JOURNAL OF ECONOMIC THEORY 49, 266217 (1989)

The Nash Program: Non-convex Bargaining Problems*

MARIA Jo& HERRERO

Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburg, Pennsylvania 15213

Received August 14, 1987; revised December 15, 19%

We detine and characterize an extension of the Nash bargaining solution to non-convex bargaining problems and, in the spirit of the Nash Program of implementing cooperative solutions non-cooperatively, we show that this solution is supported by Rubinstein’s (Econometrica SO (1982), 97-l 10) model of bargaining and its corresponding set of stationary subgame perfect equilibria. Journal of Economic Literature Classification Numbers: 020, 022, 026. 0 IPSP Academic PKS, IIIC.

1. INTRODUCTION

In this paper we defme and characterize an extension of the Nash bargaining solution to non-convex bargaining problems and, in the spirit of the Nash Program’ of implementing cooperative solutions non- cooperatively, we show that this solution is supported by Rubinstein’s [ 121 sequential model of bargaining and its corresponding set of equilibria.

The bargaining problem considered here is the Nash bargaining problem. The so-called Nash bargaining problem refers to a situation in which two players are faced with the following choice. They can either reach an agreement on a feasible pair of utilities or else fail to agree and obtain a prespecitied pair of disagreement utilities.

Cooperative solutions to Nash’s problem have been characterized by various authors for the case m which the set of feasible utility pairs is convex (i.e., convex problems). Some of these are included in papers of Nash [lo] himself, Kalai and Smorodinsky [7], Kalai [5], Roth [ll], Binmore [l], and Myerson and Thomson [9]. The aim of these and similar works is to predict the outcome of the bargaining problem when the negotiations are conducted by rational players. This is accomplished in

* This work has been partly supported by NSF Grant SES-8609986. I thank Ken Binmore for numerous and illuminating discussions. Also, the comments and suggestions of an unknown referee and an associate editor are greatly appreciated.

’ See Binmore [ 1] for a description of the Nash Program.

266 0022-053 l/S9 83.m Copyright 0 1989 by Academic F’ress, Inc. Ail rights 01 reproduction in any form reserved.

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the above papers through a set of axioms which uniquely cbaracteri~e their proposed solution and which capture different aspects of (i~divid~aI or collective) rationality.

Roughly, the cooperative solution proposed here is as follows. Let (XT denote a bargaining problem in which X refers TV the set of feasible utility pairs and d denotes the disagreement point. Then our solutio set of constrained stationary points of the Nash product (ul for the constraint Eff(X) ( i.e., the Pareto efliciency of X)~’ Therefore, if the set X is convex, the solution consists of the unique point in X which

izes (pi -d,)(uZ -dZ),3 i.e., the Nash bargaining soluti.on is obt owever, if 3’ is not convex, our proposed solution may select points in

I%(X) which do not maximize the Nash product. This solution is uniquely characterized by a set of six axioms which refer

to individual rationality, efliciency, invariance with respect to linear trans- formations of the utility functions, symmetry, continuity, and iude~e~dc~ce of irreievant alternatives.

Clearly, our extension of the Nash bargaining solution to non-convex problems is not the only plausible one at the outset (e.g., see Kaneko For this reason, it is particularly important to support onr solution cooperatively by using models in which the bargaining process is desc explicitly. In this way? one is able to assess the circumstances un our cooperative solution can be applied.

We will consider the alternating offers model of bargaining introduce by Rubinstein [12] in which both players discount time at We will show that the set of stationary subgame p model converges to our proposed solution as t concerning time intervals between proposals vanishes.

We would like to conclude this introduction by mentioning t work owes many insights to Binmore’s [ 1 ] paper in which he offers a non- cooperative analysis of the Nash bargaining solution. Snch analysis is also

rried out by Binmore, Rubinstein, and Wolinsky [ 3] in the context of binstein’s model. The paper is organized as follows. Section 2 contains the axiomatic

model of bargaining together with the defmition and characterization of the cooperative solution. The non-cooperative model of bargaining is described in Section 37 which also provides a strategic aracterization of the solution. Section 4 contains a summary and concl ng remarks.

‘u>u, CEX, and ui#uz (i= 1,2) imply u$Eff(X), i.e., E&X) denotes the set of strong Pareto effkient points of X.

’ We are implicitly assuming the existence of u E X such that ui > di (i = I, 2). Qbviously, if no such point exists, the disagreement point is obtained.

268 MARIA .TOSfi HERRERO

2. THE AXIOMATIC MODEL

A two-person bargaining problem will be described by the following two objects: (i) a bargaining set XG R2 (hereafter Rn denotes the n-dimensional Euclidean space) which consists of all utility pairs that the players can achieve by reaching a binding agreement; and (ii) a pair of disagreement utilities dER2 that would result if the players failed to agree.

For simplicity, we will assume that d = (0,O) so that each bargaining problem is uniquely characterized by its corresponding bargaining set. The following assumption will also simplify our analysis.

Assumption S.1. X= {U 6 R2: u2< $.Ju~), u>O}, where the function tiX: [O, mJ --+ [O, MX] is continuous, strictly decreasing, and such that @JO) = MX and tix(rnx) = 0, where 0 c mx, &lx c a.

Hereafter we will omit the subscript x unless confusion arises and use the notation B = {XC R2: X satisfies S.1). Also, i,K(~i) (respectively $‘+(~i)) will denote the left (respectively right) derivative of $ at or. Finally, $‘(~i) will be used to refer to the set of real numbers which lie i.n between $Y(u~) ad V+ h 1.

It may be worth recalling that solutions to bargaining problems are meant to capture general optimization principles; consequently, one must allow for multiple values when proposing a solution to bargaining problems in B (since these need not be convex). So, in this section, we will first be concerned with defining a non-empty, set-valued solution function N: B + P?(R2). Second, we will show that N is an extension of the Nash bargaining solution and, in particular, that N(X) is a singleton whenever 2’ is convex. We will conclude this section by characterizing N axiomati- cally.

DEFINITION D. 1. Let XG B. Then N(X) # @ and T e N(X) if and only if TE Eff(X) and for any cx > 0 there exist P’, Q’e Eff(X) such that

(i) O-cQ~<Tl<P~

(ii) O-C ~~P”-Qz~j <u (iii) ll,Y-- Pm11 = IlK- Qal/ where S’ and R’ are as follows. Let

L(Pa, Q*) denote the line through P’ and QU. Then Sa = (0, sa) e L(P’, Q’) and R’= (F, 0) E L(P’, Qa).

This detinition is illustrated in Fig. 1.

2.1. Preliminary Remarks about N

Remark R.1. Note that TEN(X) if and only if there exists H a sub- tangent line to X through T such that l/,S - T/l = 11 R - T 11, where S and R

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“2 I

FIGURE 1

are the intersections of H with pi = 0 and uZ = Q3 respectively (see Fig. I ). (I-Iere, a subtangent line to X through TG X is simply a set {U G ?42 = TZ--y(~i-Ti)J in which -y~$‘(Ti).)

Remark R.2. From the previous remark, it follows that if X is convex then TEN(X) only if the line H supports the set X at T so that N(X) consists of the unique point which maximizes the product niul subject to the constraint UE X. That is, N is an extension of the Nash bargaining solution to non-convex problems.

2.2. AXIOMATIC CHARACTERIZATION OF

Let f: B -+ CP(R2) be such that @ #f(X) G X for an following axioms refer to several rationality principles to set-valued function f.

Axiom A.1 (Strict individual rationality). TE f(X) implies Ti > 0 (i= 1,~2).

Axiom A.2 (Efficiency). f(X) G Eff(X), where u > u together with v # u and u G Eff(X) imply u $ X.

Axiom A.3 (Invariance). f(AX) = Af(X) for any positive, linear func- tion A: RR2 -+ R2.

Axiom A.4 (Symmetry). If 37 is convex and symmetric, then Te f(X) implies Tr = TZ. (X is symmetric if and only if (pi ~ ti2) cz X implies cu23 %)~~I.

Axiom A.5 (Lower semi-continuity). If TEE and a singleton, then there exist sequences { Tnl and {.Y] Tn e f(P), $&( Tr) is a singleton, Tn -+ T and J? -+ X (in topology).

54214912.5

270 MARIA JO& HERRERO

&ium A.6 (Independence of locally irrelevant alternatives). If (al T E fV1 n WV, @I vKCTl) . is a singleton, and (c) $i(Tl) G $h(Tl), then T E f(Y).

Note that XG B implies the existence of u e X satisfying uj > 0 (i = 1, 2). Consequently, A.1 says that both players must profit from the existence of positive gains. Axiom A.2, on the other hand, requires all gains to be materialized (i.e., collective rationality). Also, the players’ utility functions will be assumed to be unique only up to positive and linear transforma- tions (recall that the origin of the functions is fixed at the disagreement point); so A.3 simply requires the solution to be invariant with respect to such transformations. Axiom A.4 is identical to Nash’s symmetry since it requires the solution to a symmetric problem to give equal utility to the players only if the bargaining set is convex. Axiom A.5 is a weak lower semicontinuity property. Finally, A.6 means that if a problem does not change too much near one of its solutions T (and T is still efficient in the new problem), then T remains a solution. This is stronger than (but similar to) requiring f to be Zocul (Kalai [6] delines f as being local if for any X, YE B such that there exists a neighborhood of TE f(X) on which X and Y coincide, then TE f( Y)) (see Section 3.1 for a more detailed analysis of A.6).

THEOREM 1. The function N is uniquely characterized by axioms A. l-A.6.

ProoJ Clearly, N satislies A.l-A.4. In order to check that N satislies A.5, let TEN(X) and suppose $‘(Tl) is not a singleton (i.e., X is not smooth at T). We will construct the sequences {I,,} and {T”} as follows. For any a > 0, let Ya = {U G X u1 > Q:, z+ > P;}, where the points Pa and Q’ together with S’ and RN are as in D.1. Next, deline Z’= {ueR2: (Qy, P;) < u < ,lRa + (1 -A) ,S=, some J G [O, l]}. Finally, for any integer a=~‘, let JP=(X\Ym)uP and T”=$(P’+Q’). Since Eff(J?) is flat between Pa and Qa, and because ii,!? - PEi] = ijRE - Qal/, it follows that @L(T;) is a singleton and T”E N(r). Also Tn -+ T and J? + X (in the Hausdorff topology) as PZ + co. So N satisfies A.5.

To see that N satislies A.6, first let TEN(X) n Eff( Y). So there exists H a subtangent line to X through T such that l\,S - Tjl = 11 R - Tji, where S and R are the intersections of H with u1 = 0 and u2 = 0, respectively. Also, suppose H is tangent to X through T and subtangent to Y through T. It follows from remark R.1 that TE N( Y) and so N satislies A.6.

Next, we show that N is the unique function satisfying A.l-A.6 in the domain B.

Suppose the function f B -+9(R2) satislies A.l-A.6 together with @#f(X)zX and let ~={uER~: O<U~+U~<~, uj>O}. Then A.2 and

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A.4 imply f(A) = {(i, ;)I= N(A). Next,.for any XE we want to show that TEN(X). First, A.l-A.2 imply Ti > 0 and TZ = $( ). Second, we consider two cases separately: (i) $'(Tl) is a singleton and (ii) $‘(T1) is not a singleton.

(i) If $‘( T1) is a singleton for TE f(X), then there exists H a tangent line to X through T such that jlS-Til =flljR-Tll for ~=(O,~)EH~

E H, and some /I > 0. In order to show that TE N(X) (i.e., fi = l), -+ R2 denote the linear transformation satisfying A,Y = (0, I ) and

, 0). Then A.3 implies ATef(AX). Also, AH is a tangent line to both AX and A through T; hence, by axiom A.6., TEE= {($i)j. “Thus, fi = 1 and so TEN(X).

(ii) If $‘(Ti) is not a singleton for Tgf(X), then the anal part (i) together with A.6 implies the existence of sequences (Y) G {T*} such that T” E N(P), t,bLn(T;) is a singleton, and T” -+ T, Y E -+ co. So there exists W a tangent line to Y through T” s IlP - Tnl/ = 11 R" - T*ll, where S” and R" are the intersections of til = 0 and u2 = 0. I,et S and R be limiting points of the sequences {El, respectively. Then H= {u: ~=X5’+(1-2) R a subta~ge~t line to X through T and Iis-- Tjl = IIR- Tli, where sections of H with u1 = 0 and u2 = 0. So, by Remar

3. THE NON-COOPERATIVE MODEL

This section contains a non-cooperative defense of the proposed exten- sion of the Nash bargaining solution. Roughly? for any XEB, one can show that the set of equilibria of the alternating offers bargaining game in which both players discount time at the same rate converges to the set N(X) as the bargaining frictions go to zero. Following is a description of Rubinstein’s [ 121 alternating offers model.

Two players, labelled 1 and 2, alternate in making proposals in X E times 1= 0, 1, 2, . . . . with player 1 proposing at t = 0. After a proposal is made at time t, the responder can either accept or reject If she accepts the game concludes with the implementation of the accepted proposal. IIowever, a rejection leads to her becoming the proposer at time t + 1.

An early agreement is encouraged in this model by the fact tha players attach probability 1 - 8 E (0, 1) to an exogenous breakdown i negotiations and that such a breakdown leads to t Therefore, an agreement on u E X at time t gives player i a payotf equal to lYuj.

By considering stationary subgame perfection for t e analysis of this game we obtain an analogue to Rubinstein’s [I2] characterization theorem

212 MARIA JOSkHERRERO

for bargaining problems in which X need not be convex (see Binmore [2] for a complete characterization of the set of subgame perfect equilibria, i.e., without the stationary requirement). Next, we formally detine stationary subgame perfect equilibria for the alternating offers game.

DEFINITION D.2. A stationary subgame perfect equilibrium (SSPE) for the above model is a pair (e, j’) E X2 such that the following strategy prolile is subgame perfect:

At time 0 and at every even time:

Player 1 proposes e = (ei, eZ). Player 2 accepts any proposal in which she is offered at least eZ

At every odd time:

Player 2 proposes (fi, f*). Player 1 accepts any proposal in which she is offered at least fi.

Hereafter, let J?(& X) denote the set of SSPE’s.

THEOREM 2 (Rubinstein’s [12]). ,?~?(a, 3’) = { (ei, e2, j-i, f*): e2 = +(ei) =W~~~l~,.fl=~~l~ andf2=WlJ).

Proof Consider any SSPE (ei, eZ,i1,f2). Observe that (a) e2= $(ei) because e2 > t&i) implies e$X and player 1 must increase her demand whenever e2 < t,k(ei); (b) similarly, f2 = $(fi); (c) e2 = 6j2, or else player 2 is using a suboptimal acceptance rule; and (d) fi = 6ei, or else player 1 is using a suboptimal acceptance rule. Hence, e2 = 6f2 = d$(fi) = J$(dei) = bYelI.

Conversely, suppose $(ei) = be(bei). Then (ei, &j(hei), 8ei, $(dei)), together with the strategy prolile in D.2, can be seen to be a SSPE, 1

The next two results will be used in the main characterization theorem. In these corollaries, kr = {U e R’: 0 < g < ~(0, s) + (l- y)(r, O), some JJ c R 1 for s, r > 0.

COROLLARY 1. E(& LV) = {(r(l + 6)-i, ki(l + b)-‘, r8(1 + 6)-l, 3(1+8)-l)}.

ProojI It follows immediately from Theorem 2. 1

COROLLARY 2. (e, f)~E(i?, X) ij” und only $ {(e, f)J = I??(& A”) for some s, r > 0.

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Proof Let e,f~Eff(X) be such that e # f (otherwise (e,f) $E(& X)) and let H= {u: ~=ye+(l-y)L some ~GR} denote the line through the points c and j Then

are such that (Q, s)e H and (r, 0)~ H. So, since ijbV,(~i) = $.Ju~) for u1 G {el, fi 1, the required result follows from Theorem 2.

Let Ei(8, X) = {e: (e, f) E E(6, X), some f 1 and &(a, X) = {f: (e, f) e E(6, X), some ej.

THEOREM 3. Ej(h, X) converges to N(X) (in the Huusdorff topology) as is--+ I.

hoof We proceed in two steps. First, we show that if a sequence {(e’, f’)) satisfies (e’, f’) CZ,~(C!~, X) and e* + T, f’-+ T as 8 + 1, then Te N(X). Second, we show that if Te N(X), then for any u > 0 there exists 8 cz (0, I) and (e’,f’) E E(& X) such that es -+ T, f’ -+ T, md 6 -+ 1 2s fY. -+ Q.

Step 1. Consider any pair (e’, f ‘) E E(& X) and suppose e’ + T as 8 -+ 1. Note that f’ + T as C? + 1. Also, TE Eff(X) and, by Theorem 2,

Therefore? H = {u: uz = 2Tz - ( TJTl) ul, some ui 6 1 is a subtangent line to X through T such that /IS’- Tll = /jR- Tjl for S=(t)> 2Tz)~ R = (2Tl, 0) E H. So T E N(X).

Step 2. Let Te N(X). Then for any CY > 0 there exists Pa, Qa, S RN as in D.1. Next, defme 8 as follows: (l+8)//S-PXj/= i/S’- Since R; > P; > Q; >O, it follows that 6 E (0, I). Thus, by D~I and Corollaries l-2, (Pz, Q’) e E(J, X) and P” 4 T, Qz -+ T as 8 -+ 1 (i.e., as Lx+O). 1

3.1. Irrelevant Alternatives In this section we compare Axiom A.6 with local versions of two well

known axioms, namely Nash’s independence of irrelevant alternatives and Myerson and Thompson’s [9] inpendence of undominated alternatives.

observe that by restricting our attention to solution functions which are strongly individually rational, Pareto eflicient? and invariant, we immediately identify two types of alternatives which are (globally) irrele

274 MARIA JOSk HERRERO

vant, namely non-individually rational points and Pareto inefficient points. Such utility pairs are irrelevant in the sense that their deletion or addition does not affect the solution of a given problem.

Besides these two types of utility pairs, additional alternatives may also be irrelevant if the solution satisfies Nash’s independence of irrelevant alter- natives or independence of undominated alternatives. These axioms are as follows.

Axiom I.I.A. (Nash’s independence of irrelevant alternatives). TE f(X) n Y and YG X imply TE f( Y).

Axiom I.U.A. (Independence of undominated alternatives). TE f(X) n Eff( Y) and X5 Y imply TE f( Y).

The hypotheses Tef(X)n Y and T~f(x)n Eff(Y) respectively, are needed here to fix the scale of the players’ utility representations (recall that the origin is tixed at the disagreement point). This is because if XG Y and Y s X represent changes of scales (rather than a reduction and an improve- ment of the bargaining opportunities, respectively), then invariance (i.e., A.3) alone implies the conclusions of Axioms I.I.A. and I.U.A.

Axiom I.I.A. together with A.1, A.3, and A.4 characterizes Nash’s solu- tion (see Roth [ 111). Nevertheless, our extension N does not satisfy I.I.A. given that the problem need not be convex. To see this, consider the sets X= {u: O<z+<max[2-2u1, l-iul], O<ui<2j and Y={K O<uZ< 2 - 2ar, 0 < u1 < 11 Routine computations show that ($, i) e N(X) n Y and YcX but ($, +)&N(Y) = {($, l)}. However, N satisfies I.I.A. on the smooth subclass of B which motivates the following axiom on which A.6 is based.

Axiom A.7 (Local independence of irrelevant contractions). If (a) TE f(X) n Y, (b) $k( Tl) is a singleton, and (c) YG X, then TE f( Y).

It is well known that Nash’s solution only satislies Axiom I.U.A. on the smooth .subclass of B and so N does not satisfy Axiom I.U.A. either. This motivates the following axiom on which A.6 is also based.

Axiom A.8 (Local independence of irrelevant expansions). If (a) Tczf(X) n Eff( Y), (b) t,b!JTl) is a singleton, and (c) XL Y, then TE f( Y).

It is easy to check that A.7 and A.8 together imply and are implied by A.6. So this later axiom can be replaced by the former two axioms without changing any of our results.

We conclude this section with a brief comment about the meaning of the role played by A.6 in the equivalence result. Roth [ 111 and Binmore [ 1] have suggested that Nash’s axioms can be regarded as a specification of the

NASH PROGRAM 275

bargaining procedure and the equilibrium notion being implicitly modeled. The analysis of this paper confirms this interpretation by showing that there is a strong relationship between A.6 in the class of solution functions satisfying A.l-A.5 and stationary subgame perfection in the alternating offers model. This is because, for any positive g, one can think of the points P’ and Q’ of D.1 as being the credible (equilibrium) threats used by the players in order to sustain their claims to TEN(X) as E + 0. The referre relationship, however, is not true of non-stationary subgame perfect equilibria, whose threats are not close to the solution they implement and therefore cannot be captured by Axiom A.6.

3.2. Indeterminacy of the Agenda

This paper also illustrates the well known result that equilibrium need not be unique once one abandons the assumptions on which ~ubinstei~‘s [ 121 result is based, such as complete information and the bilateral nature of the problem. EIowever, unlike incomplete information models of bilateral bargaining, the multiplicity of equilibria here cannot be attributed to off equilibrium conjectures. Also, unlike I-Ierrero’s [4] three-player version of the alternating offers model, the multiplicity of equilibria here does not disappear when attention is restricted to stationary subgame perfection. In the present paper, the multiphcity of equilibria which arises because of non-convexities of the bargaining set can be attributed to the indeterminacy of the agenda, so that the non-uniqueness result is illustrated from a different perspective.

TQ see this, consider a process in which the two players first alternate in proposing an agenda in A(X) = {B G B: B s X, E(c’$ B) is a singleton 1 and3 after an agreement has been reached on an agenda & they alternate in proposing an outcome in B until a final agreement is reached or else until one or both players decide to reopen the negotiations over the agendas

By requiring stationary subgame perfection throughout the process, one can show that the set of equilibrium agendas consi subsets L3ef G X satisfying {(e, f)l= E(&, Bef) 5 E(J, X) (and since E(8> X) is generically tinite,4 the indeterminacy is generically reduced to a finite rmmber of agendas.).

In certain situations, the indeterminacy of the agenda can be elimi by appealing to Nash’s independence of irrelevant alternatives (i.e., I For example, suppose {T l= N(Co(X)) YZ N(X), where Co(X) denotes the convex hull of X. In this case, any agenda B G X satisfying N(B) = {T 1 will be a natural focal point for the players to favor, based on I.I.A~? so that the tinal outcome will be unique. IIowever, if T# N(X)? then the indeterminacy car-mot be resolved by appealing to I.I.A. This is because T $ X and so T

’ N(X) is finite whenever the set Eff(X) n {u: uLu2 = kj is finite.

276 MARIA JOShHERRERO

cannot be implemented by the players alone (i.e., without the intervention of a third party with the ability to transfer utility from one of the players to the other).

4. SUMMARY AND CONCLUDING REMARKS

In this paper we have offered an extension of Nash’s solution to non- convex bargaining problems and provided two characterizations of it, namely an axiomatic and a non-cooperative. The former gives a set of general principles on which the solution is based and the later offers a well founded strategic analysis of the range of applicability of the solution.

In relation to the strategic characterization, we should mention that the alternating offers model is only an example of the kind of non-cooperative models which implement the solution proposed here (Binmore’s [2] model with random selection of proposers is another example). What is essential for the characterization result are the following three features: (a) that no take it or leave it proposals are possible, (b) that the process is symmetric except JOY the first mover advantage (which vanishes as the frictions go to zero) and (c) that stationary subgame perfection is used for the analysis.

Following Roth [ 111, one can axiomatically characterize an asymmetric version of the solution by dropping the axiom of symmetry. This is obtained by replacing condition j[P-P’ij = i/R=- Qa/l in D.1 (ii) by /IS’ - P’il = PI/P - Q’l/, where p is a positive real number. Moreover, following Binmore [ 1 ] and Binmore et aZ. [3], one can characterize this assymmetric solution non-cooperatively by considering differences in the parties’ beliefs in respect to the probability of a breakdown. That is, in the alternating offers model, let 1 - ~5~ represent the probability assigned by player i to an exogenous breakdown and let Ji = eP’lu for ri, e > 0. Then, by allowing r2r;i to vary, one can show that, as e + 0, the set of equilibria of the model converges to the asymmetric version of our solution provided p=T&.

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