DECENTRALIZED ONE‐TO‐MANY BARGAINING

• Author: Chiu Yu Ko,Duozhe Li
• DOI: http://doi.org/10.1111/iere.12452
• Date: 01 August 2020
• Published date: 01 August 2020

INTERNATIONAL ECONOMIC REVIEW

Vol. 61, No. 3, August 2020 DOI: 10.1111/iere.12452

DECENTRALIZED ONE-TO-MANY BARGAINING∗

BYCHIU YUKOAND DUOZHE LI1

Department of Decision Sciences and Managerial Economics, Chinese University of Hong

Kong, Hong Kong; Department of Economics, Chinese University of Hong Kong, Hong Kong

We study a one-to-many bargaining model in which one active player bargains with every passive player on

how to share the surplus of a joint project. The order of bargaining is not fixed and the active player decides

whom to bargain with in each period. Our model admits a rich set of equilibria and we identify the upper and

lower bounds of equilibrium payoffs. We also examine whether two natural ordering protocols often assumed

in existing studies can sustain endogenously. Although the queuing protocol may indeed arise in an equilibrium,

the rotating protocol is in general not self-enforcing.

1. INTRODUCTION

We study the bargaining process that occurs between one active player and multiple passive

players regarding how to share the surplus of a joint project, which requires the cooperation

of all parties. Relevant real-life situations include a real-estate developer buying pieces of land

from multiple owners (Cai, 2000, 2003; Xiao, 2018), an employer trying to reach deals with

several labor unions (Horn and Wolinsky, 1988b; Stole and Zwiebel, 1996), and a downstream

firm acquiring inputs from several upstream firms (Horn and Wolinsky, 1988a). A common

feature of these situations is that the active player has to reach agreements with all of the

passive players, who do not bargain with one another.

Such one-to-many bargaining situations have been extensively studied. Existing studies usu-

ally assume an exogenously fixed bargaining protocol: (1) bargaining consists of consecutive

sessions; (2) in each session, the active player bargains with one of the passive players; and (3)

the ordering of bargaining sessions and the duration (i.e., offers and counteroffers allowed) of

each session are exogenously fixed. A crucial element of a bargaining protocol is its ordering

protocol for determining which passive player to bargain with the active player in each ses-

sion. In the literature, two natural ordering protocols are most commonly assumed, namely,

the queuing protocol (e.g., Stole and Zwiebel, 1996) and the rotating protocol (e.g., Horn and

Wolinsky, 1988b; Cai, 2000, 2003). In the queuing protocol, passive players form a queue. The

active player starts with the first passive player in the queue and does not switch to the next

without reaching an agreement. In the rotating protocol, passive players form a ring. After each

bargaining session, the active player switches to the next regardless of whether an agreement

has been reached.

In reality, although we may sometimes observe fixed ordering protocols in centralized negoti-

ations, for example, the accession negotiation of international organizations, such as the World

∗Manuscript received January 2018; revised March 2019.

1We thank the coeditor and three anonymous referees for their insightful comments and constructive suggestions. We

are also grateful to Hongbin Cai, Shinsuke Kambe, Jihong Lee, George Mailath, Martin Osborne, Satoru Takahashi,

Quan Wen, and many others for helpful discussions at different stages of this work. Chiu Yu Ko acknowledges

the financial support from Singapore Ministry of Education Academic Research Fund Tier 1 FY2017-FRC2-012.

Duozhe Li acknowledges the financial support from Hong Kong Research Grants Council General Research Fund

(Project No. CUHK492013). Please address correspondence to: Duozhe Li, Department of Economics, The Chinese

University of Hong Kong, Shatin, New Territories, Hong Kong. Phone: +852 3943 8183, Fax: +852 2603 5805. E-mail:

duozheli@cuhk.edu.hk.

1139

C

(2020) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1140 KO AND LI

Trade Organization and the European Union, there are also many decentralized one-to-many

bargaining situations without fixed protocols. Consider the situation in which the representative

of a real-estate developer can visit one landowner each day. If she does not reach a deal with

the one she visits, she is not obliged to return the next day, nor is she obliged to visit a different

one. It is rather her own choice, which may well be strategic. When analyzing such decentralized

bargaining situations, if we were to assume a specific ordering protocol, we must at least ensure

that it is self-enforcing. That is, the players have no incentive to deviate from it. Alternatively,

we can study a model with endogenously determined order of bargaining, and then investigate

whether any ordering protocol can arise in an equilibrium.

We develop a model of decentralized one-to-many bargaining. In each bargaining period, the

active player first decides which passive player to bargain with. Then, either the active player

or the chosen passive player is randomly selected to make a proposal and the other responds.

A binding agreement is reached if the proposal is accepted. If no agreement is reached in the

current period, the active player can either continue to bargain with the same passive player

or switch to any other player in the following period. Immediately after the active player

reaches agreements with all of the passive players, the joint project is implemented and each

player receives his or her agreed-upon share of the surplus.1In our model, passive players are

heterogeneous in terms of bargaining power, which is determined by two factors: the payoff

discount factor and the probability of being chosen as the proposer.

We aim to achieve two goals in our analysis. The first goal is to identify the upper and lower

bounds of each player’s equilibrium payoff in the limit in which the players become infinitely

patient. It provides the range of outcomes that one can expect from a decentralized one-to-many

bargaining situation. The second goal is to investigate whether queuing and rotating protocols

are self-enforcing, that is, whether they can arise as a part of the active player’s equilibrium

strategy. The conclusion here has important implications for whether an assumption on ordering

protocol for a specific bargaining situation is justifiable.

We characterize subgame perfect equilibria satisfying a weak condition of refinement. Our

bargaining game admits a rich set of equilibria. In her worst equilibrium, the active player adopts

a queuing protocol as a part of her strategy, whereas the first passive player in the queue receives

his highest equilibrium payoff. Meanwhile, any ordering of passive players can be sustained in

an equilibrium with a queuing protocol. However, a queuing protocol leads to a highly unequal

equilibrium division. In the symmetric case in which all players are equally patient and have

the same proposing probability, the first passive player receives one half of the surplus, the next

one receives one half of the remaining half, and so forth.

In contrast, a rotating protocol leads to a much more plausible equilibrium division. In the

symmetric case, it leads to an equal division of the surplus among all of the players. This

may explain why the rotating protocol is often assumed in existing studies. However, we find

that in generic situations, the rotating protocol is not self-enforcing. More precisely, if and

only if all of the passive players are equally patient, our model admits equilibria with rotating

protocols. Note that the asymmetry of proposing probabilities plays no role here. Therefore,

although the proposing probability and the discount factor together determine the players’

relative bargaining powers, there are subtle differences between the two factors.

Next, we construct a class of divide-and-conquer equilibria in which the active player pits one

passive player against the others. The active player obtains her highest payoff in a divide-and-

conquer equilibrium, whereas all but one of the passive players receive their lowest equilibrium

payoffs, which converge to 0 as the discount factors tend to 1.

We also characterize the unique Markov equilibrium. When all of the passive players are

equally patient, in the Markov equilibrium, the active player always randomizes with equal

probability when choosing a bargaining opponent. The payoff vector of the Markov equilibrium

converges to the same limit as that of any equilibrium with a rotating protocol.

1Henceforth, the active player is referred to as “she” and a passive player is referred to as “he.”

ONE-TO-MANY BARGAINING 1141

It is assumed in our main analysis that after reaching a bilateral agreement, the two players

sign a contingent contract, by which the passive player receives his payoff after the project is

implemented. There are many real-life situations in which only cash-offer contracts are feasible.

Hence, we also consider a modified model in which any bilateral agreement is enforced through

an immediate cash payment. We again obtain a class of equilibria with queuing protocols, but

in this case, every passive player prefers to be the last to reach an agreement, in contrast to the

case with contingent contracts. Again, the rotating protocol is not self-enforcing. A new finding

here is that an impasse (i.e., perpetual disagreement) may arise in an equilibrium when there

are at least three passive players.

Related Literature. The one-to-many bargaining game is a natural extension of the classic

bilateral bargaining game in Rubinstein (1982). It has been incorporated into models in various

contexts. Many authors have assumed a fixed ordering protocol. Horn and Wolinsky (1988b)

assume a rotating protocol in the wage negotiation between a firm and two groups of workers.

In their seminal paper on intrafirm bargaining, Stole and Zwiebel (1996) propose a bargaining

game (henceforth, the SZ game) with a queuing protocol to support the Shapley value of

the corresponding coalitional game in an equilibrium.2Westermark (2003) assumes a random-

matching protocol in a similar intrafirm bargaining model. In each period, the firm bargains with

one worker that is determined by a fixed randomization process. In equilibrium each worker

receives only a share of his marginal product, instead of a share of the average marginal product,

as specified by the Shapley value.

De Fontenay and Gans (2014) study a class of pairwise bargaining games with imperfect

information, that is, each binding bilateral agreement is only observed by the two involved

parties. By imposing a reasonable restriction on the off-equilibrium beliefs (namely, passive

beliefs), they obtain a unique perfect Bayesian equilibrium generating the Myerson–Shapley

value. Under the same belief restriction, the equilibrium payoff of the imperfect-information

version of the SZ game coincides with the Shapley value. Br ¨

ugemann et al. (2019) show that

the Shapley value cannot be sustained as an equilibrium outcome of the perfect-information

version of the SZ game and they propose a bargaining game with perfect information and

a rotating protocol to achieve that goal. In this article, we also study a perfect-information

bargaining game.

Cai (2000) studies a one-to-many bargaining problem with a fixed rotating protocol and cash-

offer contracts. An important message of his paper is that inefficient stationary equilibria may

arise in a complete information setting.3More specifically, although bargaining order is fixed in

his model, the order of reaching bilateral agreements is endogenously determined. Hence, delay

occurs when the two orders are different. Without a fixed ordering protocol, our model does

not admit inefficient stationary equilibria, with either contingent or cash-offer contracts. Cai

(2003) obtains similar results in a model with contingent contracts; however, one should note

the subtle detail that the session protocol, alternating offers with the firm making the last offer

in each session, is the crucial element of this model generating inefficient stationary equilibria.4

This leads to another question whether and how session protocol should also be endogenized.

We will briefly discuss it in Section 6.

A number of studies have attempted to endogenize bargaining order in various settings. Noe

and Wang (2004) consider the negotiation between one buyer and two sellers and investigate

2Two crucial differences between our model and that of Stole and Zwiebel (1996) should be noted. First, they consider

a more general value structure, in which the firm can settle by reaching agreements with a subset of workers. We study

a pure bargaining problem, which requires the active player to reach agreements with all of the passive players. Second,

they assume nonbinding bilateral agreements and thus allow renegotiation, whereas we assume binding agreements. In

Section5, we discuss the extension to nonbindingagreements.

3Although many other papers have obtained inefficient equilibria in various bargaining games with complete in-

formation, their inefficient equilibria are usually sustained by credible punishment schemes as in the repeated game

literature, and thus are highly nonstationary (e.g., Haller and Holden, 1990; Fernandez and Glazer, 1991; Busch and

Wen, 1995).

4If the session protocol in Cai (2003) is changed to the one with random proposer as that in our model, there will be

a unique efficient equilibrium even under a fixed orderingprotocol.