Equivalent conditions for the existence of an efficient equilibrium in coalitional bargaining with externalities and renegotiations

doi:10.1016/j.orl.2017.06.007

Operations Research Letters

Volume 45, Issue 5, September 2017, Pages 427-430

https://doi.org/10.1016/j.orl.2017.06.007 Get rights and content

Abstract

We consider a noncooperative coalitional bargaining game with externalities and renegotiations. We provide the necessary and sufficient condition for an efficient stationary subgame perfect equilibrium to exist. This condition states that a Nash bargaining solution is immune to any blocking.

Introduction

Gomes [3] provided a sufficient condition for an efficient stationary subgame perfect equilibrium (SSPE) to exist in coalitional bargaining with externalities and renegotiations. Gomes’s [3] model has four features: (i) players repeat negotiations about forming coalitions; (ii) players obtain an instantaneous payoff in every bargaining round; (iii) externalities occur among coalitions (the bargaining situation is described as a partition function game); and (iv) a proposer is randomly selected for each round. An efficient SSPE is one where a grand coalition forms immediately. Okada [10] derived a necessary and sufficient condition without externalities. However, a necessary and sufficient condition with externalities and renegotiations has not been shown.

Given an arbitrary discount factor, we provide a necessary and sufficient condition for an efficient SSPE to exist in coalitional bargaining with externalities and renegotiations. From the necessary and sufficient condition, an efficient SSPE can be fully characterized for any discount factor. Gomes [3] provided a sufficient condition for an efficient SSPE to exist for any discount factor. Even if Gomes’s sufficient condition is not satisfied, an efficient SSPE always exists when a discount factor is small enough. We also show that Gomes’s condition is necessary and sufficient for an efficient SSPE to exist for any discount factor.

Moreover, we give a cooperative-game-theoretic interpretation to the condition for an efficient SSPE to exist for any discount factor. The condition states that a Nash bargaining solution (NBS) is “bargaining-blocking-proof” in the following sense. Under a coalition structure (partition of the set of players), each coalition bargains over the worth of the grand coalition, its disagreement payoff being its worth, and an NBS under the coalition structure is given by the payoff tuple that maximizes the weighted product of net payoffs for coalitions over their disagreement payoffs. Here, by integrating and forming a new coalition, some coalitions can induce a new coalition structure as well as a new NBS under this coalition structure. If the sum of initial NBS payoffs for these coalitions is less than the new NBS payoff for the integrated coalition, these coalitions block the initial NBS. The NBS (tuple of NBSs under coalition structures) is said to be bargaining-blocking-proof if an NBS is not blocked by any coalition under any coalition structure.

We refer to other related literature on noncooperative coalitional bargaining, where renegotiations are allowed except for Kawamori and Miyakawa [8]. Gomes [4] analyzed the same noncooperative bargaining game as ours for a three-player case. He found four patterns of dynamic processes to the grand coalition and characterized the SSPE payoffs in the limit as a discount factor tends to unity. Seidmann and Winter [12] were the first to present a noncooperative coalitional bargaining game with renegotiations (they called it the “reversible actions” model), which is a rejector-propose model. They provided some examples of gradual coalition formation as well as immediate move toward grand coalition in a model without externalities. Gomes and Jehiel [5] considered a general case where coalitions may break up and have externalities. They provided a necessary and sufficient condition for convergence to the efficient state. Bloch and Gomes [2] considered a repeated coalitional bargaining game where the coalitions endogenously choose whether or not to exit under externalities among coalitions. Hyndman and Ray [6] considered nonstationary subgame perfect equilibria for a bargaining game. Kawamori and Miyakawa [8] provided a necessary and sufficient condition for an efficient SSPE to exist with externalities and rejecter-exist partial breakdowns but without renegotiations. Owing to the partial breakdowns, the condition of Kawamori and Miyakawa [8] is quite different from the condition of the present paper. We leave the characterization of inefficient SSPEs where the grand coalition does not form immediately to a future study. Gomes [4] attempted it in a three-player case.

The paper is organized as follows. Section 2 defines a noncooperative coalitional bargaining game. Section 3 provides necessary and sufficient conditions for an efficient SSPE to exist and gives a cooperative-game-theoretic interpretation of the conditions. Section 4 presents some applications. All proofs are relegated to the supplement (Kawamori and Miyakawa [9]).

Section snippets

Partition function game

Let $(N, v)$ be a partition function game; that is, a pair $(N, v)$ such that $N$ is a nonempty finite set and $v$ is a function from $C ≔ \{(S, π) \in 2^{N} \times Π | S \in π\}$ to $R_{+}$ , where $Π$ is a set of partitions of $N$ . An element of $N$ is called a player, a nonempty subset of $N$ is called a coalition, and a partition of $N$ is called a coalition structure. $v (S, π)$ represents the worth of coalition $S$ under coalition structure $π$ . For convenience, for any function $f$ from $C$ and any $(S, π) \in C$ , we write $f_{S}^{π}$ instead of $f (S, π)$ . We assume that the grand

Efficient SSPEs

We provide a necessary and sufficient condition for an efficient SSPE to exist, given a discount factor.

Theorem 1

Let $δ \in [0, 1)$ . An efficient SSPE of $G (δ)$ exists if and only if for any $π \in Π$ and nonempty $ρ \subset π$ , $b_{⋃ ρ}^{π | ρ} \leq \sum_{I \in ρ} b_{I}^{π} + (1 - δ) (\sum_{I \in π ∖ ρ} (b_{I}^{π} - v_{I}^{π}) + (b_{⋃ ρ}^{π | ρ} - v_{⋃ ρ}^{π | ρ})) .$

Remark 1

Gomes’s [3] condition corresponds to our condition in Theorem 1 as $δ \to 1$ . As in the proof of Corollary 1, the condition in Theorem 1 becomes stronger as $δ$ becomes larger. Thus, our condition in

Applications

We present two applications of our model: R&D alliances and a public good economy. We use the same suppositions and notations as in Example 1 except for $v$ in the following applications.

Acknowledgments

This paper is a substantial revision of a working paper version (Kawamori and Miyakawa [7]) entitled “Nash bargaining solution, core and coalitional bargaining game with inside options.” We are grateful to an anonymous referee, Akira Okada, Kalyan Chatterjee, Atsushi Kajii, Mihai Manea and participants of the Fourth World Congress of the Game Theory Society and the Matsuyama Meeting of the Kansai Game Theory Seminar for their useful comments. Tomohiko Kawamori and Toshiji Miyakawa gratefully

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Equivalent conditions for the existence of an efficient equilibrium in coalitional bargaining with externalities and renegotiations

Abstract

Introduction

Section snippets

Partition function game

Efficient SSPEs

Applications

Acknowledgments

J. Econom. Theory

Math. Social Sci.

Endogenous structures of association in oligopolies

Rand J. Econ.

Multilateral contracting with externalities

Econometrica

Dynamic processes of social and economic interactions: Our the persistence of inefficiencies

J. Polit. Econ.