Single-Valued Nash Bargaining Solutions

with Non-Convexity∗

Cheng-Zhong Qin† Guofu Tan‡ Adam Chi Leung Wong§

May 30, 2018

Abstract

We consider two-player bargaining problems with compact star-shaped

choice sets arising from a class of economic environments. We characterize

single-valued solutions satisfying the Nash axioms on this class of bargaining

problems. Our results show that there are exactly two Nash solutions with

each being a dictatorial (in favor of one player) selection of Nash product

maximizers. We also provide an extensive form for implementing these two

Nash solutions.

Keywords Bargaining problem · Non-convexity · Nash solution · Nash product

JEL Classification C78 · D21 · D43

∗The authors gratefully acknowledge helpful comments from Youngsub Chun, Pradeep Dubey,Mamoru Kaneko, Abraham Neyman, Hans Peters, Yair Tauman, Shmuel Zamir, Yongsheng Xu,Junjie Zhou, and seminar and conference participants at the Hong Kong University of Scienceand Technology, the University of California, Santa Barbara, 2015 International Conference onIndustrial Organization at Zhejiang University, June 17-18, Hangzhou China, 2015 Asian GameTheory Conference, August 24-26, Tokyo, and the 28th International Conference On Game Theory,Stony Brook, NY, July 17-21, 2017.†Department of Economics, University of California, Santa Barbara, CA 93106.

qin@econ.ucsb.edu.‡Department of Economics, University of Southern California, Los Angeles, CA 90089-0253.

guofutan@usc.edu.§Department of Economics, Lingnan University, Hong Kong. adamwong@ln.edu.hk.

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1 Introduction

Bargaining theory in the spirit of the seminal work of Nash (1950, 1953) postulates

that a group of players face a set of feasible outcomes in payoff space, and the

implementation of an outcome requires unanimous agreement among the players. In

cases of disagreement, the players end up getting some pre-specified outcome known

as the disagreement point or the threat point. A bargaining problem in the sense

of Nash is thus represented by a choice set along with a disagreement point in the

payoff space.

Nash considered bargaining solutions as rules that assign a feasible payoff allocation

to each problem in a class of bargaining problems. He postulated a set of axioms that

are deemed to be natural for a solution to satisfy. They are Pareto Optimality (PO),

Symmetry (SYM), Invariance to Affine Transformations (INV), and Independence

of Irrelevant Alternatives (IIA). It is remarkable that Nash showed that these axioms

jointly determine a unique solution, known as the Nash bargaining solution (Nash

solution in short), on the class of compact and convex problems. Furthermore, the

Nash solution assigns to each problem the unique maximizer of the (symmetric)

Nash product over the choice set of the problem. When SYM is removed, Kalai

(1977) showed that an asymmetric Nash solution, unique up to (henceforth unique)

the specification of a solution for Divide-the-Dollar problem, is characterized by the

remaining axioms on the class of compact convex problems.1 The Nash bargaining

theory has become one of the most fruitful paradigms in economics.

In applications, the choice set is not always directly given but often derived from

more primitive data, and as such, it needs not be convex. Indeed, non-convexity often

arises in duopoly, employer-employee, and principal-agent bargaining problems.2

1A payoff allocation for Divide-the-Dollar problem has been customarily interpreted asrepresenting players’ bargaining powers (this interpretation can be traced to Shubik (1959, p. 50)).Given players’ bargaining powers, the asymmetric Nash solution assigns to each problem the uniquemaximizer of the generalized Nash product – the Nash product weighted by the bargaining powers.

2The non-convexity of duopoly bargaining problems with asymmetric constant marginal costsand concave monopoly profit functions has been well recognized in the industrial organizationliterature due to work of Bishop (1960, p. 948), Schmalensee (1987, p. 354–356), Tirole (1988, p. 242,271), and Qin, Shi and Tan (2015), among others. As discovered in Aoki (1980), McDonald andSolow (1981), and Miyazaki (1984), among others, non-convexity also arises in employer-employeebargaining problems. It is shown in Hougaard and Tvede (2003) that bargaining problems arisingfrom the principal-agent problems need not be convex unless randomized contracts are allowed.

2

The prevalence of non-convexity raises serious theoretical and applied issues about

the Nash bargaining theory. For instance, there can be multiple Nash product

maximizers for a non-convex problem. Which maximizers can be selected as single-valued

solutions satisfying the Nash axioms and whether the solutions can be implemented

via non-cooperative games are among the questions that naturally arise.

We analyze these issues relative to bargaining problems arising from a class of

economic bargaining environments. These problems are compact and star-shaped,

in the sense that any convex combination of a feasible choice and the disagreement

point is also feasible. Both convex and comprehensive (relative to the disagreement

point) problems are star-shaped but not conversely.

Our analysis proceeds as follows. First, we characterize all possible two-player

single-valued Nash solutions satisfying INV, IIA, and Strict Individual Rationality

(SIR) on the class of regular and star-shaped bargaining problems.3 There are exactly

two Nash solutions with each being a dictatorial (in favor of one player) selection

of Nash product maximizers up to specifications of bargaining powers.4 This result

implies that the Nash solution for a bargaining problem is unique whenever the

corresponding Nash product maximizer of the problem is unique. More surprisingly,

our result also implies that if a problem has more than two Nash product maximizers,

those in the “middle” cannot be selected as solutions for the problem. An immediate

consequence of this property is that the SYM axiom is no longer compatible with

the other Nash axioms. This exact structure of Nash solutions greatly enhances the

applicability of the Nash bargaining theory in the presence of non-convexity.5

Next, we formulate an extensive form based on the representing economic environments

of bargaining problems and show that the Nash solutions described above can be

implemented as subgame perfect equilibrium outcomes. The extensive form has a

nomination phase and a trial phase. The basic building block of the trial phase is

the two-period alternating offer game (i.e., the two-period version of Stahl’s (1972)

game). The trial phase endogenizes the delay time between two offers by letting the

3By a result of Roth (1977, p. 65), PO is implied by IIA, INV, and SIR, and can be replaced bySIR when INV and IIA are imposed. Since our class of problems satisfies all the assumptions inRoth (1977) except for the convexity, we will consider IIA, INV, and SIR as the Nash axioms.

4We focus on the case of two players for simplicity. Our approach can be extended to allow formore than two players but the solutions would involve greater multiplicity.

5We also show that our characterization holds for a larger class of admissible domains ofbargaining problems.

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players sequentially bid in terms of delay time for being the first to make offer.6 In

the nomination phase, the players simultaneously nominate candidate alternatives.

Each player can only nominate one feasible alternative, and only the status quo and

the nominated candidate choices are the alternatives on the table in the subsequent

trial phase. We show that the resulting nomination-trial game implements the Nash

solution in favor of the player who submits bid first in the trial phase. As usual,

the differential discount factors of the players provide a measure of their relative

bargaining powers.

Our paper contributes to bargaining theory by characterizing the exact Nash

solutions under the Nash axioms and by offering an extensive form for implementing

the solutions.

The rest of the paper is organized as follows. Section 2 presents preliminaries.

Section 3 contains our characterization of two-player bargaining solutions satisfying

INV, IIA, and SIR on the class of compact and star-shaped bargaining problems.

Section 4 presents results on implementation of Nash solutions in Section 3. Section 5

links our analysis with the relevant literature and Section 6 concludes. The proofs

we do not provide in the text are in an Appendix.

2 Preliminaries

A two-player bargaining problem (or simply a bargaining problem) is composed of a

choice set S ⊆ R2 of feasible payoff allocations the players can jointly achieve with

agreement, and a disagreement point (or threat point) d ∈ S the players end up

getting in case of disagreement. The bargaining problem is comprehensive (relative

to the disagreement point) if u ∈ S and d ≤ v ≤ u imply v ∈ S. Bargaining

problem (S◦,0) where S◦ = {u ∈ R2+ : u1 + u2 ≤ 1} and 0 = (0, 0) is known as the

Divide-the-Dollar problem.

Definition 1. A bargaining problem (S, d) is regular if S is compact and there

exists u ∈ S such that u � d. It is star-shaped if for all u ∈ S and p ∈ [0, 1],

6If the two players submit bid simultaneously, this trial phase can be regarded as a variant ofMoulin’s (1984) game, which implements the Kalai-Smorodinsky (1975) solution with convexity.It does not implement a single-valued solution in the presence of non-convexity, unless sequentialbidding is considered.

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pu+ (1− p)d ∈ S.

Being star-shaped means a kind of free disposability of players’ payoffs that

is directional toward the disagreement point; hence, it is a weaker kind of free

disposability compared to what is implied by comprehensiveness. Alternatively,

being star-shaped can be understood as allowing for mixtures between individually

rational feasible payoff allocations and the disagreement point only. Let B denote

the class of two-player regular and star-shaped bargaining problems.7

Definition 2. A bargaining solution on B is a mapping f : B → R2 such that

f(S, d) = (f1(S, d), f2(S, d)) ∈ S for all (S, d) ∈ B.

By defining a bargaining solution on a class of bargaining problems, it is possible

to impose, among other properties, consistencies on payoff allocations across different

problems. Indeed, some of Nash axioms are specifications of consistencies via affine

transformations. A positive affine transformation for player i’s payoff is a mapping

τi : R→ R such that for some two real numbers ai > 0 and bi, τi(ui) = aiui + bi for

all ui ∈ R. Given τ1 and τ2, set τ(u) = (τ1(u1), τ2(u2)) for all u ∈ R2. We can now

state Nash axioms.8

Invariance to Positive Affine Transformations (INV): For any (S, d) ∈ B and

for any positive affine transformation τ : R2 → R2, f(τ(S), τ(d)) = τ(f(S, d)).

Independence of Irrelevant Alternatives (IIA): For any (S, d), (T, d) ∈ B with

S ⊆ T , f(T, d) ∈ S implies f(S, d) = f(T, d).

Strict Individual Rationality (SIR): For any (S, d) ∈ B, f(S, d)� d.

The INV is generally understood as a unit-freeness property. With this understanding,

it is natural to consistently require the property to be satisfied universally on the

whole class of problems. In addition, as elaborated in Remark 3 on p. 11, in the

economic environments that generate our class of bargaining problems, players’

7Here we focus on the class B of regular star-shaped bargaining problems. We show in Section 4that this class of problems are generated by a natural economic environment, upon which weprovide our implementation result. However, our main characterization result in Section 3 holdsfor alternative domain specifications, a point to be made explicit in Subsection 4.2.

8See Footnote 3 for reasons why the axiom of PO is not needed.

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preferences over outcomes 〈x, t〉 reached by agreement to choose alternative x at

time t are represented by a discount rate and an instantaneous utility function.

Positive affine transformations of the instantaneous utility functions do not change

the preferences. As such, it is natural to require that the INV axiom be satisfied on

the whole class. It turns out that the global imposition of the INV axiom enables us

to pin down a tight structure of Nash solutions.

3 Characterization of Nash bargaining solutions

3.1 Main result

Our main result in this section is the following characterization of single-valued

bargaining solutions on B that satisfy Nash axioms.

Theorem 1. A bargaining solution f on B satisfies INV, IIA, and SIR if and only

if there exists i ∈ {1, 2} such that for any (S, d) ∈ B,

{f(S, d)} = argmaxu∈Σ(S,d)

ui

where

Σ(S, d) ≡ argmaxu∈S,u≥d

(u1 − d1)α1 (u2 − d2)α2 ,

α ≡ f(S◦,0)� 0.

Theorem 1 implies that, up to the bargaining powers specified by the payoff

outcome for (S◦,0), there are exactly two solutions with each being a dictatorial

selection of Nash product maximizers. If Nash product maximizer for a bargaining

problem is unique, then the two solutions coincide for this problem. If there are

more than two Nash product maximizers, however, all non-polar maximizers cannot

be selected as Nash solutions. The example below provides an illustration.

Example 1. Let f be a bargaining solution on B satisfying INV, IIA, and SIR.

Suppose f(S◦,0) = (12, 1

2) so that the two players have equal bargaining powers.

Consider the choice set S = {u ∈ R2+ : min{2u1 + 1

2u2, u1 + u2,

12u1 + 2u2} ≤ 2} and

disagreement point d = 0. Then, the bargaining problem (S,0) has three maximizers

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Figure 1: An illustration of Example 1

of (symmetric) Nash product u1u2, which are u = (12, 2), v = (1, 1), and w = (2, 1

2).

However, the middle one, v, cannot be Nash solution for problem (S,0). To see this,

suppose on the contrary that f(S,0) = v. Let T be the comprehensive hull of {u, v}and T ′ the comprehensive hull of {v, w}, as indicated in Figure 1. Then IIA implies

f(T,0) = f(T ′,0) = v. However, by INV, f(T,0) = v implies f(T ′,0) = w, which

contradicts f(T ′,0) = v.

Remark 1. Given a solution f satisfying the axioms and given the bargaining powers

revealed by f(S◦,0), it is easy to identify whose payoff should be maximized for the

selection of Nash product maximizer. In Example 1, in which f(S◦,0) = (12, 1

2),

Theorem 1 holds with i = 1 if f(T,0) = (1, 1) and i = 2 if f(T,0) = (12, 2). More

generally, suppose f(S◦,0) = (α1, α2) � 0. Let T = {u ∈ R2+ : u ≤ (1, 1) or u ≤

(2−α2/α1 , 2)}. Then Theorem 1 holds with i = 1 if f(T,0) = (1, 1) and i = 2 if

f(T,0) = (2−α2/α1 , 2).

Remark 2. The SYM axiom for Nash solution states that f1(S, d) = f2(S, d) whenever

(S, d) is symmetric (i.e., whenever d1 = d2 and (u1, u2) ∈ S if and only if (u2, u1) ∈S). However, since our domain contains non-convex problems, SYM is no longer

compatible with the other axioms, in that none of the two single-valued solutions for

non-convex symmetric problems are guaranteed to be symmetric. Example 1 provides

an illustration.

3.2 Proof

Function f characterized in Theorem 1 is well defined and satisfies INV, IIA, and SIR.

Thus, it only remains to prove the necessity. To this end, we begin by establishing a

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few auxiliary results. We first show that INV, IIA, and SIR together imply Pareto

optimality. The method of proving Pareto optimality as implied by INV, IIA, and

SIR in Roth (1977) for convex problems can be adapted to star-shaped problems.

Given S ⊆ R2, we let Par(S) denote the strict Pareto frontier of S and com(S) =

(S −R2+) ∩R2

+ the comprehensive hull of S. When S is finite such as S = {u, v, w},we may write Par({u, v, w}) and com({u, v, w}) as Par(u, v, w) and com(u, v, w).

Lemma 1. Let f be a bargaining solution on B satisfying INV, IIA, and SIR. Then

f(S, d) ∈ Par(S) for all (S, d) ∈ B.

Lemma 1 implies in particular that, under the three axioms, f(com(S),0) ∈ Sfor S ⊆ R2

++. Now we use solution f to generate a binary relation � on R2++ by

u � v iff u 6= v and f(com(u, v),0) = u.

Lemma 2. Let � be generated by bargaining solution f on B satisfying INV, IIA,

and SIR. Then, (i) for any u, v ∈ R2++, exactly one of u � v, v � u, and u = v

holds, (ii) for any u, v, w ∈ R2++, u � v and v � w imply u � w, and (iii) for any

u, v ∈ R2++, u ≥ v and u 6= v imply u � v.

Lemma 2 states that the binary relation � generated by a bargaining solution f

satisfying the axioms is a strongly monotone strict linear order. It can be further

shown that f(S,0) is the unique maximizer of � on S ∩ R2++.

Lemma 3. Let � be generated by bargaining solution f on B satisfying INV, IIA,

and SIR. Then, for (S,0) ∈ B, we have f(S,0) ∈ S ∩ R2++ and

f(S,0) � u for any u ∈ (S ∩ R2++)\{f(S,0)}.

Lemma 3 shows that the order � summarizes all the information about f (·,0).

By INV, f (·,0) in turn summarizes all the information about bargaining solution

f . It follows that we can analyze f by analyzing �. To introduce our next auxiliary

result, let log u ≡ (log u1, log u2) for u ∈ R2++.

Lemma 4. Let � be generated by bargaining solution f on B satisfying INV, IIA,

and SIR. Then, for λ ∈ R and u0, u1, v0, v1 ∈ R2++ with u1 � u0 and log v1− log v0 =

λ (log u1 − log u0), we have v1 � v0 if λ > 0 and v0 � v1 if λ < 0.

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We are now ready to prove Theorem 1.

Proof of Theorem 1. As mentioned before, we only need to prove the necessity. To

this end, let bargaining solution f on B that satisfies INV, IIA, and SIR be given

and let � be generated by f . Let α ≡ f(S◦,0). In particular, we have α � 0 from

SIR and α1 + α2 = 1 from Lemma 1. We complete the rest of the proof in three

steps.

Step 1: For any u ∈ R2++ with α · u < 1, 1 ≡ (1, 1) � u.

Note that INV together with the definition of α implies

f({u ∈ R2+ : α · u ≤ 1},0) = 1.

For any u ∈ R2++ with α ·u < 1, com(1, u) ⊆ {u ∈ R2

+ : α · u ≤ 1}; hence, IIA implies

f(com(1, u),0) = 1 or equivalently, 1 � u.

Step 2: u � v for u, v ∈ R2++ such that uα1

1 uα22 > vα1

1 vα22 .

Pick any u, v ∈ R2++ with uα1

1 uα22 > vα1

1 vα22 . Since

limλ→0

(α1 (v1/u1)λ + α2 (v2/u2)λ

)= 1

and

limλ→0

d

dλ

(α1

(v1

u1

)λ+ α2

(v2

u2

)λ)

= limλ→0

(α1

(v1

u1

)λlog

(v1

u1

)+ α2

(v2

u2

)λlog

(v2

u2

))= log (vα1

1 vα22 )− log (uα1

1 uα22 ) < 0,

we have α1 (v1/u1)λ + α2 (v2/u2)λ < 1 for all small positive scalar λ. From step 1,

1 �(

(v1/u1)λ , (v2/u2)λ)

when λ > 0 is small enough. Thus, by Lemma 4, u � v.

Step 3: For any u, v ∈ R2++ with uα1

1 uα22 = vα1

1 vα22 and u1 > v1, u � v if and only

if (eα2 , e−α1) � 1.

Note that the vectors log u− log v and log(eα2 , e−α1)− log 1 are both pointing to

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the same direction as (α2,−α1). Hence, a direct application of Lemma 4 establishes

the claim.

Combining Step 2 and Step 3 with Lemma 3 shows that Theorem 1 holds with

i = 1 if (eα2 , e−α1) � 1 and with i = 2 if 1 � (eα2 , e−α1). �

Theorem 1 completely characterizes the exact single-valued Nash solutions on

our broad class of problems. The exact structure of Nash solutions is valuable for

applications in the presence of non-convexity.

4 Implementation of Nash solutions

We begin this section by identifying economic environments resulting in compact and

star-shaped bargaining problems, in which agents have common discount rate. Based

on these environments, we then construct non-cooperative games with a common

extensive form that can implement Nash solutions characterized in Section 3. Finally,

we make an extension to allow for asymmetries in discount rates.

4.1 Common discount rate

Two players, 1 and 2, are endowed with a compact set X (in a topological space) of

alternatives that they can jointly achieve with agreement and a status quo q ∈ X,

which the players will end up getting in case of disagreement. We assume that no

randomization device is feasible for the players. However, delays are possible and

time is continuous. A feasible bargaining outcome is a pair 〈x, t〉 with x ∈ X and

t ∈ [0,∞], which is understood as an agreement for choosing alternative x has been

reached after a delay of time t. The preferences of player i ∈ {1, 2} over outcomes

〈x, t〉 are represented by the utility function

e−rtUi(x) + (1− e−rt)Ui(q),

where r > 0 is the common instantaneous discount rate and Ui : X → R is player i’s

instantaneous utility function. We will relax the common discount rate assumption

in the next subsection. Note that for x ∈ X, 〈x,∞〉 is equivalent to the outcome

representing “bargaining breakdown” or “staying at the status quo forever”. With a

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little abuse of notation, denote this outcome by q. We assume that U is continuous

and U(x)� U(q) for some x ∈ X , where U(x) = (U1(x), U2(x)).

Remark 3. Player i’s preferences over feasible bargaining outcomes are represented

by a discount rate r and an instantaneous utility function Ui. Consider another

instantaneous utility function V . It can be proved that (r, Vi) represents the same

preferences as (r, Ui) does if and only if there exist αi, α′i ∈ R and βi, β

′i ∈ R++ such

that, for all x ∈ X,

Vi(x) =

{αi + βiUi(x) if Ui(x) ≥ Ui(q)

α′i + β′iUi(x) if Ui(x) ≤ Ui(q).

Note that there are only three degrees of freedom for choosing αi, α′i, βi, β

′i since αi +

βiUi(q) = α′i + β′iUi(q).

Remark 4. An alternative in X can also be interpreted as a physical outcome that

yields a flow utility at every instant of time for each player. Let the flow utility

for player i generated by alternative x ∈ X be given by a continuous function ui(x).

Reinterpreting a feasible outcome 〈x, t〉 as representing the choice to stay at the status

quo q until time t and then switch to alternative x after time t, player i’s payoff

generated by outcome 〈x, t〉 will be∫ t

0

e−rτ ui(q)dτ +

∫ ∞t

e−rτ ui(x)dτ = e−rtUi(x) +(1− e−rt

)Ui(q)

where Ui(·) ≡ ui(·)/r.

A quadruple (X, q, r, U) that satisfies the above assumptions is called a bargaining

environment. Given a bargaining environment (X, q, r, U), the set of feasible payoff

allocations S and disagreement point d in payoff space are:

S ≡{e−rtU(x) +

(1− e−rt

)U(q) : x ∈ X and t ∈ [0,∞]

}= {pU(x) + (1− p)U(q) : x ∈ X and p ∈ [0, 1]} ,

d ≡ U(q).

Note that (S, d) is regular and star-shaped. Conversely, any regular and star-shaped

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bargaining problem (S, d) can be derived from some bargaining environment (X, q, r, U).

Given a bargaining environment (X, q, r, U), we now proceed to construct a

non-cooperative game in extensive form for implementing its Nash bargaining solutions.

Without loss of generality, we assume d ≡ U(q) = 0. Let δ = e−r ∈ (0, 1) be the

discount factor per unit time. Next, let Γ(i, t, Y ) denote the two-period alternating

offer game as in Stahl (1972), where i ∈ {1, 2} is the first mover, t ∈ [0,∞] is

the length of time delay between two offers, and Y ⊆ X with q ∈ Y is the set

of alternatives on the table. Negotiation takes place according to the following

nomination-trial game.

Phase 1: The Nomination Phase

• Players nominate candidate alternatives in X simultaneously. Denote player

1’s and 2’s nominations by x1 and x2 respectively.

Phase 2: The Trial Phase

• Player 1 chooses a delay time t1 ∈ [0,∞].

• Then player 2 observes player 1’s choice and chooses another delay time t2 ∈[0,∞] with t2 6= t1.

• If t1 < t2, then the two players play Γ(1, t1+t22, {x1, x2, q}).

• If t2 < t1, then the two players play Γ(2, t1+t22, {x1, x2, q}).

A natural solution concept for the preceding game is pure strategy subgame perfect

equilibrium (hereafter SPE).

Player 1 has a timing advantage in the preceding nomination-trial game. We

show in Theorem 2 below that this sequential bargaining game implements the

Nash bargaining solution in favor of player 1. By reversing the timing order in

the trial phase, a different nomination-trial game will be resulted which implements

the bargaining solution in favor of player 2.

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Theorem 2. Given any bargaining environment (X, q, r, U), the nomination-trial

game has at least one SPE and all SPE payoff allocations are given by the Nash

bargaining solution characterized in Theorem 1 with i = 1 and α = f(S◦,0) = (12, 1

2).

That is, the nomination-trial game SPE-implements the Nash bargaining solution in

favor of player 1 with equal bargaining powers.

Osborne and Rubinstein (1990) describe approximate implementations by various

strategic models, including the alternating offer game with the risk of breakdown

and with time preference due to Binmore, Rubinstein, and Wolinsky (1986), and

perturbed Nash demand game due to Nash (1953). These strategic models were

primarily constructed for dealing with convex bargaining problems. They do not

implement single-valued bargaining solutions when allowing for non-convexity. But

they are all consistent with our characterization of Nash solutions: each of the two

bargaining solutions in Theorem 1 is a selection of the correspondences implemented

by those strategic models.

4.2 Asymmetric discount rates

We now relax the common discount rate assumption. Let ri > 0 be player i’s

discount rate and set r ≡ (r1, r2). According to the classical analysis of Fishburn and

Rubinstein (1982), with constant discount rate, the assumption r1 = r2 is without

loss of generality because any asymmetry in the discount rates can be incorporated

in the instantaneous utility functions. More specifically, let player i’s preferences be

represented by the utility function e−ritUi(x). For any r > 0, player i’s preferences

can also be represented by the utility function

[e−ritUi(x)

]r/ri = e−rt (Ui(x))r/ri = e−rtUi(x),

where

Ui(x) ≡ (Ui(x))r/ri .

It follows that r can be regarded as the common discount rate, if we let Ui be player

i’s instantaneous utility function. Assuming r1 = r2 is merely a normalization.

However, there are at least two reasons why one might explicitly allow for asymmetric

discount rates. First, for the bargaining problem in question, there might be a natural

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way to choose the instantaneous utility functions. For example, the players might be

two firms bargaining over how to coordinate their actions. In this case, the natural

way to represent the firms’ preferences is to use the present values of profit streams,

possibly with asymmetric discount rates.

Second and more importantly, for the bargaining problem in question, the discount

rates might not be constant. For example, for the case with two firms, one would like

to make predictions on how the firms’ actions depend on the set of feasible profits

and the discount rates. In this case, various pairs of discount rates need to be taken

as possible. Thus, a two-player bargaining problem (S, d) in payoff space is more

generally required to be regular and satisfy

u ∈ S and t ∈ [0,∞]⇒(e−ritui +

(1− e−rit

)di)i∈{1,2} ∈ S

for some r ≡ (r1, r2) ∈ R2++. Let Br denote the collection of all such bargaining

problems. Then, the choice sets of bargaining problems in Br need not be convex

nor star-shaped. Nonetheless, the key properties of domain B for Theorem 1 to hold

are:

B1: All problems in B are regular.

B2: B is closed under positive affine transformation: (τ(S), τ(d)) ∈ B for all (S, d) ∈B and positive affine transformations τ : R2 → R2.

B3: B is closed under individual rationality restriction: (S, d) ∈ B implies (S ′, d) ∈ Bwhere S ′ = {u ∈ S : u ≥ d}.

B4: B includes all regular and comprehensive problems that contain only individually

rational choices: (S, d) ∈ B whenever (S, d) is regular and comprehensive

(relative to the disagreement point), and S − d ∈ R2+.9

Observe that domain Br satisfies B1 – B4. This continues to hold even if the set

of possible pairs of discount rates r is restricted to be any subset of R2++. All the

results in Section 3 are still valid. The proofs of Lemmas 1 – 4 and Theorem 1 rely on

properties B1 – B4 only. It follows that Theorem 1 can be generalized as follows.10

9We use S − d to denote the set {u− d : u ∈ S}.10As will be discussed in Section 5, the flexibility of domain specifications in Theorem 3 also

allows us to compare our characterization result with those in other papers.

14

Theorem 3. For any domain B of bargaining problems satisfying B1 – B4, a bargaining

solution f on B satisfies INV, IIA, and SIR if and only if there exists i ∈ {1, 2} such

that for any (S, d) ∈ B,

{f(S, d)} = argmaxu∈Σ(S,d)

ui

where

Σ(S, d) ≡ argmaxu∈S,u≥d

(u1 − d1)α1 (u2 − d2)α2 ,

α ≡ f(S◦,0)� 0.

Proof. Completely the same as the proof of Theorem 1. �

Since discount rates are constant, we can obtain equivalent utility transformations

by normalizing the discount rates as mentioned in the beginning of this section.

Thus, by considering equivalent utility functions with symmetric discount rates, we

can establish the following implementation result by applying Theorem 2.

Theorem 4. Given any bargaining environment (X, q, r, U) with r = (r1, r2) ∈ R2++,

the nomination-trial game has at least one SPE, and all the SPE payoff allocations

are given by the Nash bargaining solution characterized in Theorem 3 with domain

Br, i = 1, α1 = r2/(r1 + r2), and α2 = r1/(r1 + r2).

Theorem 4 shows that the implementation result is robust with respect to asymmetries

in bargaining powers. Moreover, it also shows that players’ discount factors provide

a specific determination of bargaining powers.

5 Literature review

Several extensions of single-valued Nash bargaining solution allowing for non-convexity

have been proposed and analyzed in the literature. Conley and Wilkie (1996)

established an extension which can be found via a two-step procedure. First, given

a non-convex problem, convexify the choice set via randomized choices and consider

the Nash solution for the convexified problem. Second, the intersection point of the

original Pareto frontier with the segment between the disagreement point and the

15

Nash solution for the convexified problem is the solution for the problem.11 The

modifications of the Nash axioms required for Conley and Wilkie’s approach makes

their extension fundamentally different from ours.

Qin, Shi, and Tan (2015) identified a class of bargaining problems characterized

by log-convexity and a regularity condition, to be referred to as the log-convex class.

A problem is log-convex if the log transformation of the portion of the choice set lying

above the disagreement point is strictly convex. Familiar examples of log-convex but

not convex problems include duopoly bargaining problems with twice continuously

demand functions, constant asymmetric marginal costs, and concave profit function

for the more efficient firm. A result in Qin, Shi, and Tan (2015) states that the

solution satisfying Nash axioms of INV, IIA, and SIR on the log-convex class of

problems is unique up to solutions for (S◦,0). The log-convex class is tight in the

sense that the result is no longer valid for any class of problems containing but not

equal to the log-convex class. The log-convex class is strictly contained in but not

equal to any class of problems satisfying B1 – B4.

Most related to ours is the extension in Zhou (1997). In terms of axioms, his

paper and ours differ only in the treatment of the INV axiom in the presence of

non-convexity. While our paper applies the axiom to non-convex as well as convex

problems, Zhou (1997) restricted the application of the axiom to the subclass of

convex problems only. Besides the difference in the treatment of the INV axiom,

bargaining problems considered in Zhou’s (1997) paper are regular and comprehensive

(relative to the disagreement point). These problems consist of a class, which we

denote by Bcom, that satisfies B1 – B4. For the purpose of comparison, we denote

Zhou’s (1997) restricted imposition of the INV axiom as INV-C. Zhou’s (1997) main

result states that each single-valued solution on Bcom satisfying INV-C, IIA, and SIR

must be a selection of Nash product maximizers. We show in Theorem 5 below that,

conversely, each Nash product maximizer of a problem can be selected as a Nash

solution for the problem.

Theorem 5. Suppose f is a bargaining solution on Bcom. Define

α ≡ f(S◦,0). (1)

11Mariotti (1998) provided an alternative characterization of Conley and Wilkie’s (1996)extension.

16

For λ > 0 and u, d ∈ R2 with u ≥ d and (S, d) ∈ Bcom, define

g(u, d) ≡ (u1 − d1)α1 (u2 − d2)α2 ,

λ∗(S, d) ≡ maxu∈S,u≥d

g(u, d), Σ(S, d) ≡ argmaxu∈S,u≥d

g(u, d), (2)

Iλ,d ≡{u ∈ R2 : u� d, g(u, d) = λ

}.

Then, f satisfies INV-C, SIR, and IIA if and only if α � 0 and there exists a

collection of strict linear orders {�λ,d}λ>0, d∈R2 with each �λ,d defined on Iλ,d such

that f(S, d) is the maximizer of �λ∗(S,d),d on Σ(S, d) for all (S, d) ∈ Bcom. Moreover,

the choice of the collection {�λ,d}λ>0, d∈R2 that characterizes solution f is unique.

Theorem 5 states that a positive vector α of bargaining powers and a collection

of linear orders �λ,d indexed by positive scalars and disagreement points always yield

a Nash solution satisfying INV-C, IIA, and SIR on Bcom. The vector of bargaining

powers determines the Nash product and in comparison, the linear orders determine

how to select among Nash product maximizers and across problems. An important

implication of Theorem 5 is that the Nash product maximizers of a given bargaining

problem can all be realizable as Nash solutions for the problem under the weaker

INV-C axiom, In this sense, Nash solutions under INV-C are loose, which stands in

sharp contrast to the much tightly structured solutions under INV.12 Nevertheless,

it is not true that any selection of the Nash product maximizers across problems

constitutes a Nash solution, since certain consistency properties across different

problems are in play.13

Peters and Vermeulen (2012) considered n-person multi-valued extensions under

the axioms of weak Pareto optimality (WPO), INV (unrestricted as in the present

paper), and IIA.14 A compact problem in the sense of Peters and Vermeulen (2012)

consists of a non-empty compact choice set S of vectors in the n-dimensional Euclidean

space with strictly positive components and a disagreement point at origin, which

12Note that the argument in Example 1 that precludes the Nash product maximizer in the middleinvolves applying INV for non-convex problems.

13For example, if S 6= T and the problems (S, d) and (T, d) have the same set of Nash productmaximizers, then any Nash solution f has to exhibit f(S, d) = f(T, d).

14Other multi-valued extensions appeared in the literature include Kaneko (1980), Herrero (1989).Since the focus of the present paper is on single-valued solutions, we refer the interested reader tothose papers for details and discussions on multi-valued extensions.

17

does not belong to the choice set. Given a non-zero n-dimensional vector α, the

assignmentNα(S) of maximizers of Nash product uα11 · · ·uαn

n to each compact problem

S was termed as an α-bargaining solution in Peters and Vermeulen’s (2012) paper.

They defined an iterated Nash solution as one for which there are non-zero vectors

α1, · · · , αk ∈ Rn with α1 ≥ 0 such that the solution for each compact problem S is

given by Nαk ◦ Nαk−1 ◦ · · · ◦ Nα1(S). Among other results, Peters and Vermeulen

(2012) showed that iterated Nash solutions are the only solutions satisfying WPO,

INV, and IIA on the class of compact bargaining problems defined above. Their

method can be adapted to prove our Theorem 1. However, the single-valuedness of

the solutions and the strict Pareto optimality instead of WTO implied by SIR for our

domains enable us to establish our Theorem 1 through a more elementary and direct

method. We provide our proof for these reasons as well as for self-completeness.

6 Conclusion

The non-convexity of bargaining problems often arises in applications. A corresponding

technical problem is the lack of uniqueness of solutions under reasonable axioms such

as those for Nash bargaining solutions. One can still hope to be able to characterize

solutions under Nash axioms. Indeed, as shown in Zhou (1997), single-valued solutions

(with the restriction of the INV axiom to convex problems only) are characterized as

selections of Nash product maximizers. This is a powerful result as it narrows down

the range of single-valued solutions. Nonetheless, the characterization is rough in

the sense that it does not specify what selections can be regarded as Nash solutions.

This paper contributes to the literature on bargaining by completely characterizing

single-valued solutions satisfying Nash axioms on a large class of two-player bargaining

problems, including convex or comprehensive problems. With the global imposition

of the INV axiom, it was shown that there are two and only two solutions with each

characterized as the dictatorial Nash product maximizer selection in favor of a given

player across all problems. This tight structure greatly enhances the applicability of

the Nash bargaining theory in the presence of non-convexity. It was also shown that

the solutions can be implemented non-cooperatively. Our paper offers a novel revisit

to the Nash bargaining theory with non-convexity.

18

Appendix

Our domain B in Section 2 (i.e., the class of regular and star-shaped bargaining

problems) clearly satisfies properties B1 – B4 introduced in Subsection 4.2. Properties

B1 – B4 are all we need to prove Lemmas 1 – 4 and Theorem 1.

Proof of Lemma 1. By B2 and INV, it suffices to consider bargaining problems with

d = 0. Fix (S,0) ∈ B and set S+ ≡ S ∩ R2+. Then, S+ ⊆ S, Par(S+) ⊆ Par(S). By

B3, (S+,0) ∈ B and by SIR, f(S,0) ∈ S+. Hence, by IIA, f(S,0) = f(S+,0).

Let T ≡ com(S+). Then S+ ⊆ T ⊆ R2+, Par(T ) = Par(S+), and (T,0) ∈ B

by B4. Suppose v ≡ f(T,0) 6∈ Par(T ). Then there exists w ∈ T such that w ≥ v

and w 6= v. Since v � 0 by SIR, we also have w � 0. Set βi ≡ vi/wi > 0 for

i = 1, 2. Then β ≡ (β1, β2) ≤ (1, 1) and β 6= (1, 1). Let T ′ ≡ {(β1u1, β2u2) : u ∈ T}.By construction T ′ ⊆ T and f(T,0) = v ∈ T ′. Consequently, by IIA, f(T ′,0) = v.

However, by INV, f(T ′,0) = (β1v1, β2v2) 6= v, a contradiction. Therefore, f(T,0) ∈Par(T ).

Finally, since S+ ⊆ T and f(T,0) ∈ Par(T ) = Par(S+) ⊆ S+, IIA implies

f(S+,0) = f(T,0). In summary, f(S,0) = f(S+,0) = f(T,0) ∈ Par(T ) =

Par(S+) ⊆ Par(S). �

Proof of Lemma 2. Property (i) follows from Lemma 1. To show property (ii), fix

u, v, w ∈ R2++ such that u � v and v � w. Note that, by property (i), u � v and

v � w jointly imply u 6= w. Thus, again by property (i), it suffices to show that w � u

cannot hold. Suppose on the contrary w � u. By Lemma 1, f(com(u, v, w),0) ∈{u, v, w}. If f(com(u, v, w),0) = u, then IIA implies u � w, contradicting w �u. Similarly, f(com(u, v, w),0) = v contradicts u � v and f(com(u, v, w),0) = w

contradicts v � w. Therefore, we must have u � w. This completes the proof of

property (ii). Finally, property (iii) automatically follows from the Pareto optimality

established in Lemma 1. �

Proof of Lemma 3. Let S++ ≡ S ∩ R2++ and S+ ≡ S ∩ R2

+. That f(S,0) ∈ S++

directly follows from SIR. Next, set u∗ ≡ f(S,0). We need to show u∗ � u, or

equivalently, u∗ = f(com(u, u∗),0), for u ∈ S++\ {u∗}. By B3, (S+,0) ∈ B. By IIA,

u∗ = f(S+,0). By B4, (com(S+),0) ∈ B and (com(u, u∗),0) ∈ B. By Lemma 1,

19

we have f(com(S+),0) ∈ S+. Thus, by IIA, we have f(com(S+),0) = f(S+,0) =

u∗ ∈ com(u, u∗). By IIA again, f(com(S+),0) = f(com(u, u∗),0). This shows

u∗ = f(com(u, u∗),0). �

Proof of Lemma 4. For notational convenience, define ut for t ∈ R by

log ut ≡ log u0 + t(log u1 − log u0

).

Step 1: The result holds for λ = 1 and λ = −1.

Note that with λ = 1, log v1 − log v0 = log u1 − log u0, or equivalently v1i /u

1i =

v0i /u

0i , for each i = 1, 2. Thus, by INV and u1 � u0,

v1 =

(v1

1

u11

u11,v1

2

u12

u12

)=

(v0

1

u01

u11,v0

2

u02

u12

)�(v0

1

u01

u01,v0

2

u02

u02

)= v0.

For the case of λ = −1, the result can be proved by simply switching the roles of v0

and v1.

Step 2: The result holds when λ = 1/m for any positive integer m.

By the assumptions on u0, u1, v0, v1 and Step 1, it suffices to show u1/m � u0.

Suppose on the contrary u0 � u1/m. Since log u(i−1)/m − log ui/m for i = 1, 2, . . . ,m

are all equal to 1m

(log u0 − log u1), it follows from Step 1 that u0 � u1/m � · · · �u(m−1)/m � u1. The transitivity of � shown in Lemma 2 then implies that u0 � u1,

which contradicts u1 � u0.

Step 3: The result holds for all λ > 0.

Given that λ > 0, it follows from the assumptions on u0, u1, v0, v1 and Step

1 that it suffices to show uλ � u0. Consider the “log-convex hull” of{u0, uλ

}defined by T = {ut : 0 ≤ t ≤ λ}. From B4, (com(T ),0) ∈ B. Suppose on the

contrary f(com(T ),0) = ut for some t ∈ [0, λ). Then we can pick a large enough

positive integer m such that t + 1m< λ. By construction, ut+1/m ∈ T and by IIA,

f(com(ut, ut+1/m),0) = ut. It follows that ut � ut+1/m. Hence, from Step 1 it follows

that u0 � u1/m. This contradicts Step 2. Therefore, f(com(T ),0) 6= ut for any

t ∈ [0, λ). But f(com(T ),0) ∈ T from Lemma 1. Consequently, f(com(T ),0) = uλ.

Thus, by IIA, f(com(u0, uλ),0) = uλ or equivalently, uλ � u0.

20

Step 4. Step 1 with λ = −1 and Step 3 together imply that the result also holds

for all λ < 0. �

The proof of Theorem 2 requires the following lemma.

Lemma 5. Given t ∈ [0,∞] and a, b ∈ X, the game Γ(i, t, {a, b, q}) has at least one

SPE. If U(a), U(b) ≥ 0, then (i) Γ(i, t, {a, b, q}) has 〈a, 0〉 as its unique SPE outcome

when Uj(a) > δtUj(b) and Ui(a) > Ui(b); (ii) Γ(i, t, {a, b, q}) has 〈b, 0〉 as its unique

SPE outcome when Uj(a) < δtUj(b) and Ui(b) > 0; (iii) the set of SPE outcomes of

Γ(i, t, {a, b, q}) is {〈a, 0〉 , 〈b, 0〉} when Uj(a) = δtUj(b) and Ui(a) ≥ Ui(b).

Proof. Given t ∈ [0,∞] and a, b ∈ X, Γ(i, t, {a, b, q}) is finite and of perfect information;

hence, it has at least one (pure strategy) SPE as can be shown by backward induction.

With Uj(a) > δtUj(b) and Ui(a) > Ui(b), it is optimal for player j to accept a

whenever it is offered in Γ(i, t, {a, b, q}). Thus, since Ui(a) > Ui(b), it is optimal for

player i to offer a in SPE. This establishes (i). With Uj(a) < δtUj(b) and Ui(b) > 0,

player j always rejects any offer other than b in Γ(i, t, {a, b, q}). Thus, since Ui(b) > 0,

it is optimal for player i to offer b which will be accepted by player j in SPE. This

establishes (ii).

Now suppose Uj(a) = δtUj(b) and Ui(a) ≥ Ui(b). The following describes a SPE

in Γ(i, t, {a, b, q}) leading to outcome 〈a, 0〉: player i offers a and player j accepts

a and b but rejects q in the first period; player j offers b and player i accepts any

offer in the second period regardless of what happened in the first period. Similarly,

the following describes a SPE in Γ(i, t, {a, b, q}) leading to outcome 〈b, 0〉: player i

offers b and player j accepts b but rejects a and q in the first period; player j offers

b and player i accepts any offer in the second period whatever happened in the first

period. It is straightforward to show that there can be no other (pure strategy) SPE

outcome. �

Proof of Theorem 2. Consider the trial phase following nominations a, b ∈ X. Without

loss of generality we assume U(a), U(b) ≥ 0. Candidate alternatives that are not

individually rational can be replaced by the status quo q. We complete the remaining

proof in five steps.

21

Step 1: If U(a) = U(b), the set of SPE outcomes of the trial phase is {〈a, 0〉 , 〈b, 0〉},but the SPE payoff allocation is unique and equal to U(a).

Step 2: If U1(a)U2(a) > U1(b)U2(b), the trial phase has 〈a, 0〉 as the unique SPE

outcome.

First note that the result holds if in addition U1(a) > U1(b) and U2(a) > U2(b).

Now suppose that Uj(a) ≤ Uj(b) and Ui(a) > Ui(b). Then

0 ≤ Ui(b)

Ui(a)<Uj(a)

Uj(b)≤ 1.

From player i’s point of view, 〈a, 0〉 is the unique best possible outcome in the trial

phase. Player i can guarantee outcome 〈a, 0〉 by specifying ti such that

0 ≤ Ui(b)

Ui(a)< δti <

Uj(a)

Uj(b)≤ 1.

If player j chooses tj > ti, then two players will subsequently play Γ(i, t, {a, b, q})where t = (ti + tj)/2 > ti. In this case, δtUj(b) < Uj(a). Thus, since Ui(a) > Ui(b),

Lemma 5(i) implies that the unique SPE outcome of Γ(i, ti, {a, b, q}) is 〈a, 0〉. If

player j chooses tj < ti, then two players will subsequently play Γ(j, t, {a, b, q})where t = (ti+ tj)/2 < ti. In this case, δt > δti which in turn implies Ui(b) < δtUi(a).

Next, since U1(a)U2(a) > U1(b)U2(b), U(a) ≥ 0, and U(b) ≥ 0, it must be Uj(a) > 0.

Thus, by interchanging i with j and a with b, Lemma 5(ii) implies that the unique

SPE outcome of Γ(j, t, {a, b, q}) is also 〈a, 0〉. Therefore, if the trial phase has any

SPE, the SPE outcome must be 〈a, 0〉. Furthermore, SPE in the trial phase exists.

In summary, the SPE outcome is unique and equal to 〈a, 0〉.

Step 3: If U1(a)U2(a) = U1(b)U2(b) > 0 and U1(a) > U1(b), the trial phase has

〈a, 0〉 as the unique SPE outcome.

First note that the assumptions imply

0 <U1(b)

U1(a)=U2(a)

U2(b)< 1.

From player 1’s point of view, 〈a, 0〉 is the unique best possible outcome in the

trial phase. Player 1 can guarantee outcome 〈a, 0〉 by specifying t1 such that δt1 =

22

U2(a)/U2(b). Then, depending on player 2’s choice of t2 ∈ [0,∞] the players play

either Γ(1, t, {a, b, q}) with t = (t1 + t2)/2 > t1 or Γ(2, t, {a, b, q}) with t = (t1 +

t2)/2 < t1. In the first case, it follows from Lemma 5(i) (with i = 1 and j = 2) that

the unique SPE outcome of Γ(1, t, {a, b, q}) is 〈a, 0〉. In the second case, it follows

from Lemma 5(ii) (with a, b interchanged, i = 2, and j = 1) that the unique SPE

outcome of Γ(2, t, {a, b, q}) is again 〈a, 0〉. Therefore, if the trial phase has any SPE,

the SPE outcome must be 〈a, 0〉. It is also easy to see that the trial phase has SPE.

In summary, the SPE outcome is unique and equal to 〈a, 0〉.

For the remaining proof, we consider the overall game. First, let u = f(S,0) be

the payoff allocation given by the solution for (S,0) in favor of i = 1 and choose

a ∈ X such that U(a) = u� 0.

Step 4: Payoff allocation u is a SPE payoff allocation.

Consider a strategy profile, in which each player selects a ∈ X in the nomination

phase and following each pair of nominations, they play a SPE strategy profile in the

trial phase. By Step 1, the outcome of this strategy profile is 〈a, 0〉 and the resulting

payoff allocation is u. To complete the proof, we need to show that no one has any

incentive to deviate from a in the nomination phase. Since U(a) = f(S,0) and the

solution is in favor of player 1, if a player unilaterally deviates from a to any b ∈ Xwith U(b) ≥ 0, the conditions in one of the previous three steps must be satisfied

and thus the SPE payoff allocation of trial phase is still U(a). Therefore, no player

has no incentive to deviate.

Step 5: Payoff allocation u is the only (pure strategy) SPE payoff allocation.

First note that player 1’s payoff level in any SPE is at least u1, because player 1

can guarantee herself this payoff level by nominating a. Similarly, player 2’s payoff

level is at least u2. Since u = f(S,0) is on the strict Pareto frontier of S, u must be

the only SPE payoff allocation. �

Proof of Theorem 4. Without loss of generality, assume U(q) = 0. Next, normalize

discount rates so as to transform player i’s utility function into e−rtUi(x), where

Ui(x) ≡ (Ui(x))r/ri . By Theorem 2, the nomination-trial game has at least one SPE,

and all SPE payoff allocations are given by Nash bargaining solution in favor of

23

player i = 1. Thus, the objective function in the first-round maximization is

U1(x)U2(x) = (U1(x))r/r1 (U2(x))r/r2 = [(U1(x))α1 (U2(x))α2 ]r(r1+r2)/r1r2 .

Maximizing U1(x)U2(x) is equivalent to maximizing (U1(x))α1 (U2(x))α2 . �

Proof of Theorem 5. Necessity. Suppose that f satisfies INV-C, SIR, and IIA. By

SIR, α � 0. From Zhou (1997), f(S, d) ∈ Σ(S, d) for all (S, d) ∈ Bcom. For λ > 0

and d ∈ Rn, define a binary relation �λ,d on Iλ,d by

u �λ,d v iff u 6= v and f(com(u, v; d), d) = u,

where com(u, v; d) ≡ {w ∈ Rn : u ≥ w ≥ d or v ≥ w ≥ d} denotes the d-comprehensive

hull of {u, v}.The proof of Lemma 2 can be adapted to prove that �λ,d is a strict linear order

on Iλ,d. Suppose, for some (S, d) ∈ Bcom, f(S, d) is not the maximizer of �λ∗(S,d),d

on Σ(S, d). Let v denote the maximizer of �λ∗(S,d),d on Σ(S, d). Note that both

f(S, d) and v belong to Σ(S, d) and hence, they belong to Iλ∗(S,d),d. By the definition

of v, we have v �λ∗(S,d),d f(S, d) or equivalently, f(com(f(S, d), v; d), d) = v, which

contradicts com(f(S, d), v; d) ⊆ S and IIA. This proves that f(S, d) is the maximizer

of �λ∗(S,d),d on Σ(S, d).

Next, if f(S, d) is the maximizer of �λ∗(S,d),d on Σ(S, d) for all (S, d), then the

collection {�λ,d}λ>0, d∈Rn of strict linear orders has to be chosen as above. Otherwise,

there would be some λ > 0, d ∈ Rn, and u, v ∈ Iλ,d such that u �λ,d v and

f(com(u, v; d), d) = v, which is impossible because f(com(u, v; d), d) is the maximizer

of �λ,d on Σ(com(u, v; d), d). This proves that the choice of the collection {�λ,d}λ>0, d∈Rn that characterizes solution f is unique.

Sufficiency. Suppose α � 0 and there exists a collection of strict linear orders

{�λ,d}λ>0, d∈Rn with each �λ,d defined on Iλ,d, such that f(S, d) is the maximizer of

�λ∗(S,d),d on Σ(S, d) for all (S, d) ∈ Bcom. Since α � 0 and f(S, d) ∈ Σ(S, d), f

satisfies SIR. Next, the uniqueness of the Nash product maximizer over every convex

choice set automatically implies that f satisfies INV-C.

To prove IIA, fix (S, d), (T, d) ∈ Bcom with S ⊆ T and f(T, d) ∈ S. Then, S ⊆ T

implies λ∗(S, d) ≤ λ∗(T, d). On the other hand, f(T, d) ∈ S and f(T, d) ∈ Σ(T, d)

24

implies λ∗(S, d) ≥ λ∗(T, d). Thus, λ∗(S, d) = λ∗(T, d), which we simply write as

λ∗. Now, since f(T, d) ∈ S ∩ Iλ∗,d, we have f(T, d) ∈ Σ(S, d). Note also that

Σ(S, d) ⊆ Σ(T, d) ⊆ Iλ∗,d. Finally, since f(S, d) and f(T, d) are the maximizers

of �λ∗,d on Σ(S, d) and Σ(T, d), respectively, it follows that Σ(S, d) ⊆ Σ(T, d) and

f(T, d) ∈ Σ(S, d) together imply f(S, d) = f(T, d). �

References

[1] Aoki, M.: A model of the firm as a stockholder-employee cooperative game.

American Economic Review 70, 600-610 (1980).

[2] Binmore, K., A. Rubinstein, and A. Wolinsky: The Nash bargaining solution in

economic modelling. RAND Journal of Economics 17, 176-88 (1986).

[3] Bishop, R.: Duopoly: collusion or warfare? American Economic Review 50,

933-61 (1960).

[4] Conley, J. and S. Wilkie: An extension of the Nash bargaining solution to

nonconvex problems. Games and Economic Behavior 13, 26-38 (1996).

[5] Fishburn, P.C. and A. Rubinstein: Time preferences, International Economic

Review, 23 (3), 677-694 (1982).

[6] Herrero, M.: The Nash program: non-convex bargaining problems. Journal of

Economic Theory 49, 266-77 (1989).

[7] Hougaard, J. L. and M. Tvede: Nonconvex n-person bargaining: efficient

maximin solutions. Economic Theory 21, 81-95 (2003).

[8] Kalai, E., and M. Smorodinsky: Other solutions to Nash’s bargaining problem,

Econometrica, 43 (3), 513-518 (1975).

[9] Kalai, E.: Nonsymmetric Nash solutions and replications of two-person

bargaining. International Journal of Game Theory 6, 129-33 (1977).

[10] Kaneko, M.: An extension of the Nash bargaining problem and the Nash social

welfare function. Theory and Decision 12, 135-48 (1980).

25

[11] McDonald, I. M. and R. M. Solow: Wage bargaining and employment. American

Economic Review 71, 896-908 (1981).

[12] Mariotti, M.: Extending Nash’s axioms to nonconvex problems. Games and

Economic Behavior 22, 377-83 (1998).

[13] Miyazaki, H.: Internal bargaining, labor contracts, and a Marshallian theory of

the firm. American Economic Review 74, 381-393 (1984).

[14] Moulin, H.: Implementing the Kalai-Smorodinsky bargaining solution, Journal

of Economic Theory, 33 (1), 32-45 (1984).

[15] Nash, J.F.: The bargaining problem. Econometrica 18, 155-62 (1950).

[16] Nash, J.F.: Two-person cooperative games. Econometrica 21, 128-40 (1953).

[17] Osborne, M. and A. Rubinstein: Bargaining and markets. Academic Press

(1990).

[18] Peters, H. and D. Vermeulen: WPO, COV and IIA bargaining solutions for

non-convex bargaining problems. International Journal of Game Theory 41,

851-884 (2012).

[19] Qin, C.-Z., S. Shi, and G. Tan: Nash bargaining with log-convex problems,

Economic Theory 58 (3), 413-440 (2015).

[20] Roth, A.: Individual rationality and Nash’s solution to the bargaining problem.

Mathematics of Operations Research 2, 64-65 (1977).

[21] Schmalensee, R.: Competitive advantage and collusive optima. International

Journal of Industrial Organization 5, 351-68 (1987).

[22] Shubik, M.: Strategy and market structure: competition, oligopoly, and the

theory of games. New York: Wiley 1959.

[23] Stahl, I.: Bargaining theory. Stockholm: Economics Research Institute,

Stockholm School of Economics (1972).

[24] Tirole, J.: The theory of industrial organization. The MIT Press 1988.

26

[25] Zhou, L.: The Nash bargaining theory with non-convex problems. Econometrica

65, 681-5 (1997).

27