A generalization of Scarf's theorem: An α-core existence theorem without transitivity or completeness

JOURNAL

OF ECONOMIC

THEORY

56, 194205

(1992)

A Generalization of Scarf’s Theorem: An a-Core Existence Theorem without Transitivity or Completeness ATSUSHI KAJII * Department of Economics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6297 Received

March

6, 1990; revised

May

15, 1991

Consider an n-person non-cooperative game. A coalition is said to a-block a given social state if it has a feasible strategy with which the coalition can ensure a social state preferred by all the agents in it regardless of the strategies the other agents outside the coalition may choose. The a-core is the set of social states that cannot be cl-blocked. This paper shows the non-emptiness of the a-core without the assumption of transitivity or completeness on the agents’ preference relations. Journal of Economic Literature Classification Numbers: C60, c72. 0 1992 Academic Press, Inc.

1.

INTR~DUCTT~N

The core is one of the most natural and important ideas in economics. There have been many discussions on its non-emptiness, its relation to other solution concepts such as Walrasian equilibrium, and its limit properties as the number of agents goes to infinity. First economic applications of the core consider the case where agents’ preferences do not depend on the actions of other agents. However, agents’ preferences are interdependent; that is, they are dependent not only on the agent’s own action but also on the actions of other agents, whenever there is an externality in the economy. And this interdependence is allowed in the majority of noncooperative games analyzed in the literature. In this context, a question arises as to what is the most natural extension of the idea of the core. Since actions of the agents outside a “blocking coalition” affect the welfare of the members of the coalition, it is necessary to consider the way the agents outside the coalition react in order to define the core. * The author thanks Andreu Mas-Colell, and a referee for their comments. They omissions.

Kazuhiko Ohashi, Ben Polak, are, of course, not responsible

194 OO22-0531/92

$3.00

Copyright 0 1992 by Academic Press, Inc. All rights of reproductmn in any form reserved.

Nicholas Yannelis, for any errors or

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If we suppose that the agents outside the coalition stick to a given strategy while the coalition attempts to improve its welfare, the corresponding notion is the strong (Cournot-Nash) equilibria. Strong equilibria are clearly very stable against coalitional deviation, but they often fail to exist; it is too easy for a coalition to block if the other agents stick to a given strategy. On the other hand, it is often sensible to suppose that the outsiders might try to take revenge. Aumann [l] suggests a possible way to define “blocking” in this context as follows: a coalition is said to a-block a given social state if it has a feasible strategy with which the coalition can ensure a social state preferre by all the agents in it regardless of strategies the other agents may choose. That is to say, by using the blocking strategy each member of the coalition gets a higher utility by blocking than in the social state in question, no matter how harmfully the outsiders react to the blocking strategy. The x-core is the set of social states that cannot be a-blocked. Since the outsiders are allowed considerable freedom to react against the coaltion, it is difficult to g-block social states and hence the a-core is relatively large. Indeed, in contrast to the strong equilibria, Scarf [ 15 ] was able to prove the non-emptiness of the a-core in the case where the agent’s preference relation is representable by a continuous quasi-concave utility function, where the quasi-concavity is required with respect to not only the agent’s own strategy but also the entire strategy profile. See Mas-Cole11 [I3] for an overview of related solution concepts. In Section 2 we generalize this result to the case where the agent’s preferences may not be ordered; that is, they may not satisfy the transitivity or completeness axioms. The convexity and continuity of preferences however remain crucial in our proof. Those readers who are used to working with utility functions may ask why we should care about failures of transitivity or completeness when thinking about the a-core. There are three motivations, the first which are relevant to uses of the E-core in applied work and the t which is of theoretical interest. The first motivation is simply that there is evidence that preferences do not obey transitivity. This perhaps is less surprising than it first seems, at least in the descriptive sense. Fishburn [7] argues that violations of transitivity are not necessarily inconsistent with rationality. t psychological models of bounded rationality and incomplete information such as Herrnstein and Vaughan’s [9] notion of “melioration’ (see also Herrnstein and Prelec [lo]) tend to produce non-transitive behavior. wish our models to be applicable in situations involving real agents, it is useful to show that our results survive the removal of some possibly unrealistic assumptions. Secondly, game theoretic ideas are increasingly eing used in the a

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literature to analyze contexts in which the “players” do not represent individual agents but groups or parties. F’or example, in models of labor bargaining they might be unions, and in models of legislatures they might be political parties or interest groups. When the players represent groups, we should not in general expect their “group preferences” to satisfy transitivity or completeness. For instance, if the group’s preferences are generated by the Pareto criteria then the completeness will fail, and if they are generated by the majority voting rule then transitivity will not be satisfied. Since the idea of the coalitional stability is especially interesting in these contexts, it is appropriate to investigate if the a-core is non-empty without transitivity or completeness. Thirdly, there has been considerable theoretical interest in determining if the transitivity or completeness of preferences is essential for the existence of game theoretic or market solutions such as Walrasian equilibrium, Nash equilibrium, and the core. Transitivity and completeness have been shown to be inessential for the existence of Walrasian equilibrium by Mas-Cole11 [12] and for the existence of Nash equilibrium of generalized games by Shafer and Sonnenschein [16]. Border [4] has shown that they are inessential for the non-emptiness of the core of market games when there is no interdependence in preferences; i.e., there are no externalities. However, the interdependence of preferences complicates matters and, interestingly, it has not been known if the a-core is non-empty for general non-cooperative games, although Yannelis [17] provided a partial result by showing that the a-core is non-empty for 2-person non-cooperative games. Our result indicates that the transitivity or completeness of preferences is inessential for the non-emptiness of the a-core, hence it fills an existing theoretical gap. The technique of our proof is a modification and simplification of the proof provided in Border [4] and our result generalizes that of Border in two aspects. First, and mainly, since the a-core coincides with the core when there is no interdependence of preferences in generalized games, Border’s result can be shown to be a special case of our result. Secondly, our result allows the strategy space to be a subset of any normed vector space; Border’s proof is not, at least not directy as Yannelis [17] points out, extended to cover infinite dimensional strategy spaces. This paper is organized as follows. In Section 2 we prove the nonemptiness of the a-core for n-person non-cooperative games. We show the non-emptiness result can be extended to cover generalized games or abstract economies in Section 3. Section 4 contains remarks and discussions on related works.

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Let us consider an n-person non-cooperative game. The set of agents is denoted by N = { 1, 2, .... 72>. Let J1/’ be the set of all non-empty subset of N. Each member of JY will be called a coalition. Each agent in N has a set of strategies X’. Let X= ni, N Xi, and an element of X may be thought as a social state. For each BE M, X, denotes the strategy space of a coalition I?, namely, X, = Hi, B X’. Also, if z E X and B E N, zB denotes the natural projection of z onto X,. Each agent has a preference reiation on X which is described by a correspondence Pi from X to X. Note that agent a”~ preferences depend on the strategies of other agents. We shall make the following assumptions: (A-l) Each X’ is a non-empty compact convex subset of a norme vector space V. (A-2) Each P’ is an irreflexive, i.e., xq! P’(x) for any XE X, and convex-valued correspondence with open graph in Xx X. Let x E X be a given strategy. A coalition B is said to a-block x if there is y E X, such that for all in B and for all z E X with zB = y we have In words, a coalition B a-blocks x if it has a strategy which is to x by its members, no matter what strategy the agents outside ion B may choose. The a-core of the game is the set of strategies which no coalition can cc-block. We shall prove: PR~PQSITI~N

1. Under (A-l) and (A-2) the N-core is non-empty.

First of all, let us introduce the main mathematical

tool for the proof

FAIGBROWDER THEOREM. Let K be a non-empty compact convex subset of V and let F be a convex-valued upper-hemicontinuouscorrespondencejkom K to V. If F is inward pointing, that is, for any x E K there is a positive number 1% such that x + A(v -x) EK for some v E F(x), then there is x* EK such that 0 E F(x*).

The proof for this theorem can be found in Fan [6] or theorem is a generalization of the well-known result th pointing continuous vector field on a compact convex subset of zero. Note that if a correspondence x ++ x + F(X) is a correspondence into K itself, then F is inward-pointing so that the theorem is also a generahzation of Kakutani’s fixed-point theorem. Now let us start proving the proposition. First, for each agent iE N define a pseudo-utility function U’ on Xx X as constructed in Shafer and Sonnenschein [ 16 ] by the rule u’(x, y) = distance[(x, y), (Graph of Pi)‘],

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where ( )’ stands for the complement and the distance is derived from the product norm. Note that ui is norm-continuous and non-negative, and u’(x, y) > 0 iff y E P’(x). Moreover, as it is pointed out in Border [4], u’(x, .) is quasi-concave for any x E X. To see this, it is enough to consider the case where u’(x, yi)> u’(x, y2) = c > 0, since u’>O. Pick any t, 0 < t 6 1, and put y = tyl + (1 - t) y2. We want to show that u’(x, v) > c. Let (x’, z) E Xx X such that dist[(x’, z), (x, JJ)] < c. Then, dist [ (x’, Y,,, + (z-y)), (x, y,)] = /I(x’ -x, z - JJ)/~< c for m = 1,2. This implies that y, + (z - JJ) E P”(x’) for m = 1,2. Since P’(x’) is convex, we have z = t(y, + (z-y)) + (1 - t)(yZ + (z-y)) E Pi(x’), hence u”(x, v) > c by the construction of ui. Since Xx X is assumed to be norm-compact and each ui is norm-continuous, there is a number M > 0 such that ui d M on Xx X. For every i, put a, = -&fei, where ei is the ith unit coordinate vector of R”. For each BEN, define mB=(l/#B)CisB ai. Let p be a collection of members of JV. We say /? is balanced if there is a set of scalars { llB > 0: BE /I} such that for each iEN, &EBCil AB= 1, where p(i)= (BED: icB}. {LB: BE/~} is called a balancing weight for p. It is easy to see that /I is balanced iff mN E conv{m,: BE p}. (“conv A” stands for the convex hull of the set A.) For each coalition B, define a mapping VB: X--H R” by the rule V”(x) = {w E R”: there is y E X, such that (Vie B) (Vz E X such that zs = yB) wi < u’(x, z) for all ie B}. The pseudo-utility u’(x, y) can be interpreted as the increase in utility of the agent i when the strategy x is replaced with y. Then V”(x) can be thought as the set of (net) utility allocations feasible to the coalition B. Note that if (UBEM V”(x))nR’!+ + = @, then x is in the a-core by construction. We shall denote the non-cooperative game with strategy sets Xi and utility functions u’(x, .), i= 1, .... N, by ‘&. As ui(x, .) is quasi-concave, we can apply Scarf’s argument for gX to show that the g-core of C$ is nonempty. For the sake of completeness and a further discussion in Section 3 we sketch his proof briefly. Let /I be a balanced family of coalitions with the balancing weights {AB: BE/~). Pick any WER” and XEX. Suppose {yBKYB: BE/~} satisfies (VIE N) (VB E p) (Vz E X such that zg = v),

wi < u’(x, z).

Define y! = CBS BCij,?B(yB)i, where (Y,)~ is the component of ye corresponding to the player i. Let y’ = (y;, .... $J. Since X’ is convex, y’ E X. It can be shown that w’< u’(x, y’) for all ig N; we refer the reader to Scarf [15] for a detailed proof. Consequently, we have r)BEfl V”(x) c V”(x); that is, the characteristic function form game {V”(x)} is a balanced game

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for every XEX. Therefore the a-core of ~3~is non-empty because the core of the characteristic function form game (VB(x)}, which is non-em since (V”(x)} is balanced, corresponds to the a-core of G&.. Now consider the correspondence j’(x) = {x’ EX: there is no B such that ( 3 YEX~) (Vi~63) (Vz E X such that zg = y) u’(x, z) > 2(x, x’)>. That is, t(x) is the a-core of the game gX. Note that the set of the fixed points of < coincides exactly with the a-core of the original game because u’(x, z) > 0 implies z E P’(x) and u’(x, x) = 0 by construction. Therefore we need to solve this fixed-point problem in order to show the non-emptiness of the a-core. Kakutani’s theorem is not directly applicable because 4 is not convex-valued in general. We need an additional step. Let A = conv(ai: i= 1, .... n}. For each (s, x) E d x X, define f(s,x)=max

t>O:s+te~ i f*(s,x)=max{t30:s+~e~V~(x)), As, x) = 3 +.I+, p*(s, x) = s +f*(s,

u BE-C

V”(x)

, I

and

x)e, x)e,

where e = C ei. These mappings are easily seen to be well-defined ,u(s, x) 3 0. Define a correspondence F, : A x X ++ X by the rule

and

where U(X, y) = (u’(x, y), .... LP(x, v)) E R”. The correspondence F2 is nonempty by the construction of V”(x) and h*(s, x), and it is convex-valued by the quasi-concavity of ufx,. ). Note that if ,M(s, x) = p*(s, x) then F2(s, x) is a convex subset of t(x), since if ~(s, x) = ,u*(s, x) then p*(s, x) lies on t frontier of IJBE .+- V”(x). Also note that ~(s, x) = p*(s, x) if p(s, x) E V”(x). Therefore if we can find an x* such that x* E F2(s, x*) and ~(s, x*) E VN(x*) for some S, then we have x* E 5(x*) and we are done. For each (s, x) consider a family of coalitions j?(s, x) = {BE JV: ,u(s, x) E V”(x)}. By construction, ~(s, x) E fl,,,,,,,, V”(x). Therefore, if /?(s, x) is a balanced family, or equivalently, mN E conv(mB: BE~(s, x)3, then As, 4 E nBtBcs.xj V”(x) c V”(x), hence ~(s, x) E V”(x). Define F, : A xX -H R” by the rule F,(s, x) = mN - conv(m,: By construction

,M(s,x) E V”(x) >.

if s* is a point such that 0 E F,(s*, x), then p(s*. x) =

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200

p*(s*, x) E V”(x). by

Finally

define a correspondence

Y: d x X ++ R” x Vk”

m, x)= (F,(s,xl,(F&Yxl -xl). LEMMA. The correspondence !I? A xX + Vk” is an inward-pointing, convex-valued, and upper-hemicontinuous correspondence,

The proof of this lemma is straightforward and can be found in the appendix. This lemma asserts that Y satisfies the conditions for the Fan-Browder Theorem. Therefore there exists (s*, x*) E A x X such that OE Y(s*, x*); i.e., 0 ~Fr(s*, x*) and x* EF~(s*, x*). We have already seen that this implies x* is in the a-core. This completes the proof.

3. EXTENSION

TO GENERALIZED

GAMES

The notion of the cc-core can be defined for so-called generalized games (or abstract economies). Let N, Jlr, X’, and Pi be defined as in Section 2. Furthermore, for each BE J consider a correspondence GB: X -P+ X,. GB(x) is the set of feasible strategies for the coalition B given strategy x EX. A generalized n-person non-cooperative game is a tuple (N; (xi, Pi)ieN; (GB)BEwK) consisting of the set of agents N, the set of strategy Xi, the preference relation Pi, and the correspondence GS which determines the set of feasible strategies. For instance, an n-person noncooperative game is a generalized game where GB(x) = X, for all x E X and BEN. The E-core of a generalized game (N; (Xi, Pi)ieN; (GB)BEM) is defined as follows: a coalition B u-blocks x iff there is y E GE(x) such that for all i E B and for all z E X with zg = y we have z E P’(x). An x E X is in the a-core provided x E GN(x) and no coalition B can a-block x. We shall assume, in addition to (A-l) and (A-2): (A-3) GB is continuous (i.e., both lwoer-hemicontinuous and upperhemicontinuous), and for all x E X, GN(~) = GN, where GN is a non-empty closed convex subset of X. (A-4) {G”G41 is outcome-balanced for all XEX; that is, for any balanced family of coalition p with the balancing weight (1% BE /I), if yBgGB(x) for all BEG then y’6GN where JJ~=C~~~~~,~~(JI,),. The assumption (A-4) is originally production economies. (See also [4,5].)

introduced by Boehm Now we can show:

[3]

for

PROPOSITION 2. Under (A-l ), (A-2), (A-3), and (A-4), the a-core of the generalized game is non-empty.

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Proojf Let V”(x) = { w E R”: there is y E GB(x) such that (Vi E B) (VZ E X such that zB = y), wi d u’(x, z)}. Due to (A-4), one can show that the game ( YB(x): BE JV’> is balanced similarly as in the proof of Proposition 1. (V”(X) may be empty since GB(x) may be empty, but this does not matter.) Note that if (UBEoY. V”(x)) n R”, + = 0, then x is in the a-core by construction. Thus, we are solving essentially the same problem as in Proposition 1. Define d, ~(s, x), and p*(s, x) as before. By (A-3), G,,,(x) = GN, hence V”(x) is non-empty and the functions ~(s, x) and p*(s, x) are well defined and continuous on A x GN. Now define correspondences 46: : A x X ++ and F2: A x GN -H GN by the rule F,(s, x) = mN - conv{m,:

~(s, x) E V”(s));

E;(s, x) = { y E GN: u(x, y) 3 p*(s, x)). The correspondence F2 is non-empty by the construction of V”(x) p*(s, x), and it is convex-valued by the quasi-concavity of u(x,.) and convexity of GN. Therefore, by the same argument as in Proposition 1, there is (s”, x*) E A x GN such that OEF,(.S*, x*) and x* EI;;(s*, x*). I? is straightforward to show that x* is in the a-core. QED. In fact, since the n-person non-cooperative game we considered in the last section can be seen as an outcome-balanced generalized game, we could assert Proposition 1 as a corollary to Proposition 2. stated Proposition 1 as an independent assertion to avoid unnecessary complication. We can relax the convexity assumption slightly when V is a finite dimensional vector space as follows: (A-l)’ Each X’ is a non-empty compact convex subset of R”; (A-2)’ Each P’ satisfies x 4 conv P’(x) for any x E X, and has open graph in Xx X. COROLLARY. Under (A-l)‘, (A-2)‘, generalized game is non-empty.

(A-3), and (A-41, the se-coreof the

ProoJ: This is straightforward since conv P’(x) has open graph in XX X if P’(x) has open graph in the case where X is a convex subset of a finite dimensional vector space.

Now we shall demonstrate that our results imply Border’s [4] result. Suppose that the preference relation Pi is a correspondence from Xi to itself. Given a strategy x E X, we say a coalition B, BE Jf, blocks x if there is a strategy y E G”(x) such that yi E Pi(xj) for all ie B. A strategy x is said to be in the core if there is no BE JV which blocks x. Border’s theorem

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proves for this model that the core is non-empty. For each i E N, define the correspondence P*i: X tf X by P*‘(x) =X1 x . . . x P’(x,) x . . . x X,. If Pi is an irreflexive non-empty convex-valued correspondence with open graph, so is P*‘. It is easy to check that BE N can block a given strategy x if and only if B can a-block x in the modified game with the preference map P*‘. Therefore the core of the original game is the same as the a-core of the modified game which satisfies all the conditions for Proposition 2, thus the core is non-empty. Border assumes (A-l)’ and (A-2)’ instead of (A-l) and (A-2). Since we allow infinite dimensional topological vector spaces, Proposition 2 and its corollary generalize Border’s result.

4. REMARKS

First, let us comment on the technique we used in our proof. The key fact we rely on is that the non-transferable utility cooperative game (V”(x)} derived f rom the pseudo-utility functions u’(x, .) is well defined and balanced for all x. The other conditions for our fixed-point technique follow straightforwardly from the continuity (open graph) and the convexity of preferences and the compactness of the strategy sets. On the other hand, even if preferences are represented by utility functions, it is common in the literature to reduce the existence problem of cooperative solutions to the problem of showing the non-emptiness of the core of some nontransferable utility cooperative games. Therefore, one can expect that our technique can be applied to show the existence of other cooperative solutions of non-cooperative games without ordered preferences. The B-core is an example which conforms to this idea, but the characteristic function form (or coalitional form) game corresponding to the p-core is not balanced in general even if preferences are represented by quasi-concave utility functions. (See [13] for the definition of the &core.) Another example of interest is the strong equilibrium, or more generally, Ichiishi’s [ 111 social coalitional equilibrium. However, the existence of the strong equilibrium is not reduced to the non-emptiness of the core of nontransferable utility games in characteristic function form; the utility set for the blocking coalition will depend on the outsiders’ strategy for the strong equilibrium while it does not depend on the outsiders’ strategy for the a-core. We conjecture that our technique can be modified so as to be applicable for the existence problem of the strong equilibrium or the social coalitional equilibrium, but we have not been able to do so. Secondly, let us discuss whether we can relax the assumption that preference relations have open graph. Florenzano [8] generalizes Border’s result to the case where the strategy space is possibly infinite dimensional

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but without externalities in preferences. It is assumed that preference relations satisfy a weaker continuity hypothesis than ours. In our notation, the hypothesis is that (x E X: y E P’(x)} is open in X for any y E X; i.e., Pi has open lower section. Florenzano’s basic idea is as follows First, for each coalition B define the (group) preference relation PB by the rule PB(x) = { y E X, : yi E P’(x) for all i E B} and consider the game with 2” - 1 players with preferences described by PB, BE M; then show that it has a equilibrium, which is clearly in the core by construction. Note that i Pi has open lower section so does PB for every B, hence the group pre ces PB behave just as well as individual preferences. Although it is ass there is no interdependence in preferences, it seems to be possible to Florenzano’s technique to prove our Proposition 1 by replacing preference relation PB of B by P*B(x) = (YE X,: for all ZE X that zB = y we have ZE P’(x) for all in B). The continuity requ Florenzano’s result is that Pi has open lower section which is weaker than P’ has open graph, thus one might hope that we can relax (A-2). IJnfortunately, it is not clear that the modified P*B has open lower section when each Pi has open lower section. (P*B does have open lower section if each P’ has open graph and X is compact, which is what we assumed.) This because there is a uniformity involved in the definition of P*! conclusion, we are uncertain if we can relax our continuity assumption. Finally, we comment on the compactness of the strategy sets. Although we allow infinite dimensional strategy sets, the compactness assumption on the strategy sets puts a serious limitation on the applicability of our result in some contexts. For example, it is well known that the set of feas trades is in general just weakly compact but not norm-compact in exchange economy model when the commodity space is infinite dimensional. Mas-Cole11 and Zame [14] contains a detailed discussion on t problem. Therefore, it would be desirable if we could relax the normcompactness of the strategy set X to weak-compactness. This problem is sometimes not serious because the combination of the quasi-concavity a the norm-continuity implies the weak-continuity of utility functions. However, in our case the pseudo-utility u(x, y) is weakly continuous in JJ but not in x. Therefore, we may not obtain an appropriate co~ti~~~t~ property for V”(x), and subsequently for p. A possible way to avoid the discontinuity resulting from relaxin compactness is to follow the argument developed by Bewley [2]: prove t c+core is non-empty for the finite dimensional case and show that the limn of the net consisting of points in the a-core of games restricted to dimensional subspaces of V is in the cl-core. Indeed, this method works for Florenzano’s case. If it works for our case as well then, as a by-product, one could extend the Corallary to the case where V is intinite dimensio~a~~ IIowever, again, the uniformity involved in the definition of the ~-core 642i56+14

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prevents us from following the limit argument; a subnet of the net of a-core strategies converges but the convergence is not uniform with respect to potential blocking strategies which may arise in the limit.

APPENDIX

Proof of the Lemma. Note that for any BE JV and any iE B, the function g: Xx X, -+ R given by g(x, y) = min{ ui(x, z): z E X and zs = v} is uniformly continuous since ~2 is continuous and Xx X, is compact. So it is easy to see that the graph of the correspondence VB is closed, which implies b and ,u* are continuous and that both F, and F2 are upperhemicontinuous. The convexity of Y follows since each ui(x, .) is quasiconcave. To see that Iy is inward-pointing it is sufficient to show that F,( ., x) is inward-pointing for any x E X, since F2(s, x) + x is a subset of X for all (s, x) E A x X. Pick any s E A. Since ~(s, x) E UBE M V”(x), we can find BE JV such that ~(s, x) E V”(x), hence mgg F,(s, x). We claim that if sj = 0, then j+! B. Indeed, if si = 0, there must be some k such that sk < -M since C si = -n&f. Because ~(3, x) = s + f(s, x)e > 0, this implies f(s, x) > M. So we have ,u(s, x)~ = si + f(s, x) =f(s, x) > M, hence j $ B by the construction of M and V”(x). Note that j$ B implies (mN - mB),. < 0. Therefore we have s + I(m,--m,) EA for sufficiently small Iz >O, hence F,( ., x) is inward-pointing. This completes the proof of the lemma. Q.E.D.

The core of a cooperative game without side payments, Trans. Amer. (1961), 5399552. T. BEWLEY, Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514540. V. BOEHM, The core of an economy with production, Rev. Econ. Stud. 41 (1974), 429436. K. C. BORDER, A core existence theorem for games without ordered preferences, Econometrica 52 (1984), 1537-1542. K. C. BORDER, “Fixed Point Theorems with Applications to Economics and Game Theory,” Cambridge Univ. Press, Cambridge, 1985. K. FAN, Extensions of two fixed point theorems of F. E. Browder, Math. .k. 112 (1969), 234-240. P. C. FISHBURN,“Nonlinear Preference and Utility Theory,” Johns Hopkins Univ. Press, Baltimore, 1988. M. FLORENZANO,On the non-emptiness of the core of a coalitional production economy without ordered preferences, J. Math. Anal. A& 141 (1989), 484 490. R. HERRNSTEIN AND W. VAUGHAN, JR., Melioration and behavioral allocation, in “Limits to Action: The Allocation of Individual Behavior” (J. E. R. Staddon, Ed.), Academic Press, New York, 1980.

1. R. J. AUMANN, Math. Sot. 98

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5. 6. 7. 8. 9.

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