ANNALS OF ECONOMICS AND FINANCE 8-2, 217–227 (2007)

Bargaining and Destructive Power*

Partha Dasgupta

University of Cambridge

and

Eric S. Maskin

Institute for Advanced Study, Princeton

E-mail: Maskin@ias.edu

We explore the effect that the power to destroy the feasible set has on two-person bargaining outcomes.

Key Words: Bargaining; Axioms; Destruction.

JEL Classification Number: C78.

1. MOTIVATION

The theory of bargaining as formulated by Nash (1950, 1953) has developed along two routes. One is axiomatic (e.g., Nash 1950; Kalai and Smorodinsky 1975; Roemer 1988). Here, the negotiation process underlying the bargaining is only implicit. The idea is to try to characterize the negotiated outcome (the solution) through a set of axioms without formally modeling the process.

The advantages of the axiomatic route are clear enough. A fruitful solution concept will be applicable to a wide class of negotiation procedures.Partly for this reason the axiomatic approach has been adopted not only in the classical bargaining problem but more generally in cooperative game theory, where axiomatizations have been devised for such solution conceptsas the Shapley value and the nucleolus.

The second route formulates the problem of bargaining in strategic terms. Here, the negotiation procedure is described explicitly as a noncooperative game, and its equilibrium points are studied (see, e.g., Nash 1953). This approach has become increasingly popular in recent years. But it has limitations. Because the negotiation procedure is modeled explicitly, one may not always be sure which properties of equilibrium are sensitive to particular (and perhaps arbitrary) features of the game, and which are not.

Accordingly, Nash (1950, 1953) regarded the two avenues as complementary. In this view, modeling the negotiation process explicitly is a way of testing a solution concept. This two-sided perspective on cooperative games is sometimes referred to as the Nash program.1

The focal point of the bargaining theory literature continues to be Nash’sown solution. However, the perspective has shifted. For years, Nash’sstrategic (1953) formulation was largely ignored. In that model bargainers simultaneously make utility demands. They obtain these utilities if the demands are jointly feasible and otherwise remain at the status quo (threatpoint). Nash showed that any noncooperative equilibrium of a smooth approximation of this game approximates the utilities of his bargaining solution. But, until recently, the profession concentrated almost entirely on his (1950) axiomatic approach, perhaps in part because of the Luce-Raiffa (1957) interpretation of Nash’s axioms as normative properties of an arbitration scheme (an interpretation that Nash himself did not put forward).

The strategic approach to bargaining was revived in the early 1980’s (see, e.g., Fudenberg-Tirole 1983, Rubinstein 1982, and Sobel-Takahashi 1983). In particular, Rubinstein considered bargaining without time limit over a “cake.” Players make alternating proposals about the cake’s division untilsome proposal is accepted.2 Rubinstein showed that, if there is a finite time between offers and players discount the future, such a game has a unique subgame perfect equilibrium. In fact, if one reinterprets the cake as a feasible set of utility pairs, the equilibrium outcome approximates the Nash bargaining solution if the time between successive proposals is small and the players have nearly the same discount rate (see Binmore 1987). Rubinstein’s game is therefore another negotiation procedure that (approximately) yields the Nash bargaining solution as an equilibrium outcome.

In Nash’s and Rubinstein’s models, as well as in other strategic bargaining work that we know of, a bargainer exercises power over the other party only by the threat that a deal will not occur. Even in Nash’s variable-threat version (1953) of his model, the threat is carried out only if a bargain is never struck. The emphasis on delay and recalcitrance as a bargainer’s For extensive discussion of these complementary approaches to cooperative game theory in general and to the theory of bargaining in particular, see the essays in Binmore and Dasgupta, ed. (1987).

2Stahl (1972) analyzed the same model but with a finite time horizon.

BARGAINING AND DESTRUCTIVE POWER 219

only instruments ignores the common possibility that he might take active

steps to affect the bargain by destroying part of the feasible set.

In practice, bargaining frequently entails destruction in this sense and

sometimes even violence: A terrorist threatens to kill one of his hostages if

his demands are not met, and then does so when the deadline has passed;

the United States calls for Japan to surrender and then bombs Hiroshima

and Nagasaki before Japan concedes. Indeed, war, in general, can be viewed

as the exercise of destructive power within a framework of political bargaining.

In this paper we explore the effect of such power on bargaining. In Section

2 we provide an example of negotiation between a firm (management)

and its (unionized) workers in which each party is capable of inflicting some

damage on the other party’s interest during bargaining: the workers, by

neglecting to maintain capital and equipment; the management, by replacing

the existing production technology by ones that are less advantageous

to the union. The example is designed to illustrate that in many situations

each party has the ability to direct its destructive power toward the

other party; that is, in damaging the other party it need not hurt itself.

In Section 3 we present the strategic form of a negotiation process that

idealizes the destructive abilities in this example. We show that if the

parties have equal power to delete portions of the feasible set outcomes,

the negotiation process, although otherwise similar to that of Rubinstein

(1982) results in a unique subgame perfect equilibrium quite different from

the Nash bargaining solution. In Section 4 we propose a set of axioms for

the classical bargaining problem that yield the equilibrium outcome of our

negotiation process as the unique solution. In brief, while retaining Nash’s

other axioms, we replace his “independence of irrelevant alternatives” (or

the Kalai-Smorodinsky 1975 “monotonicity” axiom) by a “deletion” axiom

that formalizes the idea that parties have the power to affect the size and

shape of the ‘cake’ being bargained over. Finally, in Section 5 we discuss

the robustness of our solution concept to alterations in the negotiation process,

extend the analysis of Section 3 to cases where the parties differ in

their potential to destroy, and briefly examine how introducing imperfect

information would change the results.