Bargaining solutions and relative guarantees

Mathematical Social Sciences 17 (1989) 285-295

North-Holland

285

BARGAINING SOLUTIONS AND RELATIVE GUARANTEES

Youngsub CHUN

Department of Economics, Southern Illinois University, Carbondale, IL 62901.4515, U.S.A

William THOMSON*

Department of Economics, University of Rochester, Rochester, NY 14627, U.S.A.

Communicated by E. Kalai

Received I December 1987

Revised 3 August 1988

We propose to evaluate solutions to abstract bargainmg problems on the basis of the extent

to which they differentially affect the fortunes of two individuals initially present, when the num-

ber of agents increases while their opportunities do not. We introduce the concept of the relative

guarantee structure of a solution to quantify this possibility. The Kalai-Smorodinsky and

Egalitarian solutions offer maximal guarantees in the class of anonymous solutions, and, in par-

ticular, they strictly dominate the Nash solution. In that class, the Kalai-Smorodinsky solution

is the only vveakly Pareto-optimal solution to offer maximal guarantees and to satisfy scale

invariance, whereas the Egalitarian solution is the only weakly Pareto-optimal solution to offer

maximal guarantees and to satisfy contraction independence.

Ke_v words: Relative guarantee structure; Kalai-Smorodinsky solution; Egalitarian solution;

Nash solution.

1. Introduction

We consider abstract bargaining problems in circumstances in which the number

of agents may increase while their opportunities do not. We propose to evaluate

solutions to such problems on the basis of the extent to which they differentially

affect the fortunes of two individuals initially present. We suggest that the stability

of society is more likely to be preserved by solutions for which such changes remain

small. We define the notion of the relative guarantee structure of a solution as a way

of quantifying these changes and we use this measure to rank solutions.

We show that the Kalai-Smorodinsky (1975) and Egalitarian (Kalai, 1977) solu-

tions are best among all solutions satisfying the minimal requirement of Anonymity,

whereas the Nash (1950) solution is strictly inferior (we derive explicit formulas for

these three solutions). Moreover, the requirement on a solution that it offers maxi-

* Gratefully acknowledges support from NSF under grant No. 85 11136. The authors thank a referee

for helpful comments.

0165-4896/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

286 Y. Chun, W. Thomson / Bargaining solutions

ma1 guarantees can be used to characterize the Kalai-Smorodinsky and Egalitarian

solutions. In addition to this requirement, these characterizations involve two alter-

native sets of standard conditions, differing only in that Scale Invariance is used for

the first result, whereas Contraction Independence (Nash, 1950) is used for the

second one.

2. Preliminaries

Our analysis is placed within the context of Nash’s bargaining theory, generalized

so as to accommodate variations in the number of agents (Thomson, 1983a,b).’

There is an infinite set of potential agents, I= { 1,2, . . . ). W is the set of all finite sub-

sets of I, with generic elements P, Q, . . . . Given P E 9, IRc is the utility space per-

taining to the group P. 2’ is the class of problems that P may face: each SE Zp is

a convex, compact, and comprehensive (Vx, y E RR/I, if ylx and XE S, then y E S)

subset of RT containing at least one x>O.*,~ A solution F is a mapping defined on

.Z= U,Y Zp, associating with each PEY and each SEX’, a unique point of S, F(S). F(S) is interpreted as the predicted, or recommended, depending upon the context,

compromise for S. For the Nash (1950) solution, N(S) is the maximizer of np x,

for x E S; for the K&i-Smorodinsky (1975) solution, K(S) is the maximal point of

S on the segment connecting the origin to a(S), where for each ie P, a;(S) = max{x; 1 x~S}; for the Egalitarian (Kalai, 1977) solution, E(S) is the maximal

point of S with equal coordinates.

The following properties of solutions will be useful.

Weak Pareto-Optimality (W.P.O.): For all PE .Y, for all SE Zp, for all XE lRc, if

x>F(S), then x$S.

Anonymity (AN): For all P, P’E 9 with 1 P 1 = 1 P’ 1, for all SE .Zp, S’ E Zp’, for all

one-to-one functions y : P+ P’, if S’= {X’E fRp’ ) AXES with x_Jcj, =x, V’~E P}, then

FY(;)(S’) = F, (S) for all i E P.

,I : Rp + Rp is an independent person by person, positive linear transformation if

there exists (x E Rc+ with A, (x) = a,xj Vi E P and Vxc Rp. Let A’ be the class of

these transformations.

’ For a study of bargaining theory with a variable population, see Thomson and Lensberg (1988).

’ Vector inequalities: x>y, xry, xzy.

3 The usual specification of a bargaining problem involves a distinguished element of S, usually called

the disagreement point and denoted d. Here, utilities are implicitly assumed to be normalized so that

a’= 0. This normalization allows us to eliminate all references to d in the notation, with no essential loss

of substance.

Y. Chun, W. Thomson / Bargaining solutions 287

Scale Invariance (S.INV): For all PE 9, for all SE 2”, for all ,4 g/1’, F@(S)) =

WV)).

Contraction Independence (C.Z.): For all PE.??‘, for all S,S’EC’, if S’CS and

F(S) ES’, then F(S’) = F(S).

W.P.O. says that it should not be feasible to make all agents better off than they

are at the compromise; ANsays that no other information is available than that con-

tained in the mathematical description of the problem so that if two agents enter

symmetrically in that description, they should be treated symmetrically by the solu-

tion; S.ZNV says that the theory should be unaffected by positive linear transforma-

tions acting independently on each agent’s utility scale; C.Z. says that if an alterna-

tive is thought to be the best compromise for a particular problem, it should still

be thought best for any subproblem that still contains it.

The Nash solution satisfies all four properties, the Kalai-Smorodinsky solution

satisfies all but C.I. and the Egalitarian solution satisfies all but S.ZNV.

Other notation: ep is the vector of all ones in Rp. Given S, S’C I?‘, cch{S, S’} is

the smallest convex and comprehensive set containing S and S’.

3. The problem

Although our work is formulated in the abstract framework of bargaining theory,

the concrete problem of resource allocation constitutes an important motivation for

it.” Consider the classical problem of fair division: there is a vector of resources

available to a group of consumers who have equal claims on it. How should these

goods be allocated? And how should they be re-allocated if the number of claimants

happens to increase, while the resources at their disposal remain fixed? Given a solu-

tion to the problem of fair division, let z, and z,! be the consumption bundles

assigned to agent i, one of the agents initially present, before and after the arrival

of the new agents. The ratio u,($)/u,(z;) of his final to initial utilities is a measure

of how he is affected by this event. A small ratio indicates that he greatly suffers.

But, the ratio could be greater than 1, indicating that he has actually gained, in spite

of the fact that the number of claimants on the fixed resources has increased. In

order to evaluate the solution, we propose to determine how any two members of

the initial group fare relative to each other. If one of them is very negatively affected

while the other is hardly affected, or perhaps even gains, one might expect resent-

ment on the part of the disadvantaged agent and his opposition to the solution. On

the other hand, a solution for which agents are similarly affected is more likely to

preserve the stability of society.

4 A literal interpretation of the model along those lines creates difficulties, discussed by Roemer

(1986). The reference to resource allocation here is only meant to be suggestive.


This is the issue with which we will be concerned here, but we will operate in the

abstract framework of bargaining theory.

Let P be the initial group and SC Rp be the image in utility space of the set of

alternatives available to P initially. The group P enlarges to Q and TC RQ is the

new feasible set. Since the opportunities bpen to P remain fixed, S= Tp (T, is the

projection of T onto I?‘). Given two individuals i and j in P, the change in their

relative fortunes is given by the ratio

fi(T)/C(S)

FjFj(WF, (S) ’ if this ratio is well defined on the extended real line. In order to evaluate the extent

to which F is likely to preserve the stability of the group, we determine the smallest

value of the ratio as S and T vary. Let then

eF(i,j,P,Q) = inf / SEE’, TE.Z~, Tp=S, Y= FiGV’4(S) 5

1 Fj(T)/Fj(S) ’

If F satisfies AN, +(i,j, P, Q) depends on the cardinalities of P and Q \ P only,

denoted m and n respectively, and it can be written as a,?. Then, necessarily,

$!“rl for-all (m,n)E(N\l)x IN. Let +={a: / (m,n)E(N\l)x N} be the relative guarantee structure of F. &pn measures the maximal change in the relative for-

tunes of two agents initially part of a group of cardinality m upon the arrival of n

additional agents. Seen positively, a high value of cF mn indicates that each original

agent is guaranteed not to be very differentially affected from any other agent as

new agents come in.

4. The results

Our aim here is to compare solutions on the basis of their relative guarantee struc-

tures. We will in fact be able to rank the major solutions. First, we consider the

Kalai-Smorodinsky and Egalitarian solutions.

Theorem 1. E;” = E;” = 1 for all (m,n)E(N\l)xN.

Proof. Given any i, j, P, Q as in the definition of cpn, note that K;(S)/Kj(S) =

5 For some pathological solutions, the ratio

F,(T)/F,(W

may be well-defined for no pair {S,T} satisfying S= rP and as a result &l(i,j,P,Q) may not be well-

defined either. However, most solutions, and in particular the three that we examine in detail here, are

such that for all PE 9 and for all SEZ ‘, F(S)>O. Therefore, the ratio under study is always well-

defined.


a,(s)/~,(S). Also, K,(T)/K,(T)=a,(T)/aj(T). Since S=T, implies ai(S)=ai(T)

and aj(S)=a,(T), we have cpn=l.

We omit the straightforward proof that E?” = 1. c1

Together with our earlier observation that if F is anonymous, then $“‘ll for all (m, n) E (tN \ 1) x N, Theorem 1 says that the Kalai-Smorodinsky and Egalitarian

solutions are best among all anonymous solutions. Moreover, it is worth noting that

for both of them the ratio under study is equal to 1 for all pairs {S, T} satisfying

S= Tp.6 This means that these two solutions dominate all anonymous solutions for

all such pairs. Although the notion of guarantee structure is based on worst case

analysis, the domination of the Kalai-Smorodinsky and Egalitarian solution is uni-

form across all pairs. Of course, if T happens to be a symmetric problem (invariant

under all exchanges of agents), the ratio

is equal to 1 for any weakly Pareto-optimal and anonymous F. However, if T is not

a symmetric problem the ratio will usually be strictly smaller than 1. This is in par-

ticular the case for the Nash solution, to which we now turn.

Theorem 2. EN”” = [B2-2-~v]/2 where B=ll’m-m+2 for all

(rn,R)E(n\l\l)X N.

Proof. Let P= (1, .,., m} with rnr2 and Q={l,..., m+n} with nrl. We have to

solve the following problem:

Pl: Find inf N,(T)/NI(S)

SEX’, Te.ZQ, S=T, . ~2GV~2(S) 1

Since N satisfies XINV, we can assume that N(T) = eQ and since N satisfies C.I.,

that S = cch{x, ep} with x= N(S). Then, N(T) = eQ if and only if x lies below the

hyperplane in RQ supporting at eQ the set { y E I/?: 1 flQ y, 2 1). The equation of this

hyperplane is CQ y, = m + n. Also, x= N(cch{x, e,}) if and only if ep lies below the

hyperplane in fRp supporting at x the set {X’E Ry 1 np x: 2 HP xi}. The equation of

this hyperplane is Cp (x:/x,) = tn. Then Pl reduces to

P2: Cx;~m+n, c (l/x,)sm . P P

This problem takes place entirely in Rp.

First, we note that we can impose Cp x, =m + n. Indeed, starting from x with

6 In fact, the ratio for the Egalitarian solution is equal to one for all SEC’ and TEZQ, even if

.S#T,.


Cpxi<m+n and C,, (l/x,)cm, let R=(x~+E,x~,...,x~) where E>O is small

enough SO that C,, Ki 5 m + n. Then,

; (l/X;) = l/X, +pF, (l/X;) = l/(x, +F)+& (l/x;) 5 c (l/x,) 5 tn. P

Since x~/x, <x,/x,, we are done.

Next, we can require Cp (l/xi)=m. Indeed, given x with Cp xi =m+n and

Cp (l/x;)<m, let X=(x1,x1-a ,..., xm), where E>O is small enough so that

Cp(l/x,)<m. Since Cpf;ilCpx,=m+n andxz/xl<x,/x,, we are done.

Finally, we can set x, =xj for all i, je P’= P\ { 1,2}. (Of course, P’= 0 if m = 2). Indeed, supposing m > 2, let XE RT with Cp xi = m + n and Cp (1 /xi) = m be given.

Let x be such that x1 =x,, R, =x2 and X, = ( Cp, x,)/(m - 2) for all iE P’. Note

that ~p~i=~1+x~+(m-2)(~p,x,)/(m-2)=~pxj=m+n. Also, Cp(l/Xi)=

1/X~+1/X2+(m-2)2/~p~Xj5~p (l/x;)=m since

1 <L C (l/Xj)

Cp,x,/(m-2) - m-2 P'

by convexity of the function h(t)= l/t. Finally, since x2/x1 =x2/x1, we are done.

After eliminating from the two constraints the common value of x1 for je P’, we are led to solving the following problem:

P3: Find inf{x,/x, 1 l/x, + l/~~+(m-2)~/(m+n-x, -x2)=m}.

Forming the Lagrangian

L(x,,x,, A) = $ + A /

1+1+ (m - 2)2 -my

XI X2 m+n-xl-x2 I

we obtain after differentiation

I

~Ux,,xz,~)

[

-1 =?+A -+

(m - 2)2

axI xl xiz (m+n-x,-x2)2 =O’ 1 awIyx2,4 =i,, A+ L (m - 2)2

ax2 x1 xi (m+n-x,-x2)2 =O. 1

Since x>O, the coefficients of A in these two equations are non-zero. Then, elimi-

nating J. gives

X2 (-l/X:)+A where A =

L

m-2 1 2

-_=

Xl (l/x;) -A m+n-x,-x2 ’

and after cross multiplication,

1 --x2/l =-I+x,A. X2 Xl


Setting (Y=x] +x2 and /3=x,x2, this equality becomes CC//~ = aA, and since a # 0,

$= [m:;Za12.

Inserting this expression into the equality constraint written as

2+ (m-2J2 =m

P mfn-a yields

‘y-t m+n-a

P P = m,

i.e. p = (m + n)/m. Using this expression for p in (m + n - a)/(m - 2) = fiwe obtain

a=m+n-(m-2)

The expression we are looking for is the smallest root of the equation in f:

t2- a2-2/I -----t+1 =o,

P

which, after replacing a and j3 by their values as functions of m and n, becomes

t2-(B2-2)t+l where B=llm(m+n)-m+2.

The smallest root is [B2 - 2 - 1/(B2 - 2)2 - 4]/2, the desired expression.’ 0

It can be shown that for each fixed m, E,,, ‘,ln is a decreasing and convex function

of n and that E;” -+ 0 as n + 03. This means that the arrival of a group of in-

creasing size may have an increasingly disturbing impact on the relative payoffs of

the members of an initial group of fixed size. Also, for each fixed n, &,fn is a de-

creasing and convex function of m and E:” + 8/[n2 + 8n + 8 + (n + 4) dm] as

m - co. This means that the arrival of a group of fixed size also may have an in-

creasingly disturbing impact on the relative payoffs of an initial group of increasing

size. This is somewhat paradoxical.*

An intuitive reason why the Nash solution does not offer very good relative

guarantees is that, as opposed to the Kalai-Smorodinsky and Egalitarian solutions,

which keep the agents’ utilities tied together, for that solution certain ‘stretchings’

of the feasible set may affect different agents quite differently. It is also for that

reason that it violates Population Monotonicity (which says that as a result of an

increase in the number of agents, unaccompanied by an expansion of opportunities,

all agents initially present weakly lose; see Thomson (1983a,b)), a property that is

’ The derivation of the second-order conditions and the calculations needed to establish the properties of CK!” listed in the next two paragraphs can be obtained from the authors upon request.

s However, a similar conclusion was obtained by Thomson and Lensberg (1983) in their study of

absolute guarantee structures (see concluding comments for a discussion).


satisfied by both the Kalai-Smorodinsky and Egalitarian solutions. Not sur-

prisingly, the minimum of the ratio appearing in the definition of E:” is attained

precisely for a pair {S, T} for which the Nash solution violates the property. Indeed,

using the expressions for x1 +x2 and x,x2 obtained in the proof of Theorem 2, it

can be shown that x2 = N,(S) < 1 while N,(T) = 1. Agent 2 has gained in spite of the

fact that the group of claimants has enlarged from P to Q.

Theorems 1 and 2 together imply that the Kalai-Smorodinsky and Egalitarian

solutions are both strictly superior to the Nash solution from the viewpoint of rela-

tive guarantees. We have already noted that these two solutions offer maximal

relative guarantees among all anonymous solutions. We show next that they can be

characterized with the help of two alternative sets of standard conditions together

with the requirement of maximal relative guarantees. The proofs, which are based

on constructions that are somewhat similar to those used in the characterizations of

these two solutions appearing in Thomson (1983a,b), can be found in the Appendix.

Theorem 3. The Kalai-Smorodinsky solution is the only solution satisfying W. P. 0.) AN, and S.INV that offers maximal relative guarantees.

Theorem 4. The Egalitarian solution is the only solution satisfying W.P. O., AN and C.I. that offers maximal relative guarantees.

5. Concluding comments

This study is intended as a complement to an earlier contribution by Thomson

and Lensberg (1983), where solutions were evaluated on the basis of the extent to

which an agent could see his own situation deteriorate, upon the arrival of new

agents unaccompanied by an expansion of their opportunities. A notion of absolute

guarantees was introduced and used to rank solutions. It was found that the Kalai-

Smorodinsky solution performed strictly better than the Nash and Egalitarian solu-

tions, and that no solution satisfying Weak Pareto-Optimality and Anonymity

could do better.

Two main differences should be noted between the two studies. First, according

to the criterion used there to rank solutions, the Egalitarian solution performed ex-

tremely badly (it offers no absolute guarantees at all), whereas for the criterion

under consideration here this solution is just as good as the Kalai-Smorodinsky

solution. The second difference is that while we are able here to characterize the

Kalai-Smorodinsky solution by using the condition that the solution offers maximal

relative guarantees, the condition that the solution offers maximal absolute guaran-

tees, together with the same list of complementary axioms, was not sufficient there

to yield a characterization. Although the Kalai-Smorodinsky solution was more

easily distinguishable from its main competitors, isolating it in the class of solutions

satisfying Weak Pareto-Optimality and Anonymity was actually more difficult.

The Thomson-Lensberg and current studies offer new viewpoints from which to

evaluate solutions to the bargaining problem. The differences between the conclu-


sions of the two papers should reinforce what is perhaps the main lesson to be drawn

from the developments that have taken place over the last ten years in the axiomatic

theory of bargaining: no unique solution has emerged as the best solution to the bar-

gaining problem, but a few solutions have kept reappearing as major actors in study

after study. These solutions are the Nash, Kalai-Smorodinsky and Egalitarian solu-

tions. In the present study, the spotlight has been on the latter two.

Appendix

Here, we characterize the Kalai-Smorodinsky and Egalitarian solutions.

First, we formally introduce the requirement that a solution offer maximal rela-

tive guarantees.

Maximal Relative Guarantees (M.R.G.). For all P, Q E 9 with PC Q, for all i, j E P,

+(i, j, P, Q) = 1.

Now, we characterize the Kalai-Smorodinsky solution with the help of M.R.G. The proof is similar to, but more direct than, Thomson’s (1983a) characterization

of that solution based on Population Monotonicity.

Proposition 1. The Kalai-Smorodinsky solution satisfies W.P. 0.) AN, S.INV and M.R.G.

Proof. Straightforward.

Proposition 2. If a solution Fsatisfies W.P.O., AN, S.INV and M.R.G., then it is the Kalai-Smorodinsky solution.

Proof. First, we show that F= K on Zp for (P I= 2. To fix the ideas, we let P= { 1,2) and we observe that by S.INV it is enough to prove that F(S) = K(S) for all

SE Cp with a(S) = e,. We introduce a third agent whom, without loss of generality,

we take to be agent 3, and we construct a problem TeZQ, where Qz { 1,2,3), such

that

and

S’ = {(Y?,Y,) E R {2,31 1 2(x1,x2) ES with y2 = , x and y,=x2},

S2 = C(Y%Y,) E R 13,‘1 1 ?I(xr,x,)~S with y3= 1 x and yr =x2},

T= cch{S,S’,S2}.

By W.P.O. and AN, we have F(T) = ,le, for i >O.

Since by W.P.O. there exists ieP such that F,(S) >O, then by M.R.G. applied to


{1,2}, we find

By W.P.O. and the fact that K satisfies W.P.O., we obtain F(S) = K(S). Given QEg with IQ[>2, and TEZQ, we show that F(T) = K(S) by considering

PC Q with 1 P 1 = 2 and noting that F(S) = K(S) = La(S) for some h > 0 by the first

step. Therefore, by M. R. G., Fp(T) =,uua(S) for some ,U > 0. Since ap(T) = a(S), we obtain, by repeated application of this argument, F(T) =pa(T) and by W.P.O., F(T) = K(T). 0

Finally, we characterize the Egalitarian solution. The proof bears some similarity

to Thomson’s (1983b) characterization of the solution based on Population Mono-

tonicity. It essentially involves showing that if SEE’, then S is the intersection

with lap of a problem in RQ, for some Q>p, whose solution outcome can be

shown to have equal coordinates. An application of M.R.G. then implies that the

solution outcome of S also has to have equal coordinates.

Proposition 3. The Egalitarian solution satisfies W.P.O., SY, C.I. and M.R.G.

Proof. Straightforward.

Proposition 4. If a solution F satisfies W. P. O., AN, C. I. and M. R. G., then it is the Egalitarian solution.

Proof. Let PE 9 and S E_Z’ be given. Without loss of generality, assume that

E(S)=e,. Let m=max(C.xiIxES}. Note that mz/PJ. Let QE~’ with PcQ and IQ1 = q where q is the smallest integer such that q>n, and TEZ~ be defined

by T= {xEIR$ / CQ x;(q). Finally, let T’=cch{S,eQ}. By W.P.O. and AN, F(T) = eQ. Since T’C T and F(T) E T’, by I.I.A., F(T’) =F(T). It can be checked

that TL= S. By W.P.O., there exists ie P such that K(S) >O. Therefore, for all

jEP, j#i, by M.R.G., we find that

F;VVF,(S) E;(T)/F,(S) F;(S) 1 Z-Z F,(T)/F,(S) = E,(T)/F,(S) F;(S) ’

Therefore, t;;(S) =5(S) for all i, j E P. Finally, by W. P. 0.) and the fact that E satisfies W. P. 0.) we obtain F(S) = E(S). 0

References

E. Kalai, Proportional solutions to bargaining situations: Interpersonal utility comparisons, Econo-

metrica 45 (1977) 1623-1630.

Y. Chun, W. Thomson / Bargaining solurions 295

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

513-518.

J.F. Nash, The bargaining problem, Econometrica 18 (1950) 155-162.

J. Roemer, The mismarriage of bargaining theory and distributive justice, Ethics 97 (1986) 88-110.

W. Thomson, The fair division of a fixed supply among a growing population, Math. Oper. Res. 8

(1983a) 319-326.

W. Thomson, Problems of fair division and the Egalitarian solution, J. Econ. Theory 31 (1983b)

211-226.

W. Thomson and T. Lensberg, Guarantee structures for problems of fair division, Math. Sot. Sci. 4

(1983) 205-218.

W. Thomson and T. Lensberg, Axiomatic Theory of Bargaining with a Variable Number of Agents

(Cambridge University Press, 1988).

Bargaining solutions and relative guarantees

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