Introduction

Bargaining occurs whenever two or more parties can share a surplus if an agreement can be reached on how the surplus should be shared, with a status-quo point that will prevail in the event of disagreement. Until recently, bargaining has been analyzed using the cooperative approach, which typically consists of specifying a set of axioms that the bargaining outcome should satisfy, and then proving that a solution satisfying these axioms exists and is unique. More recently, a second approach has emerged, which relies on the theory of noncooperative games. The typical paper of this type specifies a particular extensive form for the bargaining process, and solves for the noncooperative equilibria. Thus, the noncooperative approach replaces the axioms of the cooperative approach with the need to specify a particular extensive form.

Although this chapter is based on the noncooperative approach, which we believe has considerable power, we should point out that the reliance of the noncooperative approach on particular extensive forms poses two problems. First, because the results depend on the extensive form, one needs to argue that the chosen specification is reasonable – that it is a good approximation to the extensive forms actually played. Second, even if one particular extensive form were used in almost all bargaining, the analysis is incomplete because it has not, at least to-date, begun to address the question of why that extensive form is used.

There are two distinct reasons why the study of bargaining is of fundamental importance to economics. The first is that many aspects of economic activity are influenced directly by bargaining between and among individuals, firms, and nations. The second is that bargaining occupies an important place in economic theory, since the “pure bargaining problem” is at the opposite pole of economic phenomena from “perfect competition.”

It is not surprising that economic theory has had less apparent success in studying bargaining than in studying perfect competition, since perfect competition represents the idealized case in which the strategic aspect of economic interaction is reduced to negligible proportions by the discipline of a market that allows each agent to behave as a solitary decision maker, whereas pure bargaining is the case of economic interaction in which the market plays no role other than to set the bounds of discussion, within which the final outcome is determined entirely by the strategic interaction of the bargainers. The fact that the outcome of bargaining depends on this strategic interaction has led many economists, at least since the time of Edgeworth (1881), to conclude that bargaining is characterized by the indeterminacy of its outcome. In this view, theories of bargaining cannot, even in principle, do more than specify a range in which an agreement may be found; to attempt to accomplish more would be to introduce arbitrary specificity.

Introduction

Beginning with the observation of Ronald Coase, it has long been held that private bargaining can provide an antidote to the inefficiencies caused by externalities. Coase (1960) argued that

1. A pair of agents, by striking a mutually advantageous agreement, would obtain an efficient economic solution to the externality, and

2. A change in the assignment of property rights or in the liability rule would not affect the attainment of efficient agreements.

The Coase “theorem” relies on a number of assumptions, some explicit, some implicit, among which are that: agents have perfect knowledge of the economic setting including each other's utility function; in the absence of transaction costs, the agents will strike mutually beneficial agreements; and there exists a costless mechanism (a court system) for enforcing such agreements.

As many observers have pointed out, the presumptions that the bargainers have perfect knowledge of, and pursue, mutually beneficial agreements – assumptions borrowed from the theory of cooperative games – are crucial for the Coase results. The usual argument is that rational bargainers would (should) never settle on a given set of agreement terms if instead they could agree on alternative terms that were preferred by both sides. The conclusion, according to this argument, is that any final agreement must be Pareto-optimal.

Although this argument seems compelling, it leaves a number of questions unanswered.

Introduction

The purpose of this chapter is to consider some recent experimental evidence that existing models of bargaining, both axiomatic and strategic, are incomplete in ways that make them unlikely candidates from which to build powerful descriptive models of bargaining. After reviewing some of this evidence, a direction will be proposed that seems to offer some promising possibilities, and this will be briefly explored with the aid of an extremely simple preliminary model.

The plan of the chapter is as follows. Section 12.2 reviews some experiments in which certain kinds of information that are assumed by existing game-theoretic models not to influence the outcome of bargaining were nevertheless observed to have a dramatic effect. The data from these experiments make it plausible to suggest that bargainers sought to identify initial bargaining positions that had some special reason for being credible, and that these credible bargaining positions then served as focal points that influenced the subsequent conduct of negotiations, and their outcome. Section 12.3 explores this idea by investigating a simple model of coordination between two well-defined focal points. This model exhibits some of the same qualitative features observed in the bargaining data, concerning the frequency of disagreements as a function of the focal points. The section concludes with a brief discussion.

Review of four experiments

To test theories that depend on the expected utilities of the players, it is desirable to design experiments that allow the participants' utility functions to be determined.

Introduction

A fundamental problem in economics is determining how agreements are reached in situations where the parties have some market power. Of particular interest are questions of efficiency and distribution:

• How efficient is the agreement?

• How can efficiency be improved?

• How are the gains from agreement divided among the parties?

Here, I explore these questions in the context of bilateral monopoly, in which a buyer and a seller are bargaining over the price of an object.

Two features of my analysis, which are important in any bargaining setting, are information and impatience. The bargainers typically have private information about their preferences and will suffer some delay costs if agreement is postponed. Information asymmetries between bargainers will often lead to inefficiencies: The bargainers will be forced to delay agreement in order to communicate their preferences. Impatience will tend to encourage an early agreement and will make the parties' communication meaningful. Bargainers with high delay costs will accept inferior terms of trade in order to conclude agreement early, whereas patient bargainers will choose to wait for more appealing terms of trade.

Some authors have examined the bargaining problem in a static context, focusing solely on the role of incomplete information and ignoring the sequential aspects of bargaining. Myerson and Satterthwaite (1983) analyze bargaining as a direct revelation game.

Introduction

The notion of reputation found in common usage represents a concept that plays a central role in the analysis of games and markets with dynamic features. The purpose of this exposition is to describe how mathematical constructs roughly interpretable as reputations arise naturally as part of the specification of equilibria of sequential games and markets. In addition, several examples will be sketched, and a few of the economic applications surveyed.

The main theme here is that reputations account for strong intertemporal linkages along a sequence of otherwise independent situations. Moreover, from examples one sees that these linkages can produce strategic and market behavior quite different from that predicted from analyses of the situations in isolation. The economic applications, for instance, indicate that a firm's reputation is an important asset that can be built, maintained, or “milked,” and that reputational considerations can be major determinants of the choices among alternative decisions.

The key idea is that one's reputation is a state variable affecting future opportunities; moreover, the evolution of this state variable depends on the history of one's actions. Hence, current decisions must optimize the tradeoffs between short-term consequences and the longer-run effects on one's reputation. As the discussion proceeds, this general idea will be shown to have a concrete formulation derived from the analysis of sequential games.

Semantics

In common usage, reputation is a characteristic or attribute ascribed to one person (firm, industry, etc.) by another (e.g., “A has a reputation for courtesy”).

Introduction

In the traditional formulation of the bargaining problem, it is typically assumed that a fixed number of agents are involved. The possibility that their number varies has recently been the object of a number of studies, which it is the purpose of the present chapter to review in a unified way, with emphasis on the main results and on the main lines of their proofs.

I propose to evaluate solutions by focusing on their behavior in circumstances where new agents come in without their entry being accompanied by an expansion of opportunities. The standard economic problem that motivated much of the work presented here is that of dividing fairly a bundle of goods among a group of agents. The number of agents involved in the division is allowed to vary while the resources at their disposal remain fixed. (Technically, this implies that the intersection in utility space of the set of alternatives available to the enlarged group with the coordinate subspace corresponding to the original group coincides with the set of alternatives initially available to that group.) Of course, this does not mean that new agents are required never to bring in additional resources nor that their presence itself may not affect the alternatives available to the original group. I simply want to allow for the case of fixed resources, and I claim that a study of this special situation yields important insights into the relative merits of solutions.

Introduction

In analyzing a cooperative game with incomplete information, three kinds of solution concepts should be considered. First, we should characterize the set of coordination mechanisms or decision rules that are feasible for the players when they cooperate, taking account of the incentive constraints that arise because the players cannot always trust each other. Second, we should characterize the mechanisms that are efficient within this feasible set. Efficiency criteria for games with incomplete information have been discussed in detail by Holmström and Myerson (1983). Third, we should try to identify equitable mechanisms on the efficient frontier that are likely to actually be implemented by the players if they are sophisticated negotiators with equal bargaining ability. (We might also want to consider cases where one player has more bargaining ability than the others, as in principal – agent problems.) For this analysis, a concept of neutral bargaining solution has been axiomatically derived by Myerson (1983, 1984).

In this chapter, two bilateral trading problems with incomplete information are analyzed in terms of these three solution concepts. Sections 7.2 through 7.4 consider the symmetric uniform trading problem, a simple problem in which the buyer and seller each have private information about how much the object being traded is worth to him. This problem was first studied by Chatterjee and Samuelson (1983), and was also considered by Myerson and Satterthwaite (1983).

Introduction

The simplest bargaining situation is that of two persons who have to agree on the choice of an outcome from a given set of feasible outcomes; in case no agreement is reached, a specified disagreement outcome results. This two-person pure bargaining problem has been extensively analyzed, starting with Nash (1950).

When there are more than two participants, the n-person straightforward generalization considers either unanimous agreement or complete disagreement (see Roth (1979)). However, intermediate subsets of the players (i.e., more than one but not all) may also play an essential role in the bargaining. One is thus led to an n-person coalitional bargaining problem, where a set of feasible outcomes is specified for each coalition (i.e., subset of the players). This type of problem is known as a game in coalitional form without side payments (or, with nontransferable utility).

It frequently arises in the analysis of various economic and other models; for references, see Aumann (1967, 1983a).

Solutions to such problems have been proposed by Harsanyi (1959, 1963, 1977), Shapley (1969), Owen (1972), and others. All of these were constructed to coincide with the Nash solution in the two-person case. Unlike the Nash solution, however, they were not defined (and determined) by a set of axioms.

Recently, Aumann (1983b) has provided an axiomatization for the Shapley solution. Following this work, further axiomatizations were obtained: for the Harsanyi solution by Hart (1983), and for a new class of monotonic solutions by Kalai and Samet (1983).

Introduction

This chapter represents the first of several putative papers on bargaining among a small number of players. The problem treated in the current paper may be thought of as the “three-player/three-cake” problem. Each pair of players exercises control over the division of a different cake, but only one of the cakes can be divided. Which of the cakes is divided and how much does each player receive? This problem is, of course, a paradigm for a much wider class of problems concerning the conditions under which coalitions will or will not form.

The general viewpoint is the same as that adopted in our previous papers on bargaining (e.g., [3], [4], and [5]). Briefly, we follow Nash ([15], [16], and [17]) in regarding “noncooperative games” as more fundamental than “cooperative games.” Operationally, this means that cooperative solution concepts need to be firmly rooted in noncooperative theory in the sense that the concept should be realizable as the solution of at least one interesting and relevant noncooperative bargaining game (and preferably of many such bargaining games).

The cooperative concept that we wish to defend in the context of the three-person/three-cake problem is a version of the “Nash bargaining solution.” A precise statement of the version required is given in Section 13.3. For the moment, we observe only that the notion can be thought of as synthesizing to some extent the different approaches of Nash and von Neumann and Morgenstern.

Introduction

Recent years have seen the parallel but largely independent development of two literatures with closely related concerns: the theory of arbitration and the theory of incentives. Most of the theoretical arbitration literature seeks to predict and compare the allocative effects of the simple compulsory-arbitration schemes frequently used to resolve public-sector bargaining disputes in practice. Crawford (1981) provides a general introduction and a brief survey of this area. Sample references include Donn (1977), Crawford (1979, 1982a, 1982b), Farber and Katz (1979), Farber (1980), Bloom (1981), Hirsch and Donn (1982), Brams and Merrill (1983), and Donn and Hirsch (1983). These papers draw on, and give some references to, the large empirical and institutional arbitration literature. The incentives literature concerned directly with bargaining focuses instead on the theoretical limits of mechanism design in environments with asymmetric information. The papers by Kalai and Rosenthal (1978), Rosenthal (1978), Myerson (1979, 1983), Holmström (1982), Holmström and Myerson (1983), Myerson and Satterthwaite (1983), Samuelson (1984), and Crawford (1985), are a representative sample.

Although both literatures have the same goal – achieving better bargaining outcomes – this difference in focus, together with differences in style and analytical technique, has almost completely shut down communication between them, with discernible adverse effects on both. On the one hand, a large body of ad hoc, application-oriented analysis and measurement has developed in arbitration without recourse to carefully specified and worked-out models.

The readings in this volume are revised versions of papers presented at the Conference on Game-Theoretic Models of Bargaining held June 27–30, 1983, at the University of Pittsburgh. Support for the conference was provided by the National Science Foundation and by the University of Pittsburgh.

The conference would not have been possible without the support at the University of Pittsburgh of Chancellor Wesley Posvar, Dean Jerome Rosenberg, and my colleague Professor Mark Perlman. Michael Rothschild was instrumental in arranging the NSF support.

The purpose of this paper is to study the effect of a change in an individual's degree of risk aversion on the perfect Bayesian Nash equilibrium in a simple model of bargaining. I find that, contrary to the results in the axiomatic model with riskless outcomes due to Nash, an opponent may be made worse off by such a change. Further, an individual may want to take an action that identifies him as more, rather than less, risk averse than he really is. In the course of the analysis, I fully characterize the equilibria of a class of “wars of attrition” with incomplete information, and single out one as “perfect” in a certain sense; this result may be of independent interest.

Introduction

The role of risk aversion in bargaining has been widely studied within the axiomatic framework of Nash (1950) (see, for example, Roth (1979), Perles and Maschler (1981)). It has been found that if the negotiation concerns riskless outcomes, then the more risk averse an individual is, the higher the payoff of his opponent. Related results show that in this case it is to the advantage of an individual to “pretend” to be less risk averse than he really is (Kurz (1977, 1980), Thomson (1979), Sobel (1981)). These results have some intuitive appeal: Given any (probabilistic) beliefs about the behavior of his opponent, it seems that an individual should behave more cautiously, the more risk averse he is.

Introduction

This chapter discusses the role of a third party in settling disputes. A judge is responsible for deciding which of two bargainers should win a dispute. There is a social value associated with giving the disputed item to a particular bargainer. This is the bargainer's claim to the item. Each bargainer knows the strength of his claim and can provide evidence that proves his claim to the judge. Presenting proof is costly and distortion is impossible. However, it is possible to refuse to present evidence. The judge has a prior distribution about the strength of the claims, but does not know them exactly. The judge uses the information provided by the bargainers' disclosures (or decision not to provide evidence) and then rules in favor of the bargainer who has the best expected claim. When the bargainers must decide whether to provide evidence simultaneously, there are typically two types of equilibria. In one, each bargainer has a positive probability of winning if he does not provide evidence. In the other, one of the bargainers wins only if he proves his claim. Thus, rules about which bargainer has the burden of proof, that is, who must provide evidence in order to win the dispute, serve to select an equilibrium outcome. This discussion compares the welfare obtainable from different rules. The results are ambiguous.

Introduction

This essay serves as an introduction to recent work on noncooperative game-theoretic models of two-player bargaining under incomplete information. The objective is to discuss some of the problems that motivated formulation of these models, as well as cover some of the issues that still need to be addressed. I have not set out to provide a detailed survey of all the existing models, and I have therefore discussed only certain specific aspects of the models that I believe to be especially important. The reader will find here, however, a guide to the relevant literature.

The title of this chapter was chosen to emphasize the phenomenon of disagreement in bargaining, which occurs almost as a natural consequence of rational behavior (i.e., equilibrium behavior) in some of these models and is difficult to explain on the basis of equilibrium behavior using the established framework of bargaining under complete information. Disagreement, of course, is only one reflection of the problem of inefficient bargaining processes. I also spend some time on the general question of efficiency and its attainment. Whereas in most models classical Pareto-efficiency is not attainable in equilibrium, it may be obtained by players who deviate from equilibrium, as will be shown.

The chapter is organized as follows. Section 2.2 lays out the problem and discusses the important modeling approaches available. Section 2.3 focuses on a particular group of models, each of which specifies a strategic (i.e., extensive) form of the bargaining process.

Introduction

The axiomatic approach to bargaining may be viewed as an attempt to predict the outcome of a bargaining situation solely on the basis of the set of pairs of utilities that corresponds to the set of possible agreements and to the nonagreement point.

The strategic approach extends the description of a bargaining situation. The rules of bargaining are assumed to be exogenous, and the solution is a function not only of the possible agreements but also of the procedural rules and the parties' time preferences.

The aim of this chapter is to show that in the case of incomplete information about the time preferences of the parties, the bargaining solution depends on additional elements, namely, the players' methods of making inferences when they reach a node in the extensive form of the game that is off the equilibrium path.

The solution concept commonly used in the literature on sequential bargaining models with incomplete information is one of sequential equilibrium (see Kreps and Wilson (1982)). Essentially, this concept requires that the players' strategies remain best responses at every node of decision in the extensive form of the game, including nodes that are not expected to be reached. The test of whether a player's strategy is a best response depends on his updated estimation of the likelihood of the uncertain elements in the model. For nodes of the game tree that are reachable, it is plausible to assume that the players use the Bayesian formula.