Bargaining with ordinal and cardinal players¤

Emilio Calvoy Hans Petersz

June 2004 (This version)

Corresponding author: Hans Peters, Department of Quantitative Eco-

nomics, University of Maastricht, 6200 MD Maastricht, Netherlands. E-

mail address: h.peters@ke.unimaas.nl. Telephone: +31-43-3883288. Fax:

+31-43-3884874.

¤We thank the associate editor and two referees for their elaborate comments, which

led to a substantial revision of this paper.yDepartamento de Analisis Economico, Campus dels Tarongers, Avinguda dels

Tarongers s/n, Edi¯cio Departamental Oriental, 46011 Valencia, Spain. E-mail address:

emilio.calvo@uv.es. Grants from the University of the Basque Country (UPV/EHU

00031.321-HA7924/2000) and the Spanish Ministry of Science and Technology (BEC2000-

0875 and (BEC2000-1429) are gratefully acknowledged.zDepartment of Quantitative Economics, University of Maastricht, 6200 MD Maas-

tricht, Netherlands. E-mail address: h.peters@ke.unimaas.nl.

1

Abstract

We consider bargaining problems with at least one cardinal player and with

ordinal players, and provide a complete description of utility invariant solu-

tions of such problems for two players. For the n-person case we provide a

procedure that: (i) returns a given cardinal solution if there are only cardinal

players; (ii) is based on the ordinal solution for gradual bargaining problems,

introduced by O'Neill et al. (2001), for the ordinal players. Finally, we in-

troduce the so-called cardinal concession solution as another example of a

utility invariant solution.

Keywords: Bargaining, cardinal invariance, ordinal invariance, gradual bar-

gaining

2

1 Introduction

We consider n-person pure bargaining problems, that is, situations where the

only possible outcomes involve either complete cooperation of all players or

complete breakdown of cooperation.

Most of the game theoretic literature on bargaining assumes that play-

ers are cardinal. Speci¯cally, it is assumed that they have von Neumann-

Morgenstern preferences over outcomes: their preferences are represented by

expected utility functions, which are unique up to positive a±ne transfor-

mations. Therefore, it is natural to impose the requirement that bargaining

solutions be invariant under such transformations. The most prominent so-

lutions in the literature (Nash, 1950; Kalai and Smorodinsky, 1975; Perles

and Maschler, 1981) satisfy this property of cardinal invariance.

In contrast, traditional economic theory considers bargaining problems

in which the players are ordinal, for instance the exchange of commodities

between consumers. In that case, a bargaining solution should be invari-

ant under all monotonically increasing transformations of the players' utility

functions. This property is called ordinal invariance.

A remarkable consequence of the fact that bargaining theory abstracts

from the physical environment and considers only the image in utility space,

is the impossibility, observed by Shapley (1969), of ordinally invariant two-

person bargaining solutions that are nondictatorial and do not assign the

disagreement point. This impossibility result, however, is no longer true if

there are more than two bargainers. Shubik (1982, pp. 90{98) describes a

three-person ordinally invariant solution, attributed to Shapley. Sprumont

(2000) uses the associated construction to characterize ordinally equivalent

bargaining problems by deriving a complete set of representative problems.

3

Basically, this characterization solves the question of describing all ordinally

invariant bargaining solutions. Kibris (2001) gives an axiomatic characteriza-

tion of the solution in Shubik (1982), based on the construction in Sprumont's

paper. See also Safra and Samet (2001, 2004) for extensions.

The assumption that bargaining theory should only use the information

available in the feasible set of utility tuples is sometimes called welfarism.

Dropping this assumption makes it much easier to ¯nd natural ordinally

invariant solutions, as recently demonstrated in Nicolµo and Perea (2000), but

at the same time makes the theory less general. Also in the present paper we

relax the welfarist assumption: we assume that at least one player is cardinal

and the other players are ordinal. (Obviously, if the names of the ordinal and

cardinal players are ¯xed and known, we are back in the welfarist framework.)

We show by an example of a bargaining problem between a manager who is

paid in risky stocks and a worker who is paid in riskless money, that such a

situation can make a lot of sense.

A bargaining solution will be called utility invariant if it is ordinally

invariant for the ordinal players and cardinally invariant for the cardinal

players. We give a complete description of utility invariant solutions for

bargaining problems with one cardinal and one ordinal player, also involving

additional conditions like individual rationality or Pareto optimality. This

is easy since all such problems are equivalent under cardinal-ordinal utility

transformations.

There are many ways to extend these two-person solutions to n-person

problems with at least one cardinal player. Here, we describe one such pro-

cedure. This procedure returns a given cardinal solution (e.g., Nash, Kalai-

Smorodinsky,...) if there are only cardinal players. If there are also ordinal

4

players, we proceed, roughly, as follows. The cardinal players receive a ¯xed

percentage of the gains associated with the cardinal solution payo®s they

would receive if the ordinal players would receive their disagreement payo®s.

The line segment connecting the disagreement payo®s for the cardinal players

and their cardinal solution payo®s determines a gradual bargaining problem

for the ordinal players in the sense of O'Neill et al. (2002). Then, we apply

the ordinal solution proposed by these authors to determine the payo®s for

the ordinal players.

Finally, we present an example of a solution, the so-called cardinal con-

cession solution, that arises in a somewhat di®erent way. It is de¯ned by

considering integral curves of a speci¯c vector ¯eld, coincides with the Perles-

Maschler solution (for two players) and an extension thereof (for n players)

if there are only cardinal players, and with the solution produced by the

above procedure if there is exactly one cardinal player. In the other mixed

cases, however, it treats cardinal players in a more sophisticated way. We

include this example to illustrate that there are many ways, apart from the

procedure above, to obtain reasonable utility invariant solutions.

The organization of the paper is as follows. In Section 2 we present a

two-person manager-worker problem as an example of a situation with a

cardinal and an ordinal player. After the preliminaries in Section 3, we treat

the two-person case in Section 4 and the extension to the n-person case in

Section 5. Section 6 is concerned with the cardinal concession solution, and

Section 7 concludes.

5

2 A two-person example

Suppose that the owner of a ¯rm wants to distribute a reward between a

worker (player 1) and a manager (player 2). The total amount to distribute

is one monetary unit. The payo®s for the worker are in money (m units),

and for the manager in stocks of the ¯rm (s units). Denoting the stock price

by p, we thus have m+ ps · 1.We assume that the worker's preferences are ordinal, represented by a

continuous and strictly increasing utility function u1 for money. We also

assume that the manager must wait until the next period to sell the stocks,

and that there is uncertainty about the future price: it will be either p1

or p2, each with probability one half, and the manager has expected utility

of the form Eu2(s) =12u2(p1s) +

12u2(p2s), where u2 is again a continuous

strictly increasing function. The disagreement outcome corresponds to the

worker and the manager both receiving zero, and the disagreement vector is

d = (u1(0); u2(0)). The bargaining problem is summarized by the pair (S; d),

where S is the feasible utility set

S = fx 2 IR2 : x1 = u1(m); x2 = Eu2(s); m+ ps · 1g:

A bargaining solution is a function à that assigns to a bargaining problem

(S; d) a feasible point Ã(S; d) 2 S: In our example a reasonable solution

should satisfy Ã(S; d) ¸ d (individual rationality) and Ã(S; d) 2 @S (Paretooptimality), where @S denotes the boundary of S. Let I(S; d) be the `ideal

point', as indicated in Figure 1. (All concepts will be introduced more for-

mally in the next section.) For every real number r 2 [0; 1], de¯ne the solution'r such that 'r(S; d) 2 @S and 'r2(S; d) = rI2(S; d) + (1 ¡ r)d2 (see Figure1). It is straightforward to check that this solution is utility invariant: more

6

I(S; d)

d

u'r(S; d)

'r1(S; d)

'r2(S; d) = rI2(S; d) + (1¡ r)d2

Card

Ord

Figure 1: A two-person example

precisely, if u1 is transformed by a strictly increasing continuous function

(player 1 is ordinal) and u2 by a positive a±ne transformation (player 2 is

cardinal) then to obtain the solution of the transformed bargaining problem

we can just apply the two transformations to the point 'r(S; d). In other

words, the underlying agreement in terms of money for the worker and stocks

for the manager has not changed.

In Section 4 we show that any individually rational, Pareto optimal, and

utility invariant solution must be equal to 'r for some r 2 [0; 1]. Hence,where there is an abundance of solutions if the two players are cardinal,

and only two dictatorial solutions (namely, '0 and '1) if the two players

are ordinal (cf. Shapley, 1969), there is basically a one-dimensional family of

solutions in the mixed case.

7

3 Preliminaries

Let N = f1; : : : ; ng (with n ¸ 2) be the set of players, and let ; 6= C µ N .Members of C are called cardinal and members of NnC are called ordinal.

A bargaining problem (for N) is a pair (S; d) where:

(i) the feasible set S µ IRN is closed and comprehensive, i.e., if x 2 S andy 2 IRN with x ¸ y then y 2 S;1

(ii) S is unbounded in every coordinate;

(iii) the set fx 2 S : x ¸ yg is bounded for every y 2 S;

(iv) the boundary @S of S is non-level , that is, if x 2 @S and y 2 IRN withy ¸ x and y 6= x, then y =2 S;

(v) the disagreement point d is an element of int(S) (the interior of S).

The interpretation is that the players either agree on some point x in S,

yielding utility xi to player i, or disagree, in which case each player i ends

up with utility di.

Observe that there is no convexity condition on the feasible set: convexity

does not make sense if one or more of the players are ordinal, since it is not

preserved under arbitrary monotonic transformations.2

We denote by BN the family of bargaining problems (S; d) for N .For (S; d) 2 BN and every i 2 N we de¯ne

Ii(S; d) := maxfyi : y 2 S and yj = dj for all j 2 Nnfigg:

The point I(S; d) := (Ii(S; d))i2N is called the ideal point of (S; d).

8

A bargaining solution à is a map à : BN ! IRN such that for every

(S; d) 2 BN , Ã(S; d) 2 S.An order-preserving transformation is a strictly increasing bijection on

IR. Such a function is necessarily continuous, and its inverse is again an

order-preserving transformation. Let f = ffi : i 2 Ng be a set of order-preserving transformations. For x 2 IRN and (S; d) 2 BN denote f(x) =

(f1(x1); : : : ; fn(xn)), and f(S) = ff(x) : x 2 Sg. Observe that (f(S); f(d)) 2BN .A subclass of order-preserving transformations is the class of the positive

a±ne transformations of the form ¿ 7! b+ a¿ for all ¿ 2 IR, where a; b 2 IR,a > 0.

We say that a bargaining solution à is utility invariant if for every set

f of order-preserving transformations such that fi is positive a±ne for all

i 2 C, we have Ã(f(S); f(d)) = f(Ã(S; d)) for all (S; d) 2 B.A bargaining solution à is Pareto optimal if Ã(S; d) 2 @S for every

(S; d) 2 BN , and individually rational if Ã(S; d) ¸ d for every (S; d) 2 BN .

4 One cardinal and one ordinal player

Let N = f1; 2g and C = f2g, so player 1 is ordinal and player 2 is cardinal.We start with the following observation.

Lemma 4.1 Let (S; d); (T; e) 2 BN . Then there is a pair of order-preservingtransformations f = (f1; f2), with f2 positive a±ne, such that (T; e) =

(f(S); f(d)).

Proof. Let g2 be a positive a±ne transformation such that e2 = g2(d2) and

I2(T; e) = g2(I2(S; d)). Next, for every ¿ 2 IR let h(¿) 2 IR be de¯ned by

9

(¿; h(¿)) 2 @S. Then h : IR ! IR is a strictly decreasing function. Also, for

every ¿ 2 IR de¯ne j(¿) by (j(¿ ); ¿ ) 2 @T . Again, j : IR ! IR is a strictly

decreasing function. Now the desired transformations are f1 := j ± g2 ±h andf2 := g2. 2

Fix an arbitrary bargaining problem ( ¹S; ¹d) 2 BN , and let x 2 ¹S be an

arbitrary point. By Lemma 4.1, there is a unique utility invariant bargaining

solution that assigns x to ( ¹S; ¹d). Denote this solution by Ãx, then we have

just derived:

Corollary 4.2 A bargaining solution à : BN ! IRN is utility invariant if

and only if there is an x 2 ¹S with à = Ãx.

Obviously, the family of all utility invariant Pareto optimal [individually

rational] bargaining solutions is obtained by restricting x to @ ¹S [to fx 2¹S : x ¸ dg]. A nicer way to describe the intersection of these families is asfollows. For every r 2 [0; 1] let 'r be the bargaining solution with for every(S; d) 2 BN : 'r(S; d) 2 @S and 'r2(S; d) = rI2(S; d) + (1¡ r)d2.

Corollary 4.3 A solution à on BN is utility invariant, Pareto optimal, andindividually rational if, and only if, à = 'r for some r 2 [0; 1].

Proof. Obviously, every 'r satis¯es the three conditions in the corollary.

For a solution à satisfying these conditions, by Pareto optimality and in-

dividual rationality there is an r 2 [0; 1] such that Ã( ¹S; ¹d) = 'r( ¹S; ¹d). By

utility invariance and Lemma 4.1, Ã = 'r. 2

Corollary 4.3 provides a simple but complete description of all utility in-

variant, Pareto optimal, and individually rational bargaining solutions. If

10

N = C, then of course any of the well-known two-person bargaining solutions

(Nash, Kalai-Smorodinsky, Perles-Maschler,...) satis¯es these conditions. In

the next section we provide an extension to more than two players.

5 More than two players

In this section we extend the family of bargaining solutions described in

Corollary 4.3 to more than two players. There are many ways to do this but

here we concentrate on a solution (or rather, class of solutions) that satis-

¯es the following requirements: (i) if there are only cardinal players, then

the solution should coincide with a given `cardinal' solution; (ii) the cardinal

players receive a ¯xed fraction of the gain associated with the cardinal solu-

tion outcome that would result if the ordinal players would be kept down to

their disagreement payo®s; (iii) subject to the constraint imposed by (ii), the

ordinal players receive payo®s according to the ordinal solution of O'Neill et

al. (2002). Implicit in this procedure is that the obtained solution should be

utility invariant, individually rational, and Pareto optimal.

Concerning the ¯rst requirement, let C be any set of cardinal players, let

DC µ BC , and let ' : DC ! IRC be a utility invariant, individually ratio-

nal, Pareto optimal, and symmetric solution. (Symmetry means that the

players should receive equal payo®s in a symmetric bargaining problem, i.e.,

a bargaining problem that is invariant under any permutation of the coor-

dinates.) For brevity, we call such a solution classical. For example, DCcould be the set of all bargaining problems with (cardinal) player set C and

with convex feasible sets, and ' could be the jCj-person3 Nash bargainingsolution, assigning to each (S; d) 2 DC the unique maximizer of the product

11

Qi2C(xi ¡ di) over the set fx 2 S : x ¸ dg.4

In order to proceed with requirements (ii) and (iii), we need to be explicit

about the ordinal solution of O'Neill et al. (2002). Let O be a (¯nite) set

of (ordinal) players (possibly, jOj = 1). An agenda is a function ® : IRO !IR that is continuously di®erentiable, with (strictly) positive gradient that

satis¯es a local Lipschitz continuity condition. (See O'Neill et al. (2002) for

a discussion of these and other assumptions, and for more details about the

ordinal solution.) If ® is an agenda and d 2 IRO, then the pair (®; d) is calleda gradual bargaining problem. Note that in a gradual bargaining problem

there is a sequence of increasing feasible sets (fx 2 IRO : ®(x) · tg)t2IR. Theordinal solution ­ associates with each gradual bargaining problem (®; d) a

(continuously di®erentiable) function (`path') ! : IR ! IRO satisfying the

system of di®erential equations

!0i(t) = [jOj®0i(!(t))]¡1 ; i 2 O; (1)

(where ®0i denotes a partial derivative) with initial condition

!(®(d)) = d:

For every gradual bargaining problem this path is unique. Denote this path

by ! = ­(®; d). Then, ®(!(t)) = t for every t 2 IR (Pareto optimality),

!(t) ¸ d for every t 2 IR with t ¸ ®(d) (individual rationality), and

­ is ordinally invariant: if f = (fi)i2O is a collection of order-preserving

transformations5, then for every gradual bargaining problem (®; d), we have

­(®; d) = f(­(¯; e)), where d = f(e) and ¯ = ® ± f . In spite of its technicalde¯nition the ordinal solution has a simple interpretation: the ratio of player

i's and j's marginal increments of utility at `time' t, !0i(t)=!0j(t), is equal to

12

the marginal rate of substitution of i's and j's utilities along the boundary

fx 2 IRO : ®(x) = tg of the feasible set at time t.

Before we continue we need some notation. For a vector y 2 IRN and a

proper subset M µ N , let yM denote the restriction of y to the coordinates

in M . For S µ IRN and ¹x 2 IRNnM , let

SM;¹x = fy 2 IRM : there is an x 2 S with y = xM and xNnM = ¹xg;

the `slice' of S for coalition M where the complement NnM receives ¹x.

Let now N be a player set, with ; 6= C µ N the subset of cardinal players,

and O = NnC the subset of ordinal players. Let ' : DC ! IRC be a classical

solution for some DC µ BC . We will de¯ne a utility invariant, individuallyrational, and Pareto optimal solution Ã';­ on a subset B0N of BN . We de¯nethis subset of bargaining problems along the way. If N = C, then Ã';­ := ';

hence, for this case we de¯ne B0N = DN . Otherwise, let (S; d) be a bargainingproblem for N and assume that (SC;dO ; dC) 2 DC (hence, this is a restrictionon BN), so that '(SC;dO ; dC) is well-de¯ned. Let zC := '(SC;dO ; dC). Hence,zC is the outcome for the cardinal players if the ordinal players would receive

their disagreement payo®s. For every t 2 IR, de¯ne the point xC(t) 2 IRC byxC(t) := (1¡ t)zC + tdC . Under appropriate conditions on BN (to guaranteesmoothness and local Lipschitz continuity|so this is another restriction on

BN ) the points xC(t) generate a gradual bargaining problem with feasible

sets SO;xC(t) (t 2 IR) and disagreement point dO. Let the ordinal solution forthis gradual bargaining problem be given by the path !. Let t¤ := jOj=jN jand de¯ne

Ã';­(S; d) := (xC(t¤); !(t¤)):

13

Hence, the cardinal players receive the fraction jCj=jN j of the gain zC ¡dC , which they would obtain if the ordinal players were kept down to their

disagreement utilities dO. The number t¤ plays a similar role as the number r

in Corollary 4.3. The choice for t¤ proposed here guarantees that the solution

is symmetric at least on the bargaining problem (fx 2 IRN : Pi2N xi · 1g; 0).Given that the cardinal players receive xC(t

¤), the ordinal players play the

induced gradual bargaining problem and receive utilities according to the

ordinal solution ­.

The solution Ã';­ proposed here is utility invariant (obvious for the cardinal

players, and for the ordinal players this follows from ordinal invariance of

the ordinal solution6), Pareto optimal, and individually rational. It can be

varied by choosing di®erent classical solutions ', or other solutions instead

of ­ for the gradual bargaining problem.

If there is exactly one cardinal player, say player n, then according to

Ã';­ this player receives (1=n)(In(S; d) ¡ dn) (this is independendent of theclassical solution ' since for only one player all these solutions coincide). The

gradual bargaining problem for O = Nnfng then consists of all slices of Swith ¯xed utility for player n. (See Figure 2 in Section 6 for an illustration.)

Thus, in this gradual bargaining problem the feasible sets are endogenously

determined by the utility levels of the cardinal player. In this sense, adding

one cardinal player results in an `endogenization' of the feasible sets in a

gradual bargaining problem.

14

6 The cardinal concession solution

The cardinal concession (cc) solution is constructed in a di®erent way than

the solution(s) in the previous section. If there are only cardinal players, then

the solution coincides with the Perles-Maschler solution (for two players) or

an extension thereof (for more than two players). If there is exactly one

cardinal player, then the solution coincides again with Ã';­ (for any classical

solution '). In the remaining mixed cases, however, the cardinal players are

treated in a more sophisticated way. Thus, this shows that the procedure for

¯nding utility invariant solutions, discussed in the preceding section, certainly

does not exhaust the possibilities.

The cc-solution will be de¯ned on the class BNs of bargaining problems, char-acterized as follows. A bargaining problem (S; d) 2 BN is in BNs if and only ifthere is a function g : IRN ! IR with continuously di®erentiable and positive

partial derivatives, such that @S = fx 2 IRN : g(x) = 0g. The i-th partialderivative of g is denoted by g0i.

Let (S; d) 2 BNs . For a point x 2 S and j 2 N , let xj := (xNnfjg; Ij(S; x)).(In this notation we suppress dependence of the point xj on the set S.) Hence,

xj is the best point for j if the other players i are kept to xi. We de¯ne the

vector ¯eld v on the interior of S, int(S), as follows:

vS(x)i :=

264Yj2C

24Yk2C

g0k(xj)

g0i(xj)

35 1jCj375

1jCj

; for all i 2 N , x 2 int(S). (2)

The vector ¯eld v can be interpreted as assigning to an interior point x a

kind of geometric mean of the exchange rates of utilities at associated points

xj on the boundary of S.

15

We proceed by showing that through each point of the interior of S there

is a unique integral curve of v. First, we construct a new vector ¯eld ¹v on

all of IRN , as follows. Extend the de¯nition of the `ideal point' I(S; x) to all

of IRN (we can take exactly the same de¯nition as in Section 3, noting that

the feasible set S is assumed unbounded in every coordinate), and de¯ne the

distance function DS(x) :=Pi2N [Ii(S; x) ¡ xi] for all x 2 IRN . Obviously,

DS(x) > 0 if x 2 int(S), DS(x) = 0 if x 2 @S, and DS(x) < 0 if x =2 S.Then let

¹vS(x) :=

8>>>><>>>>:DS(x)vS(x) if x 2 int(S),0 if x 2 @S,DS(x)vS(I(S; x)) if x =2 S.

Lemma 6.1 Let (S; d) 2 BNs , let x0 2 IRN , and let T be a real interval with0 2 int(T ). Then there is a unique di®erentiable function »(¢; x0) : T ! IRN

with »(0; x0) = x0 and with (d=dt)»(t; x0) = ¹vS(»(t; x0)) for all t 2 T .

Proof. This lemma follows from standard results in the theory of ordinary

di®erential equations. See, for example, Chapter 8 of Hirsch and Smale(1974).

It is su±cient that the map x 7! ¹vS(x) is locally Lipschitz continuous on IRN .

If g is the function associated with S, then by assumption, the partial deriva-

tives g0i(xj) never vanish and have continuous partial derivatives. Therefore,

¹vS(x) has continuous ¯rst-order partial derivatives, and this implies Lipschitz

continuity. 2

It is now straightforward to de¯ne the cardinal concession solution · : BNs !IRN . Let (S; d) 2 BNs and let »(¢; d) be the unique integral curve of the vector¯eld ¹vS passing through d. Note that (the graph of) this curve is strictly

increasing, since the partial derivatives of the function g describing @S are

16

positive. This curve must cross the boundary @S at some point »(t¤; d): if

not, then it would converge to a point z 2 int(S), but then it can be uniquelyextended by the integral curve of ¹v passing through z, a contradiction. Note

that »(¢; d) is also an (hence, the) integral curve of the vector ¯eld vS onthe interior of S, since on int(S) these vector ¯elds do not di®er in direction.

De¯ne ·(S; d) := »(t¤; d). The solution · de¯ned this way is obviously Pareto

optimal and individually rational. Further, ·(S; e) = ·(S; d) whenever e is

a point on the integral curve »(¢; d). Another direct consequence of Lemma6.1 is that di®erent curves never intersect and, in particular, never cross @S

at the same point.

We show now that · is utility invariant with respect to those order-

preserving transformations that map bargaining problems in BNs to problemsin the same set. Therefore, we restrict attention to transformations f =

ffi : i 2 Ng such that each fi is twice continuously di®erentiable, withpositive ¯rst derivative everywhere. Of course, positive a±ne transformations

have these properties. In the remainder of this section utility invariance is

understood to be invariance with respect to transformations of this kind.

The following lemma gives the simple relation between the partial deriv-

atives associated with a bargaining problem and the transformed problem.

Lemma 6.2 Let (S; d) 2 BNs and let f be a collection of order-preserving

transformations. Let g and h describe @S and @f(S), respectively. Then, for

all i 2 N and x 2 @S,h0i(f(x)) =

g0i(x)f 0i(xi)

:

Proof. Note that we can write h = g ± f¡1 (where f¡1 = ff¡1i : i 2 Ng),hence for i 2 N and x 2 @S, we have h0i(f(x)) = (g ± f¡1)0i(f(x)) = g0i(x) ¢(f¡1i )

0(fi(x)) = g0i(x)=f0i(xi). 2

17

Theorem 6.3 · is utility invariant.

Proof. In order to show that · is utility invariant, we have to show that the

integral curves of the vector ¯eld vS are utility invariant. Let (S; d) 2 BNs ,let f = ffi : i 2 Ng be a set of order preserving transformations, and let»(¢; d) and »f (¢; f(d)) be the integral curves of vS passing through d and ofvf(S) passing through f(d), respectively. Then it is su±cient to show that

d

dtfi(»i(t; d)) = ¹(f(d))

d

dt»fi (t; f(d)) for all i 2 N; (3)

where ¹ : int(f(S)) ! IR is some positive and continuously di®erentiable

function (this function may change the speed through the integral curve, but

does not change its direction). Note that for every i 2 N ,d

dtfi(»i(t; d)) = f

0i(»i(t; d)) ¢

d

dt»i(t; d) = f

0i(»i(t; d)) ¢ vS(»(t; d))i: (4)

Denoting for any two vectors x; y 2 IRN , x ¤ y = (x1y1; : : : ; xnyn), and

f 0(x) = (f 01(x1); : : : ; f0n(xn)), (3) can be rewritten as

f 0(d) ¤ vS(d) =

Ãd

dtfi(»i(t; d))

!i2N

= ¹(f(d))

Ãd

dt»fi (t; f(d))

!i2N

= ¹(f(d)) ¢ vf(S)(f(d)); (5)

where the ¯rst equality follows from (4). Hence, we are left to show that (5)

holds.

First note that the fi are positive a±ne for all i 2 C, say fi(xi) = bi+aixi.Recall that dji = di if i 6= j, and dii = Ii(S; d), for all i. Hence, f 0i(dji ) = f 0i(di)for all i 6= j. Moreover, if i 2 C, f 0i(Ii(S; d)) = ai = f 0i(di). By this and

Lemma 6.2 we have h0i(f(dj)) = g0i(d

j)=f 0i(di) (where, as in Lemma 6.2, g

18

describes @S and h describes @f(S)) if i 6= j or if i = j and i 2 C. Hence,for all i 2 N ,

vf(S)[f(d)]i =

264Yj2C

24Yk2C

h0k(f(dj))

h0i(f(dj))

35 1jCj375

1jCj

=

264Yj2C

24Yk2C

g0k(dj)=ak

g0i(dj)=f 0i(di)

35 1jCj375

1jCj

=f 0i(di)

[Qk2C ak]

1jCj¢ vS(d)i:

So [Qk2C ak]

1jCj ¢ vf(S)[f(d)] = f 0(d) ¤ vS(d), and (5) holds. 2

For the case of exactly one cardinal player the solution · coincides with the

solution Ã';­ for any classical solution '. See Figure 2.

Corollary 6.4 Let jCj = 1 and let ' be a classical solution. Then ·(S; d) =Ã';­(S; d) for every (S; d) 2 BNs .Proof. For (S; d) 2 BNs , let »(¢; d) be the integral curve of vS throughd. Then ·(S; d) is the point of intersection of the curve with @S. We may

assume d = »(0; d) and ·(S; d) = »(1; d). Let 1 be the cardinal player and

let O = Nnf1g. For 0 · t · 1, the slices SO;»1(t;d) form (part of) a gradual

bargaining problem, and projected on these sets the curve f»(t; d) : 0 · t ·1g determines a curve f»¡1(t; d) : (»2(t; d); : : : ; »n(t; d)) : 0 · t · 1g, wherefor each t 2 [0; 1], the point »¡1(t; d) is on the boundary of the feasible set attime t. Let g be the function describing @S, then by de¯nition of the vector

¯eld v in (2) it follows that

d

dt»i(t; d) ¢ g0i(»(t; d) =

d

dt»j(t; d)) ¢ g0j(»(t; d)) for all i; j 2 N:

But by (1), this implies that the curve f»¡1(t; d) : 0 · t · 1g is exactly thecurve that determines the ordinal solution the gradual bargaining problem

for O.

19

¡¡¡¡¡¡¡¡¡¡¡¡

¡¡

¡¡

¡¡

¡¡

¡¡¡

¡¡

¡¡

¡¡

¡¡

¡¡ u uuu

P

P

d = »(0; d)

-»¡1(t; d)

-»(t; d)

¡¡ª

·(S; d) = »(1; d)

¾ @SO;»1(1;d)¾ @SO;»1(t;d)

Ord

Ord

Card (player 1)

*µ

¸

u·(S; d)

d

S

S

»

Ord

Card

Figure 2: The upper diagram illustrates the proof of Corollary 6.4. The lower

diagram illustrates the case n = 2 with one cardinal player: the graph of » is

obtained by rotating the part of @S above ·(S; d), so ·(S; d) = 'r(S; d) with

r = 1=2.

20

We are left to show that ·1(S; d) = d1 +1n[I1(S; d) ¡ d1]. Consider the

paths » and P as in Figure 2. We ¯rst construct a set of order preserving

transformations f such that the transformed path »f is a straight line.

Let f1 be a positive a±ne transformation such that f1(d1) = 0 and

f1 (I1(S; d)) = 1. For all i 6= 1, de¯ne fi such that fi (»i(t; d)) = f1 (»1(t; d)).Because » is a di®erentiable function, the corresponding fi are twice contin-

uously di®erentiable, with positive ¯rst derivatives everywhere. De¯ned in

this way, and because · is utility invariant, the integral curve of vf(S) through

0 is a straight line passing through f1 (·1(S; d)) ¢ (1; :::; 1) . Hence

vf(S)³»f (t; 0)

´i=d

dt»fi (t; 0) =

d

dt»fj (t; 0) = vf(S)

³»f(t; 0)

´j; (6)

for all i; j 2 N .Now we show that the corresponding P f is also a straight line. Denote by

h the function that describes @f(S). To simplify notation, denote by pf (t)

the corresponding point in P f such that pfi (t) = »fi (t; 0), for each t 2 [0; 1]

and i 6= 1. Recall that, by de¯nition of ·, we have

vf(S)³»fi (t; 0)

´i=h01 (p(t))h0i (p(t))

; (7)

for all i 2 N . Then (6) and (7) imply h0i (p(t)) = h0j (p(t)) for all i; j 2 N . Thefact that the normal vector to @f(S) in every pf(t) has all its components

equal and positive implies that P f is a straight line through (1; 0; :::; 0) and

f1 (·1(S; d)) ¢ (1; :::; 1). Moreover, P f is included in the hyperplane H = fx 2IRN :

Pi2N xi = 1g, which ¯nally implies that f1 (·1(S; d)) = 1

n, and thus

·1(S; d) = d1 +1n[I1(S; d)¡ d1]. 2

The lower diagram in Figure 2 illustrates the two-player case with one cardi-

nal player. In this case, the graph of the curve » is obtained by rotating the

21

upper part of @S along the horizontal line through '1=2(S; d) (cf. Section 4),

as can be inferred directly from (2).

We conclude this section with three further remarks on the cc-solution.

Remark 6.5 (Two players, cardinal) If there are only two players, both

cardinal, then the cc-solution · coincides with the Perles-Maschler (1981)

solution. This can be seen as follows. Denoting by ³ the Perles-Maschler

solution, the di®erential equation for ³ using our notation is given by

³S(x)1 :=

"g02(x

2)

g01(x2)

#12

; ³S(x)2 :=

"g01(x

1)

g02(x1)

#12

; for all x 2 int(S); (8)

where, as before, g is the function describing @S. (See the de¯nition of

the solution by Procedure 2 in Perles and Maschler (1981), or Section 2 in

Calvo and Guti¶errez (1994).) Applying de¯nition (2) to the case in which

C = N = f1; 2g, vS reduces to

vS(x)1 =

24"g02(x1)g01(x1)

# 12

¢"g02(x

2)

g01(x2)

# 12

3512

; vS(x)2 =

24"g01(x1)g02(x1)

# 12

¢"g01(x

2)

g02(x2)

# 12

3512

;

for all x 2 int(S). Therefore, it is straightforward to check that

vS(x)2vS(x)1

=³S(x)2³S(x)1

=

"g01(x

1)

g02(x1)¢ g

01(x

2)

g02(x2)

#12

:

Hence, both vector ¯elds, vS and ³S, are collineal and therefore produce the

same integral curves, so · and ³ coincide.

Remark 6.6 (More than two players, cardinal) In Perles and Maschler

(1981) the super-additive solution was supported by a set of axioms: ef-

¯ciency, symmetry, continuity, cardinal invariance and superadditivity. In

22

Perles (1982) it is proved that for three-person problems a solution that sat-

is¯es e±ciency, cardinal invariance, symmetry and superadditivity does not

exist. In Calvo and Guti¶errez (1994) it was shown that (8) implies a prop-

erty that can be used to give a suitable extension of this solution to n-person

problems. It can be shown that this extension is a solution di®erent from ·.

Thus, · restricted to cardinal problems can be considered as an alternative

extension of the super-additive solution.

Remark 6.7 (General case) In general, according to the solution ·, car-

dinal players no longer receive a ¯xed proportion of the gains they would

obtain if the ordinal players were kept to their disagreement payo®s, as was

the case for the solutions studied in the preceding sections. This can be seen

by considering (2) for a bargaining problem with two cardinal players and

one ordinal player. The details are left to the reader.

7 Conclusion

The purpose of this paper was twofold: to extend the bargaining problem

by allowing both cardinal and ordinal players, and two ¯nd utility invariant

solutions for it. The procedure presented in Section 5 can be `axiomatized'

on a `meta-level' by the following choices: choose a classical solution for the

cardinal players; choose the percentage of the gains they would receive if

the ordinal players were held down to their disagreement payo®s; and choose

a solution for the gradual bargaining problem. The ordinal solution to the

gradual bargaining problem is characterized by O'Neill et al. (2001). The

de¯nition of the cc-solution was inspired by the de¯nition of a (purely) ordinal

solution by Shapley (1995).

23

References

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Kalai, E., Smorodinsky, M., 1975. Other solutions to Nash's bargaining

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Kibris, ÄO, 2001. Preference-based egalitarianism in bargaining. Mimeo,

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24

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25

Footnotes

1 x · y means xi · yi and x < y means xi < yi for every i. Analogously for¸ and >.

2 In Sections 5 and 6 we impose a few additional conditions on bargaining

problems.

3 For a ¯nite set X, jXj denotes its cardinality.

4 If jCj = 1, then a bargaining problem is de¯ned as a pair ((¡1; a]; d),where a; d 2 IR with a > d.

5 Here, only those oder-preserving transformations are allowed that map

gradual bargaining problems into gradual bargaining problems.

6 It is assumed that only those order-preserving transformations are con-

sidered such that the resulting bargaining problems are again in the class

B0N .

26