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A Characterization of approximate equilibria and cores in a class of coalition economies Myrna Wooders May 1979 Preface (November 1998). This paper duplicates a 1979 revision S.U.N.Y-Stony Brook Working Paper No. 184. This revision has been frequently cited, cf. Bennett and Wooders (1979), Scotchmer and Wooders (1986), both to appear on-line at http://www.chass.utoronto.ca/~wooders2.. Further discussions of the origins of this work and some subsequent literature appears in “A history and brief summary,” on-line at http://www.chass.utoronto.ca/~wooders2. Following is a brief summary of the paper and the results. A pregame with a …nite number T of types of players is introduced; this is simply a function v : ZT ! R+ , where ZT denotes the T -fold Cartesian product of the + + non-negative integers and R+ denotes the non-negative real numbers. An element s of ZT , called a pro…le, is interpreted as a description of a coalition, where st is the + number of players of type t in the coalition, for t = 1; :::; T . The function v ascribes a worth to each possible coalition. Note that a pregame di¤ers from a game since a pregame does not specify a total player set. But, …xing a pro…le m, the pregame determines a game (N; v) where N is a player set with mt players of type t for each t (and all players of the same type are substitutes). For the main results of the paper, it is assumed that all gains to coalition formation are exhausted by groups of players (in other words, small groups are strictly e¤ective, or coalitions have minimum e¢cient scales). Fixing a player pro…le m, consider a sequence of games with the pro…les of the total player sets equal to rm, for r ¡ 1; 2; ::::. The main results1 are of the paper are: 1. There is an integer m such that, for all integers r, the rmth replica game (containing rm times as many players of each type as the initially given game) has a nonempty core. 1 These same results appear in Wooders (1977), SUNY-Stony Brook Working Paper No. 184, available through the library. 1 2. All payo¤s in the core of the rth replica game have the equal-treatment property. 3. Given there is an integer r such that for all r 0 > r, the r0 th replica game has a nonempty "-core. 4. "-cores of large replicas converge to the equal-treatment cores (in the sense that in large games most players of the same type are treated nearly equally). 5. Equal -treatment "-cores converge to the core of a continuum game having the same proportion of players of each type as the …nite games. The paper also considers entrepreneurial coalition formation by individual players, who take prices for agents of each player type as given. The core is shown to be equivalent to a subset of the set of entrepreneurial, price-taking equilibrium outcomes. The results of this paper have been extended in a number of subsequent works, alone and with co-authors. See especially Wooders (1980), also at on-line at http://www.chass.utoronto 2 A CHARACTERIZATION OF APPROXIMATE EQUILIBRIA AND CORES IN A CLASS OF COALITION ECONOMIES* Myrna Wooders** May 1979 (A revision of Stony Brook Working Paper #184) *I am indebted to Javier Ruiz-Castillo and Thomas Muench for helpful comments, and to Elaine Bennett for pointing out some errors in a previous version of this paper. ** The author is Assistant Professor, Department of Economics, State University of New York at Stony Brook, Stony Brook, New York 11794 3 The main purpose of this paper is to investigate su¢cient conditions for the existence and convergence of approximate cores and approximate equilibria in replica coalition economies. A coalition economy is an economy with the property that any subset of agents can form a coalition (excluding other agents); thus, economies with private goods, pure public goods, and local public goods are all examples of coalition economies. The class of economies, however, is restricted to those where the coalition formation aspect of the economy can be represented as a game with side payments (for example, coalition production economies). 1 The core of these economies might be empty for any given replication. Neverthe- less, it is shown that the core exists for a subsequence of economies; an approximate core, called the "-core,2 exists for all su¢ciently large economies; and, for the sub- sequences of economies with a non-empty core, the "-core converges to the core as " becomes small and the economy, large. A set of “equilibrium” prices for agents, called quasi-equilibrium prices, is de…ned. These prices might not constitute an imputation since it might not be feasible to impute to each agent her (or his) quasi-equilibrium price. However, the set of quasi- equilibrium prices is shown to be non-empty and the "-core is shown to converge to a subset of this set. For small " and large r (the replication number of the economy), an "-core imputation can be viewed as an “approximate” equilibrium since most agents receive nearly their quasi-equilibrium prices. In addition, necessary and su¢cient conditions for the non-emptiness of the core are developed and related to the set of quasi-equilibrium prices. These conditions seem more intuitive and more powerful in application than the balancedness con- dition for the non-emptiness of the core.3 Also, these conditions are related to the balancedness condition. Theorems similar to those of this paper have been obtained by this author for economies with local public goods and endogenous coalition (i.e. jurisdiction) struc- tures in [27, 28, 29]. In these economies, however, production sets and/or preferences depended only on numbers of agents in a jurisdiction and not on the characteristics of these agents. This feature enabled the author to deal with each type of agent sep- arately; there was no necessity to consider jurisdictions consisting of more than one type of agent to obtain the results. This paper extends the model of local public good economies in that the class of coalition economies considered might have the property that all imputations in the core have associated coalitions containing more than one type of agent. A technique is developed to describe imputations in the core and in approximate cores and the coalition structures associated with these imputation. Before concluding this section, we remark that many familiar economies can be viewed as coalition economies, where partitioning of agents into groups or coalitions for joint production and/or consumption is endogenous. Both Arrow-Debreu-type private good economies, as in [5], with constant returns to scale, and Foley-type pure public good economies, as in [7], are special cases of coalition economies. For both 4 these classes of economies, the coalition of the whole is consistent with optimality. Coalition production economies, as in Boehm [3, 4], Hildenbrand [9], Ichiishi [11], and Wooders [30, 31], local public good economies as in Ellickson [6], Greenberg [8], McGuire [13], and Wooders [27, 29], and economies with systems of clubs, as in Pauly [16, 17], are all coalition economies. Some of these economies might require more complex partitioning of agents than either private good or pure public good economies for optimality to attain. In the next section of this paper, examples are developed which illustrate the class of economies being investigated and some applications of our model. The third section consists of results. In the fourth section, numerical examples are developed. We then relate our work to some of the existing literature and conclude the paper in the …fth section. Since a number of fairly standard de…nitions from game theory are used, rather than include them in the main body of the paper, in Appendix 1 we state all these de…nitions. In the main body of the paper, when one of these terms is used for the …rst time, it is followed by (D.n) where n is the number of the de…nition. In addition, in Appendix 1, we include a few remarks pointing out relationships between various game-theoretic concepts, the content of which is either well known or easily veri…ed. All theorems are proven in the second Appendix. Notation The class of economies to be considered will be described with the help of the following notation and terminology. E T : Euclidean T -dimensional space; -T : the non-negative orthant of Euclidean T -dimentionsal space; j S j: the cardinal number of a set S; P j S j: the set of all partitions of a set S; S 2 P j S j: a generic element of a P j S j , called a coalition structure; an element of S is called a coalition; jj ¢ jj : the sup norm; given x = (x1 ; :::; xT ) 2 -T ; jj x jj maximum over i of the absolute value of xi : The set of agents of the r-th replica economy is denoted by Nr = f(1; 1); :::; (1; q); :::; (1; m1 r); :::; (t; 1); :::; (t; q); ::; (t; mtr); :::; (T; mT r)g where the ordered pair (t; q) represents the qth agent of type t. Let [t]r = f(t; q) : q 2 (1; :::; mt rgg; [t]r is called the type t. An element of this set is called an agent of type t. For the purposes of this model, all agents of the same type are identical.4 We observe that 5 the set of agents of the unreplicated economy, N1 , consists of mt agents of type t for each t 2 f1; :::; T g. Let m = (m1 ; :::; mt ; :::; mT ). Let I denote the T -fold Cartesian product of the non-negative integers and let Ir = fs 2 I : s · mrg. Given S µ Nr , the pro…le of S is a vector s = (s1 ; :::; st ; :::; sT ) 2 Ir where st =j S \ [t]r j : In other words, the pro…le of a set of agents S is simply a vector listing the number of agents of each type contained in S. When s is the pro…le of S, write S ´ s. For any T P T P s 2 I, it is convenient to write j s j= st since st =j S j, the cardinal number of t=1 t=1 the set S where S ´ s: Finally, de…ne I(t0 ) and Ir (t0 ) by I(t0 ) = fs 2 I : st0 6= 0; where s = (s1 ; :::; sT )g and Ir (t0 ) = fs 2 Ir : st0 6= 0; where s = (s1 ; :::sT )g: 1 An Example In this section, we develop an example of a coalition production economy, which illustrates the class of economies being investigated and some applications of our results. A competitive equilibrium for the economy is considered and the intuition underlying our results for the one-type of agent case is developed.5 We then indicate how the fundamental ideas are extended to the case where the number of types, T , is greater than one. We remark that the results described in this section have been obtained for a general equilibrium model of a competitive coalition economy in [30, 31]. The discussion in this section is intended to be informal and suggestive; the model and results are rigorously developed in later sections. In the following discussion, a coalition S is viewed as a …rm or as a potential …rm. It is assumed that there are L “marketed” goods. For each coalition, say S µ Nr , a production possibilities set is exogenously determined and denoted by Y [S] µ E L . We assume that Y [S] = Y [S 0 ] whenever S and S 0 have the same pro…le (i.e. the same number of agents of each type), so we de…ne Y [s] = Y [S] where S has pro…le s. Several interpretations of the dependence of Y [S] on S are possible. In the labour- management literature, agents are viewed as having (possibly) di¤erent skills as en- trepreneurs and/or workers and these skills are tied to the …rms of which these agents 6 are members. Also, agents could be viewed as having (possibly) di¤erent amounts of an indivisible capital good, which can only be sold to one …rm. For our competitive coalition economy, we take prices of non-coalition-speci…c commodities (i.e. marketed goods) as exogenous. Let ¼ 2 -L be these prices. Given ¼, let the maximum pro…t realizable by a …rm with pro…le s be denoted by v(s) = maxf¼ ¢ y : y 2 Y [s]g and let v(S) = v(s) when S ´ s. We de…ne a (competitive) equilibrium as a coalition structure (a partition of the ^ ^ ^ set of agents), say Nr = fS1; :::; SK g;and a pro…t distribution ¨ = (¨11 ; :::; ¨T mT r ) 2 - Nr such that: P ^ tq (a) v(Sk ) ¡ ¨ = 0 for all Sk 2 Nr ; tq2Sk P (b) v(S) ¡ ^ ¨tq <0 for all S µ Nr . tq2S Given equilibrium prices for marketed goods, ¼, this is the usual equilibrium concept for coalition production economies (see, for example, [9] and [26]). The equilibrium, however, might not exist even if each Y [S] is a closed, convex cone. The problem of the non-existence of the equilibrium is caused by the possible emptiness of the core of a certain game in characteristic function form (D.1). Let X v(S) = max ¹ v(S 0 ); S2P [S] S 0 2S v(S) represents the maximum pro…t realizable by a coalition S when it can be parti- ¹ tioned into …rms. Then the ordered pair (Nr ; v) is a game in characteristic function ¹ form and the core (D.6) of this game is equivalent to the set of equilibrium pro…t distributions. However, the core of (Nr ; v) is non-empty if and only if the game is ¹ balanced (D.12). 1.1 The One-Type Case For the case where T = 1, it is easy to describe the fundamental assumption of the model and to illustrate the theorems. For this case, the assumption is that there ¤ exists an s¤ 2 I such that v(s¤ ) ¸ v(s) for all s 2 I and s¤ · m.6 s s ¤ Any pro…le s¤ such that v(s¤ ) ¸ v(s) for all s 2 I is called a distinguished pro…le s s and if S ´ s¤, S is called a distinguished set, or, in this example, a distinguished …rm. 7 This assumption could be viewed as resulting from, at …rst, increasing average returns to …rm size and, after some point, decreasing average returns to …rm size, so that v(s) resembles a typical value of the average product curve for labour, as in Figure 1 (where indivisibilities of agents are ignored). Given m (the number of agents in N1 ) and given s¤ as above, it is immediate that there exists sequences of integers (n` ) and (r` ) such that r` m = n` s¤. For any r such ^ ^ ^ that rm = ns¤ , it is easily veri…ed that ¨ = (¨11 ; :::; ¨T mT r ) is an equilibrium pro…t distribution where ^ v(s¤ ) ¨tq = ¤ s for all (t; q) and, for all S 2 Nr , j S j= s (the equilibrium exists for a subsequence ¤ of economies). The existence of approximate equilibria and the convergence theorems are now illustrated in Figure 2 for the case where T = 1. For the purposes of the …gure, indivisibilities of agents are ignored. Given T = 1, let v(s) g(s) = ; s g(s) represents the maximum average pro…t attainable by a …rm consisting of s agents. De…ne Ã ! s gk (s) = g ; k gk (s) is the maximum average pro…t attainable by s agents evenly divided amongst k …rms. Let ³(s) denote the upper envelope of the curves gk (s); ³(s) = max gk (s) k and represents the maximum per capita pro…t attainable by s agents when they can be partitioned into …rms. It is obvious that s³ (s) = v(s). ¹ It can easily be veri…ed that when s is the unique distinguished pro…le and rm ¤ > s¤ , the core of (Nr ; v) is non-empty if and only if rm = ns¤ for some integer n. ¹ This result is not dependent on the indivisibilities of agents; it is a result of the not necessarily non-decreasing average returns to coalition size. The reader might observe that Figure 1 appears “upside down” in literature on multi-plant production and perfect competition where average cost curves are U- shaped (see, for example [19], page 131) and the assumption of the existence of distinguished …rms is much like the assumption of a minimum of an average cost curve. On Figure 1, we observe that as s becomes large, ³(s) approaches the constant v(s¤ )=s¤ .7 This observation motivates the following de…nitions and results. An "-equilibrium is a coalition structure Nr and a pro…t distribution ^ ^ ^ = (¨11 ; :::; ¨T mT r ) such that ¨ 8 Figure 1: P ^ (a) ¨tq · v (Nr ) and ¹ tq2Nr P ^ (b) v(S) ¡ ¨tq ¡ " j S j· 0 for all S µ Nr . tq2S The economic idea behind subtracting o¤ " j S jin (b) above is that cooperating to form a …rm costs " per capita or there are “…xed costs” proportional to s00 .8 ¤ If ³(mr) ¸ v(s¤ ) ¡ " , it is easy to verify that an "-equilibrium exists. s K P Let Nr = fS1; :::; SK g be such that v(Nr ) = ¹ ^ v(Sk ) and let ¨tq = ³(mr). Then if k=1 v(s¤ ) ³(mr) ¸ s¤ ¡ "; X v(S) ¡ ^ ¨tq ¡ " j S j· 0 tq"S ^ for all S µ Nr , so Nr and ¨ are an "-equilibrium coalition structure and pro…t distribution respectively. The theorem is: given "0 > 0, there exists an r0 su¢ciently large so that the "0 -equilibrium exists for all r ¸ r0 . Also, the "-equilibrium converges to the (exact) equilibrium as " becomes small, and r, large. The convergence is in the sense that for small " and large r, in an "-equilibrium, the pro…t share of “most” agents is “nearly” v(s¤ )=s¤ . We remark that this is the type of convergence conjectured by Shapley and Shu- bik([25], page 806) for the Shapley-Shubik weak "-core (D.7). It seems obvious that if agents are divisible then convergence of the strong "-core (D.8) would obtain (but we have not formally investigated this conjecture). 9 Figure 2: 10 Since for all rm > s¤, the core of (Nr ; v) is non-empty if and only if ¹ v(s¤ ) ³(rm) = ; s¤ it is obvious that the core of (Nr ; v) is non-empty if and only if ¹ ~ ³(rm) = vB (s); as shown on the diagram. The function vB (s) is the balanced closure game charac- ~ teristic function (D.13), v(S), divided by j S jwhere s ´ S (i.e. vB (s) = v (S)= j S j). ~ ~ The insight for this author’s work on local public goods is also illustrated by Figure 2, where g(s) is interpreted as the maximum utility of a representative agent of type t when the s members of a jurisdiction all pay the same price or tax for the local public good. In fact, the …gure (relabeled and without vB (s)) appears in [27], ~ where the curves have “‡at tops,” so that, with some additional conditions, the local public equilibrium (de…ned in [27]) exists for all su¢ciently large economies (as shown in [28]). In addition, for economies with a local public good, theorems similar to the existence and convergence theorems (Theorem 1) of this paper are obtained in [29]. 1.2 The T Types Case In local public good economies, as modeled in [13], [27] and [29], and in coalition production economies where T = 1, preferences and/or production possibilities sets depend only on numbers of agents rather than more general pro…les. In more general economies, we would expect that optimality might require joint consumption and/or production by agents of di¤erent types. The most fundamental new idea of this paper is the generalization of distinguished numbers (for one-type coalition production economies or local public good economies) to distinguished pro…les and distinguished sets. Informally, for a competitive coalition production economy, a price vector p = (p1 ; :::; pT ) 2 -T is taken as given, where pt represents the price (or wage) required to induce an agent of type t to join a coalition. So far, we have not required that an agent of type t be willing to join a coalition in return for pt . Just as in standard competitive equilibrium theory, where agents take prices of goods as given, here agents take prices of other agents as given. Given p, a representative agent of type t determines the maximum pro…t she (or he) could obtain by forming a …rm, after paying all the other agents in the …rm their stated prices. Let gt (p; r) denote this maximum. Formally, T X gt (p; r) = maxfv(s) ¡ pt0 st0 ¡ (st ¡ 1)pt : s 2 Ir (t)g: s t0 =1 t6=t0 11 If gt (p; r) = pt for all types of agents, p is called a quasi-equilibrium price. Given a quasi-equilibrium price, p, let s 2 I be such that v(s) = p ¢ s; v(s) is su¢ciently large so that the total pro…ts attainable by a coalition with pro…le s equals the sum of the quasi-equilibrium price for each type times the number of agents of each type in s. If v(s) = p ¢ s then s is called a distinguished pro…le relative to p and when S ´ s, S is called a distinguished set relative to p. Also, p ¢ s ¸ v(s) for all s 2 Ir . However, given a quasi-equilibrium price, it might not be possible to partition agents into distinguished sets. This is the cause of the possible emptiness of the core of (Nr ; v). ¹ The proofs of the theorems, described previously for the one-type case, for the case of T types are extensions of those for the one-type case but signi…cantly more di¢cult to prove. 2 III. The Model and Its Properties 2.1 A. The Model We take as given a function v mapping I into the non-negative reals. Given s 2 I, v(s) represents the “payo¤” to any coalition S with pro…le s. We now impose restrictions on the function v to ensure the existence of “small” distinguished coalitions. It is primarily the large economy results that depend on the following assumption. It is assumed that (*): whenever p = (p1; :::; pT ) 2 -T , for each t0 2 f1; :::; T g, there exists s0 = (s01; :::; s0T ) 2 I1(t0 ) so that P P v(s0 ) ¡ s0t pt v(s) ¡ st pt t6=t0 t6=t0 0 ¸ st0 st0 for all s = (s1 ; :::; sT ) 2 I(t0 ): Informally, it will become apparent that assumption (*) ensures that, for each type, there will be some distinguished pro…le which contains no more players of any type than those contained in N1 and limits the extent of increasing returns to coalition size.9 Given the set of agents of the r-th replica economy, Nr , we interpret the pair (Nr ; v j Nr ) as a game in characteristic function form (D.1), where v j Nr (S) = v(s) 12 for any S µ Nr and where s 2 I is the pro…le of S. For ease in notation, we will denote the game (Nr ; v j Nr ), its superadditive closure game (D.2), and its balanced closure game (D.13) by (Nr ; v); (Nr ; v ) and (Nr ; v ) respectively. ¹ ~ It is easily veri…ed that all players of the same type are substitutes (D.9) (but it is not required that players of di¤erent types are not substitutes). Given the set of players Nr , a price for a player of type t, pt , is a non-negative real number and a price vector is denoted by p = (p1 ; :::; pT ) 2 -T . Given p 2 -T , let X gt(p; r) = maxfv(s) ¡ st0 pt0 ¡ (st ¡ 1)pt : s = (s1; :::; sT ) 2 Ir (t)g: s t0 6=t The value of the function, gt (p; r) is interpreted as the maximum “payo¤” attainable by an agent of type t, given prices for other types of agents. De…ne the function g(p; r) mapping -T into -T by g(p; r) = (g1 (p; r); :::; gT (p; r)): Let Pr¤ = fp 2 -T : g(p; r) = pg: The set Pr¤ consists of “equilibrium” price vectors for types of agents, except that p¤ might not satisfy the condition that v (Nr ) = rp¤ ¢ m for any p¤ 2 Pr¤. In other words, ¹ it might not be feasible for each agent (t; q) to be paid p¤ for all (t; q). t We call Pr the set of quasi-equilibrium prices for (Nr ; v). When assumption (*) ¤ ¹ holds, Pr¤ = Pr¤0 for all r; r0 so we let P ¤ = Pr¤ and call P ¤ the set of quasi-equilibrium prices. The sets Pr¤ and P ¤ are of interest for several reasons; they are non-empty (as will be shown in Appendix II); they contain the equal-treatment imputations (D.10) in the core (D.6) of (Nr ; v); given (*), the "-core converges to a subset of P ¤ ; and they ¹ are familiar kinds of constructs to an economist. A subset of Pr¤ will be of particular interest. Let Pr¤¤ = fp 2 Pr¤ : for all p0 2 Pr¤ ; m ¢ p · m ¢ p0 g: The set Pr¤¤ will be shown to be equivalent to the equal-treatment imputations in the core of (Nr ; v) when the core is non-empty, and will also be related to the balanced ¹ closure game (Nr ; v). ~ When (*) holds, let Pr¤¤ = P ¤¤ for all r (Pr¤ = Pr¤0 implies Pr¤¤ = Pr¤¤). 0 In this paper, we consider only cores of the superadditive closure game (Nr ; v ), ¹ rather than cores of the game (Nr ; v). If, however, (Nr ; v) is superadditive (D.2), v = v. In investigations of the cores of games, it is often assumed that the game ¹ is superadditive; if a game is not partially superadditive (D.4) then it is immediate 13 that the core is empty. A rationale for the assumption of superadditivity is that one possibility open to a coalition S, is to form to disjoint subcoalitions, say S 0 and S 00 , where S 0 [ S 00 = S, so v(S) ¸ v(S 0 ) + v(S 00 ). The interest of this author in games grew out of work with economies where the properties of partitions which achieve v(N) are of interest. Consequently, rather than assuming (N; v) is superadditive, a ¹ distinction is maintained between v and v. ¹ 2.2 B. Replication Results The following two theorems show the nature of the convergence of the "-core to the set of quasi-equilibrium prices. It is for these theorems that assumption (*) is required. In the view of the author, theorem 1 is the most important result of this paper. Theorem 1 Assume (*) holds. 1.1 Given any " > 0, there exists an r0 su¢ciently large so that if r ¸ r0 , the "-core of (Nr ; v ) is non-empty. ¹ 1.2 Given any ± > 0 and any ¸ > 0, there exists an "0 and an r0 so that for all r ¸ r0 and for all " 2 [0; "0 ], if x 2 -Nr is in the "-core of (Nr ; v), then for some ¹ p¤ 2 P ¤ , 1 j f(t; q) 2 Nr :jj xtq ¡ p¤ jj> ±g j t <¸ j Nr j (given any ± > 0 and any ¸ > 0 , there exists some "0 and r0 so that if r ¸ r0 and " · "0 , then every imputation in the "-core of (Nr ; v) has the property that ¹ the percentage of agents whose imputations di¤ers by more than ± from some quasi- equilibrium price is less than ¸). 1.3 Let p¤ 2 P ¤¤ . Then there are sequences ("` ); (r` ), and (x` ), where "` ! 0 and r` ! 1 so that (a) x` 2 -Nr` is in the "` -core of (Nr` ; v), and ¹ (b) given any ± > 0 and any ¸ > 0, there is an `0 such that for all ` > `0 , j f(t; q) 2 Nr :jj x` ¡ p¤ jj> ±g j tq t <¸ j Nr` j (given any p¤ 2 P ¤¤ , there is a set of "-core imputations which converge to p¤ in the sense of theorem 1.2). All formal proofs are contained in Appendix II. It will be shown that when (*) holds, if r ¸ 2 and the core of (Nr ; v ) is non-empty, ¹ then the core and P ¤¤ are equivalent. Also, when (*) holds, it will be shown that the core of (Nr ; v) is non-empty for a subsequence of r. Consequently, theorem 1 can be ¹ 14 interpreted as showing the convergence of the "-core to the core for a subsequence of replica economies. However, this interpretation only says that the "-core is “close” to the core for economies in the subsequence of economies where the core is non-empty. Consequently, we …nd the original statement more satisfactory since it holds for all su¢ciently large replica economies. In addition, when (*) holds, the equal-treatment core imputations of the balanced closure game are equivalent to P ¤¤ . Consequently, theorem 1 can be interpreted as showing that “large” economies are “nearly” balanced, since it shows that, given ", for all su¢ciently large economies, the "-core is non-empty and, given any imputation in the core of the balanced closure game, say p¤ , there is a sequence of "-core imputations which converges to p¤.10 Theorem 2. Given a sequence of replica economies (Nr ; v ) satisfying assumption ¹ (*), there exists a subsequence of replica economies (Nr` ; v ) such that every economy ¹ in the subsequence has a non-empty core. 2.3 Relationships to the Core In this section we formally de…ne “distinguished sets,” and state conditions for the non-emptiness of the core both in terms of these sets and in terms of the quasi- equilibrium prices. The “distinguished set” condition appears to be particularly in- teresting and plays an important role in the proof and interpretation of theorem 1. Also, we relate the core of (Nr ; v) to Pr¤ and to Pr¤¤ . ¹ Given p 2 Pr , let ¤ ¹t0 (p; r) = fs = (s1; :::; sT ) 2 Ir (t0 ) : P v(s) ¡ t6=t0 st pt ¡ (st0 ¡ 1)pt0 = gt0 (p; r)g: An element of ¹t0 (p; r) is called a distinguished for type t0 relative p. As previously stated, when (*) holds, Pr¤ = Pr¤0 = P ¤ . We remark that from this conclusion, as will be shown in the appendix, it follows that for any p 2 Pr¤ there is an s 2 ¹t0 (p; r) such that s · m. Theorem 3. The core of (Nr ; v) is non-empty if and only if ¹ 3.1 there is a p 2 Pr¤ so that v(Nr ) = rp ¢ m; ¹ 3.2 there is a p 2 Pr¤ and a collection of distinguished pro…les fs1; :::sk g relative to p so that K X rm = sk :11 k=1 15 Informally, the …rst part of Theorem 3 says that the core is non-empty if and only if v (Nr ) is su¢ciently large so that each player can be imputed a quasi-equilibrium ¹ price for (Nr ; v ). ¹ The second part of the theorem relates the existence of the core to the number of players of each type. In terms of distinguished sets, the core is non-empty if and only if Nr can be partitioned into a collection of distinguished sets relative to some p 2 Pr¤ . Informally, the distinguished sets can be viewed as ‘e¤ective’ coalitions in the sense that if an imputation can be blocked by a coalition S, then it can be blocked by a distinguished set relative to some p 2 Pr¤ . Alternatively, if a coalition is considered to be a …rm, then a distinguished set is an “optimal” …rm relative to a quasi-equilibrium price for (Nr ; v ). ¹ Theorem 4. Assume that the core of (Nr ; v) is non-empty. ¹ 4.1 If x = (x11 ; :::; xTmT r ) is an equal-treatment imputation in the core of (Nr ; v ), ¹ then p = (p1 ; :::; pT ) 2 Pr¤¤ , where pt = xtq for all t; 4.2 When (*) holds, r ¸ 2, and x is in the core of (Nr ; v ), then x is an equal- ¹ treatment imputation. Informally, if the core is non-empty, (a) the equal-treatment core imputations are equivalent to Pr¤¤, and (b) when (*) holds and r ¸ 2; P ¤¤ and set of core imputations are equivalent. 2.4 The Balancedness Condition It is well known that the core of a game in characteristic function form is non-empty if and only if the game is balanced12 (D.12). However, it also seems to be generally agreed that the balancedness condition has no intuitive economic interpretation. The following theorems (and also the examples in the next section) provide additional insight into the balancedness condition, particularly for replica economies. Although we state the theorems in terms of (Nr ; v), they hold for any partially ¹ superadditive game (D.4) and assumption (*) is not required. As in de…nition 13, let P v(S) = maxf ~ wS 0 v(S) : ¯ is a balanced family of subsets of S ¯ S 0 2¯ with weight wS 0 for S 0 2 ¯g: 16 Theorem 5. 5.1 For any game (Nr ; v ), v (Nr ) = rp ¢ m for all p 2 Pr¤¤; ¹ ~ P 5.2 Whenever wS v(S) = v(Nr ) for the balanced collection ¯ with all non-zero- ~ S2¯ weights, wS , then for any p 2 Pr¤¤ the elements of ¯ are distinguished sets relative to p (the only critical balanced collections of sets are distinguished collections of sets). Given S µ Nr , let fS 1 ; :::; S K g be a balanced family of subsets of S with weights wk for S K , k 2 f1; :::; Kg. Since K X wk S k = 1; k=1 (t;q)2S k mt r X K X wk S k =j S \ [t]r j; q=1 k=1 (t;q)2S k the number of agents of type t in S. Consequently, it is natural to de…ne a subpro…le of s 2 I as a pro…le s0 where s0 · s and a balanced family of subpro…les of s 2 I as a collection of subpro…les of s, say fs1; :::; sK g, so that for some w1 ; :::; wK 2 -1 , K X wK sk = s: k=1 If fS 1; :::; S K g is a balanced family of subsets of S ´ s with weights w1 ; :::; wK , then fs1; :::; sK g, where S k ´ sk , is a balanced family of subpro…les of s with weights w1; :::; wk . In addition, if fs1; :::; sK g is a balanced family of distinct subpro…les of s 2 Ir with weights wk for sk , then there is a balanced collection of subsets of 0 S (where S ´ s), say fS 1 ; :::; S ` ; :::; S L g, with weight w` for S ` so that S ` ´ sk for 0 some k 0 2 f1; :::; Kg, and w` = wk divided by the number of members of fS 1 ; :::; S L g 0 with pro…le sk for each `.13 Also, P v (S) = maxf ~ S ws0 v(s0 ) : ¯ is a balanced family ¯ S 0 2¯ 0 of subpro…les of s ´ S with weights ws0 for s g ~ = v (s): We remark that balancing weights for subpro…les need not be less than or equal to one. 17 To prove theorem 1.1, consider the problem: minimize p ¢ m subject to p ¢ s ¸ v(s) for all s 2 Ir and p ¸ 0: The set of solutions to this problem is Pr¤¤. The dual linear programming problem is: K P maximize wk v(sk ) k=1 K P subject to wk sk = rm; k=1 and w1 ; :::; wk ¸ 0: From the fundamental duality theorem of linear programming, K X min p ¢ m = max wk v(sk ) = v (Nr ) e k=1 (subject to the constraints) so 5.1 is proven. The second part of the theorem, 5.2, follows from Theorem 3, and it is formally proven in Appendix II. Informally, suppose, for example, v(s) represents the output of a …rm with pro…le s and we want to minimize wages subject to the condition that there will be no positive net bene…t to any collection of workers in the formation of a new …rm. A solution to this problem is an element of Pr¤¤. The dual problem is to determine the optimal number of …rms with each given pro…le to maximize output. A solution to the dual exists, say w1 ; :::; wK . However, ¤ ¤ wk represents the optimal number of …rms with pro…le sk . If wk is not an integer, we ¤ ¤ require some fraction of a …rm to maximize output. When we consider the superaddi- tive closure game (or any super-additive game) it follows that the core is non-empty if and only if there exists a solution to the dual where all weights wk are integers. Con- ¤ sequently, the assumption of balancedness of a game is equivalent to the assumption that the set of agents can be partitioned into distinguished sets. When (*) holds, the assumption of balancedness is thus seen to be closely analo- gous to the assumption that an industry characterized by a large number of identical …rms with U-shaped average cost curves has a constant supply curve, and the equi- librium price of the industry output is the minimum average cost of a …rm in the industry. Just as the assumption of U-shaped cost curves limits the extent of increas- ing returns to scale, assumption (*) ensures that there is a limit to increasing average returns to coalition size. Our …nal theorem relates Pr¤¤ and P ¤¤ to the core imputations of (Nr ; v ). ~ 18 Theorem 6. Let x be in the core of (Nr ; v): ~ 6.1 If x has the equal-treatment property, then for some p 2 Pr¤¤ , for each t; xtq = pt for all (t; q). 6.2 If (*) holds and r ¸ 2, then x has the equal treatment property so, for some p 2 P ¤¤ , for each t, xtq = pt for all (t; q). 3 Numerical Examples In this section, we provide three examples illustrating our results. In the …rst example, some of the concepts and theorems are illustrated for a game where no players are substitutes. Our second example is the opposite extreme; all players are substitutes: this example highlights the limit properties of the replication economy. The third example, with two types of players and more than one player of each type is used to illustrate most of our results and in particular the result that for any game there exists a balanced collection of distinguished sets (a conclusion which follows from Theorem 5). Example 1. In this example, we consider a superadditive game with only one player of each type, to illustrate our results for this special case. Since we assume there is only one player of each type, we denote the set of players by N = f1; 2; 3g. Let v(fig) = 0 for all i 2 N ; v(f1; 2g) = 7 ; 8 v(f1; 3g) = 3 ; 4 v(f2; 3g) = 5 ; 8 v(f1; 2; 3g) = 1: It can be easily veri…ed that ¤¤ 1 3 3 P1 = f( ; ; )g: 2 8 4 Since there is only one player of each type, the pro…le of S µ N is a vector in I of zeros and ones with the t-th coordinate equal to one if the player t 2 S and zero otherwise. 19 The distinguished pro…les for each type t relative to p¤ = ( 2 ; 3 ; 1 ) are: 1 8 4 ¹1(p¤ ) = f(1; 1; 0); (1; 0; 1)g ¹2(p¤ ) = f(0; 1; 1); (1; 1; 0)g ¹3 (p¤ ) = f(0; 1; 1); (1; 0; 1)g: 3 P It is immediately apparent that p¤ > v (N) and that there does not exist a t ¹ t=1 distinguished collection of pro…les relative to p¤, say s1 ; :::; sk , so that K P sk = (1; 1; 1). k=1 To illustrate that the game is unbalanced, consider the collection of distinguished pro…les f(1; 1; 0); (1; 0; 1); (0; 1; 1)g or, equivalently, the collection of distinguished sets ff1; 2g; f1; 3g; f2; 3gg: This is a balanced collections of sets with the weight 1 for each set, and as in Theorem 2 5, T 1 9 X ¤ [v(f1; 2g) + v(f1; 3g) + vf(2; 3)g] = = p: 2 8 t=1 t Example 2. We now consider the other extreme special case where all players are substitutes (of the same type). This example provides insight into all our theorems, particularly Theorem 1, and also demonstrates the existence of a balanced collection of distin- guished sets. Since there is only one type of consumer, the pro…le of a set is simply a non- negative integer. We take as given the following function v: I ! -1. ( ³cn3=2 ´ if n · n ¤ v(n) = 1 max cn(2n ¡ n 2 ); 0 if n > n ¤ ¤ where c and n¤ are parameters. If m · n¤, then m is a distinguished pro…le and the core of (N1 ; v ) is non-empty. ¹ If m > n , then the core is non-empty if and only if m = `n for some integer `, ¤ ¤ and every imputation in the core has the equal-treatment property. We illustrate the construction of the distinguished sets and balancing weights which ‘unbalance’ the game if the core is empty. If there are m players, where ³ ´13 m m > n¤, we can construct n¤ distinct sets of players, say fS1; :::; Sk ; :::; SK g, ³ ´ n¤ m each containing n¤ players. Each player is contained in m n¤ of these sets. Let Ã ! n¤ m ¡1 w=[ ] : m n¤ 20 Then the collection fS1 ; :::; Sk ; :::; SK g is a balanced collection of distinguished sets ¹ with weights w for each set. It can be veri…ed that, as in Figure II, v (n) ! cn¤1=2 as n n ! 1. Example 3. In the preceding two examples, because of their special features, it was easy to construct balanced collections of distinguished sets. The main purpose of this example is to illustrate the construction of such a collection of sets for a more challenging case (this example, with two types of players and some distinguished set which contains both types, is su¢ciently challenging). Again we represent the characteristic function as a mapping from pro…les to -1 . Let ( n2 1 if n1 · n1¹ v(n1 ; 0) = maxfn1 (2¹ 1 ¡ n1 ); 0g if n1 < n1 n ¹ ( 2n2 2 if n2 < n2 ¹ v(0; n2 ) = maxf2n2(2¹ 2 ¡ n2); 0g if n2 < n2 n ¹ ( 3n2 + 4n2 1 2 if 0 < n1 · n¤ and 0 < n2 · n¤ 1 2 v(n1; n2 ) = maxf6n¤2 + 8n¤2 ¡ 3n2 ¡ 4n2; 0g otherwise. 1 2 1 2 where n1; n2 ; n¤ ; n¤ > 0 are given parameters. ¹ ¹ 1 2 It is immediate that minfp1 : (p1 ; p2 ) 2 P ¤ g ¸ n1 ¹ and similarly minfp2 : (p1 ; p2 ) 2 P ¤ g ¸ 2¹ 2: n Consequently, to get (n¤ ; n¤ ) to be a distinguished pro…le relative to some p 2 P ¤ , 1 2 we must select n¤ and n¤ su¢ciently large relative to n1 and n2. It is convenient to 1 2 ¹ ¹ select n¤ > n1 and n¤ > n2. Now 1 ¹ 2 ¹ ¤¤ P1 = f(p1; p2) : p1 ¸ n1 ; p2 > 2¹ 2 ; and p1 n¤ + p2 n¤ = 3n¤2 + 4n¤2g: ¹ n T 2 1 2 players ´ Given m1 ³ ´ ³ of type 1, and m2 players of type 2, where m1 ¸ n¤ , m2 ¸ n¤ , 1 2 we can form m¤1 m¤ distinct distinguished sets relative to (any) p¤ 2 P ¤ so that n n 2 1 2 21 each of these sets contains n¤ players of type 1 and n¤ players of type 2.14 Denote 1 2 these sets by S1; :::; SK1 . Each player of type 1 appears in Ã !Ã ! n¤ m1 1 m2 = c1 m1 n¤ 1 n¤ 2 of these sets and each player of type 2 appears in Ã !Ã ! n¤ m1 2 m2 = c2 m2 n¤ 1 n¤ 2 of them. Let 1 1 w1 = minf ; ): c1 c2 Suppose w1 = c11 . Then if we assign the weight w1 to each of the sets Sk , k 2 f1; :::; K1 g, the sum of the weights on the sets containing any player of type 1 is one. However, the sum of the weights on the sets containing a player of type 2 is only c2 · c1 c2 1. If c1 = 1, we’re done. If not, consider the pro…le (0; n2 ), which is a distinguished ¹ pro…le relative to p0 = (p01 ; p02 ) 2 P ¤ where 1 p01 = (3n¤2 + 4n¤2 ¡ 2¹ 2n¤ ) 1 2 n 2 n¤ 1 and p02 = 2¹ 2 : n ³ ´ We can then construct m2 distinct distinguished sets relative to p0 , containing only n ¹ 2 member of type 2, say fSk1+1 ; :::; SK2 g, so that each player of type 2 appears in ³ ´ n2 m2 ¹ m2 n2 ¹ = c3 of these sets. We solve the following equation for w2 : c2 (1 ¡ ) = w2c3 : c1 Substituting for c1, c2 and c3 , we obtain m2n¤ ¡ m1 n¤ 1 2 w2 = ³ m2 ´ : n¤ n2 1¹ ¹ n2 The reader can verify, by substituting in the following expression for K1 ; K2 ; w1 and w2, that K1 X K2 X w1 v(Sk ) + w2 v(Sk ) = m1 p01 + m2 p02 k=1 k=K1 +1 and that the collection fS1 ; :::; SK2 g is balanced. 22 4 Conclusions There is a vast literature on games in characteristic function form and the represen- tation of economies as games. It is primarily directed towards obtaining economic conditions under which the core of the associated game is non-empty and studying the properties of the associated game (see, for example, [1, 2, 12, 20, 21, 22, 23 and 24]). This paper is more closely related to [25], in that it is concerned with economies where the core might be empty for any given …nite economy and with limit prop- erties of such economies. Rather than investigating su¢cient conditions to obtain a non-empty core, or almost equivalently, to obtain “balancedness” of the economy represented as a game, it is shown that large economies are nearly balanced. Also, there are a number of papers on coalition production economies, where “balancedness” has been assumed to attain existence of an equilibrium and the core (see [3, 8, and 11]). Again, in [30] and [31], in which the results of this paper are utilized, the author shows that large coalition production economies are nearly balanced. Another paper, by Bohem [4], on replica coalition production economies is closely related to this paper and [30]. The main theorem of that paper is that if an allo- cation is in the core for all replications of the economy, then it is an equilibrium allocation, an Edgeworth-type equivalence theorem. Again, our work di¤ers in that we are concerned with economies where existence of the core can be shown only for a subsequence of economies and limit properties of approximate equilibria. Also, in terms of productions economies, this work is related to that of Rashid [10], in that Rashid shows that “large but …nite economies (with productions) must possess approximate competitive equilibria with the degree of approximation getting better as the economy gets larger.” Rashid, however, assumes additivity of the production correspondence; an assumption which is not required either in this paper (as it applies to coalition production economies) or in [29, 30]. The only assumption on the model of this paper is (*). An assumption analogous to (*) was used by this author in [27] and [29] to di¤er- entiate economies with a local public good from those with a pure public good. This assumption limited the extent of increasing returns to coalition size and permitted a convergence theorem rather than a Muench-type counterexample [14]. An assumption similar to (*) is also implicit in Boehm [4] when he requires for feasibility that the output y is produced from the technology available to the coalition of the whole, y 2 Y [Nr ], and that Y [Nr ] = rY [N1] (using the notation of Section II). Given equilibrium prices of marketed goods, as in Section II, this assumption ensures that v(Nr ) = rv(N1 ). The existence of Boehm’s equilibrium then implies that v(N1) = v(N1 ) so all possibilities for increasing average returns to coalition size ~ are exhausted by the economy N1. Consequently, when the equilibrium exists (or, alternatively, some imputation is in the core of all the replica economies), any set 23 with the same pro…le as N1 is a distinguished set and the economy, relative to the equilibrium prices for marketed goods, satis…es assumption (*). Since part of the purpose of this paper is the investigation of su¢cient conditions for the existence of approximate cores, and since it is not superadditivity but rather lack of su¢ciently strong superadditivity which prevents the existence of the core, assumption (*), limiting the superadditivity, does not seem too restrictive. In conclusion, we reiterate our comments in the introduction; in view of this author’s other work and this paper, it appears that the techniques of this paper can be extended to a large class of economies. In particular, we conjecture that all members of a “large” class of “large” coalition economies are “nearly” balanced. 5 Appendix I The following game-theoretic de…nitions (which are mostly well-known) and remarks are required. D.1. A game in characteristic function form, or simply a game, is an ordered pair (N; v) where N is a …nite set, called the set of players, and v, called the characteristic function, is a function from subsets of N to the non-negative real num- bers where v(;) = 0. D.2. Given the game (N; v), the superadditive closure game of (N; v) is the ordered pair (N; v ) where ¹ X v (S) = max ¹ v(S 0 ): S2P [S] S02S D.3. A game (N; v) is superadditive if, for all S µ N; T µ N, where S \ T = ;; v(S [ T ) ¸ v(S) + v(T ): Remark. A superadditive closure game is itself a game. Consequently, any de…nitions for a game (N; v) can equally well be applied to the superadditive closure game (N; v ). ¹ D.4. A game (N; v) is partially superadditive, if K X v(N) ¸ v(Jk ) k=1 for every partition fJ1 ; :::; Jk ; :::; JK g of N . 24 D.5. An imputation for the game (N; v), where N = f1; :::; i; :::; ng, is a vector x = (x1; :::; xn ) 2 -n such that n X xi = v(N): i=1 D.6. The core of the game (N; v) consists of the set of imputations for the game, x = (x1; :::; xn ), such that X xi ¸ v(S) i2S for all S µ N. Remarks. If (N; v) is not partially superadditive, the core of (N; v) is empty. Also, it can easily be veri…ed that x is in the core of (N; v) if and only if x is an imputation for ¹ P (N; v) and ¹ xi ¸ v(S) for all S µ N. In other words, if, for some imputation x of i2S P the game (N; v ), there exists S µ N so that ¹ xi < v(S), then there exists S 0 µ N ¹ P i2S so that xi < v(S 0 ). i2S D.7. Given " > 0, the "-core of the game (N; v) consists of all imputations x = P (x1 ; :::; xn ) such that xi ¸ v(S) ¡ "jSj for all S µ N. i2S 0 Remark. For the purposes of this paper, an alternative de…nition of the "-core (D.7 ) would have been equally fruitful. However, we have stated and proved our results using D.7, and the reader can verify that all results would continue to hold using 0 D.7 . P D.70 . Let x 2 -n satisfy v(N) ¡"jNj · xi · v(N). Then x is in the "-core of i2S (N; v) if X xi ¸ v(S) ¡ "jSj i2S for all S µ N . P P D.8. Let x 2 -n satisfy xi · v(N ). Then x is in the strong "-core if x ¸ v(S)¡ i2S i2S " for all S µ N. D.9. Players i and j are substitutes if, for all S µ N; i 2 S; j 2 S; = = v(S [ fig) = v(S [ fjg): 25 Remark. The relation “substitutes” is an equivalence relation on the set N. D.10. An imputation x = (x1 ; :::; xn ) has the equal-treatment property if xi = xj whenever i and j are substitutes. D.11. Given a set S, consider a family ¯ of subsets of S, and let ¯ i = fS 2 ¯ : i 2 Sg; i.e., we take a collection of coalitions, ¯, and denote by ¯ i all the coalitions in ¯ to which agent i belongs. A family ¯ of subsets of S is balanced if there exists P non-negative ‘balanced weights’ wS 0 , for all S 0 2 ¯ where wS 0 ; = 1 for all i 2 S. S 0 2¯ i D.12. A game (N; v) is balanced if and only if for every balanced family ¯ of subsets of N with weights wS 0 for S 0 2 ¯, X v(S 0 )wS 0 · v(N): S 0 2¯ D.13. Given the game (N; v) the balanced closure game is the ordered pair (N; v) ~ where P v(S) = maxf ~ wS 0 v(S 0 ) : ¯ is a balanced family of subsets ¯ S 0 2¯ of S with balancing weights wS 0 for S 0 2 ¯g: v is called the balanced closure characteristic function. ~ Remark. When v = v , the game (N; v) is called totally balanced. ~ D.14. Given the game (N; v) player i is null if v(S [ fig) = v(S) for all S µ N. 6 Appendix II Before proceeding with the proofs, some additional notation is introduced. Given an imputation x 2 -Nr , let X x(S) = xtq ; tq2S mt r X ¹ xt = xtq ; q=1 26 and x = (¹1; :::; xT ) 2 -T : ¹ x ¹ The vector x represents an imputation with the equal-treatment property. Given ¹ s 2 Ir (t), let s)t( = (s1 ; :::; st¡1 ; 0; st+1 ; :::; sT ): Let ¹ = (1; 1; :::; 1) 2 -T . We remark that some of the following proofs might have been more easily ob- tained using existing results concerning the core and balanced games; however, in the view of this author the results are more intuitively obtained using the concepts of distinguished pro…les and quasi-equilibrium prices. Also, we will occasionally use the fact that p 2 Pr¤ if and only if v(s) · p ¢ s for all s 2 Ir and, for each t, there exists s 2 Ir (t) such that v(s) ¡ p ¢ s = 0. This is easily veri…ed. Lemma 1. The core of (Nr ; v) is non-empty if and only if it contains an equal- ¹ treatment imputation. Proof. Let x be in the core of (Nr ; v). Now suppose x is not in the core of (Nr ; v ). ¹ ¹ ¹ Then there exists S ´ s so that x ¢ s < v(s). Select S ¤ ´ s so that for each t, if ¹ (t; q) 2S ¤ , xtq ¸ xtq0 for all (t; q0 ) 2 S ¤ (S ¤ contains the “worst-o¤” agents). But then x(S ¤ ) · x ¢ s < v(S ¤) which contradicts the initial hypothesis. ¹ Clearly, if the core contains an equal-treatment imputation it is non-empty. Q.E.D. Lemma 2. Pr¤ 6= ;. Proof. Clearly, gt (p; r) · v(Nr ) for any p 2 -T and gt (p; r) ¸ v(f(t; q)g) ¸ 0. ¹ Consequently, we can restrict P to a compact, convex subset of -T , say P 0 . Let nt (p; s) = v(s) ¡ p ¢ s)t( ¡ (st ¡ 1)pt where p 2 P 0 , s 2 Ir (t). Observe that nt (¢; ¢) is a continuous function. It follows that gt (p; r) = max nt (p; s) s2Ir (t) is a continuous function of p (see Debreu [5], page 19). From the Brouwer …xed point theorem it follows that Pr¤ 6= ;. Q.E.D. Lemma 3. When (*) holds, Pr¤0 = Pr¤00 for all r 0 ; r00 . Proof. Suppose r0 < r00 and p = (p1 ; :::; pT ) 2 Pr¤0 but p 2 Pr¤00 . It is immediate that = g(p; r 00 ) ¸ g(p; r0 ) 27 (since the maximization is carried out over a larger choice set). Since p 2 Pr¤00 , there = exists t0 so that gt0 (p; r00 ) > gt0 (p; r0 ): Also, there exists s0 2 Ir0 (t0 ) and s00 2 Ir00 (t0 ) such that (a) gt0 (p; r 0 ) = v(s0 ) ¡ p ¢ s0)t0 ( ¡ (s0t0 ¡ 1)pt0 = pt0 and (b) gt0 (p; r 00 ) = v(s00 ) ¡ p ¢ s00 0 ( ¡ (s000 ¡ 1)pt0 > pt0 . )t t Rearranging terms, we obtain v(s00 ) ¡ p ¢ s00 0 ( )t v(s0 ) ¡ p ¢ s0 )t0 ( > = pt0 : s000 t s0t0 But v(s0 ) ¡ p ¢ s0 )t0 ( v(s) ¡ p ¢ s)t( 0 ¸ st0 st for all s 2 Ir0 (t0 ) so v(s00 ) ¡ p ¢ s00 0 ( )t v(s) ¡ p ¢ s)t( > s000 t st for all s 2 Ir (t0 ), which contradicts assumption (*). If p 2 Pr¤00 and p 2 Pr¤0 , since it is still true that g(p; r00 ) ¸ g(p; r 0 ) when r0 < r00 , = the above argument can be repeated with gt0 (p; r00 ) = pt0 > gt0 (p; r0 ) and again we obtain a contradiction. Q.E.D. The proof of the proceeding lemma shows that when (*) holds, if s · m is a distinguished pro…le relative to p 2 P1 , for (N1 ; v), then s is a distinguished pro…le ¤ ¹ relative to p 2 Pr for (Nr ; v) for all r. In other words, if p 2 P1 and ¤ ¹ ¤ gt0 (p; 1) = v(s) ¡ p ¢ s)t0 ( ¡ (st0 ¡ 1)pt0 ; then gt0 (p; r) = v(s) ¡ p ¢ s)t0 ( ¡ (st0 ¡ 1)pt0 : If this were not true, then it would be the case that gt0 (p; r) > gt0 (p; 1) for some r ¸ 2, which leads to the contradiction in the above proof. Theorems 3, 4, 5.2, 2, and 6 are proven, and then Theorem 1, since the proofs of Theorems 1 and 2 require some of the preceding theorems and Theorem 1 is the most di¢cult. Proof of Theorem 3. 3.1 Assume that the core of (Nr ; v ) is non-empty and let x represent and equal- ¹ ¹ treatment imputation in the core. Then 28 (a) for all s 2 Ir , v(s) · s ¢ x, or equivalently, for each t0 , ¹ xt0 ¸ v(s) ¡ s)t0 ( ¢ x ¡ (st0 ¡ 1)¹t0 ; ¹ ¹ x (b) v (Nr ) = r¹ ¢ m, or equivalently, for each t0 , ¹ x xt0 = v(Nr ) ¡ r¹ ¢ m)t0 ( ¡ (rmt0 ¡ 1)¹t0 : ¹ ¹ x x Therefore the maximum of the expression v(s) ¡ s)t0 ( ¢ x ¡ (st0 ¡ 1)¹t0 ¹ x on s 2 Ir (t0 ) is xt0 , and x 2 Pr¤. ¹ ¹ The converse can be established by following the above logic (in (a)) in the oppo- site direction. 3.2 Assume that s1; :::; sK are distinguished pro…les relative to p 2 Pr¤ where K X sk = rm: k=1 Note that K X v(sk ) = r p ¢ m: k=1 From the de…nition of v (Nr ), there exists a collection of pro…les, say s01 ; :::; s0` ; :::; s0L ¹ such that L X v(Nr ) = ¹ v(s0` ) `=1 and L X K X v(s0` ) ¸ v(sk ): `=1 k=1 If K X v(Nr ) > ¹ v(sk ); k=1 then L X K X L X v(s0` ) > p ¢ sk = rp ¢ m = p ¢ s0` : `=1 k=1 `=1 Consequently, for some ` , v(s0` ) > p ¢ s0` ; contradicting the assumption that p 2 Pr¤ , so v(Nr ) = r p ¢ m; ¹ 29 and, from 3.1, the core is non-empty. Assume that x is an equal-treatment imputation in the core of (Nr ; v). Then, as ¹ in the proof of 3.1, x 2 Pr . Since x is in the core, ¹ ¤ ¹ r x ¢ m = v(Nr ): ¹ ¹ Let s1 , . . . sK be pro…les so that K X v(Nr ) = ¹ v(sk ) k=1 and K X sk = rm k=1 (from the de…nition of v (Nr ) such a collection of pro…les exists). Since x is in the ¹ ¹ core, v(sk ) · x ¢ sk ¹ for all k so K X K X v(sk ) · x ¢ sk = v (Nr ) ¹ ¹ k=1 k=1 and it follows that v(sk ) = x ¢ sk : ¹ Consequently, sk is a distinguished pro…le relative to x 2 Pr¤ for all k. Q.E.D. ¹ Proof of Theorem 4. 4.1 Let x be an equal-treatment imputation in the core of (Nr ; v). As in the proof ¹ of theorem 3, x 2 Pr , so we need only show that x ¢ m · p ¢ m for all p 2 Pr¤ to ¹ ¤ ¹ obtain the conclusion that x 2 Pr¤¤ . Suppose x 2 Pr¤¤, which implies that there exists ¹ ¹= p 2 Pr¤ such that p ¢ m < x ¢ m. Since x is in the core, ¹ v(Nr ) = r¹ ¢ m ¹ x K P and for some collection of pro…les, say, s1 ; :::; sK where sk = rm, k=1 K X v(Nr ) = ¹ v(sk ): k=1 Consequently, K X rp ¢ m < v(sk ); k=1 30 from which it follows that for some sk , p ¢ sk < v(sk ); which contradicts the assumption that p 2 Pr¤ . 4.2 Assume (*) holds and r ¸ 2. Let x be in the core of (Nr ; v ). ¹ From the de…nition of the superadditive closure game, there exists some coalition structure, say Nr = fS1; :::; Sk ; :::; SK g, so that K X ¹ v(Nr ) = v(Sk ): k=1 Also, it is immediate that for some such coalition structure, x(Sk ) = v(Sk ) for all k. Suppose fS1 ; :::; SK g satis…es these two conditions. We will …rst show that if xt0 q0 > xt0 q00 for some q00 , then [t0 ]r µ Sk0 , for some k 0 . Next, this is used in conjunction with (*) to obtain a contradiction. Suppose that there exists Sk 0 ; Sk00 2 Nr ; Sk0 6=Sk00 ; (t0 ; q 0 ) 2 Sk0 ; (t0 ; q00 ) 2 Sk 00 , and = xt0 q0 > xt0 q00 . Let S = Sk 0 [ f(t0 ; q 00 )g \ ~f(t0 ; q 0 )g; S is simply Sk0 , with (t0 ; q0 ) replaced by (t0 ; q 00 ). Since (t0 ; q0 ) and (t0 ; q 00 ) are substitutes v(S) = v(Sk0 ) so v(S) > x(S), which is a contradiction. Suppose [t0 ] µ Sk0 for some k 0 . From (*), there exists s¤ 2 I1(t0 ) such that v(s¤) ¡ s¤ x)t0 ¹ ( v(s) ¡ s ¢ x)t0 ( ¹ > s¤0 t st0 mtr P for all s 2 Ir (t0 ) (¹t = x xtq =mtr and x = (¹1 ; :::; xT )). As shown in the proof ¹ x ¹ q=1 of theorem 3, x 2 Pr¤ and from Lemma 3, x 2 P ¤ since (*) is required to hold for ¹ ¹ theorem 4.2. Consequently, xt0 = v(s¤) ¡ x ¢ s¤ 0 ( ¡ (s¤0 ¡ 1)¹t0 ; ¹ ¹ )t t x and rearranging terms, v(s¤ ) ¡ x ¢ s¤ 0 ( ¹ )t ¹ xt 0 = : s¤0 t 31 Select S ¤ µ Nr so that S ¤ ´ s¤ and so that, for each t, if (t; q) 2 S ¤ , (t; q) · (t; q 0 ) for all (t; q 0 ) 2 Nr \ ~S ¤ . Observe that mtr X xt s¤ ¸ ¹ t xtq q=1 t;q2S ¤ for each t and, mt0 r X x ¹ t0 s¤0 t > xt 0 q q=1 0 t ;q2S ¤ since xt0 q0 > xt0 q00 : But then v(S ¤ ) = x ¢ s¤ > x(S ¤ ); ¹ which contradicts the assumption that x is in the core. Q.E.D. Proof of theorem 5.2 Assume that ¯ = fS 1, . . . , S L g is a balanced collection of subsets of Nr with weights w1 ; :::; wL respectively, where w` > 0 for all `. From theorem 5.1, L X w` v(S ` ) = r p ¢ m `=1 for p 2 Pr¤¤ . Let S ` ´ s` for all `. If, for some `0 , 0 0 v(S ` ) ¡ p ¢ s` < 0; then for some `00 , 00 00 v(S ` ) ¡ p ¢ s` > 0; contradicting the assumption that p 2 Pr¤¤ µ Pr¤ . Therefore, v(S ` ) = p ¢ s` and S ` is a distinguished set relative to p for all `. Q.E.D. Proof of theorem 2 Recall that when (*) holds, Pr¤¤ = Pr¤¤ for all r; r0 so if v(N1 ) = p ¢ m then 0 v(Nr ) = rp ¢ m. ~ Let s1 , . . . , sK be a balanced collection of distinguished subpro…les of m with weights w1 ; :::; wK so that K X wk v(sk ) = p ¢ m k=1 32 where p 2 P ¤¤ . Since w1; :::; wK can be viewed as a solution to a linear programming problem where the constraint inequalities have all integer coe¢cients, we can assume that w1 ; :::; wK are all rational numbers. Consequently there is an r0 so that ro wk is an integer for all k. Then s1 ; :::; sK is a balanced collection of subpro…les of r0m K P with weight (r0wk ) for sk ( r0wk = r0 m). Since r0wk is an integer for all k, from k=1 theorem 3.2, the core of (Nr0 , v) is non-empty. It follows that the core of (Nr ,¹) is ¹ v non-empty for all r such that r = nr0 for some integer n. Q.E.D. Proof of Theorem 6. 6.1 Let x be an equal-treatment imputation in the core of (Nr ,~). Since x ¢ s ¸ v ¹ v(s) for all s 2 Ir , and obviously, v(s) ¸ v(s) for all s 2 Ir , x ¢ s ¸ v(s) for all s 2 Ir . ~ ~ ¹ Also, for some balanced collection of distinguished pro…les, say ¯ = fs1; :::; sK g, with corresponding weights w1 ; :::; wK , K X v(Nr ) = ~ wk v(sk ); k=1 and, since x is in the core, K X K X wk v(sk ) = r¹ ¢ m = x ¢ x ¹ wk sk : k=1 k=1 Consequently, K X wk (v(sk ) ¡ x ¢ sk ) = 0: ¹ k=1 Since wk ¸ 0 for all k and v(sk ) ¡ x ¢ sk · 0, it follows that x ¢ sk = v(sk ) for all k. ¹ ¹ Consequently, for each t, there exists s 2 Ir (t) such that v(s) ¡ x ¢ s = 0: ¹ From the remarks at the beginning of Appendix II, it follows that x 2 Pr¤ . From the ¹ property that r x ¢ m = v(Nr ); ¹ ~ and the fundamental duality theorem of linear programming, it follows that x 2 Pr¤¤ . ¹ 6.2 Assume x is in the core of (Nr ; v). Then, as in 6.1, x 2 Pr¤¤. Since (*) holds, ~ ¹ x 2 P ¤¤ and ¹ v(N1 ) = x ¢ m: ~ ¹ If x does not have the equal-treatment property, since r ¸ 2, there is a subset S ´ m so that x ¢ m > x(S), which is a contradiction. Q.E.D. ¹ Before proving theorem 1, we require some additional lemmas and de…nitions. The strategy of the proof of 1.1 is to show that given "0, we can impute pt ¡ " to each agent in a distinguished set relative to p where p 2 Pr¤ , and divide " times the 33 number of agents in distinguished sets amongst the agents in non-distinguished sets so that, for su¢ciently large r, each agent of type t is imputed at least pt ¡ ". However, if for some t, pt = 0, we cannot impute pt ¡ " to agents of type t. Consequently, we partition the set of agents into two groups; one group where pt > 0 for some p 2 P ¤ , and another where pt = 0 for all p 2 P ¤, and consider a game where only the coalitions contained in one group or the other are relevant. This is shown to not a¤ect the game in any substantive way. Let pt = maxfpt : p = (p1; :::; pT ) 2 P ¤g; ~ ¤p2P let T 0 = ft : pt > 0g; ~ and let T 00 = ft : pt = 0g: ~ We remark that t 2 T 00 does not imply that an agent of type t is null (D.14) but the converse holds; however, the following lemma shows these agents “might as well” be null. De…ne Ir (T 0 ) and Ir (T 00 ) by Ir (T 0 ) = fs 2 Ir : st00 = 0 for all t00 2 T 00 g and Ir (T 00 ) = fs 2 Ir : st0 = 0 for all t0 2 T 0 g; Ir (T 0 ) consists of all pro…les of subsets of Nr containing no agents of type t00 for all t00 2 T 00 and similarly for Ir (T 0 ). Construct another characteristic function as follows: ( 0 v(s) if s 2 Ir (T 0 ) [ Ir (T 00 ) v (s) = 0 otherwise. In other words, v0 (s) assigns v(s) to s if s 2 Ir (T 0 ) or s 2 Ir (T 00 ), and assigns 0 to all other pro…les (those pro…les, s, where for some t0 2 T 00 and t0 2 T 00 , st0 6=0 and st00 6=0). Let P 0 denote the set of quasi-equilibrium prices for the superadditive closure game (Nr ; v0 ) ¹ Lemma.4 P 0 = P ¤ . In addition, if p 2 P ¤ , for t0 2 T 0 there is an s 2 Ir (T 0 ) \ Ir (t0 ) such that v(s) ¡ p ¢ s = 0 and, for t00 2 T 00 , there is an s 2 Ir (T 00 ) \ Ir (t00 ) such that v(s)¡ p ¢ s = 0. Proof Given p0 2 P 0 and s = (s1 ; :::; sT ) 2 Ir , de…ne v00 (s) as follows: ( 00 maxf0; v(s) ¡ p0 sg when s 2 Ir (T 0 ) v (s) = 00 maxfv(s0 ) ¡ p0 ¢ s0 : s0 2 Ir and s0t00 = st00 for all t 2 T 00 g 0 S otherwise. 34 Informally, for all s 2 Ir (T 0 ), v00 (s) is the maximum of v(s0 ) over the set of all pro…les = s0 2 Ir where s0 has the same number of agents of each type t00 2 T 00 as s. If s 2 Ir (T 0 ), v00 (s) = 0. For the game (Nr ,¹00 ), for all t0 2 T 0 , all players of type t0 are null since v 00 0 v (s) = 0 for all s 2 Ir (T ) and, from the construction of v00 (¢), v00 (s + (0; :::; 0; s0t0 ; 0; :::; 0)) = v00 (s) for all pro…les (0; :::; 0; s0t0 ; 0; :::; 0) where t0 2 T 0 . Let P 00 denote the set of quasi-equilibrium prices for (Nr ,¹00 ). Given p00 2 P 00 , v 00 0 0 0 observe that pt0 = 0 for all t 2 T (since players of type t are null). We now show that given p00 2 P 00 , p = p0 + p00 2 P ¤ . First observe that for all s 2 Ir , v00 (s) ¡ p00 ¢ s · 0 so v(s) ¡ p0 ¢ s ¡ p00 ¢ s = v(s) ¡ p ¢ s · 0: Consequently, if p 2 P ¤ there is a t 2 T such that for all s 2 Ir (t), = v(s) ¡ p ¢ s < 0 (from the remark preceding Lemma 1) but v(s) ¡ p ¢ s < 0 implies that v(s) ¡ p0 ¢ s ¡ p00 ¢ s < 0 for all s 2 Ir (t), contradicting either the assumption that p0 2 P 0 or p00 2 P 00 . Therefore, p 2 P ¤. Consequently, Pt00 = 0 for all t 2 T 00 , and, since p00 = 0 for all t t 2 T 0 , p00 = 0. Then p = p0 so p0 2 P ¤ and P 0 µ P ¤ . Given p¤ 2 P ¤ , let fp 2 -T : p · p¤ ; and ¤= p ¢ s ¸ v(s) for all s 2 Ir (T 0 ) [ Ir (T 00 )g Note that ¤ 6= ; since p¤ 2 ¤. Let p0 2 ¤. We claim that p0 2 P 0 . Since p0 ¢ s ¸ v(s) for all s 2 Ir (T 0 ) [ Ir (T 00 ), if p0 2 P 0 , there is a t 2 T such that for all s 2 Ir (t), = v(s) ¡ p0 ¢ s < 0: But p¤ ¸ p0 implies that p¤ ¢ s ¸ p0 ¢ s and v(s) ¡ p¤ ¢ s · v(s) ¡ p0 ¢ s < 0 for all s 2 Ir (t), contradicting the assumption that p¤ 2 P ¤ , so p0 2 P ;. We now will show that p0 = p¤ so p¤ 2 P 0 . Suppose for some t, p0t < p¤ . But t p¤ = maxfv(s) ¡ p¤ ¢ s)t( ¡ (st ¡ 1)p¤ : s 2 Ir (t)g: t t S 35 Since p0 2 P 0 µ P ¤ (a shown above), 0 pt = maxfv(s) ¡ p0 ¢ s))t( ¡ (st¡1 )p0t : s 2 Ir (t)g: S Since p0 · p¤ , v(s) ¡ p0 ¢ s)t( ¡ (st ¡ 1)p0t ¸ v(s) ¡ p¤ ¢ s)t( ¡ (st ¡ 1)p¤ t for all s 2 Ir (t), so p0t ¸ p¤, which is a contradiction. Therefore p¤ 2 P 0 . t Since P 0 = P ¤ , the conclusion of the theorem is immediate from the de…nitions of v0 and P 0 . Q.E.D. To show that there is a p 2 P ¤¤ where pt > 0 for all t 2 T 0 , the following lemma is employed. Lemma 5. Let p(t) 2 P ¤¤ where pt (t) = pt (p(t) is a quasi-equilibrium price vector ~ where the t-th coordinate is maximal). Then any convex combination of p(1); :::; p(T ) is an element of P ¤¤ . Proof of Lemma 5. T P T P Let p = ¸t p(t) where ¸t ¸ 0, and ¸t = 1. t=1 t=1 Suppose there is an s 2 Ir so that v(s) ¡ p ¢ s > 0. But then T X T X ¸tv(s) ¡ ¸t p(t) ¢ s > 0 t=1 t=1 so, for some t0 , v(s) ¡ p(t0 ) ¢ s > 0 which is a contradiction to the assumption that p(t0 ) 2 P ¤¤ . Also, p ¢ m = p(t) ¢ m so p minimizes p0 ¢ m subject to the condition that p0 2 -T and p0 ¢ s ¸ v(s) for all s 2 Ir . Consequently, p 2 P ¤¤ . Q.E.D. It is immediate that there is a p 2 P ¤¤ where pt > 0 for all t 2 T 0 . Given the game (Nr ; v) and " ¸ 0, de…ne another game (Nr ; v" ) where v" (S) = maxfv(S) ¡ " j S j; 0g: Let (Nr ; v" ) denote the superadditive closure game of (Nr ; v" ) and let P ¤ (") denote ¹ the set of quasi-equilibrium prices for (Nr ; v" ). ¹ 0 Lemma 6. Select p0 2 P ¤¤ so that pt > 0 for all t 2 T 0 and select " > 0 so that " 0 < pt for all t 2 T 0 . Let p 2 P ¤ ("). Then p¤ 2 P ¤ where p¤ = pt + " t for all t 2 T 0 and p¤ = 0 otherwise. t 36 Proof Let p, p0 , and p¤ be as in the statement of the lemma. First, we observe that, from lemma 4, for each t 2 T 00 , there is an s0 2 Ir (t) \ Ir (T 00 ) so that v" (s0 ) = p ¢ s0 and p ¢ s ¸ v" (s) for all s 2 Ir (t). Also, for each t 2 T 0 , there is an s0 2 Ir (t) \ Ir (T 0 ) so that v" (s0 ) = p ¢ s0 and p ¢ s ¸ v" (s) for all s 2 Ir (t). From the de…nition of p¤, for all s 2 Ir (T 0 ), p¤ ¢ s = p ¢ s + " j s j : Since p ¢ s ¸ v" (s) ¸ v(s) ¡ " j s j for all s 2 Ir (T 0 ), we have p¤ ¢ s ¸ v(s) for all s 2 Ir (T 0 ). Suppose, for some t0 2 T 0 , p¤ ¢ s > v(s) for all s 2 Ir (t0 ). Then p ¢ s + " j s j> v(s) for all s 2 Ir (t0 ) \ Ir (T 0 ). This implies that p ¢ s = 0 > v(s) ¡ " j s j for all s 2 Ir (t0 ) \ Ir (T 0 ). However, since p0 2 P ¤¤ where p0t > 0 for all t 2 T 0 and p0t ¸ " for all t 2 T 0 , we have p0 ¢ s ¡ " j s j¸ 0 > v(s) ¡ " j s j for all s 2 Ir (t0 ) \ Ir (T 0 ), so p0 ¢ s > v(s) for s 2 Ir (t0 ) \ Ir (T 0 ). This is a contradiction since p0 2 P ¤ , and, from lemma 4, there is an s0 2 Ir (t0 ) \ Ir (T 0 ) so that v(s0 ) = p0 ¢ s0 . Therefore, for each t 2 T 0 , there is an s0 2 Ir (t) so that p¤ ¢ s0 = v(s0 ). 37 Again, from the de…nition of p¤ , for all s 2 Ir (T 00 ), p¤ ¢ s = p ¢ s: Also, from the observation that v(s) = 0 for all s 2 Ir (T 00 ), v" (s) = v(s) = 0 for all s 2 Ir (T 00 ). Consequently, p¤ ¢ s = v(s) for all s 2 Ir (T 00 ). Now if t00 2 T 00 and s 2 Ir (T 00 ), clearly p¤ ¢ s ¸ v(s): If t00 2 T 00 and s 2 Ir (t00 ) where s 2 Ir (T 00 ), then, since there is a t0 2 T 0 so that = s 2 Ir (t ), p ¢ s ¸ v(s) from the preceding argument. Consequently, for each t00 2 T 00 , 0 ¤ p¤ ¢ s ¸ v(s) for all s 2 Ir (t00 ). Q.E.D. Observe that if " < pt for all t 2 T 0 where p = (p1; :::; pT ) 2 Pr¤, the set of quasi-equilibrium prices for (Nr ; v" ), is equal to P ¤("), where ¹0 0 v" (s) = maxfv0 (s) ¡ " j s j; 0g and v0 (s) is as previously de…ned. Consequently, in the following, when " is su¢ciently small, P ¤ (") is used to denote the set of quasi-equilibrium prices for (Nr ; v" ) and ¹ 0 ¹ (Nr ; v" ). The following lemma shows that there is a constant C so that given any r, there is a partition of Nr where the number of agents in non-distinguished sets relative to p 2 P ¤¤ is less than or equal to T C. This lemma is required for the proof of theorem 1.1 For the following proofs, recall that ¹ is de…ned by ¹ = (1; :::; 1) 2 QT . Lemma 7. Given p 2 P ¤¤ let s1 ; :::; sK denote the collection of all distinguished pro…les relative to p, where sk · m for all k 2 f1; :::; Kg: Then there is a constant C (inde- pendent of r) so that for any Nr , there exists non-negative integers n1 (r); :::; nK (r) such that K X 0 · rm ¡ nk (r)sk · C¹: k=1 Proof. From theorem 5, the discussion of the theorem, and the remarks after the proof of Lemma 3, there exists w1; :::; wK ¸ 0 so that K X p¢m = wk v(sk ) k=1 38 where K X m= wk sk k=1 and p 2 P : Given r, let nk (r) be the largest integer such that ¤¤ rwk ¡ nk (r) ¸ 0: Then rwk ¡ nk (r) · 1 and P rm ¡ K nk (r)sk PK t k=1 P t = r k=1 wk sk ¡ K nk (r)sk t k=1 t P = K (rwk ¡ nk (r))sk k=1 P t · K sk k=1 t which is constant and independent of r. _ Q.E.D. For the following theorem, assume, without loss of generality, that "0 is su¢ciently small so that for some p 2 P ¤ , where pt > 0 for all t 2 T 0 ; "0 < pt for t 2 T 0 . Proof of Theorem 1 . 1.1 To prove the theorem, we take "0 as given and determine an r0 so that we can construct an imputation in the "0 -core for (Nr ; v) for all r ¸ r0 : For ease in notation, we assume T = T 0 and prove the theorem for this case. However, this proves the theorem in general, since when T = T [ T 00 , given x0 2 "0 - core of (Nr ; v) where 0 0 Nr = f(t; q) 2 Nr : t 2 T 0 g; it is immediate that x is in the "0 -core of (Nr ; v) where ( x0tq for all (t:q) 2 Nr0 and xtq = 0 otherwise. Let fs1 ; :::; sK g denote the collection of all distinguished pro…les sk · m relative to some p 2 P ¤¤: Given r, let K X `(r) = r(m1 ; :::; mT ) ¡ nk (r)sk k=1 where n1(r); :::; nK (r) are non-negative integers such that K X 0 · rm ¡ nk (r)sk · C¹ k=1 39 where C satis…es the hypotheses of Lemma 7. Let A = p ¢ `(r) ¡ v(`) and note that A · T C(max pt ) t so A is bounded independently of r. Select r0 su¢ciently large so that K X nk (r)"0 ¹ ¢ sk ¸ A k=1 for all r > r0 : Given r ¸ r0 , construct a partition of Nr containing nk (r) sets with pro…le sk for each k and let these sets be denoted by D1 ; :::; Dw ; :::; DW : Let L = W S Nr \ ~( Dw ); L is the set of “left-over” agents and w=1 L ´ `(r): Let xtq = pt ¡ "0 W S for each (t; q) 2 Dw . Since w=1 K X v (L) + ¹ nk (r)"0 u ¢ sk ¸ v (L) + A = p ¢ `(r); ¹ k=1 we can select xtq for each (t; q) 2 L so that xtq ¸ pt ¡ "0 ; X K X xtq ¸ v(L) + ¹ nk (r)"0¹ ¢ sk ; tq2L k=1 and X ¹ xtq = v(Nr ): tq2Nr It is immediate that x = (x11 ; :::; xtq ; :::; xT mT r ) is in the "0-core since xtq ¸ pt ¡ "0 for all (t; q) 2 Nr . Q.E.D. 1.2 Let m0 denote the pro…le of the set f(t; q) 2 N1 : t 2 T 0 g, let Z = max p ¢ m; ¤p2P 40 and let j m0 j : K = max t mt Select r0 su¢ciently large and "0 ²0 su¢ciently small so that "0(K + 1) · ±; and 1 Z "0(K + 1) + + < ¸: r0 r0 j m j ± ± Suppose x = (x11 ; :::; xT mT r ) 2 "0-core of (Nr ; v ) where r ¸ r0 . Using the same ¹ techniques as in Lemma 1, it follows that v"0 (s) · x ¢ s ¹ mt r P 1 for all s (where xt = ¹ mt r xtq for each t). Also, since x is in the "0 -core, q=1 r¹ ¢ m = v(Nr ): x ¹ We claim there is a p0 2 P ¤ ("0 ) such that p0 · x. Let ¹ ¤ = fp 2 -T : p ¢ s ¸ v"0 (s) for all s 2 Ir and p · xg; ¹ ¤ 6= 0 since x 2 ¤. Let p0 2 ¤ be such that ¹ p0 ¢ m · p ¢ m for all p 2 ¤: From the remarks at the beginning of Appendix II, p0 2 P ¤ ("0) if, for each t, there is an s 2 Ir (t) such that p0 ¢ s = v"0 (s) since p0 ¢ s ¸ v"0 (s) for all s 2 Ir . Suppose for some t0 and for all s 2 Ir (t0 ), p0 ¢ s > v"0 (s): Let v"0 (s) ¡ p0 ¢ s)t0 ( p00 = maxf0; max0 t g s2Ir (t ) st0 and let p00 = p0t for all t6=t0 : t 0 Clearly, since p00 < pt0 , t p00 ¢ m < p0 ¢ m: Also, for all s 2 Ir ; s 2 Ir (t0 ), = v"0 (s) · p00 ¢ s since, in this case, p00 ¢ s = p0 ¢ s: For all s 2 Ir (t0 ); st ¢ p000 ¸ v"0 (s) ¡ p00 ¢ s)t0 ( t 41 so p00 ¢ s ¸ v"0 (s): Consequently, p00 2 ¤ and p00 ¢ m < p0 ¢ m which is a contradiction. Therefore, p0 2 P ¤ ("0 ). Also, from Lemma 6, p¤ 2 P ¤ , where p¤ = p0t + "0 for all t 2 T 0 and p¤ t t = 0 for all t 2 T 00 . Since p0 · x, rp0 ¢ m · r x ¢ m: Since p¤ 2 P ¤ , we have ¹ v(Nr ) · v(Nr ) < rp¤ ¢ m ¹ ~ and r¹ ¢ m = v (Nr ) · rp¤ ¢ m: x ¹ We now show that for any subset S with the same pro…le as N1 , p0 ¢ m · x(S). Select S 0 so that S 0 ´ m and so that, for each t, if (t; q0 ) 2 S 0 , xtq0 ¸ xtq for all (t; q) = mT r P xtq 2 S 0 (S 0 contains the “worst o¤” agents of each type). Let x0t = ¹ mt for each t. q=1 tq2S 0 Now for all s · m, x ¢ s ¸ x0 ¢ s ¸ v"0 (s) ¹ ¹ from the assumption that x is in the "0 -core. From assumption (*), Lemma 3, and the comments following the proof of Lemma 3, x0 ¢ s ¸ v"0 (s) for all s 2 Ir . Since p0 ¹ minimizes p ¢ m on the set ¤, it follows that p0 ¢ m · x0 ¢ m. Since x(S 0 ) · x(S) for ¹ any S ´ m, it follows that p ¢ m · x(S) for all S ´ m. 0 For each t 2 T 0 , let j m0 j S 0 (t) = f(t; q) 2 [t]r : xtq < p0t ¡ "0 g: mt Observe that f(t; q) 2 [t]r : xtq < p¤ ¡ ±g µ S 0 (t) t since "0 (K + 1) · ± implies that j m0 j p¤ ¡ ± · p0t ¡ "0 t : mt If t 2 T 00 , then S 0 (t) = ; since p0t = 0 and xtq ¸ 0 for all (t; q). Assume t0 2 T 0 and that j S 0 (t0 ) j ¸ mt0 . Let S be a subset of agents so that S ´ m and S contains the “worst o¤” agents of each type (given t, (t; q) 2 S ) xtq · xtq0 for all q 0 where (t; q 0 ) 2 S). Note that = mT r X xtq · xt ¹ q=1 (t;q)2S for each t and S \ [t0 ]r µ S 0 (t0 ): 42 Then p0 ¢ m · x(S) (since S ´ m), and 0 x(S) < x ¢ m)t0 ( + mt (p0t0 ¡ "0 jm0 j ) ¹ mt · x ¢ m ¡ "0 j m0 j ¹ · p ¤ ¢ m ¡ "0 j m 0 j = p0 ¢ m from the construction of S and the properties of S 0 (t0 );and this is a contradiction since it implies that p0 ¢ m · x(s) < p0 ¢ m. Therefore j S 0 (t) j< mt0 , for all t 2 T . Let T [ S 00 = S 0 (t): t=1 Observe that X (p¤ ¡ xtq ) · Z t tq2S 00 since xtq ¸ 0 for all (t; q). Let V = f(t; q) 2 Nr : xtq > p¤ + ±g t and let S U = f(t; q) : (t; q) 2 V S 00 g = 0 jm0 = f(t; q) 2 Nr : pt ¡ "0mt j · xtq · p¤ + ±g: t P Then xtq · rp¤ ¢ m implies that tq2Nr X X X (xtq ¡ p¤) · t (p¤ ¡ xtq ) + t (p¤ ¡ xtq ): t tq2V tq2[ tq2S 00 Since, when (t; q) 2 U, j m0 j p¤ ¡ xtq · p0t + "0 ¡ p0t + "0 t · "0 (K + 1); mt X (p¤ ¡ xtq ) ·j U j "0(K + 1) · r j m j "0(K + 1): t tq2U Also X (p¤ ¡ xtq ) · Z: t tq2S 00 Consequently, it follows that X jV j±· (xtq ¡ p¤) · r j m j "0(K + 1) + Z t tq2V 43 and jf(t;q)2Nr :jjxtq ¡p¤ jj>±gj t jNr j 00 j jS · jNr j + jNrjjjV jmjr"0 (K+1) Z [jmj+j +±] · ± jNr j = 1 + "0 (K+1) + ±jmjr r ± Z < ¸: The conclusion follows from the observation that if " 2 [0; "0 ] and if x is in the "-core, then x is in the "0 -core. Q.E.D. 1.3 The proof of 1.3 follows from the proof of 1.1. In 1.1, starting with an arbitrary p 2 P ¤ and an arbitrary " > 0; we showed that there was an r0 so that for all r ¸ r0 , the "-core was non-empty. Consequently, 1.3(a) is shown. Also, given " and r ¸ r0, we constructed x 2 -Nr , x 2 "-core of Nr , so that given p, for each t, xtq = pt ¡ " for (t; q) 2 L (where L is the set of agents not in distinguished sets relative to p 2 P ¤¤ ), and xtq ¸ pt ¡ " for all (t; q) 2 L. Clearly, given ¸ and ± , we can select " < ± and r so that jLj <¸ j Nr j since j L j is bounded independently of r. Then the "-core imputation x, as constructed in the proof of 1.1, satis…es the condition that jf(t; q) 2 Nr :jj xtq ¡ pt jj> ±gj jLj j· < ¸: jmrj j Nr j Q.E.D. 44 FOOTNOTES 1. This example is investigated in the next section. 2. This is the Shapley-Shubik weak "-core [25]. 3. The balancedness condition is stated later and is also investigated in many of our references. For balancedness of games with side payments, see, for example, [22], or [10], 89-91. 4. In other words, all agents of the same type are substitutes (D.9), although it is not required that agents of di¤erent types are di¤erent (i.e. not substitutes). 5. We call the equilibrium “competitive” since prices of marketed goods are taken as given. It could be interpreted, however, as non-competitive in that the members of a coalition S can be viewed as owning the technology Y [S], which suggests a market-socialist economy. 6. It could alternatively be assumed that there is an r0 su¢ciently large so that s 2 Ir0 . However, this merely increases notational complexity without increasing the ¤ generality of the model. 7. See [19], pages 130-134 for a proof (for average cost curves). 8. In [30], a more general coalition formation cost function is used. 9. Assumptions similar to (*) have some precedents in the literature. For example, in [5], given Walrasian equilibrium prices, it is assumed that each …rm has a minimum e¢cient scale. In other words, an assumption of U-shaped average cost as a function of quantity produced for a …rm is analogous to assumption (*). 10. In [32] concepts of approximate and asymptotic balancedness are introduced and in [33], su¢cient conditions are demonstrated for the approximate and asymptotic balancedness of sequences of games. 11. When we assume (*) holds, we can, of course, replace Pr¤ by P ¤ in this theorem and in the next one. 12. See, for example, Hildenbrand and Kirman [7], page 89 or [10]. 0 13. By distinct subpro…les, we mean that if k 6=k 0 , then sk 6=sk . ³ ´ 14. n¤ = n¤!(m¡n¤)! . By “distinct sets,” we mean that no two sets contain m m! exactly the same players. 45 References [1] L. Billera, On Games Without Side Payments Arising From a General Class of Markets, Journal of Mathematical Economics 1 (1974) 129-139. [2] L. Billera and R. Bixby, A Characterization of Polyhedral Market Games, Inter- national Journal of Game Theory 2 (1973), 253-261. [3] V. Boehm, Firms and Market Equilibria in a Private Ownership Economy, Zeitschrift fur Nationalokonomie, 33 (1973), 87-102. [4] V. Boehm, The Limit of the Core of an Economy with Production, International Economic Review 15 (1974), 143-148. [5] G. Debreu, Theory of Value. New York : Wiley, 1959. [6] B. Ellickson, A Generalization of the Pure Theory of Public Goods, American Economic Review, 63 (1973), 417-432. [7] D. Foley, Lindahl’s Solution and the Core of an Economy with Public Goods, Econometrica 38 (1970), 66-72. [8] J. Greenburg, Existence and Optimality of Equilibrium in Labour Managed Economies, Discussion paper No. 216, The Centre for Mathematical Studies in Economics and Management Science, Northwestern University, 1976. [9] W. Hildenbrand, Core and Equilibria of a Large Economy, Princeton University Press, Princeton and London, 1974. [10] W. Hildenbrand and A. P. Kirman, Introduction to Equilibrium Analysis, Amer- ican Elsevier Publishing Co., Inc., New York, 1976. [11] T. Ichiishi, Coalition Structure in a Labour-Managed Market Economy, Econo- metrica 45 (1977), 341-360. [12] A. Mas-Colell, A Further Result on the Representation of Games by Markets, Journal of Economic Theory 10 (1975), 117-122. [13] M. McGuire, Group Segregation and Optimal Jurisdictions, Journal of Public Economics, 82 (1974), 112-132. [14] T. Muench, The Core and the Lindahl Equilibrium of an Economy with a Public Good: An Example, Journal of Economic Theory 4 (1972), 241-255. [15] W. Novshek and H. Sonnenschein, Cournot and Walras Equilibrium, Journal of Economic Theory 19 (1978), 223-266. 46 [16] M. V. Pauly, Cores and Clubs, Public Choice, (Fall 1970), 53-65. [17] M. V. Pauly, Clubs, Commonality, and the Core: An Integration of Game Theory and the Theory of Public Goods, Economica 35 (1976), 314-324. [18] S. Rashid, Existence of Equilibrium in In…nite Economies with Production, Econometrica, 46 (1978), 1155-1164. [19] P. Samuelson, The Monopolistic Competition Revolution, in Monopolistic Com- petition Theory: Studies in Impact, Essays in Honor of Edward H. Chamberlin, edited by Robert E. Kuenne, New York: John Wiley, 1968. [20] H. Scarf, The Core of an N-Person Game, Econometrica 35 (1967), 50-69. [21] H. Scarf, On the Existence of a Cooperative Solution for a General Class of N-Person Games, Journal of Economic Theory 3 (1971) 169-181. [22] L. Shapley, On Balanced Sets and Cores, Rand Corporation Memorandum RM- 4601-PR, June 1965. [23] L. Shapley and H. Scarf, On Cores and Indivisibilities, Journal of Mathematical Economics 1 (1974), 23-37. [24] L. Shapley and M. Shubik, On Market Games, Journal of Economic Theory 1 (1969), 9-25. [25] L. S. Shapley and M. Shubik, Quasi-Cores in a Monetary Economy with Non- Convex Preferences, Econometrica, 34 (1966), 805-827. [26] D. Sondermann, Economies of Scale and Equilibria in Coalition Production Economies, Journal of Economic Theory 8 (1974), 259-271. [27] M. Wooders, Equilibria, the Core and Jurisdiction Structures in Economies with a Local Public Good, Journal of Economic Theory 18 (1978), 328-348. [28] M. Wooders, Equilibria, the Core, and Jurisdiction Structures in Economies with a Local Public Good: A Correction (mimeographed) June (1979). [29] M. Wooders, The Tiebout Hypothesis: Near-Optimality in Local Public Good Economies, Econometrica (accepted for publication). [30] M. Wooders, Existence of Approximate Equilibria of Coalition Production Economies, Economic Research Bureau, S.U.N.Y. - Stony Brook, Working Paper (forthcoming). [31] M. Wooders, Convergence and Near-Optimality of Approximate Equilibria of Coalition Production Economies, Economic Research Bureau S.U.N.Y. - Stony Brook, Working Paper (forthcoming). 47 [32] M. Wooders, A Characterization of Games that Have Non-Empty "-cores, S.U.N.Y. - Stony Brook , Economic Researce Bureau Working Paper # 213 (July 1979). [33] M. Wooders, Asymptotic Cores and Asymptotic Balancedness of Large Replica Games, S.U.N.Y. - Stoney Brook, Economic Research Bureau Working Paper No.215 (September 1979). 48

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