Valuations and dynamics of negotiations

• Author: Armando Gomes
• DOI: http://doi.org/10.1111/jpet.12410
• Date: 01 February 2020
• Published date: 01 February 2020

J Public Econ Theory. 2020;22:245–273. wileyonlinelibrary.com/journal/jpet © 2019 Wiley Periodicals, Inc.

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Received: 21 January 2019

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Accepted: 22 October 2019

DOI: 10.1111/jpet.12410

ORIGINAL ARTICLE

Valuations and dynamics of negotiations

Armando Gomes

Olin Business School, Washington

University in St. Louis

Correspondence

Armando Gomes, Washington University

in St. Louis, One Brookings Drive, St.

Louis, MO 63130.

Email: gomes@wustl.edu

Abstract

This paper analyzes three‐party negotiations in the

presence of externalities. We obtain a closed‐form

solution for the Markov perfect equilibrium of a

multilateral non‐cooperative bargaining model, yielding

an equilibrium value and dynamics of negotiations that

are supported by experimental studies. Players’values

are monotonically increasing (or decreasing) in the

amount of negative (or positive) externalities that they

impose on others. Moreover, players’values are

continuous and piecewise linear on the worth of

bilateral coalitions, and are inextricably related to their

negotiation strategies: the equilibrium value is the Nash

bargaining solution when no bilateral coalitions form;

the Shapley value when all bilateral coalitions form; or

the nucleolus, when either one bilateral coalition among

“natural partners”or two bilateral coalitions including a

“pivotal player”form.

KEYWORDS

coalitional bargaining, externalities, mergers and acquisitions,

negotiations, value

JEL CLASSIFICATION

C71; C72; C78; D62

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INTRODUCTION

This paper studies multilateral negotiations in the presence of externalities. These problems are

important in economics, appearing in such diverse areas as mergers and acquisitions,

bankruptcy, and international treaty negotiations, as well as the formation of labor unions and

coalitional governments. For example, mergers create synergies to the firms merging, but often

also impose externalities on the remaining firms in the industry.

What are the dynamics of negotiations? How do parties involved form valuations and/or

prices at which transactions will take place? Our goal in this paper is to propose and analyze a

standard strategic model of negotiations and to derive a closed‐form solution answering the

questions above.

While bilateral negotiations have been extensively studied, much less is known about the

more complex problem of three‐player negotiations specially when the threat of coalition

formation (or partial mergers) is an integral part of the negotiations. The goal of this paper is to

fill out this gap in the literature by providing decision‐makers an intuitive off‐the‐shelf solution

for three‐player negotiations with externalities. The results obtained for three‐player games are

helpful in extending our understanding of one step beyond bilateral interactions and is useful in

applications where coalition formation and externalities play an important role. The focus on

three‐player games is natural because coalitional games with more than three‐players are too

complex to yield a simple closed‐form solution.

The bargaining model analyzed in this paper is a natural non‐cooperative dynamic

multilateral negotiation model (see literature review below). In our model, there are three

players who can either form bilateral coalitions and/or the grand coalition among three‐players.

A set of parameters describes the payoff flows generated by the grand coalition and all bilateral

coalitions, including the number of externalities they impose on excluded players. The

bargaining game evolves over time with players making offers followed by responses every

period.

We derive the closed‐form solution for the equilibrium value, referred to as the coalitional

bargaining value (CBV), and show that it is a continuous and piecewise linear function of the

parameters of the game. Specifically, the space of all games is divided into four convex regions

(eight including all permutations), and in each region, the CBV is a linear function of the

parameters of the game. For three‐player games without externalities, in one of the regions the

CBV coincides with the Nash bargaining solution, in another region with the Shapley value,

and in the remaining regions with the nucleolus. For three‐player games with externalities, a

similar treatment applies as long as an adjusted measure for the worth of pairwise coalitions is

used to generalize the Shapley value and the nucleolus. This adjustment involves measuring the

worth of a pairwise coalition by adding the amount of negative externalities (or subtracting the

amount of positive externalities) that it creates for the excluded player.

The solution proposed in this paper has been already empirically tested in the context of

mergers and acquisitions by Croson, Gomes, McGinn, and Noth (2004). They experimentally

compared the new equilibrium value and the dynamics of coalition formation proposed in an

earlier version of this paper to that of competing concepts in situations with and without

externalities. Their experimental results indicate that the CBV performs significantly better

than other leading solution concepts. Moreover, they show that the dynamics of coalition

formation is as predicted by our model in over 75% of the experiments conducted. Overall the

experimental results support the predictions of the new equilibrium concept, indicating that it

is an attractive off‐the‐shelf solution concept to use by decision‐makers for three‐player

negotiations with and without externalities.

Given the practical importance of takeover negotiations, the results should be of interest to

practitioners. Boone and Mulherin (2007) document that about half of the takeover targets are

sold by negotiations with acquirers and about half of them are sold through auctions. There are

several papers in the finance literature studying takeover bidding (such as Grossman & Hart,

1988, Fishman, 1988, and Marquez & Yilmaz, 2008) and takeover auctions (such as DeMarzo,

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GOMES