SOME PROPERTIES OF WEAK DOMINATION IN AN EXCHANGE MARKET WITH INDIVISIBLE GOODS*

Transcription

1 Vol. 42, No. 4, December 1991 SOME PROPERTIES OF WEAK DOMINATION IN AN EXCHANGE MARKET WITH INDIVISIBLE GOODS* By JUN WAKO We investigate general properties of weak domination in the Shapley-Scarf exchange model with indivisible goods. We prove that every weakly dominated allocation is weakly dominated by some competitive allocation. Furthermore, if an allocation does not weakly dominate some other allocation, it is a weakly dominated allocation or the two allocations are indifferent. By using these results, we show that every non-competitive allocation is weakly dominated by some competitive allocation. The nonempty strong core proves to be a von Neumann-Morgenstern solution. We give also a necessary and sufficient condition for the equivalence of the strong core and the competitive allocations. 1. Introduction This paper studies some general properties of weak domination in an exchange model with indivisible goods. We consider the model of Shapley and Scarf (1974). In this model, each trader owns one indivisible good and wants to exchange it for one preferable indivisible good. Every trader has no use for more than one unit of indivisible goods. A typical example in the literature is a trade of houses.1) We could also consider an exchange of tasks assigned to agents in this framework. Shapley and Scarf showed that even in such a discrete model a competitive equilibrium always exists. This paper, following Roth and Postlewaite (1977) and Wako (1984), further investigates how a competitive allocation relates to other allocations through weak domi nation,and generalizes some previous results. Given two allocations in an exchange market, we say that one allocation weakly dominates the other through a coalition of traders if some member of the coalition is better off in the former than in the latter while the other members are not worse off in the former, and the coalition can attain the same consumption as in the former allocation by redistributing their initial resources among themselves. An allocation weakly dominates some other allocation if there is such a coa lition. The set of allocations that are not weakly dominated by any allocation is called the strong core. * This research is supported by the grant of Tokyo Center of Economic Researcḥ The author wishes to thank Professors Ahmet Alkan, Jerzy Legut, Shigeo Muto, Koh Nishihara, Marilda Sotomayor, Walter Trockel and Myrna Wooders for their helpful comments and encouragement. 1) Quinzii (1984) and Wako (1991) considered an extended Shapley-Scarf model with money

2 The gcore h in Debreu and Scarf (1963) is in fact the strong core. In their standard exchange mar ketwith perfectly divisible goods, the strong core exists and contains the competitive allocations. However, Shapley and Scarf (1974) pointed out that the strong core does not necessarily exist in their indivisible good market with non-convex and satiable preferences. In the Shapley-Scarf model, the strong core may be empty, while a competitive equilibrium exists. Roth and Postlewaite (1977) then proved that if no trader is indifferent between any goods, the strong core exists and consists of a unique competitive allocation, which weakly domi natesany other allocation. In general, Wako (1984) showed that the strong core is a subset, and may be a nonempty proper subset of the competitive allocations. These results are not obtained in a standard exchange model with perfectly divisible goods. Some basic feature should be found out in the weak domination relation in the Shapley-Scarf model. We investigate this question. In the following section, we define the details of the Shapley-Scarf model, the weak domination relation and its related solution concepts. In Section 3, we introduce the competitive equilib riuminto the model and review the previous results. The main results are stated in Section 4. In the Shapley-Scarf model, every weakly dominated allocation is weakly dominated by some competitive allocation. If an allocation does not weakly dominate some other alocation, it is a weakly dominated allocation or the two allocations are equivalent in preference. These proper tiesgeneralize some previous results. Every non-competitive allocation is weakly dominated by some competitive allocation. The nonempty strong core is a von Neumann-Morgenstern solu tion.furthermore, the strong core coincides with the set of competitive allocations if and only if any two competitive allocations are equivalent in preference. Proofs are given in Section The Shapley-Scarf Model and the Concept of Weak Domination We consider the Shapley-Scarf model (1974). This model has n traders, each with one indivis iblegood initially. Each trader wants only one preferable indivisible good in exchange for his good, whereas there is no medium of exchange. Let N={1,...,n} be the set of traders. We denote trader is initial endowment by the i-th unit vector ei in Rn, namely, trader i owns one unit of the i-th indivisible good. Since each trader wants only one indivisible good, the consumption set is the set G(N)={e1,...,en}. Trader i has a preference Ri on G(N). The expression ekrieh means that trader i weakly prefers the k-th good to the h-th good. We denote ekpieh if he prefers the k-th good to the h-th good, and ekiieh if he is indifferent between the two goods. An allocation is defined as a bijection from N onto G(N). A nonempty subset S of N is called a coalition S. We extend the definition of G(N) to any coalition S, namely, let G(S)={ek k S}. The set G(S) shows the goods that coalition S can redistribute among themselves. For a given allocation x, we define a trading cycle in x to be any coalition S such that x(s)=g(s) and x(s') G(S') for any nonempty S' ¼S with S' S, where x(s)= {x(i) i S} and x(s') is defined similarly. If S={i1,...,it} is a trading cycle in x, we have, by reindexing if necessary, x(ik)+eik+1 for k=1,...,t-1 and x(it)=ei1. A trading cycle S in x is a minimal effective coalition under x in the sense that the members of S can obtain the goods assigned in x by themselves

3 J. Wako: Some Properties of Weak Domination in an Exchange Market It is defined that an allocation x weakly dominates an allocation y (through S) if and only if there exists a trading cycle S in x such that x(i)riy(i) for each i S, and x(j)pjy(j) for some j S. The strong core is the set of allocations that are not weakly dominated by any allocation. Any nonempty set K of allocations is defined to be a von Neumann-Morgenstern solution defined by weak domination if and only if 1) no allocation in K is weakly dominated by any allocation in K, and 2) any allocation that is not in K is weakly dominated by some allocation in K. The con ditions1) and 2) are called internal stability and external stability, respectively. In the following, we simply write ga von Neumann-Morgenstern solution. h 3. Competitive Equilibrium and Previous Results on Its Properties Let p=(p1,...,pn) be a vector of positive numbers and let each pi denote a price of the i-th good. A competitive equilibrium of our model is defined as a pair <x, p> of an allocation x and a price vector p satisfying (3.1) p x(i) p ei for each i N, and (3.2) for each i N, x(i)rieh for any eh G(N) with p eh p ei. Condition (3.1) is a budget constraint. (3.2) is a requirement for utility maximization within a budget. An allocation x is said to be competitive if there is a price vector p with the pair <x, p> being a competitive equilibrium. Let R be any nonempty subset of N. For trader i R, define B(R, i)={ek G(R) ekrieh for any eh G(R)}. This set shows the goods that trader i likes best among those avail ablein R. A top trading cycle for R is defined to be any coalition T ¼R for which there is a bijection xt from T onto G(T) with (3.3) xt(i) B(R, i) for each i T, and (3.4) xt(t') G(T') for any nonempty T' ¼T with T' T. There exists at least one top trading cycle for any R. In fact, starting with any trader in R, repeat the procedure: as a next trader, choose anyone among the owners of the goods that the trader chosen immediately before likes best in G(R). Since R is a finite set, we must get a cycle, namely, a top trading cycle for R. The method of top trading cycles2) is the procedure that constructs an allocation by using top trading cycles. In this procedure, we first find any sequence of disjoint top trading cycles {T1,...,Tt} such that (3.5) N= ¾tk=1Tk and Tk is a top trading cycle for N _ ¾k-1h=1Th for k=1,...,t, where N _ ¾k-1h=1Th=N for k=1. To do this, take any top trading cycle for N as T1; take any top trading cycle for N _T1 as T2; proceed in the same manner until there is no remainders. Since N is a finite set, we get a required sequence and also a bijection xtk with (3.3) and (3.4) for each Tk. Then define an allocation x by putting x(i)=xtk(i) for each i in each Tk {T1,...,Tt}. 2) This procedure is due to Professor David Gale

4 Theorem A (Shapley and Scarf 1974). In the Shapley-Scarf model, there is at least one com petitiveallocation, and the set of competitive allocations coincides with the set of allocations obtainable by the method of top trading cycles. Let x be any allocation obtained by the method of top trading cycles, and {T1,...,Tt} be the associated sequence of top trading cycles. Take any numbers Ĕ1, Ĕ2,...,Ĕt with Ĕ1> Ĕ 2> c>Ĕt>0. Then from (3.5), x proves to be competitive under prices p=(p1,...,pn) with pi=Ĕk for each i in each Tk {T1,...,Tt}. Thus any allocation obtained by the method above is competitive. This also shows the existence. Conversely, let x be any competitive allocation and p=(p1,...,pn) be its competitive prices. Let T1,...,Tt be all the trading cycles in x. Then for each Tk, there must be a constant Ĕk with pi=Ĕk for each i Tk. Arrange the trading cycles in such an order that Tk1, Tk2,...,Tkt andĔ k1 Ĕk2 c Ĕkt. Since x is competitive, this sequence is that of top trading cycles satisfy ing(3.5). Thus we can obtain any competitive allocation by the method of top trading cycles. From Theorem A, if no trader is indifferent between any goods, there is a unique competitive allocation, since we get a unique allocation by the method of top trading cycles under this assumption of strict preferences. Assuming strict preferences, Roth and Postlewaite (1977) showed external stability of a unique competitive allocation and its equivalence to the strong core. Wako (1984) proved the general inclusion relationship between the competitive allocations and the strong core. These results show distinctive properties of the Shapley-Scarf model. Theorem B (Both and Postlewaite 1977). If no trader is indifferent between any goods, then a unique competitive allocation weakly dominates any other allocation. Theorem C (Roth and Postlewaite 1977). If no trader is indifferent between any goods, then the strong core is nonempty, and contains exactly one allocation, which is the unique competitive allocation. Theorem D (Wako 1984). The strong core is included in the set of competitive allocations. 4. Weak Domination by Competitive Allocations In the Shapley-Scarf model, the competitive equilibrium has close relation to the concept of weak domination. By investigating an underlying property of the weak domination relation, we could tell the feature of the Shapley-Scarf model more lucidly. Then we generally investi gatehow given allocations are related to a competitive allocation through weak domination. As basic results, we can prove the following propositions for the case that an allocation x weakly dominates an allocation y, and the other case that an allocation x does not weakly dominate an allocation y. Proposition 1. If an allocation x weakly dominates an allocation y, then y is weakly dominated by some competitive allocation z

5 J. Wako: Some Properties of Weak Domination in an Exchange Market Proposition 2. If an allocation x does not weakly dominate an allocation y, then (1) x(i)iiy(i) for any i N, or (2) x is weakly dominated by some allocation z. Proposition 1 means that every weakly dominated allocation is weakly dominated by some competitive allocation. Let us say that allocations x and y are equivalent in preference if x(i)iiy(i) for any i N. Proposition 2, together with Proposition 1, means that if an allocation does not weakly dominate some other allocation and if they are not equivalent in preference, the former is weakly dominated by some competitive allocation. Note that Proposition 2 is not trivial, since the fact that allocation x does not weakly dominate allocation y does not imply that x is weakly dominated by y. Using these propositions and Theorems A and D, we can generalize Theorems B and C for the case that a trader may be indifferent between some goods. Firstly, Proposition 1 generally proves external stability of the set of competitive allocations, since Theorem D implies that every non - competitive allocation is a weakly dominated allocation. Theorem 1. Every non-competitive allocation is weakly dominated by some competitive allo cation.3) The set of competitive allocations is stable externally, and not necessarily stable internally. Since we allow indifferences in traders' preferences, the set of competitive allocations does not necessarily coincide with the strong core. In the case of non-equivalence, some competitive allo cationis weakly dominated by another competitive allocation (Theorem D and Proposition 1). The strong core has internal stability by its definition. Furthermore, Proposition 2 proves its external stability. Theorem 2. (1) Any strong core allocation weakly dominates all the other allocations that are not contained in the strong core; (2) Any two strong core allocations are equivalent in preference. Note that when the strong core is empty, the statements hold in the trivial sense. Result (1) shows external stability of the strong core. Thus when the strong core is nonempty, it is a unique4) von Neumann-Morgenstern solution. Note that external stability here holds in a strong sense that any allocation outside the strong core is weakly dominated by every strong core allo cation.furthermore, by result (2), the nonempty strong core gives a unique point in terms of preference. The strong core has the remarkable properties as above. We can guarantee its existence when 3) Shapley and Scarf (1974, p. 32) raised a question: Can there be a non-competitive allocation that is not weakly dominated by a competitive allocation? Theorem 1 gives a negative answer to this question. (See also Roth and Postlewaite, 1977, p. 136.) 4) If the strong core is a von Neumann-Morgenstern solution, there is no other set of allocations with internal and external stability

6 it coincides with the set of competitive allocations. In addition, the equivalence of the two sets is necessary for internal stability of the set of competitive allocations. From our results, we can tell in what situation the equivalence holds. A necessary condition derives from Theorem 2-(2), and it proves to be sufficient by Proposition 1 and Theorem D. Theorem 3. The strong core coincides with the set of competitive allocations Wand only if any two competitive allocations are equivalent in preference. It is not yet known how the condition above can be completely represented by preference profiles. However, we can prove Theorem C as a special case of our result. Theorem 3 implies that if a unique competitive allocation exists, the strong core is a singleton containing the com petitiveallocation. Theorem C derives from this fact, since under the assumption of Theorem C we have a unique competitive allocation by Theorem A. Note that Theorem 3 gives also a necessary and sufficient condition for the set of competitive allocation to become a von Neumann-Morgenstern solution. However, when the condition is violated, a von Neumann-Morgenstern solution may contain a non-competitive allocation (see Example 1). The general existence of a von Neumann-Morgenstern tion. Finally, we show how our results work by examples. solution is an open ques Example 1.5) Let N={1, 2, 3} and the preferences of the traders be as follows: (1) e2p1e3i1e1 (2) e1i2e3p2e2 (3) e2p3e1p3e3 Let x=(h, i, j) denote an allocation x with x(1)=eh, x(2)=ei, and x(3)=ej. By the method of top trading cycles, we can find two competitive allocations x=(1, 3, 2) and x=(2, 1, 3). Since these allocations weakly dominate each other, the strong core is empty by Theorem D. Allocations y=(2, 3, 1) and y=(3, 1, 2) do not weakly dominate each other, and are not equivalent in preference. Thus by Propositions 1 and 2, they must be weakly dominated by some competitive allocations. In fact, y and y are weakly dominated by competitive alloca tionsx and x, respectively. Further, x is weakly dominated by y through N. The remaining allocations z=(1, 2, 3) and z=(3, 2, 1) are weakly dominated by the other four allocations. As Theorem 1 shows, the set of competitive allocations has external stability. However, it does not have internal stability, nor does it coincide with the strong core. According to Theorem 3, these phenomena arise from the fact that the two competitive allocations are not equivalent in preference. Although the strong core is empty in this example, the set {x, y} is the unique von Neumann-Morgenstern solution. Example 2. Let N={1, 2, 3} and the preferences of the traders be as follows: 5) This example is due to Koh Nishihara

7 (1) e3i1e2p1e1 J. Wako: Some Properties of Weak Domination an Exchange Market (2) e1i2e3p2e2 (3) e2i3e1p3e3 In this example, all the allocations except z=(1, 2, 3) are competitive. Since x=(1, 3, 2) and x=(2, 1, 3), for example, are not equivalent in preference, the strong core does not coincide with the set of competitive allocations. The strong core of this example is the set of allocations y=(2, 3, 1) and y=(3, 1, 2). As Theorem 2 shows, both y and y weakly dominate the other allocations except them, and y and y are equivalent in preference. This nonempty strong core is the unique von Neumann-Morgenstern solution. Remark. Wako (1990) considered what sort of competitive allocation a strong core allocation is. It was proved that allocation x is a strong core allocation if and only if x is a competitive allocation for which there exists a price vector with conditions (3.1), (3.2) and for each i N, x(i)pieh for any eh G(N) with P Eeh<p Eei. Theorem 2-(1) was proved in that paper by using this fact. 5. Proofs of Propositions and Theorems We will prove Propositions 1 and 2, and Theorems 2 and 3. Theorem 1 follows from Proposi tion1 and Theorem D immediately Proof of Proposition 1 Suppose that an allocation x weakly dominates an allocation y. If x is competitive, the proposi tionis clear. Thus we assume that x is not competitive. Since x weakly dominates y, there exists a trading cycle S* in x with x(i)riy(i) for all i S*, and x(j)pjy(j) for some j S*. We will make top trading cycles from allocations x, y, and trading cycle S*, and then show that those cycles give a competitive allocation which weakly dominates y. For any given nonempty subset R of N, a nonempty subset T of R is called a top trading cycle for R under (x, y, S*) if and only if there exists a bijection zt from T onto G(T) satisfying (5.1) zt(i) B(R, i) for each i T, (5.2) zt(i)=y(i) if i T _S* and y(i) B(R, i), (5.3) zt(i)=x(i) if i T S* and x(i) B(R, i), and (5.4) zt(t') G (T') for any nonempty proper subset T' of T. A top trading cycle for R under (x, y, S*) is a top trading cycle for R satisfying the conditional requirements (5.2) and (5.3). Since N is a finite set, for any nonempty subset R ¼N, we can find at least one top trading cycle for R under (x, y, S*). Then let {T1,...,Tt} be a partition of N such that (5.5) Tk is a top trading cycle for N _ ¾k-1k=1Th under (x, y, S*) for k=1,...,t, where N _ ¾k-1k=1Th=N for k=1. Let ztk be the bijection associated with top trading cycle Tk under (x, y, S*) for each k=1,...,t. Then define the allocation z by (5.6) z(i)=ztk(i) for each i Tk and k=1,...,t

8 Since T1,...,Tt satisfy condition (5.5), z is a competitive allocation. We will show that the competitive allocation z weakly dominates allocation y. At first, we will prove the following claim: Claim 1. Let k be a number with 1 k t. If z(i)=y(i) for each i ¾kh=1Th, then Th S* =_??_ for each Th {T1,...,Tk}. From Claim 1, it is impossible that z(i)=y(i) for all i N= ¾th=1Th, since _??_ S* ¼N. Thus there exists a Tk {T1,...,Tt} with z(j) y(j) for some j Tk. Then we will prove Claim 2: Claim 2. Let Tk* {T1,...,Tt} be the trading cycle whose index is minimum among those cycles Th {T1,...,Tt} with z(j) y(j) for some j Th. Then allocation z weakly domi natesallocation y through Tk*. Since allocation z is competitive, Proposition 1 is shown if we prove these claims. (Proof of Claim 1) Assume that Claim 1 does not hold, namely, (5.7) { z(i)=y(i) for each i ¾kh=1Th, and Th S* _??_ for some Th {T1,...,Tk}. Let Tm be the trading cycle with the minimum index among those Th {T1,...,Tk} with Th S* _??_. For a general description of a proof, we prove the case of m>1. The claim can be proved in the same manner for m=1. By the definition of Tm, we have S* ¼N _ ¾m-1h=1Th. Since S* is a trading cycle in x, we have x(j) G(N _ ¾m-1h=1Th) for each j S*. Since we have assumed that z(i)=y(i) for each i Tm, it must hold that y(i) B(N _ ¾m-1h=1Th, i) for each i Tm by (5.1), (5.5) and (5.6). However, x(j)rjy(j) for each j Tm S*. Thus by (5.3), we must have (5.8) z(j)=x(j)=y(j) for each j Tm S*. The first equality of (5.8) means z(tm S*)=x(Tm S*) ¼G(Tm S*), since z(tm)=g(tm) and x(s*)=g(s*). Then the fact that z is a bijection from N onto G(N) implies (5.9) z(tm S*)=x(Tm S*)=G(Tm S*). Since Tm and S* are trading cycles in z and x, respectively, we have z(t') G(T') for any nonempty proper subset T' of Tm, and x(s') G(S') for any nonempty proper subset S' of S*. Since Tm S* _??_, equation (5.9) holds only if (5.10) Tm=S*. Thus it follows from (5.8) that x(i)=y(i) for each i S*. However, this is impossible, since allocation x weakly dominates y through S*. This contradiction is due to assumption (5.7). Hence Claim 1 holds. (Proof of Claim 2) For a general description of a proof, we prove the case of k*>1. The claim -310-

9 J. Wako: Some Properties of Weak Domination in an Exchange Market can be proved in the same manner for k*=1. From the definition of Tk*, (5.11) z(i)=y(i) for each i Th with 1 h<k*. Thus T1,...,Tk*-1 are also trading cycles in y, and so y(i) G(N _ ¾k*-1h=1Th) for each i Tk*. Since Tk* is a top trading cycle for N _ ¾k*-1h=1Th, (5.12) z(i)riy(i) for each i Tk*. Regarding Tk* and S*, we have the following three possible cases: (case 1) Tk* S*=_??_, (case 2) Tk* S* _??_ and z(j) x(j) for some j Tk* S*, or (case 3) Tk* S* _??_ and z(j)=x(j) for all j Tk* S*. Then we will prove Claim 2 for each case. (case 1) By the definition of Tk*, there exists a j Tk* with z(j) y(j). Since Tk* S* =_??_, if y(j) B(N _ ¾k*-1h=1Th,j), we must have z(j)=y(j) by (5.2), (5.5) and (5.6). Thus y(j)_??_b(n _ ¾k*-1h=1Th,j) for j Tk* with z(j) y(j). Since y(j) G(N _ ¾k*-1h=1 Th) and z(j) B(N _ ¾k*-1h=1Th,j), z(j)pjy(j) for j Tk* with z(j) y(j). This together with (5.12) imply that z weakly dominates y through Tk*. (case 2) It follows from Claim 1 and (5.11) that S* ¼N _ ¾k*-1h=1Th. Since S* is a trading cycle in x, we have x(j) G(N _ ¾k*-1h=1Th) for each j Tk* S*. Since x(j)rjy(j) for each j S* and since Tk* is a top trading cycle for N _ ¾k*-1h=1Th under (x, y, S*), we have z(j)pjx(j)rjy(j) for j Tk* S* with z(j) x(j) by (5.3). This together with (5.12) imply that z weakly dominates y through Tk*. (case 3) If z(j)=x(j) for each j Tk* S*, we have Tk*=S* by the same argument given to show (5.10). This immediately means that z weakly dominates y through Tk*, since x weakly dominates y through S*. Claim 2 holds for every case. The proof of Proposition 1 is complete Proof of Proposition 2 Proposition 2 is equivalent to Propostion 2' below, which we will prove. Proposition 2'. If x and y are allocations satisfying i) not x(i)iiy(i) for some i N, and ii) x does not weakly dominate y, then there exists an allocation z that weakly dominates x. Let x and y be any allocations such that i) not x(i)iiy(i) for some i N, and ii) x does not weakly dominate y. By using these allocations, we partition N into the three sets X, Y and E defined by X={i N x(i)piy(i)}, Y={i N y(i)pix(i)}, E={i N x(i)iiy(i)}. By condition i), X ¾Y _??_. If Y=_??_, x weakly dominates y, since N=E ¾X and X _??_ in this case. This contradicts condition ii). Thus the set Y is nonempty

10 Define the mapping z0 from N to G(N) as follows. z0(i)={ y(i) for i N with T(i x) ¼E or i Y, x(i) for i N with T(i x)_??_ and i_??_y, where T(i x) denotes the trading cycle in x in which i is contained. T(i x) is determined uniquely. Note that mapping z0 is not necessarily a bijection from N onto G(N), and that z0(i)rix(i) for each i N. Since z0(n) ¼G(N) and N is a finite set, there exists a nonempty subset S of N with z0(s)=g(s) and z0(s') G(S') for any nonempty proper subset S' of S. We call such a set S a cycle in z0, and denote by J the set of all cycles in z0. Proposition 2' is then proved by showing the following claims: Claim 1. If there exists a cycle S J with S Y _??_, then there exists an allocation z that weakly dominates x. Claim 2. The exists a cycle S J with S Y _??_. (Proof of Claim 1) Let S be a cycle in z0 with S Y _??_. We have z0(i)rix(i) for each i S, z0(j)=y(j) and z0(j)pjx(j) for each j S Y, and z0(s)=g(s). Then define the mapping z from N onto G(N) by z(i)={ z0(i) for i S ei for i N _S. This mapping z is an allocation that weakly dominates x through S. (Proof of Claim 2) At first, we will show that (5.13) T(i* x)_??_e for some i* ¾S JS. To prove this, assume on the contrary that (5.14) T(i x) ¼E for each i ¾S JS. This implies ¾S JS ¼E, because i T(i x) for any i N. Since Y E=_??_, we have Y ¼N _ ¾S JS. Since Y is nonempty, the set N _ ¾S JS is nonempty. Here, if z0(i) G(N _ ¾S JS) for any i N _ ¾S JS, there must be a cycle S' in z0 with S' ¼N _ ¾S JS. However, this is impossible since S' must be contained in J. Thus there exist some j* N _ ¾S JS and S* J with z0(j*) G(S*). From assumption (5.14), we have T(i x) ¼E for any i S*. By the definition of z0, z0(i)=y(i) for each i S*, and so y(s*)=g(s*). Since y is a bijection from N onto G(N), we must have z0(j*) y(j*). Again by the definition of z0, z0(j*)=x(j*) and T(j* x)_??_e. Here, let i* be the element or N with x(j*)=ei*. Then we have T(i* x)_??_e and i* S*EJ, since T(i* x)=t(j* x)_??_e and x(j*)=z0(j*)=ei* G(S*). However, this contra dictsassumption (5.14). Thus claim (5.13) holds

11 J. Wako: Some Properties of Weak Domination in an Exchange Market Now we will prove Claim 2. Assume that it does not hold, i.e., (5.15) S Y=_??_ for any S J. From claim (5.13), there is a cycle S* J with T(i x)_??_e for some i S*. We may sup posethat S*={i1,...,it} and T(i1 x)_??_e, z0(it)=ei1, and z0(ik)=eik+1 for k=1,...,t-1. It follows from assumption (5.15) that ik_??_y for each ik S*. Since T(i1 x)_??_e, we have z0(i1)=x(i1)=ei2 by the definition of z0. Equation x(i1)=ei2 implies T(i1 x)=t(i2 x), and so T(i2 x)_??_e. Thus we have z0(i2)=x(i2)=ei3. Repeating the same argument, we have z0(i)=x(i) for each i S*. This means that S* is a trading cycle in x, since S* is a cycle in z0. Thus S*=T(i1 x), and sos *_??_E. Further, since S* Y=_??_ by (5.15), we have S* ¼E ¾X and S* X _??_. This means that x weakly dominates y through S*. However, this contradicts condition ii). Thus Claim 2 holds. The proof of Proposition 2 is complete Proof of Theorem 2 To see (1), assume that a strong core allocation x does not weakly dominate an allocationy out sidethe strong core. Since x and y are not equivalent in preference, x must be a weakly dominated allocation by Proposition 2. However, this contradicts that x is in the strong core. To see (2), assume that there are two strong core allocations x and y that are not equivalent in preference. Since x cannot weakly dominate y, x must be a weakly dominated allocation by Pro position2. This contradicts that x is in the strong core. Hence Theorem 2 holds Proof of Theorem 3 (Necessity) Suppose that the strong core coincides with the set of competitive allocations. Then any competitive allocation is also a strong core allocation. Thus by Theorem 2-(2), any two competitive allocations are equivalent in preference. (Sufficiency) Suppose that any two competitive allocations are equivalent in preference. Here, assume that there is a competitive allocation x that is not contained in the strong core. By Proposition 1, x must be weakly dominated by some competitive allocation z. However, this is impossible, since any two competitive allocations are assumed to be equivalent in preference. Thus any competitive allocation is contained in the strong core. Since the strong core is a subset of the competitive allocations by Theorem D, the strong core coincides with the set of com petitiveallocations. (K omazawa University) REFERENCES Debreu, G. and H. Scarf (1963) ga Limit Theorem on the Core of an Economy, h International Economic Review, Vol. 4, pp Quinzii, M. (1984) gcore and Competitive Equilibria with Indivisibilities, h International Journal of Game Theory, Vol. 13, pp Roth, A. E. and A. Postlewaite (1977) gweak versus Strong Domination in a Market with Indivisible Goods, h Journal of Mathematical Economics, Vol. 4, pp

12 Shapley, L. and H. Scarf (1974) gon Cores and Indivisibility, h Journal of Mathematical Economics, Vol. 1, pp Wako, J. (1984) ga Note on the Strong Core of a Market with Indivisible Goods, h Journal of Mathematical Economics, Vol. 13, pp (1990) gstrong Core and Competitive Allocations of the Shapley-Scarf Exchange Market with Indivisible Goods, h Komadai Keiei Kenkyu, Vol. 21, No. 4, pp (in Japanese). - (1991) gstrong Core and Competitive Equilibria of an Exchange Market with Indivisible Goods, h forthcoming in International Economic Review