In international treaty negotiations, sovereign countries voluntarily decide whether to participate in an agreement. Examples are abundant. A prominent one is climate negotiation. As we reviewed in Subsection 6.3.1, developed countries participated in the Kyoto Protocol, and developing ones did not. Almost all countries participated in the Paris Agreement. Many international treaties have an ‘open access’ rule in that countries are free to participate. They should approve each other’s memberships. In contrast, the proposal–response game in Part II had a ‘closed access’ rule in that players could join a coalition only when a proposer invited them and they accepted the invitation.

So far, we have assumed that once players form a coalition, they allocate the coalition worth and exit the game. After that, the other players continue the game. The basic model of coalition formation in Section 5.1 describes a situation where players cannot change their coalitions. This assumption is reasonable when the costs of changing coalitions are high. For example, firms make irreversible investments in building factories and R&D to participate in a joint project. When investments are project-specific, the cost of changing them for a new project may be prohibitively high for firms.

This chapter presents non-cooperative bargaining models for n-person characteristic function games. We extend the sequential bargaining models studied in Chapter 4 so that proposers can choose coalitions and payoff allocations. We explore how various cooperative solutions, the equal allocation (the Nash bargaining solution), the core, the stable matching, and the Shapley value, can be attained as a non-cooperative equilibrium of the bargaining models.

An n-person game in strategic form is defined by a triplet G = (N, {Ai}i∈N, {ui}i∈N) with the following elements. N = {1,...,n} is a set of players where n ≥ 2. Each Ai (i ∈ N) is a finite set of pure strategies (or actions) for player i. The Cartesian product A = Πi∈NAi is the set of pure strategy profiles a = (a1,...,an). The real-valued function ui on A is a payoff function of player i. A game in strategic form is abbreviated as a strategic game. In what follows, we denote by Δ(X) the set of probability distributions on a finite set X.

In Part I, we introduce basic concepts, definitions and terminologies used in the main part of the book. Chapter 2 presents basic tools of non-cooperative games. Chapter 3 presents those of cooperative games. Textbooks on game theory provide further properties and proofs. Readers with knowledge of game theory can skip Part I and go to Part II.

Coalitional bargaining arises in many political and economic situations. This chapter presents three applications of the random proposer model for cooperative TU games and pure bargaining problems: bargaining in legislatures, labour markets, and international climate agreements. In Part III, we will provide further applications of coalitional bargaining to efficient renegotiations (Chapter 8), climate cooperation and free-riding (Chapter 9), and trading under uncertainty (Chapter 10).

Part II consists of four chapters and presents n-person non-cooperative coalitional bargaining theory and its applications to legislative, wage, and climate bargaining.

Part III consists of three chapters, each extending the basic model of non-cooperative coalition formation in Part II in three directions: renegotiations, participation, and incomplete information. Chapter 8 presents a coalitional bargaining model with renegotiations and explores whether and how renegotiations can enhance the efficiency of coalition formation. Chapter 9 considers a new participation rule called an open access rule, under which players are free to participate in coalitions without invitations from proposers. Participation needs the approval of incumbent members of coalitions. The new model is motivated for the study of international environmental negotiations. Chapter 10 presents the core theory of n-person cooperative games with incomplete information. Coalition formation becomes a complicated process when players have private information. The incomplete informational core considers endogenous information leakage during negotiations. We extend a non-cooperative bargaining model in Chapter 5 to the case of incomplete information.

So far, we have assumed that players have complete information about economic environments. This assumption is stringent when we want to analyse bargaining situations in the real world. There are many uncertain events in actual bargaining problems. Players have only imperfect information on uncertainties. Furthermore, they have different information. In such situations, it is said that players have incomplete information, which is also called asymmetric information when we want to emphasise the asymmetry of players’ information.

This section presents non-cooperative bargaining models for n-person pure bargaining games in Section 3.1. Initiated by the seminal work of Rubinstein (1982), the bargaining models have a sequential structure in which some player proposes a payoff allocation, and all other players sequentially respond to it. Bargaining continues until an agreement is made. A rule that selects a proposer is called a bargaining protocol. The relationship between non-cooperative equilibrium and the Nash bargaining solution is considered.

In this postscript, we summarise the book and discuss some topics for future research.

We first introduce a cooperative game where players cannot form any coalition other than the grand coalition. The only possible outcome is that all players cooperate or that no players cooperate.

This book aims to present some of the recent developments in the studies of coalition formation in game theory through bridging non-cooperative and cooperative approaches.