resource-monotonic solutions to the problem of fair division when preferences are single-peaked

Soc Choice Welfare (1994) 11:205-223

Social Choice .dWelfare

© Springer-Verlag 1994

Resource-monotonic solutions to the problem of fair division when preferences are single-peaked

William Thomson

Department of Economics, University of Rochester, Rochester, NY 14627, USA

Received: 9 January 1992 / Accepted: 5 August 1993

Abstract. We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We search for solutions satisfying resource-monotonicity, the requirement that all agents be affected in the same direction when the amount to divide changes. Although there are resource-monotonic selections from the Pareto solution, there are none satisfying the distributional requirements of no-envy or individual rationality from equal division. We then consider the weakening of resource-monotonicity obtained by allowing only changes in the amount to divide that do not reverse the direction of the inequality between the amount to divide and the sum of the preferred amounts. We show that there is essentially a unique selection from the solution that associates with each economy its set of envy-free and efficient allocations satisfying this property of one-sided resource-monotonicity : it is the uniform rule, a solution that has played a central role in previous analyses of the problem.

1. Introduction

This paper is devoted to an axiomatic analysis, based on considerations of monotonicity, of the problem of allocating an infinitely divisible commodity among a group of agents who have single-peaked preferences. By this, we mean that each agent has a preferred consumption; the further away he moves from it, in either direction, the worse off he is. The objective is to achieve a fair division. 1 We search for desirable methods, or solutions, of performing it. The ideal situation is when the amount to divide is equal to the sum of the preferred consumptions. Then, every agent can be given his preferred consumption. If the amount to divide is less than the sum of the preferred consumptions, the situation is the usual one since having less of the commodity to divide will be socially undesirable: there is "not enough". If the amount to divide is greater than the sum of the

Alternatively, we could imagine agents to be endowed with possibly different amounts of the commodity. The problem then would be to reallocate it in some equitable way.

206 W. Thomson

preferred consumptions, the opposite holds and it is having more of the commodity to divide that will be socially undesirable. Then, there is "too much". What should be done in these cases?

This model was recently considered by Sprumont (1991) who searched for strategy-proof solutions, and by Thomson (1990) who looked for solutions satisfying a certain property of consistency, pertaining to economies of variable size. 2 It can be given a variety of interpretations: rationing in a two-good economy in which prices are in disequilibrium; allocation of a task among the members of a team, paid an hourly wage and whose disutility of labor is a convex function of labor supplied; allocation of a commodity when preferences are satiated at some point and free disposal is not allowed.

Our objective is to identify which, if any, solutions respond well to changes in the amount to divide. We are particularly interested in solutions that in addition satisfy appealing distributional requirements. On "classical" domains, where preferences are monotone, the property that has been considered is that all agents benefit from an increase in the amount to divide. Obviously, in situations such as the one considered here, when at some point the commodity becomes less desirable, this requirement does not make sense. A weaker one that is natural then is that all agents be affected in the same direction: they gain together or they lose together as a result of an increase in the amount to divide, in fact, as a result of any change in the amount to divide, whether it be an increase or a decrease. We refer to this property as resource-monotonicity.

Our first results reveal that the property is actually quite strong. When com- plemented with natural distributional requirements, it cannot be met. The distributional requirements are "individual rationality from equal division - every agent prefers what he receives to an equal share of the total amount to divide - and "no-envy" - every agent prefers what he receives to what any other agent

receives. Indeed, we establish the non-existence of resource-monotonic selections from either the individually rational solution from equal division or the no-envy solution. However, resource-monotonic selections from the Pareto solution, the solution that selects for each economy its set of efficient allocations, do exist, and we characterize all such selections.

Then we examine the weaker requirement that agents be affected in the same direction if the change is "not too disruptive" - in the following sense: if there is not enough to the commodity initially, (that is, if the amount to divide is less than the sum of the preferred amounts), there is still not enough after the change, and if there is too much initially, (this is when the sum of the preferred amounts is less than the amount to divide), there is still too much after the change. When efficiency is imposed, the direction in which agents are affected is of course unambiguous. We name this property one-sided resource-monotonicity. This weaker requirement is compatible with our main distributional requirements and there is a sense in which it is the most that one can obtain.

Indeed, our next results are the following. We first characterize the class of one-sided resource-monotonic selections from the Pareto solution, and we introduce a variety of examples in the class, motivated by intuitive considerations of fairness; these solutions are based on the special features of the model and would not be well defined in more general models. We show the existence of an infinite

2 Other studies of this model are due to Ching (1992a, b), Gensemer, Hong and Kelly (1992a, b) and Thomson (1991, 1992).

Resource-monotonic solutions to the problem of fair division...-peaked 207

class of one-sided resource-monotonic selections from the intersection of the Pareto solution with the individually rational solution from equal division, but all such selections coincide over whole intervals of values of the amount to divide. Our main result here is that, on a large subdomain of our principal domain, there is a unique one-sided resource-monotonic selection from the intersection of the Pareto solution with the no-envy solution. This selection is the uniform rule, a solution that plays a prominent role in the Sprumont and Thomson papers mentioned earlier. We also offer a characterization without the domain restriction. Then, uniqueness is lost but the admissible solutions can stiU be completely described: they are obtained by "piecing together" the uniform allocations of certain subeconomies.

These results confirm the importance, already established in previous studies, of the uniform rule in solving the problem of fair division in economies with single-peaked preferences. They should also be seen in the wider context of the recent literature on the design of allocation rules in concretely specified economic environments, a literature whose objective is to identify the tradeoffs one faces in this design. The paper is intended as a contribution to this larger program.

2. The model

An amount M E IR+ of some infinitely divisible commodity has to be allocated among a set N = { 1 .. . . . n} of agents, indexed by i, each agent i being equipped with a continuous preference relation R i defined over IR+. Let Pi denote the strict preference relation associated with Ri, and Ii the indifference relation. These preference relations are single-peaked: for each Ri, there is a number p (R~) E IR + such that for all xi, x: ~IR+, if x: < x i ' < p ( R i ) , or p ( R i ) < = x i < x ' , then x~ Pi x; . The preference relation R~ can be described in terms of the function ri: IR+---,lR+u{oo} defined as follows: given xi<=p(Ri), ri(x~)>=p(Ri) and x i I i r i (x~) if such a number exists and r~ (xi) = 0o otherwise; given x~ _>_ p (R~), r i (x i) <= p (Ri) and x~ I i r~ (x~) if such a number exists and ri (x~)= 0 otherwise. (The number r~. (x~) is the consumption on the other side of agent i 's preferred consumption that he finds indifferent to xf, if such a consumption exists; it is 0 or oo otherwise.) Let r,. (oo) = lim r~ (x~). Let ~ be the class of all such preference

x i ~ o o

relations. We write R = (Ri)i ~ N and p (R) = (p (Ri))i ~ N" An economy is a pair (R, M) ~ ~"x IR+.

A feasible allocation for (R, M) ~ ~ " x IR + is a vector x = (x i)i ~ N ~ IR~+ such that Xx~ = M. Note that free disposal of the commodity is not assumed. Let X ( M ) be the set of feasible allocations of (R, M).

We wish to achieve equitable distributions. A solution is a mapping ~o: ~ '~ x IR+ -+ IR'~ which associates with each economy (R, M) ~ ~ x 1R+ a non- empty subset of X ( M ) . Each of the points in ~0 (R, M) is interpreted as a re- commendation. Apart from several solutions derived from standard economic notions, we will introduce others that are specific to the model. A number of them are single-valued, a desirable property that is difficult to obtain on "classical" domains, that is, domains of economies with infinitely divisible goods and continuous, convex, and monotone preferences.

208 W. Thomson

Note that in our formulation, solutions may be required to depend only on the restriction of preferences to [0, M ] , 3 or they may be allowed to depend on the whole of the preferences. With only small changes in the exposition, we could have assumed preferences to be defined over some fixed interval [0, M0]. Such a specification of the domain would be particularly appropriate in the case of dividing a task among workers, M 0 being the maximal amount of time each worker can work.

The intersection of two solutions ~0 and ~0' is denoted ~0~0', I f ~o is single- valued and {x} = ~0 (R, M), we slightly abuse notation and write x = ~0 (R, M).

3. Basic solutions

In this section, we present the two solutions that have been most often advocated in the literature on the problem of fair division, and variants of them. Unfor- tunately, these solutions typically allow for a wide range of allocations. One of our objectives later on is to identify selection procedures that are well-behaved from the viewpoint of monotonicity. In the course of this investigation, we are led to introducing a variety of solutions whose definitions make use of the particular features of the model under study. First, however, we take care of efficiency:

Pareto solution, P: x ~ P ( R , M ) if x ~ X ( M ) and there is no x" ~ X ( M ) with x" R e x i for all i E N and x" Pi xi for some i e N.

It is easy to check that at an efficient allocation, all agents consume less than their preferred amounts if M < = X p ( R i ) , and more if M>=Zp(R~) (Sprumont 1991). These conditions are also sufficient for efficiency.

Our first fundamental concept with meaningful distributional implications is individual rationality from equal division. I t is often suggested that in order to solve problems of fair division, each agent be given ownership rights on an equal share of the available resources; in the present model, in order to allow for cases when there is so much of the commodity that additional units are not welcome, we should say "equal rights or equal responsibilities". F rom that idea, the requirement that all agents end up better off than at equal division follows naturally.

Individually rational solution f r o m equal division, led : x ~ Ie~ ( R, M ) if x ~ X ( M ) and x i R i ( M / n ) for all i ~ N.

A refinement of the above notion is that no group of agents on its own be able to make all of its members better off, assuming each initially receives M / n . The set of allocations passing this test is "the core f rom equal division".

Our second fundamental concept says that no agent should prefer anyone else's consumption to his own. 4 This idea has played a central role in recent

3 In his analysis, where M is fixed, Sprumont assumes preferences only to be defined on the interval [0, M]. 4 The following definition due to Pazner and Schineidler (1978), as well as variants and exten- sions of it, have been very useful in other contexts: the allocation x ~ X ( M ) is egalitarian- equivalent for (R, M) if there exists a reference amount Xo such that xi Ii Xo for all i ~ N. Let E = (R, M) be the set of these allocations. Here, this concept will not play a role, since E=P(R,M) will typically be empty. Suppose for instance that Xp(Ri)=M. Then, { (p (R1) ..... p (Rn))} = P (R, M). As a result, if for at least one pair {i, j } ___ N, p (Ri) g:p (R j), then E ~ P (R, M) = 0.


developments in the theory of fair allocation. In addition to its direct intuitive appeal, it has the advantage of being meaningful in a wide variety of situations, for example when indivisible goods are present (assigrmaent of jobs; Svensson 1983), and when "equal" division is not well defined (division of a heterogeneous good such as land or time; Berliant, Dunz and Thomson 1992).

No-envy solution, F (Foley 1967): x ~ F(R , M ) if x ~ X ( M ) and for all i , j ~ N, X i R i x j .

We could additionally require that no group be able to make all of its members better off by redistributing among themselves the resources received by any other group of the same cardinality. An allocation passing this test is "group envy- free" (Vind 1971; Varian 1974).

A general discussion of these criteria and of their relation to each other appears in Thomson (1993). When applied to the present model, a few important facts are (Thomson 1990): envy-free and efficient allocations, and individually rational from equal division and efficient allocations, always exist; the "uniform allocation", introduced later on, enjoys all of these properties. In fact, there typically is a large set of allocations (a continuum) satisfying each of these conditions and the need arises then for more precise selections. Since the set of group envy-free allocations and the core from equal division may be empty, these two concepts do not help in this regard: They will not be discussed any further. Moreover, and mainly, we would like our selections to be well behaved from the viewpoint of monotonicity.

4. Resource-monotonicity

We now consider changes in the amount to divide, limiting our attention to single-valued solutions. Suppose that it increases. If agents had monotone preferences, and given that the objective is to treat them "similarly", it would be desirable that all be made to gain. This requirement has been the object of much attention recently in the context of classical economies (Roemer 1986; Chun and Thomson 1988; Moulin and Thomson 1988).

Of course, in our context, where having more of the commodity may be socially undesirable, it would make no sense. A condition that is natural however is that agents all lose together or all gain together when the amount to divide increases, in fact when it increases or decreases. The general requirement that all agents be affected in the same direction "as their environment changes" is the essence of solidarity. An application of this idea to the theory of bargaining appears in Thomson and Myerson (1980). 6 Its usefulness in the theory of quasi-linear social choice when it is population that may change was observed by Chun (1986). Here, it takes the following form:

Resource-monotonicity. For all R ~ ~ n , for all M, M ' ~ IR+, either ~o i (R, M ' ) R i (pi (R, M) for all i ~ N, or (p~ (R, M) R~ (o i (R, M' ) for all i ~ N. Strict resource-monotonicity holds if, in addition, whenever one of the preferences is strict, then they all are.

s Note however that the weaker definitions obtained by requiring that all the members of the objecting group be made strictly better off are satisfied by the uniform rule. 6 Any change in the feasible set affects all agents in the same direction.

210 W. Thomson

Although we argued that this is the "right" requirement for our domain, note that it remains meaningful for classical economies. When combined with efficiency, it does give the condition that has been analyzed on that domain. 7 There, it was shown to be very strong and in fact, to be incompatible with even very undemanding distributional requirements (Moulin and Thomson 1988). Unfor- tunately, in spite of the fact that we have only one commodity here, it retains much force due to the lack of monotonicity of preferences. When combined with individual rationality from equal division, it cannot be met, and even if efficiency is not imposed. This result, which is proved by way of a two-person counterex- ample, is actually implied by a similar impossibility stated in Proposition 2, in which no-envy is used. This is because for n = 2, led c F (Thomson 1990). How- ever, we give the direct proof, which is very simple. It will also be needed in establishing a claim stated in Remark 3 following Theorem 2.

Proposition 1. There is no resource-monotonic selection from the individually rational solution from equal division.

Proof Let ~0 ~Iea. Let N = { 1, 2}, and R e ~ 2 be such that p ( R ) = (1, 2), and let M = 2 and M ' = 4. Let x ~ Ied (R, M). Since M/2 = 1 = p (R 0, then x I = 1 and therefore x = ( 1 , 1). Let y ~ Ied(R,M" ). Since M ' / 2 = 2 = p ( R 2 ) , then y 2 = 2 and therefore y = (2, 2). Since ~0 c led , x = ~0 (R, M) and y = ~0 (R, M ' ) . In the change from M to M ' , agent 1 loses and agent 2 gains, in contradiction with resource- monotonicity. Q.E.D.

In the next proposition, another impossibility is established, based on the distributional requirement of no-envy (instead of individual rationality from equal division). Note that it still makes no use of efficiency.

Proposition 2. There is no resource-monotonic selection from the no-envy solution.

Proof Let ~0~F. Let N={1 ,2} , and R E ~ . ~ 2 be such that p (R) = (2, 7), 1/14, 5 I 2 8, with rl linear in the interval [0, 2] and r 2 linear in the interval [0, 7], and let M = 5 and M ' = 13. 8 Let x ~ F(R, M). For agent 1 not to envy agent 2 at x, we need 1 _<_x~ =< 2.5 or 4 =< x~ =< 5. Since agent 2's preferences are monotone in the interval [0, 5], he does not envy agent 1 at x if and only if x 1 __< x 2. Altogether, we have 1 <=xa _<2.5.

Now. let y e F(R, M') . Agent 2 does not envy agent 1 at y if and only if 0 =< Y2 =< 5 or 6.5 _<_ Y2 =~ 8. For any Y2 in the second interval, p (R 0 =< Yl _-< Y2 so that agent 1 does not envy agent 2. But for any Y2 in the first interval, agent 1 envies agent 2 at y. Altogether, we have 6.5 =< Y2_-< 8.

We conclude by noting that any x ~ F(R, M) is preferred by agent 1 to any y ~ F(R, M ' ) whereas the reverse holds for agent 2, and since ~0 _ F , we obtain a violation of resource-monotonicity. Q.E.D.

7 It is really because we have efficiency in mind that on the classical domain we require that all agents gain from an increase in resources. So, on that domain, and in order to keep considerations of efficiency separate from considerations of monotonicity, one could argue that the "right" condition is also the one above. In the formulation of one-sided resource-monotonicity below, we also chose to keep considerations of efficiency separate from considerations of monotonicity. 8 This means that R 1 can be given a representation that is linear in the intervals [0, 2] and [2, 6], with I I 14.


A noteworthy property of several of the rules that have been examined in the literature is that they depend only on preferred consumptions (Sprumont 1991; Thomson 1990). This property plays an important role in obtaining strategy- proofness (Sprumont 1991). Unfortunately, it essentially prevents resource- monotonicity : this is the content of the next proposition, which also involves the very mild requirement that identical agents be treated identically. We state the requirement for solution functions.

Symmetry. For all (R,M)~,9~J~×]R+, for all i, j e N , if R i = R i, then (oj (R, M)I~ ~o j (R, M).

Again, note that efficiency plays no role in the next impossibility.

Proposition 3. There is no resource-monotonic and symmetric solution that depends only on preferred consumptions.

Proof Let (p be a symmetric solution that depends only on preferred consumptions. Let N={1 ,2} , R e ~ 2 be such t ha tp (RO=2, 1.5/14.5, r 1 being linear on the segment [0, 2], and R 1 = R z, and let M = 2, and M" = 6. Let x = ~0 (R, M). By symmetry, x 11~ x 2. This is possible only if x = (1, 1). Let y = (0 (R, M ' ) . Again, by symmetry, Yl Ii Y2. By the choice of rl, this is possible only if y = (3, 3), y=(1 .5 ,4 .5 ) or y = (4.5, 1.5). If y = ( 3 , 3) or y = (1.5, 4.5), let R ' e ~ 2 be such that R{ = R1, p ( R ~ ) = 2 and 1 I~ 2.5. Since ~0 depends only on preferred consumptions, ~o (R, M) = ~ (R' , M) and ¢ (R, M ' ) ---- ¢ (R ' , M ' ) . I f preferences are R', in the change from M t o M ' , agent 1 gains and agent 2 loses. If y= (4 .5 , 1.5), we repeat the same argument by changing agent l 's preferences in the same way we just changed agent 2's preferences. In either case, we obtain a violation of resource-monotonicity. Q.E.D.

Next, in the light of the above impossibilities, we simply look for resource- monotonic selections from the Pareto solution. Fortunately, such selections do exist. The class all such solutions can even be characterized, in spite of the fact that it is large. This characterization is given in Theorem 1 below, which essentially says that any of them can be obtained as follows: first, fix R e :~n. Then choose any n non-decreasing functions (oi(R , .):[O, XP(Ri)]--~]R + such that 27~0 i (R, M) = M for all M e [0, 27p (R~)] and ~oi (R, 27p (R i)) = p (R~) for all i e N. Once this choice is made, the solution is also specified on [Xp (R~),Xr i (0)[. This is because with any M ' in that interval can be associated a unique M ~ [0, 22p (Ri)] and a unique x ' e P ( R , M ' ) such that x; I~oe(R,M) for all i e N . Resource- monotonicity requires x' = ( o ( R , M ' ) . This completes the construction if X r i ( 0 ) = o o . If not, choose any n non-decreasing functions o~(R,- ) : [2 ; re(0) ,oo[~lR+ such that X ~ o i ( R , M ) = M for all M~[Xre(O),oo[ and ~o~ (R, Xr~ (0) )= r~ (0) for all i ~ N. The reason for this freedom of choice if the amount to divide lies in that interval is that by Nving to each agent i ~ N more than r~ (0), we ensure that each is worse off than at any M e [O, Xr~ (0)]. Putting these observations together, we have:

Theorem 1. A subsolution ~o of the Pareto solution is resource-monotonic i f and only if for each R ~ ~ " ,

(i) for each i ~ N , ~o~(R,.):[O, Xp(Ri )]~IR+ is non-decreasing with Oi (R, 2;p (Ri)) = p (R~), and the list (q~i ( R , . ) ) ~ N satisfies X~0 i (R, M ) = M for all M e [0,2;p (Rg)],

212 W. Thomson

(iia) if Xri (0) < oo, (*) for all M ' e [Xp(Ri),Zr,(O)], let Me[O, Zp(Ri)] be such that for some x" e X ( M ' ) , x" I i ~ (R, M) for all i e N. (The amount M and the vector x' exist uniquely.) Then ~o (R, M ' ) = x'. (**) for each i e N , ~oi(R,.):[Xr,(0),oo[~IR+ is non-decreasing with {o; (R, Xr, (0))= r~ (0) and the list (~o~ (R , . ) )~N satisfies Xq~, (R, M ) = M for all M e [Xr, (0), oo [. (iib) if Xr~ (0)= oo, ~o is extended to the interval [Xp (R~), oo [ by the operation described in (*).

It is important to note that the specification of ~0 in steps (i) and (**) can be made to depend on features of preferences other than the preferred consumptions. Also, within the class described in Theorem 1, it is easy to obtain symmetric solutions.

To illustrate these points, we conclude this section with the discussion of a solution that is based on the idea that the size of agent i 's upper contour set at x i can meaningfully be taken as a measure of the sacrifice imposed on him at x. Then the solution equates sacrifices across agents, an adjustment being made to guarantee that all consumptions are non-negative. Among standard solutions, this solution is the closest to being resource-monotonic.

Equal-sacrifice solution, Sac: x ~ Sac (R, M) if x ~ X ( M ) and (i) when M<=Xp(R~), there exists a > 0 such that ri(x~)-xi<=a for all i e N , strict inequality holding only if x~ = 0, and (ii) when Xp (R~) <= M, there exists a _> 0 such that xi - r i (x~) = a for all i ~ N.

The existence of allocations satisfying this definition is guaranteed under the domain restriction that each ri be finite. If the domain restriction is not imposed, natural generalizations of the solution can be obtained as follows: For each i e N, let x~ = lim r~ (xi). Then, let ~o = (~) i~N be a list of non-decreasing functions

Xi---+ ¢~0

¢v~(R~,.):[0,2:x~]~IR+ such that X~oi (Ri ,M)=M for all M e [ 0 , X x i ] and ~o~(R~,Xxi)=x_i for all i e N . For each M e ]Xx~, oo[, let x e X ( M ) be such that r~ (x~) - x~ = r: (x: ) - x: for all i, j e N. The examples below show that the equal-sacrifice solution satisfies neither individual rationality from equal division nor no-envy.

Example 1. Let N={I ,2} and R e ~;2 be such that 2/14 and 4/26, and M = 6 . Then, Sac (R, M) = (2, 4). Since M / 2 = 3 and 3P 12, Sac(R,M)¢Ied(R,M).

Example 2. Let N={1,2} and R e ~ 2 be such that 2I~4 and 3/25, and M = 5 . Then Sac (R, M) = (2, 3). Since 3 P12, Sac (R, M) q~ F(R, M).

The next example shows that the equal-sacrifice solution is not resource- monotonic:

Example 3. Let N={1,2} and R e ~ 2 be such that 0 I l l , 1/23, M = 0 , and M' = 5. Then Sac (R, M) = (0, 0) and Sac (R, M ' ) = (2, 3). As M increases to M ' , agent 1 loses and agent 2 gains.

However, on the domain of economies such that ri(O)--oo and lim rl (x~)= 0 for all i e N, the solution is resource-monotonic, and on the do-

X i ~ O O


main of economies such that r~ (0) = oo for all i ~ N, the generalizations defined above also are. 9

Here are related solutions that have essentially the same properties: let f . IR+ ~ I R + + be an integrable function; then, pick x ~ P ( R , M ) such that the

ri(x~) "f-weighted sacrifices" ~ f (u) du be equal across agents (again, since boundary

X i

problems may occur, some agents may have to be given 0; this is as in the definition of the equal-sacrifice solution).

5. One-sided resouree-monotonicity

In the previous section we discovered the incompatibility of our basic distributional requirements of no-envy and individual rationality from equal division with resource-monotonicity. However, an examination of the proofs of these negative results reveals that violations of resource-monotonicity occur only when the change in the amount to divide is so disruptive that it turns the economy from one in which there is too much to divide into one in which there is too little, or conversely. It is therefore natural to consider the weakening of resource-monotonicity obtained by applying it only in situations where such reversals do not occur: in this section, we only require that agents be similarly affected if initially there is not enough of the commodity and after the increase, there still is not enough, so that each agent initially receives less than his preferred amount and still does after the increase, as efficiency demands; and if initially there is too much of the commodity, so that all agents have already passed their preferred amounts - again, this is as efficiency demands - we ask that all agents lose as a result of a further increase : altogether, if M <= M" < X p (R i) or 27p (Ri) =< M ' < M, then ~0~ (R, M ' ) R i ~0; (R, M) for all i e N. However, since it is important fully to separate out considerations of monotonicity from considerations of efficiency, I° we will adopt the following formulation:

One-sided resource-monotonicity, n For all R ~ . ~ n, for all M , M ' ~IR+, if M<=Zp(Ri ) and M" <__Zp(R~) or if Zp(R~)<=M and Zp(R~)<=M' , then either ~0i (R, M ' ) R; ~o i (R, M) for all i ~ N or Oi (R, M) R; ~0 i (R, M ' ) for all i ~ N. Strict one-sided resource-monotonicity holds if, in addition, whenever one of the preferences is strict, then they all are.

We find that many solutions are one-sided resource-monotonic. Our first example is the "proportional solution". This solution (mentioned by Sprumont) is the natural expression on our domain of a fundamental principle that underlies much of the theory of allocative fairness (Young 1988, quotes Aristotle: "What is just. . , is what is proportional and what is unjust is what violates the propor- tion").

9 The variant of the equal-sacrifice solution obtained by using the restrictions of the preferences to the interval [0, M] (equating "constrained sacrifices") is not resource-monotonic for any n. 10 See the discussion of this point of footnote 6. 11 The choice of the phrase one-sided resource-monotonicity should be clear. We only consider situations in which the amounts Mand M' are "on the same side" ofXp (Ri). Since the condition of Sect. 4 applies to situations in which M and M' may be on either side of Xp (Ri), we referred to it in an earlier version of this paper as two-sided resource-monotonicity.

214 W. Thomson

Proportional solution, Pro" x= Pro (R, M) if x ~ X ( M ) and there exists )t ~ IR+ such that x i = Xp (Ri) for all i ~ N; if no such 2 exists, x = (M/n ..... M/n).

Note that ,~ in the definition exists as soon as the preferred consumption of at least one agent is positive. In the rare case when all preferred consumptions are zero, we propose equal division. Since then all agents have identical preferences, this choice is quite natural. Unfortunately, it causes the rule to be discon- tinuous with respect to preferences. 12 Unless 27p (R i) = M, at a proportional allocation, no agent with a positive preferred consumption reaches it. Clearly, a proportional allocation is necessarily efficient.

The following variant of the proportional solution is continuous with respect to preferences. Moreover it treats units of the good above or below the preferred consumptions symmetrically (as do all of the other solutions discussed in this paper). This is desirable for some of the interpretations of the model, such as rationing. 13 Of course, it remains efficient.

Symmetrically proportional solution, Pro*." x = Pro* (R, M) if x ~ X (M) and ( i) when M<=X,p (Ri) , there exists ~. e IR+ such that xi = ).p (Ri) for all i e N, and (ii) when Zp(R~)<=M, there exists 2 e IR+ such that M - x ~ = 2 [M-p(R~)] for all i e N.

The symmetrically proportional solution is also one-sided resource-monotonic. Both the proportional solution and its symmetricized version are strictly one- sided resource-monotonic if all preferred consumptions are positive.

A version of the proportional solution that depends only on the restrictions of the preference relations to the interval [0, M] is obtained by requiring pro- portionality to the "constrained" preferred consumptions: PM (R~) = p (R~) if p (R~) __< M and p ~ (Ri) = M otherwise. The resulting solution as well as the solution obtained in a similar way from the symmetrically proportional solution are one-sided resource-monotonic, and strictly so if all preferred consumptions are positive.

These properties should be compared with those of the next solution. When it is not possible to give every agent his preferred consumption, and sacrifices have to be imposed on them, it is natural that all agents be required to sacrifice and the proportional solution achieves this (again, except in the special case when some of the preferred consumptions are equal to zero). But the idea of proportional sacrifice is not the only one that merits attention. The solution below is based on comparing distances from preferred consumptions unit for unit as op- posed to proportionally. It selects the allocation at which all agents are equally far from their preferred consumptions, except when boundary problems occur, in which case those agents whose consumptions would be negative are given zero instead:

Equal-distance solution, Dis: x = Dis (R, M) if x ~ X(M) and (i) when M<=Xp (R~), there exists d = 0 such that x~ = max{0, p (R~) - d} for all i e N, and (ii) when Xp (R i) <_ M, there exists d>= 0 such that x~ = p (Ri) + d for all i e N.

~z A natural topology on the space of preferences is proposed by Sprumont (1991). It is easy to construct examples showing that the rule is not continuous when this topology is used. Note that continuity with respect to the amount to be divided does hold however. 23 Aumann and Maschler (1985) use a similar requirement in their study of bankruptcy problems.


P(R3)- P(RI) + P(R2) 3

2p(R2)- P(R1) 3 p(R~)

3 2p(R1)- P(R 2) /

3

X 2 X2=X 3

, ~ 2.y.._x3 ---"

P(R1 ) P(R2 ) P(R3 ) Zp(R

J

J

J

Fig. 1. An illustration of the variant of the equal-distance solution based on the constrained preferred consumptions. The figure pertains to a three-person example such that p(R1) < p(R2) < 2p(R1) < p(R3)

The equal-distance solution is single-valued and it produces efficient allocations. It is one-sided resource-monotonic. However, the variant of the solution obtained by using the restrictions of the preference relations to the interval [0, M] (measuring distances to the constrained preferred consumptions p~¢ (Ri)) is not, except for the two-agent case. This is illustrated in Fig. 1 which depicts the amount received by each of three agents as a function of the amount to divide. Note that as this amount increases from the smallest preferred consumption to the second smallest preferred consumption, agent 1 is made progressively worse off whereas agents 2 and 3 are made better off.

It follows from Proposition 3 that neither the proportional nor equal-distance solutions are resource-monotonic. Direct proofs might be useful:

Example 4. Let N={1,2} and R e 2 2 be such that p (R) = (2, 4), 1/12.5 and 2/27, and let M = 3 , M ' =9. Then Pro (R, M) = (1, 2), Pro (R, M ' ) = (3, 6). In the change from M to M ' , agent 1 loses and agent 2 gains.

Example 5. Let N={1,2} and R e ~2 be such that p (R) = (1, 2), 0 I 11.5 and 1124, and let M = I , M ' =5. Then Dis (R, M) = (0, 1), Dis (R, M ' ) = (2, 3). In the change from M to M ' , agent 1 loses and agent 2 gains.

The examples below, which should not be surprising, allow us to relate the above rules to our primary distributional criteria. 14 First of all, neither the proportional solution nor the equal-distance solution necessarily selects allocations that are individually rational from equal division.

Example 6. Let N = { 1,2} and R e ~ 2 be such that p (R) = (3, 6) and let M = 6. Then Pro (R, M) = (2, 4). Since M/2 = 3 and 3 P~ 2, Pro (R, M) ¢ lea (R, M).

Example 7. Let N={1,2} and R e ~ 2 be such that p(R)=(3,6), 2/14, and let M = 7 . Then Dis(R,M)=(2,5). Since M / 2 = 3 . 5 and 3.5P12 , Dis(R,M)

Ied (R, M).

14 The results also hold for the versions of the solutions obtained by using the restrictions of the preferences selection to the interval [0, M].

216 W. Thomson

Also, neither solution necessarily selects envy-free allocations:

Example 8. Let N = { 1,2} and R ~ ~ 2 be such that p (R) = (2, 4), 1.5 I 1 4, and let M=4 .5 . Then Pro ( R , M ) = (1.5, 3). Since 3P 1 1.5, Pro(R ,M)¢F(R ,M) .

Example 9. Let N = { 1, 2} and R e ~,~2 be such that p (R) = (3, 4), and let M = 5. Then Dis (R, M) = (2, 3). Since 3 P1 2, Dis (R, M) ~ F(R, M).

In light of these observations, it is natural to attempt redefining the solutions so as to recover either individual rationality from equal division or no-envy. An appealing way to do this is to perform lexicographic operations. Such operations are standard in game theory and social choice. Taking the differences d e (x )= I xi - P (Ri) [, for i ~ N, as a point of departure, for example, we would look for allocations x in/_od P (R, M) or in FP (R, M) at which these differences are not equal, as in the definition of the equal-distance solution, but as equal as possible, according to the lexicographic ordering.

Formally, given t ~ IR n, let t" be obtained by rewriting the coordinates of t in decreasing order. Given t and t' E IR ~, say that t is lexicographically greater than t[, written t >~t ' , if[/1 >/1' ] .or [/1 =/1' and ~ > ~ ], or for some k ~ { 1,..., n} [t 1 = t ; , . . . , t k_ 1 = t~_l, and t k = t[ ]. Also, let d(x) = (d I (x) .... , d, (x)). Then, we define selections from /ee P and FP by {x e Ied P (R, M) I d(y) >_ zd(x) for all y e/~d P (R, M)} and {x e FP (R, M) I d(y) >_ rd(x) for all y e FP (R, M)}.

Similarly, we could define selections from Ied P and FP in the spirit of the proportional solution by associating with each x~IR~_ the vector

( Xp~ x~ ) (where 0/0 = oe by convention), and looking for allocations 1) ' " " p

in leaP(R, M) and FP(R, M) whose associated vectors are lexicographically minimal.

Unfortunately, so adapting the solutions will cost us one-sided resource-monotonicity, as will follow from Theorem 2. In Sect. 4, we saw variants of the equal- sacrifice solution that are resource-monotonic, and therefore one-sided resource- monotonic, but again, they do not necessarily select envy-free or individually rational from equal division allocations. 15 It is therefore with some relief that we encounter our next solution, which is one-sided resource-monotonic and always selects an allocation that is efficient, envy-free and individually rational from equal division. This solution, called the uniform rule, already played an important role in Sprumont (1991) and Thomson (1990), who characterized it on the basis of strategy-proofness and consistency, respectively.

Uniform rule, U: x = U ( R , M ) if x ~ X ( M ) and (i) when M<=Xp(Re), xi=min{p(Ri) ,) .} for all i e N , where 2 solves X m i n { p ( R i ) , 2 } = M , and (ii) when E p ( R i ) < M , xi=max{p(R~),)~ } for all i ~ N , where 3~ solves 2:max{p (R,),,~ } = M .

If M < 27p (R e), the uniform allocation is obtained by successively making the agents who receive the least as well-off as possible, and if Zp (R~) < M, by successively making the agents who receive the most as well-off as possible. Here are the payments as a function of M (Fig. 2 illustrates the rule for n = 3). For M small, all agents receive the same amount; this holds until all have received an

is The variant of the equal-sacrifice solution obtained by using the restriction of the preference to the interval [0, M] (see footnote 6) is one-sided resource-monotonic if n = 2, but not if n > 2.


P(R 3)

P(R2)

P(R 1)

X 1 = X ~

Xl 21> -x2=×;I

3p(R 0 3p(R 1) + 2[p(R2)- p(R~)]

/4 x 1 = X 2 ~

/

Z P(Ri) I ZP(Ri) + P(R2) - p(R1)

3p(R 3)

Fig. 2. An illustration of the uniform rule in the three-agent case

amount equal to the smallest preferred consumption. Then, the agent with the smallest preferred consumption does not receive anything for a while. Instead, any increase in M is divided equally among the remaining agents until each of them has received an amount equal to the second smallest preferred consumption. Then, the agent with the second smallest preferred consumption does not receive anything for a while... This process continues until each agent has received his preferred consumption. Any increase beyond 27p (Ri) goes first to the agent with the smallest preferred consumption until he has received an amount equal to the second smallest preferred consumption. A further increase is divided equally among the agents with the two smallest preferred consumptions until they have received an amount equal to the third smallest preferred consumption. . . This goes on until all agents have reached the largest preferred consumption. Any further increase is divided equally. 16

As noted above, the uniform allocation is efficient, envy-free, and individually rational from equal division. Also, it depends only on preferred consumptions. Moreover we have the following:

Lemma 1. The uniform rule is the only selection from the envy-free and efficient solution to depend only on preferred consumptions. 17

Proof Let (o c_FP be a solution that depends only on preferred consumptions, and suppose by contradiction that for some (R ,M) e ~ ' n × I R + , x = (o (R, M)--l= U(R,M). Suppose, without loss of generality, that M<=Z,p(Ri). Then, for some i, j e N with x i < x j , x; ~ p (R i). Since e - P, xi < p (R,.). Now, let R; e R be such that p (R;) = p (R~) and xy P; x i. Since (0 depends only on pre-

16 We chose this way of presenting the uniform rule because it will facilitate understanding its one-sided resource-monotonicity and also because it reveals its similarity to the rule advocated in particular by Maimonides for the adjudication of conflicting claims (O'Neill 1982; Aumann and Maschler 1985). Indeed, the algorithm describing that solution is identical up to the point where each agent has received his preferred consumption, by replacing the vector of preferred consumptions by the vectors of claims. 17 We note however that the uniform rule is not the only selection from the individually rational from equal division and efficient solution to depend only on preferred consumptions.

218 w. Thomson

ferred consumptions, ¢0 (R;, R_~, M) = ¢0 (R, M). But x¢ F ( R ' , R_~, M) since agent i now envies agent j at x.~S Q.E.D.

Since the proportional and equal-distance solutions are efficient solutions that depend only on preferred consumptions, it follows that they violate no-envy, as we established earlier by means of two-person examples.

The fact that there exists a one-sided resource-monotonic selection from the envy-free and efficient solution, the uniform rule, should be emphasized. We repeat that on classical domains, there is no solution that satisfy even considerably weakened versions of these distributional requirements and respond well to changes in resources.

Next, we ask whether there are one-sided resource-monotonic selections from the envy-free and efficient solution other than the uniform rule. The answer is no, provided the fairly weak domain restriction is imposed that for each agent, there be a finite consumption indifferent to 0. As the consumption of the commodity increases, it eventually becomes a "bad": consuming more than some critical amount is worse than consuming none of it. (Technically, this restriction prevents the consumption of the agent with the largest preferred consumption to become infinite while the consumptions of the others remain finite, without his becoming envious of them.) We describe in Remark 2 following the theorem the class of additional solutions that become admissible when the domain restriction is removed.

Theorem 2. Let ~ ~× IR+ be the domain of preference profiles such that each r~ is bounded. On the domain ~ ~ × IR +, the uniform rule is the only one-sided resource- monotonic selection from the envy-free and efficient solution.

Proof Let ~ 0 : ~ n× IR+-~IR% be a one-sided resource-monotonic selection from FP.

(i) ¢o is continuous with respect to M. Suppose not. Then, there exist sequences {M v} in IR+ and {x v } in ]Rt , M ° ~ IR+, x ° ~ JR+ such that M V ~ M °, x V ~ x °, x v = ¢0 (R, M v) for all v ~ N and x ° 4: ¢o (R, M°). Let y = ¢o (R, M°). Since ¢o c p, M ° ~ Z p (R i) (otherwise y = p (R) and x ° = ¢o (R, M°)) . Suppose, without loss of generality, that M ° < Z p (Ri). There exists 17 such that for all v >= ~, M v <=Zp (R~). Then, since x ° ~ y , and since ¢o _ P , there are i , j ~ N such that x ° < yi<=p(Ri) and yj < x °__< p (R i) . There is ~ so that for all v >_ ~, x~ < y~ and yj < x}. Let v >__max{iX, ¢}. In the change from M ~ to M °, agent i gains and agent j loses, in contradiction with one-sided resource-monotonicity. (ii) For all M such that M<=Zp (R~), q~ (R, M) = U(R, M). Suppose, by contradiction, that for some Mwi th M<=Zp (R~), x = ¢o (R, M) ~ U(R, M). Since ~0 c p, xi _--< p (Ri) for all i E N. Then, there are i, j e N such that x i < p (R~) and x~ < xj . Since ¢o(R,M')>O for all M'~IR+, ¢Oio(R,M')_~O as M ' ~ 0 . Since c o , F, p (R;) < r~ (xi) __< xj . Therefore, by (i), there is M =< M such that _¢o j (R, M) = p (R~). Since ¢0 o F , for agent i not to envy agent j_" at ¢o (R, M), we need ¢o~ (R, _~¢) = p (R~). Therefore, in the change from M to M, agent i gains and agent j loses, in contradiction with one-sided resouree-monotonicity.

~s Note that the argument would apply to any solution simply required to be such that the consumption received by any agent depends on his own preference relation only through his preferred consumption.


(iii) For all i e N, ~o~ (R, M)--~ oo as M ~ oo. If not, there exists i e N such that 2~ = sup {~0 i (R, M) [M e IR+ } < oo. Also, since ~o (R, M) > 0 for all M e IR+, there exists j e N such that ~o] (R, M)--* oo as M--* oo. Let 2]-= max{r] (0), 2g}. Then, for M large enough, ~0]. (R, M) > 2].. We have ~0~ (R, M) Pj ~oj (R, M), so that agent j envies agent i at O (R, M), in contradiction with ~o c F. (iv) For all M such that Xp(R~)< M, (o ( R , M ) = U(R,M). Suppose, by contradiction that for some M e IR+ with Z,p(R~) < M, x=~o ( R , M ) ~ U(R,M). Since ~0 c_ p, xi _-> p (R~) for all i e N. Then, there are i, j e N such that x; > p (R i) and xl > xj. By (iii), ¢j (R,_ M)-* oo as m ~ oo. Since ¢ c_ F, xj _< r; (x~)=< p (Ri). Therefore by (i), there is M such that ~oj (R, 3)) = _p (Rg). Since ~0 c F, for agent i not to envy agent j_at ~ (R, 3~r), we need ¢~(R,M)=P(Ri) . Therefore, in the change from M to M, agent i gains and agent j loses, in contradiction with one-sided resource-monotonicity. Q.E.D.

Remark 1. Theorem 2 would still hold if the domain were restricted by the weaker condition that, assuming agents numbered so that p(RO<=.. .<p(Rn) , rn(oo ) < p(R1). The only change in the proof would occur in step (iii), which would continue after the second sentence as follows: "We claim that this implies that ~on (R, M)--* oo as M ~ o0. Indeed, either j = n, and then we are done; or if j~en, for every B > p ( R n ) , there is M e l R + large enough so that p(R~) < p (Rn) < B < ~oj (R, M) < ~o, (R, M), where the last inequality follows from the requirement that agent j not envy agent n at ~o (R,M). Now, since rn(oo ) <p(R1) , there is M e IR+ large enough so that r , (~o , (R ,M))<p(R 0 < ~o i (R, M ) < ~0~ (R, M), where the middle inequality follows from the inclusion ~0 c_ p, and then agent n envies agent i at ~o (R, M), in contradiction with ~o___F. "

Remark 2. The following example shows that Theorem 2 would not be true if the domain restriction were dropped altogether. Let N = { 1,2} and R e ~ 2 be such that p ( R ) = ( 1 , 3) and r2(oo ) =2. Let ~0 be such that ~0 (R,M) = U(R,M) for all m<=Xp(Ri) and ~o (R, M) = (p (R1) , m - p ( R 1 ) ) for all M > X p ( R i ) . It is easy to check that ~o c_ FP and that ~0 is one-sided resource-monotonic. It is also possible to modify the definition of ~o so that the amount received by agent 1 would actually increase beyond p(R1) (the main thing is that it should not increase beyond r 2 (o0)). A complete characterization without the domain restriction can be obtained by generalizing this example (Appendix A).

Remark 3. The non-existence of resource-monotonic selections from the individually rational from equal division and efficient solution (implied by Proposition 1) follows from (i) Theorem 2, (ii) the fact that the uniform rule does not satisfy this property, and (iii) the fact noted earlier (see the discussion before Proposition 1) that for n = 2, any allocation that is individually rational from equal division is envy-free. The proof of (ii) is provided in Proposition 1. It involves economies for which there is a unique allocation that is individually rational from equal division. This allocation is therefore the uniform allocation.

Remark 4. A direct consequence of Theorem 2 is that on ~.~ n× IR+, there exists no strictly one-sided resource-monotonic selection from the envy-free and efficient

220 W. Thomson

solution. 19 This is because the uniform rule does not satisfy this stronger property, as can be seen from Fig. 2.

Remark 5. What if no-envy is replaced by individual rationality from equal division? Since, as stated above no-envy is a weaker condition for n = 2, it follows from Theorem 2 that, if ~0 is a one-sided resource-monotonic selection from led P, then ~o = U on ~ 2 x IR. However, for n > 2, the uniform rule does not remain the only one-sided resource-monotonic selection from Ieu P on ~_~ n× IR. An in- formal description of all solutions satisfying these requirements is given in Appendix B for a special case.

Remark 6. We conclude with a characterization of all one-sided resource-monotonic selections from the Pareto solution: for each R ~ ~ n simply choose n non- decreasing functions ~0; (R, . ) : IR+ ~ I R + adding up to the identity function and such that ~o i (R, X p ( R i ) ) = p ( R i ) for all i~ N.

6. Conclusion

It is instructive to compare the conclusions obtained here for the problem of fairly dividing a commodity among agents with single-peaked preferences to what is known for classical problems of fair division.

The condition of monotonicity with respect to resources that has been analyzed there (all agents benefit from an increase in the available resources) would be inappropriate for the current model. Instead, the requirement that all agents be affected in the same direction by an increase, or a decrease, in the amount to divide is the natural requirement to consider. Unfortunately, we found this property of resource-monotonicity to be quite demanding. Indeed, if there are resource- monotonic selections from the Pareto solution, there are none from either the individually rational solution from equal division or the no-envy solution.

Then, we formulated a weakening of resource-monotonicity that takes into account the special features of the model, and essentially limits its application to situations in which the change in the amount to divide is not too disruptive. Although it is easy to define selections from the Pareto solution satisfying this property of one-sided resource-monotonicity, and many one-sided resource- monotonic selections from the individually rational from equal division and efficient solution exist, our main result was that there is a unique one-sided resource-monotonic selection from the envy-free and efficient solution on a large subdomain of our primary domain.

Results pertaining to the alternative distributional requirement of "average no-envy" paralleling our results pertaining to no-envy are presented in Appendix C.

Not all of our results are positive, but they are in sharp contrast with the results obtained for the classical domain. There, the property that all agents benefit from an increase in the resources available is satisfied by no selection from the envy-free and efficient solution, nor from the individually rational from equal division and efficient solution. (See Moulin and Thomson 1988). 20

19 This is true even if all preferred consumptions are positive. (Recall that this restriction had helped in the case of the proportional solution.) 20 In fact, the property is incompatible with considerably weakened distributional requirements, as is established in that paper.

Resource-monotonic solutions to the problem of fair division...-peaked

Table 1

221

Individual ratio- No-envy Resource- One-sided nality from equal monotonicity resource- division monotonieity

Uniform rule yes yes no yes (Prop 1, 2, 3)

Proportional no no no yes solution a (Ex 6) (Ex 8) (Prop 3, Ex 4)

Equal-distance no no no yes solution (Ex 7) (Ex 9) (Prop 3, Ex 5)

Equal-sacrifice no no yes c yes solution b (Ex 1) (Ex 2)

~o ~_ Iea yes may or no may or (by Def) may not (Prop 1) may not

~0 ___ F may or yes no only if may not (by Def) (Prop 2) ~ = U(Th 2) a

The same properties hold for the symmetrically proportional solution b This solution is well-defined on the domain of economies for which each r~ is bounded. Variants of this solution can be defined for the whole domain ° This result holds on the domain of economies for which r i (0)= m and lim r i (x~)=0 for all

i ~ N. It extends to some generalizations of the solution on some wider domain. It does not hold for the original definition (Ex 3) nor for the version of the solution obtained by using the restrictions of preferences to the interval [0, M], unless there are only 2 agents d This characterization of the uniform rule also involves the requirement ~0 c p. It holds on the domain of economies for which ri (0) < oe for all i e N. Otherwise, variants of the rule are obtained (Appendix A)

The results are summar ized in the fol lowing table. A "yes" in cell (a, b) means tha t the so lu t ion in row a satisfies the p r o p e r t y in co lumn b. A "no" means the opposi te .

Appendix A

Characterization of the class of one-sided resource-monotonic selections from the envy-free and efficient solution without the domain restriction of Theorem 2.

Fi r s t o f all, it fol lows f rom steps ( i ) - ( i i ) of the p r o o f o f T h e o r e m 2 , tha t even wi thou t the d o m a i n restr ic t ion, any ~0 c_ FP sat isfying one-sided resource- monotonicity is such tha t ~0 ( R , M ) = U ( R , M ) whenever M<=Xp(Ri). I f M increases b e y o n d Xp (R i), ~o (R, M) is ob ta ined by " jux tapos ing" the un i fo rm alloca t ions o f cer ta in subeconomies , as follows. (We omi t the p r o o f o f these as- sertions, mos t o f which are pa t t e rned af ter the var ious steps o f Theo rem 2.)

(i) The agents are pa r t i t i oned into groups , each g roup consis t ing o f agents wi th successive prefer red consumpt ions , two agents wi th the same prefer red consumpt ion be longing to the same group. (Therefore , there is a na tu ra l o rder for the groups , the prefer red consumpt ions o f all the members o f any g roup being strictly lower than the prefer red consumpt ions o f all the members o f the next highest g roup . ) F o r any i ~ N such tha t sup { ~0 i (R, M ) I M e IR + } < o% let si be this supremum. I f ~og (R, M)--* m as M ~ oo, let s i = oo.

222 W. Thomson

(ii) The consumptions of all the members of the highest group become infinite with M. For any agent i in that group, s~ = ~ . (iii) The consumptions of the members of any other group have a common finite supremum. For any agent i in such a group and for any agent j in the next highest group, s~ < rj (sj). (Therefore, there will be only one group if for all i e N, r i ( ~ ) < minp(Rs) , as explained in the paragraph following Theorem 2; also, the consumptions of all the members of any group other than the highest group remain finite.) (iv) What each group receives as a whole is allocated between its members by applying the uniform rule. (v) What each group receives as a whole is a continuous increasing function of M. For each M e IR+, the values of these functions add up to M.

Appendix B

Characterization of the class of one-sided resource-monotonic selections from the individually rational and efficient solution. The is an infinite dimensional class that is not easy to describe. However, in order to give the reader an idea of what they look like, we indicate the features shared by all of them for n = 3, and, after numbering agents so that p(R1)<=p(R2)<=p(R3), under the assumption that 2p (R2) => p (R1) ÷ p (R3).

Then, any one-sided resource-monotonic selection from Ied P, ~0, is such that for all M e [0, 3 p (R 0] u [3 p (R3), oo [, ~0 (R, M) = (M/3, M/3, M/3) = U(R, M); for all M e [3p(R1),Zp(R~)], ~1 (R,M)=p(R1); for all M e [2:p (Re), 3p (R2)], ~0 (R, M) = ( M - p (R2) - p (R3), p (Re), p (R3)) = U(R, M); for all M e [3 p (R2), 3p (R3)[, ~03 (R, M) = p (R3). Otherwise, some indeterminacy exists. However, precise bounds can be established. (Details are available from the author.)

Appendix C

The distributional concept of average no-envy. Here, we present results pertaining to a distributional concept intermediate in spirit between no-envy and individual rationality from equal division. It simply says that every agent prefers his consumption to what the others receive on average.

Average no-envy solution, A (Thomson 1979, 1982; Baumol 1986; Kolpin 1991):

x e A (R'M)' if x ~ X (M) and xiRi ( ~ i x J / ( n - 1 ) ) fora l l i e N.

Any allocation that is individually rational from equal division is average envy-free. There is no containment relation between the sets of envy-free allocations and of average envy-free allocations, unless of course n = 2, in which case the two concepts coincide. Therefore, Proposition 2 also establishes the non- existence of resource-monotonic selections from the average envy-free solution.

It follows from what we have established concerning one-sided resource- monotonic selections from lea P and the inclusion Ie, ~ ~__ A that the uniform rule is not the only one-sided resource-monotonic selection from AP. However, at least for n = 3, the class of one-sided resource-monotonic selections from AP is not any larger than the class of one-sided resource-monotonic selections from Iea P.


The comment s o f H. Moul in , J. Roemer , S. Sonn, T. Y a m a t o , and a referee as well as suppor t f rom N S F under G r a n t No. 8809822, are gra teful ly acknowl- edged.

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