guarantee structures for problems of fair division

Mathematical Social Sciences 4 (1983) 205-2 18 North-Holland

205

GUARANTEE STRUCTURES FOR PROBLEMS OF FAIR DIVISION*

William THOMSON Department of Economics, Harvard University, Cambridge, MA 02138, U.S.A. and

Department of Economics, University of Minnesota, Minneapolis, MN 55455, U.S.A.

Terje LENSBERG Norwegian School of Economics and Business Admrnistration, Bergen, Vorwa_v

Communicated by E. Kalai Received 1 March 1982 Revised 19 October 1982

We study the problem of fair division in situations where the number of individuals involved may vary while the resources at their disposal remain fixed. We are interested in minimizing the loss that an agent originally present may incur in such circumstances. Given a solution. i.e., a systematic method of solving any division problem in some class, we introduce the notion of its guarantee structure as a measure of the protection it offers to the original agents. We show that

the Kalai-Smorodinsky solution offers greater guarantees than any weakly Pareto-optimal and anonymous solution and in particular than the Nash solution.

Key words: Fair division; Kalai-Smorodinsky solution; guarantee structure.

1. Introduction

We are concerned here with the problem of fair division in situations where the number of agents among whom the division is to take place may vary. A division problem arises when some resources have to be distributed among a group of agents whose preferences over the possible divisions conflict. All the agents are assumed to have equally legitimate claims on the resources and the question is how to establish a fair compromise. A division principle or solution defined over a certain class of division problems provides a way of resolving all problems in that class.

We imagine here that variations in the number of participants may take place while the resources at their disposal remain fixed, and we are interested in minimizing the losses that the agents originally present may incur upon the arrival of additional claimants.

To make this idea more precise, let a solution be given, as well as some *roup P of agents, some member i of P, and some group P’ of agents disjoint from P. Wheil P' joins P, the recognition of the claims of its members will typically necessitate that

*This research was partially supported by NSF under Grant No. SES 8006284.

0165-48%/83/$3.00 0 1983, Elsevier Science Publishers B.V. (North-Holland)

206 W. Thomson, ‘I: Lensberg / Guarantee structures

some of the resources previously distributed among the members of P be transferred to them. Agent i will not necessarily lose from the reshuffling of bundles required to re-establish an equita.ble compromise, but if he does, he certainly will be interested in the mmimal proportional loss that he may experience upon the arrival of p’, To measure the ‘protection’ offered by the solution to agent i, we determine the greatest number a! with the property that agent i be guaranteed that his final utility be at least cly times his original utility. The guarantee structure of a solution is the list of all of the CXI’S so obtained, when P, i and P’ are chosen so as to satisfy the conditions listed above.

Solutions thlat offer greater guarantees are of course more appealing. If each term of the guarantee structure of some solution is not smaller than the corresponding term of the guarantee structure of some other solution, we will say that the former wecpkly dominates the latter. However, given two arbitrary solutions, their guarantee structures may or may not be comparable in this way.

A number of natural questions arises concerning guarantee structures. What are the guarantee structures of the most commonly used solutions? Is there a maximal element in the set of guarantee structures associated with solutions satisfying minimal requirements? Does any one of the well-known solutions offer these maximal guarantees (if they exist)?

The answers to these questions are established in the present paper. (i) The most commonly used solutions can be ranked as follows: the Egalitarian

and Utilitarian solutions fare the worst. In fact they offer no guarantees at all. Then comes the Nash (1950) solution. Finally comes the Kalai-Smorodinsky (1975) solution.

(ii) There is a maximal guarantee structure in the set of guarantee structures associated with solutions satisfying weak Pareto-optimality and Anonymity.

(iii) ‘This maximal guarantee structure is achieved by the Kalai-Smorodinsky solution.

(iv) Is the Kalai-Smorodinsky solution the only one satisfying weak Pareto- optimality, Anonymity and achieving the maximal guarantee structure? No, not even in the set of solutions satisfying other appealing properties such as Scale In- variant e and Continuity. Other solutions with these five properties can in particular be consrrur,ied by perturbing the Kalai-Smorodinsky solution.

The r,Aoperties of the Kalai-Smorodinsky solution described in this paper, along with several others recently discovered by various authors (Imai, 1981; Segal, 1980; Thomson, Ml), should help give this solution a more prominent place in the litera- ture on arbitration and bargaining.

This paper is organized as follows: Section 2 contains some preliminaries, Section 3 introduces the concept of a guarantee structure and contains the results, and Section 4 gives a concluding remark.

W. Thomson, T. Lensberg 1 Guarantee structures 207

In Thomson (1981) the problem of fair division is reformulated so as to allow for the possibility that the number of agents among whom the division is to take place varies. In order to deal with such situations the following concepts are introduced. I, the natural integers, is the set of ‘potential agents’. 3 is the class of finite subsets of I. Elements of .4” are denoted by P, Q, . . . with their respective cardinalities indi- cated by IPI, IQI, . . . . Given Pt9, Rr is the Cartesian product of jPI copies of R, indexed by the members of P, and Cp is the class of division problems that the group P may conceivably face. The elements S of Zp are defined as follows:

(i) S is a compact subset of Rr and there exists an XE S with x > 0. (ii) S is convex.

(iii) S is comprehensive, i.e., for all x,y~Rr if x& and xzy, then YES. (Given x and y in R ‘, x 1 y means that Xi Zyi for all i E P, x 2 y means that x 1 y

and x#:y and X>Y means that Xi>yi for ai hp.) S is interpreted as the set of utility vectors, measured in some von Neumann-

Morgenstern scales, achievable by the group P through some joint action. The assumptions made on S are standard: (i) Compactness is a technical requirz-

ment. The existence of XE S with x > 0 implies that all agents are nontrivially involved in the division problem. (ii) Convexity holds in particular if joint randomi- zation among alternatives is permitted. (iii) Comprehensiveness follows in particular from assuming free disposability of utility.

We note that the image in the utility space of the set of allocations attainable through arbitrary distributions of a fixed bundle of goods, assumed to be freely disposable, among a group P of agents with continuous, concave and nondecreasing utility functions, normalized by assigning zero utility to the zero ~:undle, is also a member of 2’.

Conversely, any element of Cp can be rationalized in this way, as demonstrated by Billera and Bixby (1973). This is an important result which indicates that there is no loss of generality in restricting one’s attention to ‘economic’ division problems.

A solution is a list F= (Fp, PE 9 ) where FP: Zp --, RP is a function associating with each &CP a unique element F’(S) of S called the solution outcome for S and interpreted as the best compromise among the conflicting preferences of the agents.

What is meapt by ‘best’ remains of course to be defined. The kind of properties that are commonly deemed desirable for a solution are the following.

Pareto-optimality (PO). For all PE 9, for ail S E Zp and for ad1 _v E Rf, ijr x 1 F “(S ), then x $ S.

Often a weaker form of this property is used.

208 W. Thomson, T. Lensberg I’ Guarantee structures

Weak Pareto-optimality (WPO). For all PE 9, for all S E Zp and for all XE Rf, if x> F’(S), then x@ S.

Anonymity (AN). For all P, P’E ‘9 with lPl= IP’I, for all one-to-one functions y:P+ P’, for all SEEP andfor all SkZP’, if

S’={X’ERpI 3XES S.t. ViEP,X~~ij=Xi),

?!wI, for all i E P, F,$) (S ‘) = Fip(S ).

Given PE 9 :ve denote by A’ the class of transformations A from RP to RP of the following form: For each k P there exists a positive real number ai such that for all XE RP, Ai = aixi . Also, given SE Zp and ;I E A’, we denote by A(S) the set

{xk RP 13~~s s.t. x’=A(x)}.

Note that J(S) is also in Zp.

Scale Invariance (S.INV). For all PE 9, for all SE 2’ and for all A E A’, FP(AsS)) = A(F’(S)).

Continuity (CONT). For all P E 9, for all sequences 1 Sk ) of elements of Cp con- verging in the HausdorJf topo,!ogy to some S E C ‘, F ‘(S k, + F ‘(S ).

These axioms are familiar axioms, appropriately reformulated for the problem at hand:

PO says that it is not feasible to increase any agent’s utility from the solution outcorn: without decreasing some other agent’s utility;

WPG says that it is not feasible to increase all agents’ utilities from the solution outcome.

AN says that the names of the agents do not matter in the sense that two division problems involving two different groups of agents but with the same geometrical structue are solved at the same point.

S.INv SZ:X that changing a division problem by independently resealing the individuis’ utility functions leads to a new division problem which is solved at the image under this transformation of the solution outcome of the original problem.

CONT says that small changes in division problems lead to small changes in solution outcomes.

We now introduce (the generalization to the present context of) what is probably the best-known solution, due to Nash (1950), and defined as follows: Given PE 9 ,and SeC”, N’(S) is the unique maximizer of the product fliEp Xi over S. Then -the Nash solution is N= {N’, PE 9). N satisfies PO, AN, S.INV and CONT.

In this paper we will in fact promote another solution, introduced by Kalai and Smorodinsky (1975), and defined as follows: Given PE 9 and S E .Xp, K’(S) is the maximal (according to the partial ordering of vectors in R ‘) point of S on the seg-

W. Thomson, T. Lensberg 1 Guarantee structures 209

ment connecting the origin to the ideul point, the point of Rr the ith coordinate of which is the maximal utility achievable in S by agent i. The ideal point is denoted by a(S) where, for each kf, aJS)=max,,, xi . Then the Kalai-Smorodinsky solution, or KS solution for short, is K= (K’, PE 9). K satisfies WPO, AN, SJNV and CONT.

We will also briefly discuss the Egalitarian solution, E, defined as follows: Given PE 9 and SEX’, E’(S) is the maximal point of S with equal coordinates, and E= (Ep, Pe 9 ). E satisfies WPO, AN and CONT but not S.INV.

Finally, we qualify as Utilitarian any solution U defined as follows: Given PE .-+J and SEC’, Up(S) is a maximizer of Cis,, xi over all points x of S. (411 utilitarian solutions coincide on division problems with a strictly convex Pareto-optimal boundary.) Each U satisfies PO and AN but not CONT nor S.INV.

Additional notations. e is the infinite list (1, . . . , 1, . ..). Given PE .9, ep is the projection of e on R!, i.e.,

it is the vector of RF that has all of its coordinates equal to 1. Given Q E .3 with Q>Pand givenyER p, yp is the projection of y on Rf. ei is the ith unit vector. Given x1 , . . . . X/E R,p, cch(x’, . . . . x’) is the convex and comprehensive hull of X1 , . . . , xi, i.e., it is the smallest convex and comprehensive subset of R! containing these points.

3. Guarantee structures

Let F= (Fp,P~3j be a solution. Let &I, PE.~ with kP and P’E 3 with Pf7 f’ = 0 be given. Assume that for some a E R, U f=] the following is true:

For all SEX’ and for all BZPUP’, if S= TnRP, then &aPUP’(T) 2 Q&,‘(S).

(1)

We start from the division problem S which involves the group P to which agent i belongs, and we determine its solution outcome; then the group P’ enters the scene and it is recognized that its members have the same rights on the resources as the members of P. We denote by T the division problem that the group P U P’ faces (r is such that S = Tfl Rp) and we determine its solution outcome. Typically, we would expect agent i to lose. This may or may not be the case. However, if he does lose, the above inequality means that he will get in the new division problem at feast the fraction a of what he originally had. Let

a&P,P’)=sup{a)(l) holds).

The number a&i, P, P’) represents the guarantee offered by the solution F to agent i in group P when group P’ comes in. The list

210 W. Thomson, T. Lensberg / Guarantee structures

of all these numbers is the guarantee structure of the solution F. If the solution satisfies AN, its guarantee structure is simply given by a sequence

(Q/?), where m and n run over the natural integers, and c$“’ represents the guarantee offered to any agent involved in any group of cardinality m upon the addition of any group of cardinality n, From here on, we will limit our attention to solutions satisfying AN, and therefore to guarantee structures of that form.

Given two solutions F and G, we say that a~ weakly dominates CTG if for all m, n E I, crj? z c@‘*. The domination relation provides a partial ordering on the set of guarantee structures.

In order to illustrate the concept of a guarantee structure and to suggest properties of guarantee structures that may be of interest, we turn to the study of a few examples.

Example 1. Tha Egalitarian and Utilitarian solutions

hgosition 1. The guarantee structures QE and CYU of the Egalitarian and Utilitarian solutions are given by cx$” z: err = 0 for all m, n E I.

Proof. we limit ourselves to showing that C$ = 0. Let P=(Q). Given c>O, we define S’~cch((l,O), (0,a)). We have

E{‘)(SnR”))=E{‘~([O, 11) = 1. However Er(S’) -+ 0 as n-, 00. Therefore &/i = 0. A similar argument can be made to establish that c-@” = 0 for any other m, n E I.

We omit the proof that CT~~ =O. 0

The zero guarantee structure is of course the worst possible one.

Example 2. The transferable utility case la departure from the rest of this paper, we assume for a moment that “utility

is transferable among the agents”, i.e., that the undominated boundary of the admissib,e division problems for any group PE 9 is the intersection with Rf of a hyperplane in Rf normal to ep (the vector of all ones in R:). This amounts to saying dsar t r.cre is a fixed amount of ‘utility’ which can be freely transferred among the agent: _ On this restricted domain, there is only one solution satisfying WPO and AN: It consists in giving to the agents equal shares of whatever amount is available. The guarantee structure CT of this solution is such that

a mn = m/(m + n) for all m, n E I.

It is of particular interest since the proportional loss incurred by an agent upon the addition of a fixed number of agents goes to zero as the number of agents originally present goes to infinity. It seems legitimate to conjecture that this asymptotic property will hold for other solutions even if the domain is not restricted as in the present example. We show below that in fact the Nash solution does not enjoy the property and we then turn to a general examination of the conjecture.

IV. Thomson, r Lensberg / Guarantee structures

Example 3. The Nash solution

ProposiG~n 2. The guarantee structure aN of the Nash solution is given by

211

ar= m(n + 2) - ymn(mn + Qm - 4)

2(m + n) for all m, n E I.

Proof. LetP,QE~~withPCQ,~~I=mandIQI=m+nbegiven,andletP~Q\~. Let k be an arbitrary but fixed member of P. We want to determine

a{ =inf{NkQ(T)/Nkp(S) 1 StGP, TECQ, S= Tn Rp}.

Given SEZ’ and TeCQ with S=TnRP, let A-N’(S) and v=NQ(T). We observe that

so that

NkQ(T)/N,&S)=NA~(cch(x,y))/N~(cch(x,~J f7 Rp)

a$” =inf(y&x&ERf, +vERF, x=NP(cch(x,y] nRP),

y = N Q(cch (x, y ] ) ) .

The search among pairs S, T is then reduced to a search among pairs A-, y. Given XER~, YER~, let S=cch(x,y) nRP and T=cch{x,y).

Because the Nash solution satisfies SJNV we can assume that x = &Then S is supported at ep by the hyperplane normal to e p. Comprehensiveness of T implies that CiEP yi s m. In order to ensure that y = NQ(cch {x, y I), it is then necessary and sufficient that the hyperplane of support to the set

c 1 Y’ER,Q II .Y;ZII Yi

1eQ ieQ

at y not go below ep, a condition that can be expressed as

Therefore, we have

C YiSmm,YkZg(Y& - IEP

Note that only the projection yp of y on Rf matters. Because g is d<:creasing in each yi for k P’, we can limit our search among yp’s E Rr such that xIEP _v, = m: and because g is continuous, we can also assume that yk = g( yp.). Finally, we claim that we can choose yi = yj for all i, j E P’. This will completely specify yp. TO see this, let yp satisfying the two equalities CiEp yi = m and yk = g( yp,) be given; we show that there exists a j+ such that yk = yk, _3;i = _yi for all i, jE P’, xrEP Ji s rn and j$gg(j$). Indeed, let &=yk and j’i = &,,vj/(m- 1) for all k P’. Then

Ci,pPi =m, which takes care of the first three conditions. The second inequality


can be written as

1 +C

1

ii --mm+n,

iEP’ pi

which, since

1 -+ c

1 --mm+,

Yk iEPp’ yi

is equivalent to

1

Ercp’ YiW - 1) s C -!- /(m-l)

( > ieP’ yi

which is true by the convexity of the function h(x)= l/x. Thus yi = (m -y&)/(m - 1) for all in P’. Fubstituting for ypt in the definition of

g( y&. we obtain the quadratic expression in y&,

the sm&st root of which is

m(n + 2) - dmn(mn + 4m - 4)

2(m + n) 9 the desired expression. Ll

This proof is illustrated in Fig. 1 for the case P= ( 1,2), P= (3) and k = 1.

P ={1,2}

F ={3)

k=l

/ Agent 3

Fig. 1.

W. Thomson, T. Lensberg / Guarantee structures 213

It is easy to check that the Nash guarantee sLructure does not satisfy the asymptotic property described in Example 2. In fact cvr s + for all m, n E I. Somewhat more disturbingly, (rr is a decreasing function of m for each n; as m-+ 00, cxEn --) +(n +2- iw). What is disturbing is of course the fact that as the number of agents originally present increases, greater and greater proportional sacrifices may be required of at least one of them upon the addition of a group of fixed size.

Are there solutions satisfying the asymptotic property? The answer is that no solution satisfying WPO (in addition to AN which is imposed throughout) does. In fact, a much stronger result holds.

Proposition 3. The guarantee structure of any sohtion F satisfying WPO and AN

is weakly dominated by & defined by &mn = 1 /(n + 1) for all m, n E I.

Proof. Let P’, QE.~ with P’CQ, lP’l=rn-1, iQI=m+n be given and let P” = Q\F’. Also, let TtGQ be defined by

T= 1 xER$?Ixpsepe and c Xisl .

ieP” 1

It follows from AN that F$(T) has equal coordinates. This, in conjunction with feasibility, implies that hQ(T) s l/(P” I= l/(n + 1) for all i E P”. Given some arbitrary kE P”, let now .P= F’U{k} and S = TnRP. We note that S= cch{ep}. By WPO and AN, F’(S) =ep. Thus FkQ(T)s F[(S)/(n + 1). Since IQ\Pi = n, we conclude that cr;“” s l/(n + 1). Cl

Although the Billera and Bixby result quoted earlier implies that the problem T

used in the proof of Proposition 3 can be rationalized by some economy, it is in- structive to provide an explicit example of such an economy. The complementarities and incompatibilities exhibited by the agents’ preferences in relation to the bundle available to them may shed some light on the nature of the result.

Assume that there are m goods, indexed 1l.y the members of P, that each agent ie P cares only about good i and that each of the agents E Q \P cares only about good k. This is achieved by the following chorce of utility functions: Given an arbitrary consumption bundle z in the m-dimensional commodity space R!, set Ui (2) = Zi for all i E P, and Ui (2) = zk for all i E Q \ P. Finally, assume that there is one unit of each good available for distribution. It is easy to check that

T= t xE RF 1 Vie Q 3& Rf s.t. c z’lep and Vie Q, u&‘)=s, .

leQ

We first distribute the aggregate bundle among the members of P, whose preferences do not conflict; each of them ends up with one unit of the good he likes, which results in a utility of 1 for agent k. The large group Q is then formed by introducing n agents whose preferences are identical to that of agent k. The only symmetric compromise consists in giving each of these n + 1 agents an equal amount of the only

214 W. Tnomson, T. Lensberg / Guarantee structures

commodity he likes; this brings agent k’s utility down to l/(n + 1). The result described in Proposition 3 certainly is a disappointment since it says

that even if there is already a large number of agents, the addition of only one agent (more generally a fixed number n of agents) may require that at least one of the original agents loses a full + (more generally a full n/(n + 1)) of what he was getting.

At this point, we do not even know whether there are any solutions offering the guarantee structure 6. Things are perhaps even worse. Fortunately we have the following.

Pmposition 4. The guarantee structure of the Kalai-Smorodinsky solution is di.

Proof. Let P, QE.~ with PCQ, IPI=m and IQI=m+n be given as well as iEP, SeZp and TEXT with S=TnRP. We will show that Kp(T)/Kr(S)g l/(n+ 1); the desired conclusion will then follow from Proposition 3 and the fact that K satisfies WPO and AN. Since K satisfies S.INV, we can assume that a(T) = eQ.

This implies that a(S) = ep and consequently K’(S) = Aep for some ,4 ~1411. Since the points Aep and ej for all je Q\P belong to T, their linear combination that has equal coordinates also does. This is the point AeQ/(nA + 1). Since KQ(T) also has equal c <ordinates, it follows from WPO that KiQ(T)e A/(nA + 1). Therefore &?(T)zKr(S)/(nL + l)zKr(S)/(n + I). 0

Propou~~ti;: 4 describes a remarkable property of the KS solution, which distin- guishes it favorably from other solutions, and in particular from the Nash solution.

An intuitive explanation of how the four solutions that we have examined compare may be useful at this point. N allows for substitutions among the agents’ utilities ar.d as the number of additional agents increases, the new division problem has more freedom to ‘stretch out’ in a direction unfavorable to a given one of the original a,Jents. By contrast, K keeps all utilities tied together and prevents a given agent to De ‘exploited’ for the benefit of the others.

The po(>r performance of E and U can be understood by the fact that these solutions involve interpersonal comparisons of utilities. If one of the new agents is a very bad ‘pie; swe :Gachine’, he will force everyone down if utilities have to be kept equal. On Cle other hand, if one of the new agents is a very good ‘pleasure machine’, he will fc;lrce everyone down if it is only the sum of utilities that matters.

In view of Propositions 3 and 4, we will refer to & as the maximal guarantee structure.

A natural question is whether the KS solution is the only solution to satisfy WPO and AN and to offer the maximal guarantee structure. This last condition can be formalized as follows.

Menimal Guarantee (MAX GUAR). For all P, QE 9 with PC Q and IQ\PI = n, fop alf SEC’, for all TEZ~ with S=TnRP and for all kP,

F;:O(T)g#‘(S)/(n + II).

W. T’hcrnson, T. Lensberg 1 Guarantee structures 215

The next proposition shows this not to be the case, even if two additional appealing requirements, S.INV and CONT. are imposed on the solutions. Solutions other than the KS solution and satisfying all five properties can be obtained by perturbing the KS solution.

It may be useful to compare this non-uniqueness result to the characterization of the KS solution provided in Thomson (1981). The KS solution is shown there to be the only one to satisfy WPO, AN, S.INV, CONT and an axiom of Monotonicity with respect to changes in the number of agents (MON), which says that if fixed resources have to be redistributed to accommodate one more agent, then sue!: 2 re- distribution should require sacrifices (in the weak sense) from every agent originally present. MON is a sort of dual of MAX GUAR, limiting agents’ gains instead of their losses. This ‘duality’ suggests that a parallel characterization of the KS solution could be obtained by substituting MAX GUAR for MON in the list characterizing it. But this is not the case, as now formally established.

Proposition 5. There are solutions different from the KS solution that satisfy WPO, AN, S.INV, CONT and MAX GUAR.

Proof. The proof consists in constructing a solution F which satisfies all the a_xioms but differs from K. To do this, we first set P= ( 1,2) and we define FP for the members S of Cp that are normalized by the requirement that a(S) =ep. The definition of FP is completed by an appeal to SJNV. Given any P’E .? with 1 P’i = 2, FP’ is defined from FP by applying AN. Finally, given any Q E .? with IQI+2, we set FQ=KQ.

The specification of FP for the ncrmalized members of Zp is illustrated in Figs. 2 and 3. Let SE Cp be given such that a(S) = ep. We first determine the arc !](S) of points that are weakly Pareto-optimal for S and weakly dominate +ep. This arc contains K’(S). Unless the Pareto-optimal boundary of S coincides with the segment connecting the two unit vectors in R ‘, II(S) is nondegenerate and K’(S) belongs to its relative interior.

Next, we introduce a third agent (by AN we can assume that it is agent 3), we set QI. (1,2,3) and T *= cch (S, e3 1. The intersection of T* with a plane containing thz third axis and K’(S) is represented in Fig. 3. It is easily checked there that K$(T*) is not less than halfway from the origin towards KP(S). This implies that the arc /z(S) of points that are weakly Pareto-optimal for S and weakly dominated by 2Kg(T+) is non-empty. This arc contains KP(S). Unless K’(S) = a(S), /z(S) is nondegenerate and KP(S) belongs to its relative interior.

We note that /,(S)nr,(S) +0 for any normalized SE Zp. We now define FP to

be a symmetric and continuous single-valued selection of II (S ) n &S )* (As an example, we could choose F’(S) to be the maximizer of the Nash product x~~Y~ over all points x of r,(s)nr,(s).)

It is easy to see that F satisfies WPO, AN, S.INV and CONT. To prove that F satisfies MAX GUAR, we first observe that, by definition of F, CT;-“” = crE” = c?““~


&gent 3

1 P= 192)

Q= 1,2,3} t

ent 2

1

/

Agent 1

a(S)=eP

Fig. 2. I,(S) = curvilinear segment [x1,x2], /z(S) = curvilinear segment [y1,y2].

1

KS (T") KP(S) J

o(T+)

Fig. 3.

for all m,nd with m=l and ng2 or with mr3. To show that a:=+, we set P = (1.2) and we observe that, for all S&‘,

This is because F’(S) belongs to I,(S). Finally, to show that # = 1 /(n + l), we let P,QE~~ with PCQ, IPI=2, iQl=2+n be given as well as SEC’, TtzZQ with S=TnRP. Since F satisfies SJNV, we can assume that a(T)=eQ. Then K’(S) = rrey for some a E JO, 11, and KQ(T) = 6eQ for some b ~1411. Given S, the minimal t, is obtained for T- cch( S; ei, k Q \P}. Then t, = a/( 1 + an). (This is as

W. Thomson, T. Lensberg / Guarantee structures 217

in the proof of Proposition 4.) Therefore, given i e P, and Ui E F;:‘(S), we want that

a 1 1 --)- or Uig

(1 +n)a h-an Ui=tt+l l+an

for all n ~1.

If n = 1, then the inequality is equivalent to ui s 2KiQ(T), which holds since F’(S) belongs to l*(S). As the function f(n)=(l + n)/( 1 + an) is a nondecreasing function of n, the inequality is in fact satisfied for all n. Cl

We note that there is nothing phthological about F. Proposition 5 shows that in order for a solution to satisfy the five properties, it need not prohibit utility substitutions altogether as the KS solution does. However, it is intuitive from thr’ construction of F that the five properties are compatible with only limited substitwf- ability.

4. Concluding remark

We conclude on a somewhat negative note by providing an example showing that the KS solution, although it does guarantee that the proportional loss incurred by an agent originally part of some group of m agents upon the arrival of some incre- mental group of n agents will never be more than l/(n + l), cannot prevent the proportional losses incurred by all m agents to simultaneously reach this upper bound, and this independently of n, even if m is large compared to n.

Example. Let P,Qc+ begiven with PCQand IQ\Pi =n, and let T=cch{ep;e,,k Q\P). Define S=TnR’ and note that S=cch(ep) and that KP(S)=ep. An elementary computation also yields that KQ( T) = eQ/(n + 1). Therefore, for all i E P, KiQ(T)/Kip(S) = 1 /(n + 1) = CY~“.

Here too, we exhibit an exchange economy that yields the division problem T used above. Assume that there are m goods, indexed by the members of P, that each agent i E P cares only about good i and that each of the agents E Q \ P cares about all the goods, provided they come in equal amounts. This is achieved by the following choice of the utility functions: Given an arbitrary consumption bundle 2 in the m-dimensional commodity space Rf’, set 14, (z)= z1 for all i E: P, and Ui(Z) e min (zk, k E P) for all i E Q \ P. Finally, assume that there is one unit of each good available for distribution. It is easy to check that

T=

In the original group, P, there is no conflict among the agents’ preferences and each of them ends up with one unit of the good he likes. In the larger group Q, the only way the utilities of the new agents can be simultaneously brought up is to give each of them equal amounts of each of the commodities. Equality of utilities is


achieved when each agent in P has given up the proportion n/(n + 1) of what he originally consumed. This leaves him with utility I/(n + 1). In the aggregate, tht: bundle nep/(n + 1) is released and each of the agents E Q\P ends up with e&n + 1).

Acknwledgment

We thank J. Green and a referee for their helpful comments.

IUeferences

L.F. Billera and R.E. Bixby. A characterization of Pareto surfaces, Proc. Amer Math. Sot. 41 (1973) 261-267.

H. Imai, Individual monotonicity and lexicographic maxmin solution, Mimeographed notes (revised August 1981).

E. Kalai and M. Smorodinsky, Other solutions to Nash’s bargaining problem, Econometrica 43 (1972) 513-21%

J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155-162. U. Segall, The monotonic solution for the bargaining problem: A note, Mimeographed notes, 1980. W. Thomson, The fair division of a fixed supply among a growing poputation, Harvard institute of

Economic Research Discussion Paper No. 812, 1981; Math. Oper. Res., forthcoming.

guarantee structures for problems of fair division

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