Aspirational Bargaining
by
Lones Smith
University of Michigan
Coauthors: Ennio Stacchetti (University of Michigan)

The classic bargaining problem presents two individuals the opportunity to share a dollar and only demands that they first agree how to partition it. An agreement consists of a proposal by one of the players and an immediate acceptance by the other.

The modern noncooperative dynamic approach to this problem begins with Stahl (1972), whose alternating offer model gave a unique prediction, but blossoms in Rubinstein's 1982 paper that found a unique subgame perfect equilibrium yielding immediate agreement in an infinite-horizon model with payoff discounting.

The tractability of this model is absolutely undeniable, and the central role it confers upon impatience is a well-praised virtue. However, its technical foundation upon a hardwired God-given offering order, or in more recent formulations on a pre-ordained delay between repeated offers of the same player, is rarely defended, and intuitively unappealing.

For any (possibly randomly) alternating offer model of bargaining is really a story of temporal monopoly: Agreement is the outcome of a process in which players in turn are exogenously given the right to ask the other party to accept an offer, who must burn payoff if he declines. The offerer is at a strategic advantage, because at that moment he can speak and his counterpart cannot. So Rubinstein's genius was to convert an insoluble inherently bilateral monopoly into a sequence of alternating monopolies (i.e. ultimatum games)!

The bargaining game and solution is not endowed by nature, but rather should reflect key structural constraints and accepted behavioral norms. While incorporating incomplete information and more recently reputational concerns has pushed this solution in this direction we suggest that it is the complete information foundation that needs new ideas.

Yet absent any structure, a Nash demand game may emerge, where any outcome at any time can happen. Rather than restrict the action space as does temporal monopoly, this paper instead restricts the solution concept. We introduce a new notion of aspirational bargaining: continuous time bargaining with unrestricted offers and counteroffers, and endogenous unrestricted timing of offers, but in Markovian strategies only depend on expected payoffs - namely, their aspirations.

Of course, once we abandon the temporal monopoly approach, we simply must forsake uniqueness. But we argue that any supported equilibrium has desirable and realistic features. One can show that it logically necessitates delay, in which players lock horns in a stationary war of attrition `waiting game' to see who will make the next concession, i.e. offers must be concessions; also, some offers may be turned down with positive chance.

A modelling innovation required by the paradigm shift is to look for a class of equilibria, and not `the' equilibrium. In each war of attrition, players concede at Poisson time intervals. If the concession is rejected, another war of attrition begins; however, the pursuant subgame is strictly worse for the spurned offerer - his aspiration level is lower, while the player being offered to is indifferent about accepting. The state space is the space of possible pairs of aspiration levels, with exogenously-given starting point.

We prove many properties of this new richer bargaining framework. For instance, under plausible conditions, the players' expected values are a submartingale, conditional on rejections. This means that the expected e^-rT, where T is the elapse time from today until final agreement, and r the interest rate, should be increasing as rejections accumulate.

We also compare and contrast what happens as a player becomes more impatient in this framework with the Rubinstein model.

Home Page of Lones Smith (A Voyage of Exploration)

Date received: June 4, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafc-78.