z i=l 1 1 
Tfz .. A 
i=O 1 
typical element of Z will be denoted (x,y), where x e z0 and 
N 
y e ITz . •  Define preference correspondences µi i=l 1 
follows. 
z -7-7 zi as 
8 
Define µ0 by
µo(x, y) = {y}, 
and for i = l, • • •  ,N set 
µi(x,y) 
J int S.(x) l co Pi ;y) n int Si(x) 
if yi J Si(x) 
if Yi e Si(x)' 
Note that µ0 is continuous and singleton-valued and that for i=l, • •  , , N
the correspondence µ. i s  convex-valued- and satisfies y .  t µ1.(x,y), 1 1 
Also for i=l, • • •  , N, the graph of µi is open. To see this for 
i = 1, ... ,N set 
and 
A. 1 {(x,y,zi) 
B. = {(x,y, zi) 1 
C. = {(x,y,zi) 1 
and note that 
zi e int Si(x)}. 
Yi t Si(x)}, 
zi e co Pi(y)}. 
Gr µi (Ai n Bi) u (Ai n Ci). 
The set Ai is open because int Si has open graph and Ci is open by 
hypothesis (iii), The set B. is also open, for if y. t S.(x) then 1 1 1 
there is a �losed neighborhood F of yi such that Si(x) c F
e and upper 
hemicontinuity of Si then gives the desired result. 
Thus the hypotheses of Theorem 1 are satisfied and so there 
exists c�.y) 8 z such that 
(a) x e µ0(x,y) 
and for i=l , ••• ,N
(b) µi (;,y) d.
Now (a) implies x = y and since Si(x) is never empty (b) becomes
co P.(;} n int S.(;) 1 1 d for i=l, ••• ,N. 
Thus P.(;) n int S.(;) = d, but S.(;) = cl [int S.(;)] and P.(;) is 1 1 1 1 1 
open so Pi(;) n Si(;) = d, i.e.,� is an equilibrium. 
9 
Remark. We can view the N+l game as a formalization of a tatonnement 
process. The N+lst player, call him the coordinator, performs in the 
abstract what a Walrasian auctioneer does in a market economy. The 
coordinator suggests a strategy vector. The players look at the 
choices they have made to see whether they are feasible given the 
coordinator's suggestion. If not, they choose a new strategy which is 
feasible. If the choice is feasible they try to improve upon it. 
Otherwise they don't change their strategy. The coordinator changes 
his suggestion to what the players were doing originally. A 
stationary point of this process is an equilibrium for the abstract 
economy. The question of under what circumstances this process is 
convergent remains to be investigated. 
� We could have defined µ0 in an alternative fashion, namely 
µ0Cx,y) = {z a z0 : lz - yl < Ix - ylJ. Then µ0 would have an open 
graph and be convex valued with x I µ(x, y) so Theorem 1 would still 
apply. 
10 
Remark. In the application of Theorem 2 to the problem of existence 
of Walrasian equilibrium we have the following interpretations. Call 
the auctioneer player 1. Except for the auctioneer, each Xi is a 
subset of the commodity space Rk. Compactness is achieved by suitably 
truncating consumption and production sets. For the auctioneer Xi is 
a subset of price space, also Rk. For producers S. is a constant 
1 
correspondence equal to their truncated convex production set. This 
satisfies (ii) as all interiors are relative to Xi. Likewise for the 
auctioneer s
1 
= x
1
• For consumers, 
S.(p,x2 •• .,xN) = {y. a X. : p • y. � p • 111. + L 9 .. (p • x.)+J. where 1 1 1 1 1 J 1J J 
"'i is his endowment and &ij is his share of firm j. If player j is 
not a firm then 9 .. = O. Under the typical assumption (Shafer [1976J, 1J 
Gale and Mas-Colell [1975], Debreu [1959]) that "'i e int Xi (or 
something like this), then Si will satisfy the hypotheses of Theorem 
2. 
11 
REFERENCES 
Azraw, K. J. and G. Debreu. 1954. "Existence of an Equilibri11111 for a 
Competitive Economy." Econometrica 22:265-290. 
Borglin, A. and H. Keiding. 1976. "Existence of Equilibrium Actions 
and of Equilibrium: A Note on the 'New' Existence 
Theorems." Journal of Mathematical Economics 3:313-316. 
Bergstrom, T. C., R. P. Parks, and T. Rader. 1976. "Preferences 
which have open graphs." Journal of Mathematical Economics 
3(3):265-268. 
Debreu, G. 1952. "A Social Equilibrium Existence Theorem." 
Proceedings of the National Academy of Sciences 38:886-893. 
1959. Theory of value. New York: Wiley. 
Gale, D. and A. Mas-Colell. 1975. "An Equilibri11111 Existence Theorem 
for a General Model without Ordered Preferences." Journal of 
Mathematical Economics 2:9-15. 
Mas-Colell, A. 1974. "An Equilibri11111 Existence Theorem Without 
Complete or Transitive Preferences." Journal of Mathematical 
Economics 1(3):237-246. 
Nash, J. 1951. "Non-cooperative games." Annals of Mathematics 
54(2): 286-295. 
Shafer, W. J. 1974. "The Nontransitive Cons11111er." Econometrica 
42: 913-919. 
Shafer, W. and H. Sonnenschein. 1975. "Equilibrium in Abstract 
Economics without Ordered Preferences." Journal of 
Mathematical Economics 2:345-348. 
Shafer, W. J. 1976. "Equilibri11111 in Economies Without Ordered 
Preferences or Free Disposal." Journal of Mathematical 
Economics 3(2):135-137. 
12 


English

Noncooperative Games, Abstract Economies, and Walrasian Equilibria

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125
NONCOOPERATIVE GAMES, ABSTRACT ECONOMIES AND
WALRASI.AN EQUILIBRIA
Kim C. Border
"""c;1\lUTf Of: 
''"" 1'1; �'to � 
� � 
� � � 0 
� Q 
� :ii 
�� � 
'Pt,.� .... � " ..L� SHALL tAP,." 
SOCIAL SCIENCE WORKING PAPER 396 
July 1981
ABSTRACT 
The introduction of an additional player to serve as 
coordinator in an N-person abstract economy leads in a natural way to 
an N+l-person noncooperative game. Sufficient conditions on the 
abstract economy are considered which lead to the existence of 
equilibrium in the resulting game and hence for the abstract economy. 
NONCOOPER.ATIVE GAMES, ABSTRACT ECONOMIES AND WALRASIAN EQUILIBRIA 
Kim C. Border 
O. Introduction. 
Dcbreu [1952] introduced the concept of an abstract economy as 
a generalization of a noncooperative game (Nash [1951]). In a 
noncooperative game each player has a set of strategies available to 
him regardless of what strategies the other players select. The 
choice of strategies by all of the players determine an outcome and 
the players are all assumed to have preferences over outcomes. These 
preferences over outcomes then generate preferences over strategy 
vec�ors. An equilibrium is a strategy vector such that no player can 
change his strategy so as to yield a more preferred outcome. In an 
abstract economy the set of strategies available to a player may be a 
proper subset of strategies which depends on the choices of the other 
players. An equilibrium for an abstract economy is a strategy vector 
which is feasible for all players given everyone's choices and no 
player can alter his strategy so as to effect a more preferred 
outcome. Axrow and Debreu [1954] used Debreu's result on the 
existence of equilibrium for an abstract economy to find sufficient 
conditions for the existence of Walrasian equilibrium for a market 
economy. The reduction of a market economy to an abstract economy is 
accomplished by introducing an "auctioneer" or "market participant", a 
new player whose strategy set is the set of prices. The auctioneer's 
choice of price restricts each consumer' s choice of consumption 
through the budget correspondence. Prof its of producers also depend 
on the auctioneer's choice of price, but their set of available 
productions does not depend on the price. 
2 
The preferences of consumers were assumed to be representable 
by real valued utility functions by Axrow and Debreu. Sonnenschein 
[1971] showed that transitivity of preferences could be dispensed with 
for proving the existence of Walrasian equilibrium. Mas-Colell [1974] 
was further able to dispense with completeness of preferences. Gale 
and Mas-Colell [1975] present a proof of existence of equilibrium for 
noncooperative games without complete or transitive preferences and 
use it to establish existence of Walrasian equilibrium. Shafer and 
Sonnenschein [1975] present a proof of existence of equilibrium for 
abstract economies without complete or transitive preferences and 
Shafer [1976] uses this to prove the existence of Walrasian 
equilibrium under fairly general conditions. 
The theorem of Gale and Mas-Colell on noncooperative games 
follows from Shafer and Sonnenschein' s theorem on abstract economies, 
but not conversely. This suggests that the Shafer-Sonnenschein 
theorem is stronger than needed to prove the existence of Walrasian 
equilibrium. By imposing additional hypotheses on the feasible 
strategy correspondences, while somewhat weakening assumptions on 
preferences, we can convert an N-person abstract economy to an N+l­
player noncooperative game. These extra assumptions were essentially 
introduced by Borglin and Keiding [1976], and are automatically 
satisfied by Walrasian budget correspondences. This is why Gale and 
Mas-Colell were able to prove the existence of Walrasian equilibrium 
with their theorem. Borglin and Keiding reduced an abstract economy 
to a 1-person game. Our technique of expanding the player set has a 
neat interpretation in terms of stationary points of a tatonnement 
procedure. 
1. Definitions. 
An N-person noncooperative game is a tuple G N (Xi, Pi) i=l' 
where for each i, 
P. 
1 x -7-7 x. 1 
N 
is a possibly empty-valued correspondence, 
3 
where X = lT X.. An equilibrium for G is a strategy vector x e X such 
j=l J 
that for each i 
Pi(;) " or x. eP.(;). 1 1 
Here Xi is the set of strategies of player i and Pi(x) is interpreted 
in one of two ways. The set Pi (x) may be the set of strategies which 
player i strictly prefers to xi given x1, 
•
• •  ,xi-l'xi+l•••••xN. Then 
Pi(x) = rJ means that there are no preferred strategies to xi given the 
others' choices. Under this interpretation it never happens that 
xi£ Pi(x). The second way we may interpret Pi(x) is as the set of 
"best replies" to x1, ••• ,xi-l'xi+l•···•xw Then Pi (x) is never empty 
and xi E Pi(x) is the appropriate equilibrium notion. Our definition 
of equilibrium allows for Pi to have one interpretation for some 
players and the other interpretation for the rest. 
An N-person abstract economy is a tuple E = (X.,S.,P.)� 1, 1 1 1 1= 
where for each i, 
and 
P. 
1 
s .  
1 
N 
lT x
J
. -7-7 X. is a possibly empty-valued correspondence 
j=l 1 
lTx. -7-7 X. is a nonempty-valued correspondence. 
j=l J 
1 
j#i 
N 
Put X lT X.. We will write S.
j=l J 1 
x -7-7 xi. with the understanding 
that Si(x) does not depend on xi. 
and 
An equilibrium for E is a vector x e X such that for each i, 
x. £ s .«;> 1 1 
Pi(x) n Si(x) ". 
Again Xi is player i' s strategy set and Si(x) is the set of 
feasible strategies for i given the choices x1, • • •  ,xi-l'xi+l•·
···�· 
4 
The set Pi(x) is the set of strategies preferred to xi given everyone 
else' s choice. The equilibrium condition is that the vector be 
feasible for everyone and no one has a strategy which is both 
preferred and feasible. 
5 
2. Theorems. 
The following theorem is a slight variation of a theorem of 
Gale and Mas-Colell [1975] . Denote by co Pi, the correspondence whose 
value at x is the convex hull of Pi(x). 
Jheorem 1. Let G (Xi,Pi)i=l be a noncooperative game satisfying for 
each i, 
( i) Xi is a nonempty compact convex subset of a finite dimensional 
euclidean space. 
(ii) Either 
(a) co Pi has an open graph in X x Xi' and for each x e X, 
xi I co Pi(x). 
or 
(b ) Pi is a continuous singleton-valued correspondence, 
i.e., a continuous function, 
Then G has an equilibrium. 
Proof, The proof is virtually identical to that of Gale and Mas-
Colell [1975, p.10] and so we only sketch the proof, If Pi satisfies 
(ii.a) let Ui = {x : co Pi(x) F d} and let f i be a continuous 
selection from co Pi lu.· Define yi : X ��Xi via 
1 
yi(x) 
{ {fi(x)} 
X. 
1 
x e ui 
x I u. 
1 
6 
If Pi satisfies (ii.b) set Yi = Pi. 
Then y = TTyi satisfies the hypotheses of Kakutani's theorem and has a 
fixed point x. By construction, if Pi satisfies (ii.a)· then 
co P.(;) = d so P.(;) = d and if P. satisfies (ii.b) then x. e P.(x}. 1 1 1 1 1 
Q,E,D, 
Theorem 2 below is a modification of a theorem of Shafer and 
Sonnenschein [1975], The assumptions on preferences have been 
slightly weakened and those on the feasibility correspondences 
strengthened, but in a way so as to remain useful for proving the 
nonemptiness of Walrasian equilibrium. The proof proceeds by 
converting the abstract economy to an N+l person noncooperative game, 
Theorem 2, Let E (Xi,Si,Pi)�=l be an abstract economy satisfying, 
for each i, 
( i) Xi is a nonempty compact convex subset of a finite dimensional 
Euclidean space, 
(ii) Si has a closed graph in X x Xi, the correspondence 
x �� int Si(x) has an open graph in X x Xi' and for each 
x e X, Si(x) is compact, convex, and has nonempty interior 
(relative to Xi).
(iii) co pi has an open graph in x x  xi. 
(iv) For each x e X, xi J co Pi(x). 
Then E has an equilibrium. 
Remark. Shafer and Sonnenschein assume that Si is lower hemi-
continuous as well as closed, We have strengthened this in (ii.a) to 
assuming int Si is a nonempty-valued correspondence with open graph. 
Note that since Si(x) is compact and convex with nonempty interior 
that Si(x) = cl (int Si(x)), i.e., Si(x) is topologically regular. 
Note that interior here is relative to Xi, not the underlying 
Buel ide an space, 
7 
Shafer and Sonnenschein assume that Pi has an open graph. We 
have weakened this to assuming co Pi has an open graph. Assuming that 
Pi has open sections (see Bergstrom, Parks and Rader [1976]), then 
co Pi will have open graph. ( In this case co Pi will be convex-valued 
with open sections. See Bergstrom, Parks and Rader [1976] and Shafer 
[1974],) The assumption of open sections for Pi is strictly weaker 
than the assumption of open graph as long as Pi is not convex-valued. 
Proof of Theorem 2. We define an N+l person game as follows. 
N N 
Put Zo = ITx . •  and for i=l, • • •  ,N put z. = xi. Set z i=l 1 1 
Tfz .. A 
i=O 1 
typical element of Z will be denoted (x,y), where x e z0 and 
N 
y e ITz . •  Define preference correspondences µi i=l 1 
follows. 
z -7-7 zi as 
8 
Define µ0 by
µo(x, y) = {y}, 
and for i = l, • • •  ,N set 
µi(x,y) 
J int S.(x) l co Pi ;y) n int Si(x) 
if yi J Si(x) 
if Yi e Si(x)' 
Note that µ0 is continuous and singleton-valued and that for i=l, • •  , , N
the correspondence µ. i s  convex-valued- and satisfies y .  t µ1.(x,y), 1 1 
Also for i=l, • • •  , N, the graph of µi is open. To see this for 
i = 1, ... ,N set 
and 
A. 1 {(x,y,zi) 
B. = {(x,y, zi) 1 
C. = {(x,y,zi) 1 
and note that 
zi e int Si(x)}. 
Yi t Si(x)}, 
zi e co Pi(y)}. 
Gr µi (Ai n Bi) u (Ai n Ci). 
The set Ai is open because int Si has open graph and Ci is open by 
hypothesis (iii), The set B. is also open, for if y. t S.(x) then 1 1 1 
there is a �losed neighborhood F of yi such that Si(x) c F
e and upper 
hemicontinuity of Si then gives the desired result. 
Thus the hypotheses of Theorem 1 are satisfied and so there 
exists c�.y) 8 z such that 
(a) x e µ0(x,y) 
and for i=l , ••• ,N
(b) µi (;,y) d.
Now (a) implies x = y and since Si(x) is never empty (b) becomes
co P.(;} n int S.(;) 1 1 d for i=l, ••• ,N. 
Thus P.(;) n int S.(;) = d, but S.(;) = cl [int S.(;)] and P.(;) is 1 1 1 1 1 
open so Pi(;) n Si(;) = d, i.e.,� is an equilibrium. 
9 
Remark. We can view the N+l game as a formalization of a tatonnement 
process. The N+lst player, call him the coordinator, performs in the 
abstract what a Walrasian auctioneer does in a market economy. The 
coordinator suggests a strategy vector. The players look at the 
choices they have made to see whether they are feasible given the 
coordinator's suggestion. If not, they choose a new strategy which is 
feasible. If the choice is feasible they try to improve upon it. 
Otherwise they don't change their strategy. The coordinator changes 
his suggestion to what the players were doing originally. A 
stationary point of this process is an equilibrium for the abstract 
economy. The question of under what circumstances this process is 
convergent remains to be investigated. 
� We could have defined µ0 in an alternative fashion, namely 
µ0Cx,y) = {z a z0 : lz - yl < Ix - ylJ. Then µ0 would have an open 
graph and be convex valued with x I µ(x, y) so Theorem 1 would still 
apply. 
10 
Remark. In the application of Theorem 2 to the problem of existence 
of Walrasian equilibrium we have the following interpretations. Call 
the auctioneer player 1. Except for the auctioneer, each Xi is a 
subset of the commodity space Rk. Compactness is achieved by suitably 
truncating consumption and production sets. For the auctioneer Xi is 
a subset of price space, also Rk. For producers S. is a constant 1 
correspondence equal to their truncated convex production set. This 
satisfies (ii) as all interiors are relative to Xi. Likewise for the 
auctioneer s1 = x1• For consumers, 
S.(p,x2 •• .,xN) = {y. a X. : p • y. � p • 111. + L 9 .. (p • x.)+J. where 1 1 1 1 1 J 1J J 
"'i is his endowment and &ij is his share of firm j. If player j is 
not a firm then 9 .. = O. Under the typical assumption (Shafer [1976J, 1J 
Gale and Mas-Colell [1975], Debreu [1959]) that "'i e int Xi (or 
something like this), then Si will satisfy the hypotheses of Theorem 
2. 
11 
REFERENCES 
Azraw, K. J. and G. Debreu. 1954. "Existence of an Equilibri11111 for a 
Competitive Economy." Econometrica 22:265-290. 
Borglin, A. and H. Keiding. 1976. "Existence of Equilibrium Actions 
and of Equilibrium: A Note on the 'New' Existence 
Theorems." Journal of Mathematical Economics 3:313-316. 
Bergstrom, T. C., R. P. Parks, and T. Rader. 1976. "Preferences 
which have open graphs." Journal of Mathematical Economics 
3(3):265-268. 
Debreu, G. 1952. "A Social Equilibrium Existence Theorem." 
Proceedings of the National Academy of Sciences 38:886-893. 
1959. Theory of value. New York: Wiley. 
Gale, D. and A. Mas-Colell. 1975. "An Equilibri11111 Existence Theorem 
for a General Model without Ordered Preferences." Journal of 
Mathematical Economics 2:9-15. 
Mas-Colell, A. 1974. "An Equilibri11111 Existence Theorem Without 
Complete or Transitive Preferences." Journal of Mathematical 
Economics 1(3):237-246. 
Nash, J. 1951. "Non-cooperative games." Annals of Mathematics 
54(2): 286-295. 
Shafer, W. J. 1974. "The Nontransitive Cons11111er." Econometrica 
42: 913-919. 
Shafer, W. and H. Sonnenschein. 1975. "Equilibrium in Abstract 
Economics without Ordered Preferences." Journal of 
Mathematical Economics 2:345-348. 
Shafer, W. J. 1976. "Equilibri11111 in Economies Without Ordered 
Preferences or Free Disposal." Journal of Mathematical 
Economics 3(2):135-137. 
12 


https://authors.library.caltech.edu/82098/1/sswp396.pdf

Noncooperative Games, Abstract Economies, and Walrasian Equilibria

Abstract

Similar works

Full text

Available Versions

Caltech Authors - Main

Caltech Authors - Main