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1992 | Buch

Preliminaries Kapitel lesen Erstes Kapitel lesen

Axiomatic Bargaining Game Theory

verfasst von: Hans J. M. Peters

Verlag: Springer Netherlands

Buchreihe : Theory and Decision Library C

Enthalten in: Professional Book Archive

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Über dieses Buch

Many social or economic conflict situations can be modeled by specifying the alternatives on which the involved parties may agree, and a special alternative which summarizes what happens in the event that no agreement is reached. Such a model is called a bargaining game, and a prescription assigning an alternative to each bargaining game is called a bargaining solution. In the cooperative game-theoretical approach, bargaining solutions are mathematically characterized by desirable properties, usually called axioms. In the noncooperative approach, solutions are derived as equilibria of strategic models describing an underlying bargaining procedure.
Axiomatic Bargaining Game Theory provides the reader with an up-to-date survey of cooperative, axiomatic models of bargaining, starting with Nash's seminal paper, The Bargaining Problem. It presents an overview of the main results in this area during the past four decades. Axiomatic Bargaining Game Theory provides a chapter on noncooperative models of bargaining, in particular on those models leading to bargaining solutions that also result from the axiomatic approach.
The main existing axiomatizations of solutions for coalitional bargaining games are included, as well as an auxiliary chapter on the relevant demands from utility theory.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Preliminaries
Abstract
The main theme of this book is axiomatic bargaining game theory, as initiated by Nash’s seminal paper of 1950. Axiomatic bargaining game theory is a mathematical discipline which studies the problem of bargaining between two or more parties by studying the mathematical properties of maps assigning an outcome to each bargaining game in some class of bargaining games. A bargaining game is a set of outcomes representing the utilities attainable by the parties or players involved, together with a disagreement outcome. The interpretation is that this last outcome results if the players are unable to reach a unanimous agreement on some other possible outcome. Maps as mentioned will be called bargaining solutions . Axiomatic bargaining game theory is concerned with a mathematical investigation of the properties of such bargaining solutions. Usually, following Nash (1950), one formulates desirable properties for these solutions, and then tries to characterize a solution or a class of solutions by its properties. Therefore, such properties are often referred to as axioms1, which is a less neutral expression. We will use both terms, axioms as well as properties. In a nutshell, this is what this book is mainly about.
Hans J. M. Peters
Chapter 2. Nash bargaining solutions
Abstract
Any book on axiomatic bargaining game theory should start with Nash’s 1950 article and with the Nash bargaining solution, and so will this one. Without any doubt the Nash bargaining solution is the most well-known and popular solution concept in bargaining — in the theoretical literature as well as in applied and empirical work. What could be the reasons for this popularity? Empirical evidence for the Nash bargaining solution certainly is not overwhelming and besides, lack of empirical results concerning other solution concepts makes any comparison difficult if not impossible. (For some empirical work see Svejnar (1986), or van Cayseele (1987).) Further, many experiments have been conducted — see Roth and Malouf (1979) for an overview — but also these are not unambiguously conclusive in favor of the Nash solution. Even, earlier experiments by Crott (1971) point in the direction of the next popular solution, the Raiffa-Kalai-Smorodinsky solution (Raiffa, 1953, Kalai and Smorodinsky, 1975; see chapter 4).
Hans J. M. Peters
Chapter 3. Independence of irrelevant alternatives and revealed preferences
Abstract
Chapter 2 dealt with the Nash bargaining solution and its nonsymmetric extensions. The first characterization of the Nash bargaining solution, by Nash (1950), was based on the independence of irrelevant alternatives (IIA) axiom. This chapter is a further exploration into the consequences of this axiom. This exploration is based on the concept of revealed preference, in the wider context of choice functions and choice situations.
Hans J. M. Peters
Chapter 4. Monotonicity properties
Abstract
Both foregoing chapters dealt with solutions depending on only local properties of the Pareto optimal subset of a bargaining game. Chapter 2 extensively discussed nonsymmetric Nash bargaining solutions, and chapter 3 led us to conclude that, at least for the case of two players and in the presence of Pareto optimality and (feasible set) continuity, the independence of irrelevant alternatives axiom implies the maximization of a strongly monotonic, strongly quasiconcave function (corollary 3.21, lemma 3.18): again a quite local phenomenon. The localization axiom (LOC) explicitly expresses this solution property (see section 2.5). As remarked at the end of subsection 2.5.2, feasible set continuity of a solution suffices to imply the equivalence of IIA and LOC. Thus, it is not surprising that the critical discussion in the literature has focussed on this localization property of the symmetric Nash bargaining solution, and on the IIA axiom.
Hans J. M. Peters
Chapter 5. Additivity properties
Abstract
Monotonicity properties and independence properties like independence of irrelevant alternatives describe the effect on the outcome assigned by a bargaining solution to a bargaining game of changing — expanding or shrinking — the feasible set. Additivity axioms contain statements concerning the outcomes assigned to specific sums of bargaining games.
Hans J. M. Peters
Chapter 6. Risk properties
Abstract
Suppose a player in a bargaining game is replaced by a more risk averse player. Intuitively, in many situations one would expect this to be advantageous for the other players. Also, one might expect the more risk averse player to envy his (less risk averse) predecessor, because that player would probably get more out of the bargaining process. It is nevertheless surprising that bargaining solutions almost generically seem to confirm this intuition.
Hans J. M. Peters
Chapter 7. Bargaining with a variable number of players
Abstract
With the exception of the replication models for nonsymmetric Nash and Raiffa-KalaiSmorodinsky bargaining solutions (subsection 2.4.4 and section 4.3) hitherto the number of players in a bargaining game was assumed to be fixed. In Thomson and Lensberg (1989) axiomatic characterizations of bargaining solutions are collected where the number of players may vary. The book shows that axioms based on such a variable population of players have proved to be powerful tools in axiomatic bargaining, leading to new characterizations of well-known solutions like the Nash, Raiffa-Kalai-Smorodinsky, and egalitarian solutions.
Hans J. M. Peters
Chapter 8. Alternative models and solution concepts
Abstract
In this chapter some alternative models of axiomatic bargaining theory are collected. In section 8.2 we consider multivalued solutions, which assign to a bargaining game a subset of feasible outcomes rather than a unique outcome. Section 8.3 deals with probabilistic solutions, which assign to each bargaining game a probability measure on the feasible set. In section 8.4 we discuss some extensions of existing solution concepts to bargaining with possibly nonconvex feasible sets. Certain applications and implications of axiomatic bargaining game theory for specific economic models are considered in section 8.5. Section 8.6 reviews a few (axiomatic) models where time is involved. Sections 8.7 and 8.8 very briefly discuss ordinally covariant solutions and continuity, respectively.
Hans J. M. Peters
Chapter 9. Noncooperative models for bargaining solutions
Abstract
Axiomatic bargaining theory started with Nash’s seminal paper “The Bargaining Problem”, which appeared in 1950. This paper is still the most important paper in the field; it introduces and axiomatically characterizes the Nash bargaining solution. In his 1951 paper, Nash proposes his equilibrium concept for noncooperative games. Also this contribution to game theory is pathbreaking. Nash’s 1953 paper on bargaining tries to combine both the cooperative and the noncooperative approach. Nash designs a noncooperative demand game of which the, in a certain sense unique, Nash equilibrium leads to the payoffs prescribed by the Nash bargaining solution. Although the argument laid out in the last paper is formally incomplete and somewhat ad hoc, it has plotted a course for what some authors have termed the Nash program (Binmore and Dasgupta, 1987). This “program” aims at constructing bargaining procedures that have an axiomatic as well as a noncooperative justification. See also the quotation from Nash (1953) in section 9.3.
Hans J. M. Peters
Chapter 10. Solutions for coalitional bargaining games
Abstract
A natural extension of an n-person pure bargaining game is a coalitional bargaining game. In a coalitional bargaining game coalitions other than the grand coalition consisting of the whole player set N, or trivial coalitions consisting of single players, may form. Such a game is described by a characteristic function assigning to each coalition M ⊂ N some subset of ℝ M . We call these games coalitional bargaining games in order to keep in line with the main subject of this book; more often, however, they are called games without transferable utility or without sidepayments — the latter expression being more general, see Aumann (1967). There are many applications of these games to economic models, see Friedman (1986), Rosenmüller (1981), or the references in Aumann (1985b).
Hans J. M. Peters
Chapter 11. Elements from utility theory
Abstract
This chapter reviews and sometimes modifies a number of concepts and results from utility theory needed elsewhere in this book. The reader already familiar with or not interested in these basics, which underly most of the other material in this book, may skip this chapter or the larger part of it. Only an understanding is required of the definition of a von NeumannMorgenstern utility function, which is presented in section 11.2. Everything else in this chapter may be read upon references in other chapters.
Hans J. M. Peters
Backmatter
Metadaten
Titel
Axiomatic Bargaining Game Theory
verfasst von
Hans J. M. Peters
Copyright-Jahr
1992
Verlag
Springer Netherlands
Electronic ISBN
978-94-015-8022-9
Print ISBN
978-90-481-4178-4
DOI
https://doi.org/10.1007/978-94-015-8022-9