Mathematical Social Sciences 15 (1988) II-27 North-Holland

11

MONOTONICITY PROPERTIES OF BARGAINING SOLUTIONS

WHEN APPLIED TO ECONOMICS*

Youngsub CHUN Department of Economics, Southern Illinois University, Carbondale, IL 62901, U.S.A.

William THOMSON** Department of Economics, Universi?v of Rochester, Rochester, New York 14627, U.S.A.

Communicated by E. Kalai Received 22 July 1985 Revised 30 December 1986

We are concerned with the application of bargaining solutions to economic problems of fair division, and, in particular, with the way they respond to changes in the amount to be divided. For instance, one may want an increase in that amount to benefit all agents. A variety of mono- tonicity properties have been studied in the abstract framework of bargaining theory. Most of the commonly studied solutions are known not to satisfy many of these properties. Here, we investigate the extent to which they do when applied to economics. We show that when there is only one good, they do in fact satisfy many monotonicity properties that they do not satisfy in general. However, this positive result fails as soon as the number of commodities is greater than 2.

Key words: Monotonicity, bargaining solutions, fair division.

1. Introduction

Bargaining theory is concerned with the formulation of rules to select, for each bargaining problem (S, d) - where S is a set of feasible alternatives and d is the dis- agreement point, both given in the utility space - a point of S, suggested as compro- mise among the agents’ conflicting interests. Various such rules, or solutions, have been proposed, starting with Nash’s solution (1950), defined in his seminal article. Other solutions were introduced later, the Kalai-Smorodinsky (1975) and Perles- Maschler (1981) solutions being important alternative ones.

A number of properties of solutions have been extensively studied. Of particular interest are monotonicity properties, an example of which is ‘strong monotonicity’: the solution F is strongly monotonic if whenever two problems (S, d) and S’, d’) are such that S’>S and d =d’, then F(S’,d’)gF(S,d). It is known that many of the

* We thank the referee for his/her helpful comments. ** Gratefully acknowledges the support of NSF:SES 83-11249.

0165-4896/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

I2 Y. Chun, W. Thomson / .Monoroniciry of bargaining solutions

monotonicity properties are not satisfied by the solutions most commonly used. In particular, none of the three mentioned above satisfies strong monotonicity. We are interested here in finding out whether these negative results are preserved when solu- tions are applied to economic problems of fair division.

An economic problem of fair division is given by specifying a list of agents, each equipped with a utility function defined over the commodity space & where I is the number of commodities,’ together with an aggregate endowment Q, a point of 6%:. The agents have ‘equal rights’ on B and the problem is that of determining allocations at which all rights have been properly recognized. A solution F to the bargaining problem can be used in a natural way to construct a solution F to the problem of fair division by (i) taking S to be the image in the utility space of the set of feasible allocations and d to be the image of the zero allocation, (ii) solving (S, d) according to F (this, of course, requires that some regularity assumptions be satisfied by the economy under consideration), and (iii) selecting the allocation(s) whose image in utility space coincides with F(S,d).

In such a context, the typical circumstance that would cause expansions of oppor- tunities, as described in the formulation of the monotonicity properties of bar- gaining theory, are increases in Q, and the interesting question is whether solutions would respond appropriately to such changes. For instance, the counterpart of strong monotonicity of F is that F recommends that all agents benefit from an in- crease in R.

For a given monotonicity property, consider the class of all pairs of bargaining problems satisfying its assumptions. Also, consider the class of all such pairs that can be obtained from two problems of fair division differing only in their aggregate resources. In general, the second class is a proper subclass of the first class. There- fore, knowing that a solution F does not satisfy the property does not allow us to make a similar statement about f? F has to be studied directly. This is what will be done here.

Note that the greater /, the larger the second class and therefore the more likely is F to behave similarly to F. In fact, we will be able to identify exact bounds on I beyond which F does behave like F.

Billera and Bixby (1973) established conditions under which a bargaining problem can be seen as the image in the utility space of some economic problem, and Roemer (1985) has recently investigated when pairs of problems can be seen as the images in the utility space of some pair of economic problems differing only in some para- meter, such as endowments. Their results are not directly relevant here however since these authors imposed no constraints on the number of commodities, while the sensitivity of solutions to variations in this number is of central concern to us.

This issue concerning I is of great relevance to the application of bargaining theory. Indeed, the most widely discussed problem of fair division is the so-called ‘divide the dollar game’ (see Lute and Raiffa (1957)): two agents receive a dollar

‘B designates the real numbers and N the positive integers.

Y. Chun, W. Thomson / ,Monoronicit_v of bargaining solutions 13

if they agree on a way to divide it and nothing otherwise.’ Can the insights gained by the study of this canonical problem help us in understanding more general situa- tions and in particular the case of more than 1 good and 2 agents?

It turns out, as we show here, that the divide the dollar game is indeed excep- tional: solutions tend to behave quite well if I= 1 but not as soon as 12 2. In particu- lar, the Nash and Kalai-Smorodinsky solutions, although they are strongly mono- tonic in the two-person case if I= 1, lose this property for any 122. The Perles- Maschler solution fares the worst; it is not strongly monotonic even if I= 1.

Similar results are obtained for two related but weaker properties: ‘individual monotonicity’ and ‘weak monotonicity.’ These properties pertain to situations in

which expansions in the feasible set are constrained, as would happen if agents were initially satiated.

The final property we study is ‘population monotonicity,’ which says that when an increase in population size, unaccompanied by an increase in resources, occurs, all of the agents originally present share the burden of supporting the newcomers. Here again, the Nash solution behaves well if I= 1 but not for any lg2. (The Kalai- Smorodinsky solution behaves well in the general class of bargaining problems, as discussed in Thomson, 1983. The Perles-Maschler solution has not been formally defined for more than 2 agents.)

The results should help clarify the usefulness of bargaining solutions in solving problems of fair division. They expose the danger of relying on the ‘divide the dollar game’ as the main paradigm, and identify I= 1 as the exact bound on the number of commodities beyond which solutions start behaving in the way they do for abstract bargaining problems. Since the Egalitarian solution (see, for instance, Kalai, 1977), trivially satisfies all of the properties, our results should reinforce its appeal.

2. Definitions. Notation

An l-commodity, n-person problem of fair division is a pair (u, Q) of a utility pro- file u=(Ur,..., u,), where for each i, ui : 3?:+ ST! is agent i’s utility function, and of an aggregate endowment Q ~92:. Let E(I, n) be the class of all problems (u, Q) such that for each i, ui is continuous, concave, non-constant and non-decreasing (for all Xi,XilE~~ with X;ZXi, Ui(X,!)lui(~;)),~ and satisfies Ui(0) ~0.

An n-person bargaining problem is a pair (S,d) of a convex, compact subset S of 9’” and of a point deS, strictly dominated by at least one point of S. Each point of S is a utility vector attainable by the n agents through some joint action, and d, the ‘disagreement point’, is the utility vector that would result if the agents

*Also, most of the strategic models of bargaining developed in the recent years involve only one good.

‘Notation for vector inequalities: xky, xry, x>y.

11 Y. Chun, W. Thomson / Monoroniciry of bargaining solutions

failed to reach a compromise among the points of S. Let Z” be the class of all such problems. An n-person bargaining solution, or simply, a solution, is a function de- fined on Z”, which associates with every (S, d) E Z” a point of S, interpreted as the recommended compromise.

Solutions can be applied to problems of fair division as follows. Given (u,.Q)E E(I,n), let Y(u,Q)={x=(x, ,..., x,,) EC%?? 1 Zjr,~R} be the set of feasible alloca- tions of (u, 52) and let S c 222 and d EZt be defined by

L

SE {DE&f I3XE Y(U,Q) S.t. Vi, Uj(X;)=iii}*

dn 0.

Note that (S, d) EZ” if S2>0 so that, given a solution F, F(S, d) can be deter- mined. The allocations XE Y(u,Q) such that u(x) = F(S,d) are called the F-alloca- tions of (u, Q). They are the allocations recommended by F for (u, Q). The set they constitute is denoted F(u, a).

Under the assumptions imposed above on economies, more can actually be said about the set S associated with any economy (u, Q) E E(i, n): S is comprehensive (i.e. for all ii, a’~.%?,“, if arzS and aza’, then D’ES). Therefore, it is natural to intro- duce the subclass Z,” of Z” of problems S, where S is a subset of &?t which is con- vex, compact, comprehensive, and contains at least one point strictly dominating the origin. The disagreement point is now omitted from the notation; it should be understood to be the origin.

The set of Pareto-optimal points of S, PO(S), is defined by PO(S)= {a E S I@‘E S

with a’r a>. The set of weakly Pareto-optimal points of S, lVPO(S), is defined by WPO(S)={(~~ES 1 jqii’~S with a’>~}.

The domain C,” is of particular interest not only because it arises naturally from an examination of economic problems but also because solutions are often better behaved on it than on C”. In particular, on this domain, all the solutions that we will consider select weakly Pareto-optimal, and, in fact, usually Pareto-optimal, points. The other properties of solutions that will be central to our analysis are: strong monotonicity (S.M.): For all S, S’EZ,” with S’>S, F(S’)zF(S). individual monotonicity (I.M.): For all S, S’ E Zz with S > S, for all i, if {iiESIai=o)={- UES’I &=O}, then F,(S’)zFi(S). weak monotonicity (W.M.): For all S, S’EZ,” with S’>S, if for all i, max{u; 1 u ES} =

max{ui) UES’}, then F(S’)zF(S). Note that S.M. implied 1-M. and W.M., and that if n = 2, I.M. implies W.M. The last property concerns solutions that are defined on UIIEd). C:.

population monotonicity (P.M.): For all n, n’E& with n >n’, for all S EZJ, S’ E c,“‘, if S’={ii’~g$I 3~7~s with a;=a,Vi~n’}, then F;(S)s;F;(S’) for all isn’.

These four properties were respectively introduced by Lute and Raiffa (1957), Kalai and Smorodinsky (1975), Roth (1979) and Thomson (1983). The first one ex- presses a strong form of solidarity among the agents; they should all benefit from any expansion of opportunities. The next one recognizes that some expansions may

Y. Chun. W. Thomson / Monotoniciry of bargaining solutions 15

PM(S)

"1

A0

Fig. 1.

be skewed in favor of a particular agent: the requirement there is only that this agent gains. The third one pertains to expansions that are skewed in no direction, and the requirement is that all agents gain. The final property is that all of the agents initially present be affected in the same direction when additional claimants enter the scene, while resources do not change.

Next, we define the solutions on which our analysis will focus. Given SEZ,“, for the Nash (1950) solution, N(S) is the unique maximizer of IIXi

for XE S; for the K&i-Smorodinsky (1975) solution, K(S) is the maximal point of S on the segment connecting the origin to the ‘ideal point’ a(S) where ai( max{a, 1 ZI E S}; for the Egalitarian solution, E(S) is the maximal point of S with equal coordinates; for a Utilitarian solution, U(S) is a maximizer of Zxi for XES; finally, for the Perles-Maschler (1981) 2-person solution, and assuming that S is polygonal,4 PM(S) is the first point in common of the sequences {Ak} and (Bk) constructed as follows: A0 and B. are the points of PO(S) with minimal second and first coordinates respectively; A, and B, are the points of PO(S) such that (i) the segments [&A,] and [Bo,B,] are contained in PO(S), (ii) they do not overlap, although their endpoints At and Bt may coincide, (iii) the areas of the right-angle triangles with hypothenuses [A,,A,] and [B,, B,] are equal and maximal; . . . ; Ak and Bk for k=2,..., are defined from Ak_t and Bk_, in a similar way. This con- struction is illustrated in Fig. 1 with an example S for which PM(S) is obtained in three steps.

4The definition of PM(S) when S is not a polygonal problem is given by a limiting argument

involving approximations of S by polygonal problems; here we will need only polygonal problems.

16 Y. Chun, W. Thomson / Monofonicify of bargaining solutions

In the economic context in which we are placing our analysis of monotonicity pro- perties of solutions, the parameter changes that are of interest are increases and de- creases in Q, keeping the number of agents constant for the first three, and increases and decreases in the number of agents, keeping Sz constant for the last one. An in- crease in Q leads to an expansion of the feasible set as described in the hypothesis of S.M. If each agent is satiated at Q, or satiated in those goods whose availability increases, then the expansion of the feasible set will be of the sort described in the hypotheses of W.M. If none of the agents, except agent i, ever cares about the goods whose availability increases, then the expansion of the feasible set will be of the sort described in the hypothesis of I.M. Finally, an increase in the number of agents accompanied by no change in 52 and no external effects will lead to expansions of the kind described in the hypothesis of P.M.

3. The results

First, we consider the Nash solution. It is known from Kalai and Smorodinsky (1975) that if n = 2 the Nash solution is not weakly monotonic and therefore not in- dividually monotonic and a fortiori not strongly monotonic. However, when applied to economic division problems with I= 1 it is strongly monotonic, indepen- dently of the number of agents. Unfortunately, this result does not extend to lg2, and the Nash solution violates even weak monotonicity for all 12 2. These negative results hold as soon as ng2.

Theorem 1. The Nash solution is strongly monotonic on E(1, n) for all n.

Proof. For simplicity we consider an economy (u,SZ)E E(l,n) for which all utility functions are differentiable and increasing. The conclusion for the other cases fol- lows directly from the fact that the Nash solution is continuous. N(u, $2) is obtained by maximizing Z7Ui(Xi) in XC@,” subject to _Zxi =R. This problem is solved by requiring that

Uf(Xi)/Ui(Xi) = Uj’(Xi)/Uj(Xi) for all i, j and ZXi = 0. (1)

Since each Ui is concave and increasing, each function gi : 9?!++ + ~8 defined by gi(Xt) = Ui(Xt)/Ut(Xi) is decreasing. If Sz increases, (1) can be preserved only if each xi increases, which implies that all agents gain. •i

Theorem 2. The Nash solution is not strongly monotonic on E(I,n) for any

(4 n) 1(2,2).

Proof. (i) The main step of the proof consists of an example with (I, n) = (2,2), illus- trated in Fig. 2. The other values of (I,n) are dealt with later.

0.8660 q == =

u,u2 = 1.67

u,uz= 1.29

17 Y. Chun. W. Thomson / Monotoniciry of bargaining solurions

Fig. 2.

Let u,,u2 :.?%?j*B+ be defined by

t

u,(x,) = x;:2x;:2+x;:4,

&(X2) = xi<‘.

At first, we assume that X2= (1,l). Z?(u, 0) is obtained by solving

max u,(xI)u2(xz) s.t. xl +x,~;(l, 1). x1.x2

Since u2 is independent of x22 and U, is increasing in x12, solving this problem

requires setting xz2 = 0 and x,~ = 1. We are led to solving

max(xt{2+1)(1-x,,)“2 s.t. O~x,,~l. XI!

After differentiating, we obtain the equation in xl1

2x,, +x;:2- 1 = 0,

whose unique positive solution is

Then x2, = + and the agents’ utilities are U, = 1.5, u2 = 1/1/2 =0.8660. Next, we assume that R increases to Q’=(l, (1.2)4). An analogous reasoning

leads to the problem

max(1.44x~{2+1.2)(1-x,r)1’2 s.t. Osxt,sl, XII

which yields the equation in xl1

2.88x,, + 1.2~;;~- 1.44 = 0,

whose unique positive solution is

x1, = 0.2796.

18 Y. Chun, W. Thomson / Monotonicity of bargaining solutions

The utilities are then U, = 1.9614 and uz = 0.8488. Agent 2 has lost in spite of the increase in the social endowment. This concludes the proof for case (i).

(ii) Next, we take care of the case n>2. Let S and S’ be the problems obtained in (i) for $2 and Q’ respectively. We now introduce n - 2 additional agents indexed by k, where 2 c ks n and with utility functions U: : 59: + S?+ defined by

@k) = E%l +x&2) if +r f&2<&,

1 otherwise,

where E>O. (The motivation for introducing these agents is that they become satiated quickly if E is small so that only small amounts of resources will ever be allocated to them by any solution satisfying Pareto-optimality, such as the Nash solution. Thus their presence will hardly disturb the problem’faced by the two agents originally present .)

First, let TECRP,” be defined by T={a~@i?P,“/(fi,,ii,)~S, &sl V/C>~}. T is a truncated cylinder with base S; it is represented in Fig. 3 in the case n = 3, for which k takes only the value k= 3.

Now it is easily checked (see Thomson (1984) for a study of related properties of solutions) that N(T) projects onto the coordinate plane pertaining to the original two agents at N(S): in fact N,(T)=N,(S), N2(T)=N2(S) and N,(T)=1 for all k> 2. Similarly, if T’ is constructed from S’ in the same way as T was constructed from S, we have that N,(T’)=N,(S’), N2(T’)=N2(S’) and N,(T)=1 for all k>2.

Let T’,T”E.Z,” be the problems derived from the economies with utility func- tions ur , u2 and uj for k> 2, and aggregate endowments Q = (1,l) and a’= (1, (1 .2)4)

Fig. 3.

Y. Chun, W. Thomson / Monoronicity of bargaining solurions 19

respectively. Note that as E * 0, TE -, T and T” + T’. Since the Nash solution is continuous, we have that N(T’) + N(T) and N(T”) -, N(T’) as E -+ 0. Given that N,(T) =N#)>N,(T’) =&(S’), we obtain that for E small enough, N,(T’)> N,(T”), the desired conclusion.

(iii) We conclude by examining the case I> 2. To that effect, it suffices to modify the example of (ii) by introducing utility functions ui:L?Zi+Se+ related to the ui defined earlier by

Oi(X;) = Ui(Xit,Xiz) for all Xie&?l and for all i,

since this choice will leave unaffected the problems T and T’, TE and T”. 0

Remark 1. The example of (i) can be modified to show that the Nash solution does not even satisfy weak monotonicity on E(2, n) for any n 12; it suffices to replace ul by wI defined by:

w,(x,)=min{u,(x,), 2) for all x1 EBj.

Indeed, the derived 9 and 9’ are such that s= S while s” is obtained from S’ by a truncation by a vertical line of abscissa al(S). (Recall that Ui(S) = max{ tii 1 a ES) .)

Obviously N(s’) = N(S) and since iVt (S’) < a,(S), IV@‘) = N(S’). The claim is proved by noting that a(S) = a@‘).

Remark 2. It is conceivable that the imposition of additional conditions on the utili- ty functions would permit to extend the positive result of Theorem 1. We have deter- mined that if the utility functions are separable additive, the Nash solution is strongly monotonic for all (I, n) with n =2. (See the appendix for a proof.) It is an open question whether this result extends to other values of (I, n).

Remark 3. Theorem 1 can easily be generalized to solutions defined by the max- imization of a function of the form

ELfi: for UES,

where each A is strictly concave. This family includes in particular all CES solu- tions. (Solutions obtained by maximizing a function with constant elasticity of substitution.)

Next, we turn to the Kalai-Smorodinsky solution. Although this solution satisfies individual monotonicity (it is essentially on this fact that Kalai and Smorodinsky based their characterization of the solution for n = 2) and weak monotonicity, it violates strong monotonicity. We show here that it does satisfy strong monotonicity on E( 1,2) but not on E(I, n) for any other pair (I, n). This is in unfavorable contrast to the Nash solution.

Theorem 3. The Kalai-Smorodinsky solution is strongly monotonic on E( 1,2).

20 Y. Chun, W Thomson / Monoroniciry of bargaining solufions

Proof. For simplicity, we consider an economy (u, 0) E E( 1, 2) for which the utility functions are twice differentiable and increasing (the other cases are dealt with by making use of the continuity of the KS solution). Then R(u, Q) is obtained by fin- ding the unique XE 6%: such that

u,(xt)/u,(R) = u~(x*)/u#2) s.t. x, +x2 = sz. (2)

Let (x1(G), x2(Q)) be the solution to this equation. By substituting into (2), we ob- tain an identity both sides of which can be differentiated with respect to Sz, yielding a second identity

where

A(Q)= ul(x, (Q))

n;(Q) xi’(s2) -

ui (xi (O))

(“i (n))2

uf(f2) for i= 1,2.

Suppose now that xi (Q) < 0 for some 52. Since x1 (52) +x2(C) = $2 for all Q, we con- clude that xi (Q) + xi (a) = 1 for all s2. Therefore xi (Q) > 1. Since u2 is concave and x,(Q) $ Q, then

n;(&(Q))/~&(Q)) h u;(Q)&(Q).

This, in conjunction with x;(Q)> 1, yields fi(Q)>O. But if x;(Q) ~0, then ft(Q)<O. These two statements on the signs of f,(O) and h(Q) are incompatible with (3). Cl

Theorem 4. The KS solution is not strongly monotonic on E(1, n) whenever

(A n)*(l, 2).

Proof. (i) We first prove that the KS solution is not strongly monotonic on (1,3). The proof is by way of an example. Let ul, u2, u3 : kZ+ +B+ be defined by

4x1 if Osxts5, u,(xt)= (1/2)x,+35/2 if 5~~~525,

30 if 25 5x1,

Uz(xz)= 2x2 if Osx,$lO, (2/3)x,+40/3 if 101x,,

u3(x3)= 2x3 if 0sx3s 10, (2/3)x3 + 40/3 if 10~~~.

First, we assume that 52 = 25. We claim that the KS allocation is (5, 10, IO). Indeed,

U,(5)4(25) = uz(10)/u2(25) = U3(1o)/U3(25) = 20130

and feasibility holds. Next, we increase Sz to Q’ = 40. Then ~~(40) = 30, ~~(40) G ~~(40) = 40. The new

KS allocation x= (x,, x2, x3) has to satisfy

Y. Chun, W. Thomson / Monotonicity of bargaining solutions 21

u,(.u,)/30 = u2(xz)/40 = u3(x3)/40 and x, +x2 +x3 = 40.

Since agents 2 and 3 are identical and the KS solution treats identical agents iden- tically, x, =x3 and therefore x, = 40 - 2x,. We can determine the KS allocation by

solving the following equation in x2:

4(40 - 2x#30 = [(2/3)x, + 40/3]/40.

The solution is xz =x3 = 17.647 and x, = 4.706. This is the desired result, since agent 1 has lost as resources increased from Q to f2’.

(ii) Next, we take care of the case (1, n) for n> 3. Starting from the 3-person economy (u, Q) defined in step (i), let S be its utility feasibility set; and let us add n - 3 new agents with utility functions of : S?+ -+S?+, for 4 5 kj n, defined by

As E --* 0, we find that the enlarged economy yields a utility feasibility set TE which converges to T={x~.?R” 1 (x,,x2,x3)eS, x&l for all k>3}. Also, K(T’)+K(T),

since the KS solution is continuous. The other steps of the argument are analogous to those in (ii) of the proof of Theorem 2.

(iii) The case of arbitrary 1 and n 2 3 is treated as in (iii) of the proof of Theorem 2.

(iv) Finally, we discuss the case (I, n) = (2,2). The proof is by way of an example, the same as the example used in step (i) of the proof of Theorem 2. First, we assume that Q = (1,l). Then u,(Q) = 2 and +(sZ) = 1. The KS allocation satisfies

(xi{2+ 1)/2 = (1 -x*t)“2,

since u2 is independent of x,~, and therefore xz2 should be set equal to 0, and xl2 to 1. Solving this equation yields xl1 = 0.36, and consequently, xzl = 0.64. The cor- responding utilities are ut = 1.6 and u2= 0.8.

Next, we assume that Q increases to Q’=(l,(l.2)4). An analogous calculation leads to the KS allocation x,, =0.3769, x21 = 0.623 1, xl2 = (1 .2)4 = 2.0736 and x22 = 0, and the corresponding utilities U, =2.0840, ~~-0.7894. Agent 2 has lost as re- sources increased from S2 to Q’. •i

Finally, we consider the Perles-Maschler solution. This solution satisfies very few of the properties that have been advocated in bargaining theory. The unsatisfactory behavior of the solution is confirmed here since, even if (I, n) = (1,2), it does not satisfy W.M.’

Theorem 5. The Perles-Maschler solution is not weakly monotonic on E(I,2) for

all I.

5 We do not consider the case of more agents since the generalization of the solution to that case has not appeared in print.

22 Y. Chun, W. Thomson / Monotonicity of bargaining solutions

"1

Fig. 4.

Proof. The proof is by way of an example, illustrated in Fig. 4. Let ul, u2 : .%+-tS?~

be given by

(9/5)x, if Osx, 55,

ut(x,) = (1/5)x1 + 8 if 5 gxt 6 15,

11 if 15$x,,

uz(x2) = (1/5)x2 if osX25 15,

3 if 155x2.

We assume first that Q= 15. Then, the utility feasibility set is given by Sr cch((11,0),(9,2), (0, 3)},6 these three points being the images of the allocations (15,0), (5,lO) and (0,15) respectively. The PM solution outcome of S is (15/2,13/6); it is obtained by noting that the areas of the triangles with vertices (1 l,O), (9,2) and (9,0) on the one hand and (0,3) (0,7/3), (6,713) on the other are equal and that (15/2,13/6) is the midpoint of the segment [(9,2),(6,7/3)].

For Q’= 20, the utility feasibility set is given by S’S cch((l1, l), (9,3)}, these two points being the images of the allocations (15,5) and (5,15) respectively. The PM solution outcome of S’ is (10,2); it is obtained by noting that (10,2) is the midpoint of the segment [(ll, 1),(9,3)].

We conclude by observing that as CI increases to Q’, agent 2’s utility decreases

6Civen a list x1 , . . . ,xk of points in .%‘:, cch(x’, . . . , x”} denotes the convex and comprehensive hull of these points, i.e. the smallest subset of .9?!: containing these points which is’convex and comprehensive (for all x.yeBJ’, if xeS and xry. then YES).

Y. Chun, W: Thomson / Monoroniciry of bargaining solutions 23

from 13/6 to 2, in violation of weak monotonicity which applies, since S’>S, and a(S) = a(S’).

This takes case of the case I= 1. The case of arbitrary I is dealt with as in the previous theorems. 0

The last property we consider is population monotonicity. Since KS satisfies the property in general (Thomson, 1983), and as pointed out earlier, the PM solution is not defined for more than two persons, we limit our attention to the Nash solu- tion. It is known that the Nash solution does not satisfy the property in general (Thomson, 1983). However, we have:

Theorem 6. The Nash solution is population monotonic on E( 1, n) for all n.

Proof. For simplicity, consider an economy (u, Q) E E( 1, n) for which utility func- tions are differentiable and increasing. Then, the Nash allocation XC&! is given

by XI, . . . . x,, E S?+ such that C Xi = Q and

Ui(X,)/Ui (Xi) = Ui(Xj)/Uj (Xj) for all i, j.

Recall that the function

Uf (Xi)/Ui (Xi)

is non-increasing for all i, and finally, note that the increase in the number of agents has the same effect as the decrease in the social endowment on the original agents. Cl

Remark 4. As discussed in a previous remark concerning the strong monotonicity of the Nash solution, the positive result of Theorem 6 can be extended to solutions defined by the maximization of a function of the form CA(q) for u ES.

Theorem 7. The Nash solution is not population monotonic on E(1, n) if 12 2 and n h 2. i

Proof. (i) We first consider the case I= 2 and we show that the Nash solution is not population monotonic when n increases from 2 to 3. The proof is by way of an ex- ample. Let u,, u2 be as in Theorem 2 and Q = (1.1); There, we already determined that at the Nash allocation x, x2] = 3 and x22 = 0.

Next, we assume that a third agent enters the scene with u3 : 8: + W, defined by z+(x~)=x~~~. Using the fact that u2 is independent of x22 and u3 of x31, the Nash solution is obtained by solving

max (~f{~x~~~+xtP)(l -~,t)“~(l --x~~)*‘~, XII.XIZ

s-t. 0_1x,,sl, 05x,251.

24 Y. Chun, WL Thomson / Monotonicity of bargaining solutions

After differentiating, we obtain the equation in x,,, xt2,

2X,, +x~;‘zx;~ l;l= 1 .’

From the feasibility condition, xtzs 1. At the Nash allocation, xt2< 1. This implies that xll<f. Therefore xzI>S and xz2=0. Agent 2 gains upon the arrival of agent 3.

(ii) In order to take care of the case when the economy enlarges from an element of E(2, n) to an element of E(2, m + n) we introduce m + n - 3 additional agents with utility functions ui : $921 *S?+ defined by

4(Xk) = c-t(xkl +xk2) for Osxk, +xk25c,

1 for cdxkl +x,,.

The argument concludes as in the proof of Theorem 2. (iii) The case of arbitrary I can be treated as it was in the proof of Theorem

2. cl

Appendix

Let E*(l,n) be the class of economies where all agents have additive separable, strictly increasing utility functions from af to 99.

Proposition. The Nash solution is strongly monotonic on E*(l,2) for all I.

Proof. Let U, u be the two agents’ utility functions, x, y E a’ being their respective consumptions. By assumption, there exists ulr . . . , uI, ul, . . . , IJ/ : W, ---t .%+ such that U(X) = C Ui(Xi) and u(y) = C oj( yi). The Nash allocation of the economy ((u, o), Q) is obtained by solving the following maximization exercise:

max(C Ui(Xi>)(C tJj(_Yj)) S.t. X+_YsQ.

Since each of the Ui and Vj is increasing, we are led to solving

max(C Ui(xi))(C uj(mj-xj)), where Q~(~j)j=t,....,*

Assuming each of the Ui and Vj to be g2, (this is without loss of generality since the Nash solution is continuous), we obtain the following first-order conditions (from here on, we omit arguments).

u;v - uv; = 0,

u;v- uv; = 0, . . . . . . . . . . . . . . . . . .

u;v-uv; = 0.

‘In order to obtain the inequality ‘XI I< f’, which is all that we need, it suffices to use this first-order condition. The precise calculation of x1, requires the use of the second first-order condition too. We omit these calculations.

Y. Chun. Ul Thomson / Monoroniciry of bargaining solutions 25

Next, we obtain the marginal change in agent l’s consumption caused by a margi- nal change in w,, by totally differentiating these equations with respect to ol.

Collecting terms and changing all signs, we obtain

(-u;tJ + 244 - uo;) d-u, -+ (U;u;+ +;, ctvz k/

dot -+...+(U;u;+U;u;)- dwt dwl

= u;u;- uu;,

<24;0; + u;u;> d4

-+ (-u;u + 2&J;- Uu;) k2 h/

dot -+ ..* +<u;u;+f4;0;>- d&l dot

= u;u;,

cu;u; + u;u;> &I duz - + (u;u;+ I.+;> do h/

dw +.*.+(-U;u+2U;u;-Uu;)-

do, = u;u;,

I

or Ax = b where

AE

u;u; + u;u; u;u;+ u;u;

i 1 , 2u;u;-u;u-uu;

Note that A is a symmetric matrix. Next, we show that A is positive semi-definite. Let A, and A, be defined by

26

and

Y. Chun, W. Thomson / Monoronicity of bargaining solurions

AZ’

i

-u;v - uv; * .

0 -u;v - uv; 1%

Since u~vj=ujv~ for all i,j=l,..., I, from the first-order conditions, we can easily show that both Al and A2 are positive semi-definite, so that A = A, + AZ is also positive semi-definite.

We will assume that A is in fact positive definite (if A is not positive definite, we approximate the utilities so that the resulting A does have this property, and we appeal again to the continuity of the Nash solution).

Then we can write

x E A-lb.

We are interested in the sign of

hl ,&2 U+-

I bl do, +u27--&+“‘+5--4= u’A-‘b

= v;u’A-‘(~‘)~ - uv;u’A-‘(e’)T,

where u’E(u;, . . . . u;), T denotes transposition and e’ is the first unit vector. Since A is positive definite and v;gO, the first term of this sum is non-negative.

Since v;$O, the second term also will be positive if u’A-‘(e’)’ is non-negative. Let B be the matrix obtained from A by replacing 2u~v~-u~v- uv; by 2u;v;.

Note that, like A, B is symmetric and positive semi-definite. In fact, without loss of generality, we can assume that B is positive definite. Also, the first columns of cofactors of A-’ and B-’ are the same. Therefore,

, PI u’A-‘(e’)T = u’B- m (el)T

PI 1 =- --;B,B-‘(e’)T, where B, is the first row of B, IA1 24

PI 1 =- ,A, gW)T

PI 1 =---s-o IAl 2v;= * I3

Y. Chun. W. Thomson / Monotonicity of bargaining solutions 27

References

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

E. Kalai, Proportional solutions to bargaining situations: interpersonal utility comparisons, Econo- metrica 45 (1977) 1623-1630.

E. Kalai and M. Smorodinsky, Other solutions to Nash’s bargaining problem, Econometrica 43 (1975) 513-518.

D. Lute and H. Raiffa, Games and Decisions (Wiley, New York, 1957). J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155-162. M.A. Perles and M. Maschler, The super-additive solution for the Nash bargaining game, Int. J. Game

Theory 10 (1981) 163-193. J. Roemer, Equality of resources implies equality of welfare, Quart. J. Economics 101 (1986) 751-784. A.E. Roth, Axiomatic Models of Bargaining, No. 170 (Springer-Verlag. Berlin, 1979). W. Thomson, The fair division of a fixed supply among a growing population, Math. Operations Res.

8 (1983) 319-326. W’. Thomson, Juxtaposition and replication of bargaining problems, University of Rochester mimeo,

1984.