Bargaining with Imperfect Commitment

We study the effect of imperfect commitment in noncooperative two-person bargaining games. By establishing the reputation for being stubborn, a player sometimes commits to her initial demand, becoming unable to change her demands or to accept an inferior offer from her opponent. When the probability of being stubborn is small, the set of equilibria is shown to be small and agreement may be reached immediately despite the possibility of stubbornness. A player has greater bargaining power when she is more patient and/or is more likely to be stubborn. Journal of Economic Literature Classification Number: C78.     

The Choquet Bargaining Solutions

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of Rⁿ₊ . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica62, 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.     

Multilateral Bargaining Problems

In many situations in economics and political science there are gains from forming coalitions but conflict over which coalition to form and how to distribute the gains. This paper presents an approach to suchmultilateral bargaining problems. Asolutionto a multilateral bargaining problem specifies an agreement for each coalition that is consistent with the bargaining process in every coalition. We establish the existence of such solutions, show that they are determined by reservation prices, and characterize these reservation prices as the payoffs ofsubgame perfect equilibrium outcomesof a non-cooperative bargaining model.Journal of Economic LiteratureClassification Numbers: C71, C72, C78.     

A Note on Multi-Issue Two-Sided Bargaining: Bilateral Procedures

This note considers a two-sided multi-issue bargaining problem in which players that belong to the same “side” may have conflicting priorities regarding the different negotiated issues. The note examines different bilateral bargaining procedures and shows the different equilibrium settlements that they yield. In particular the note examines the possibility that group heterogeneity (conflicting priorities) may be exploited in order to gain a better settlement. The different potential outcomes that are implied by the different procedures explain why we often observe such intense negotiation over bargaining procedures. Moreover, the conflict over procedure can be substantial, among parties with common interest as well as between opposing players. Journal of Economic Literature Classification Number: C7.     

Finite Horizon Bargaining and the Consistent Field

We explore the relationships between noncooperative bargaining games and the consistent value for nontransferable utility (NTU) cooperative games. A dynamic approach to the consistent value is introduced: the consistent vector field. The main contribution is to establish that theconsistent fieldis intimately related to the concept ofsubgame perfectionfor finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. We then show that in the general NTU case—unlike the transferable utility and pure bargaining cases—the finite horizon subgame perfect equilibria need not approach the consistent value.Journal of Economic LiteratureClassification Numbers: C71, C72.     

The Effect of Disagreement on Noncooperative Bargaining

A seemingly mild assumption of the standard alternating offers bargaining model under risk is that the breakdown event is not strictly worse than the worst agreement. When this assumption is relaxed the structure of the equilibrium set of agreements changes in an interesting way. We analyse the effect of disagreement on equilibrium, and relate our result to a class of outside option models. Journal of Economic Literature Classification Number: C78.     

Bargaining in a non-stationary environment

We study an alternating offers bargaining model in which the set of possible utility pairs evolves through time in a non-stationary, but smooth manner. In general, there exists a multiplicity of subgame perfect equilibria. However, we show that in the limit as the time interval between two consecutive offers becomes arbitrarily small, there exists a unique subgame perfect equilibrium. Furthermore, we derive a powerful characterization of the unique (limiting) subgame perfect equilibrium payoffs. We then explore the circumstances under which Nash's bargaining solution implements this bargaining equilibrium. Finally, we extend our results to the case when the players have time-varying inside options.     

On the Least Core and the Mas-Colell Bargaining Set

We show that the least core of a TU coalitional game with a finite set of players is contained in the Mas-Colell bargaining set. This result is extended to games with a measurable space of players in which the worth of the grand coalition is at least that of any other coalition in the game. As a consequence, we obtain an existence theorem for the Mas-Colell bargaining set in TU games with a measurable space of players. Journal of Economic Literature Classification Number: C71.     

Evolutionary Stability in Alternating-Offers Bargaining Games

This paper characterizes modified evolutionarily stable strategies (messes) in Rubinstein's alternating-offers, infinite-horizon bargaining game. We show that a mess causes agreement to be achieved immediately, with neither player willing to delay the agreement by one period in order to achieve the other player's share of the surplus. Each player's share of the surplus is then bounded between the shares received by the two players in the unique subgame-perfect equilibrium of Rubinstein's game. As the probability of a break-down in negotiations becomes small (or discount factors become large), these bounds collapse on the subgame-perfect equilibrium.Journal of Economic LiteratureClassification Numbers C70, C78.     

Trade Negotiations under Alternative Bargaining Structure

We build a bargaining model, in which a country (leader) decides whether or not to form a free trade agreement with other nations (followers), either through a sequential or a multilateral bargaining procedure. Unlike Aghion et al. ( 2007 , Journal of International Economics, 73, 1–30), in our specification of multilateral bargaining, the leader can collude with all those follower countries who agree to its offer. This has important implications for the choice of sequential and multilateral bargaining by the leader in presence of coalition externalities. Moreover, this bargaining procedure ensures that “stumbling block” equilibrium will never occur.     

Bargaining in Uncertain Environments: A Systematic Distortion of Perfect Equilibrium Demands

We present data supporting the notion that subjects distort monotonically subgame perfect equilibrium demands in two-person discount bargaining games, where, in addition to the traditional offer counter-offer sequence of moves, a player who rejects an offer may either be forced out of the bargaining or be given the option to opt out. Subjects also exhibit behavioral adaptivity—learning from past demands in a search for the highest acceptable demand. These reuslts are consistent with previous experimental evidence for these types of games, but stand in sharp contrast to those obtained in discount bargaining where side options are not available.Journal of Economic LiteratureClassification Numbers: C72, C78.     

The Evolutionary Stability of Auctions over Bargaining

This paper considers equilibrium in transaction mechanisms. In an environment with homogeneous buyers and sellers, which eliminates the advantage auctions possess of matching buyers and sellers, both auctions and bargaining are equilibria. However, only auctions are evolutionarily stable. This identifies a new advantage of auctions over bargaining, arising from the division of the gains from trade.Journal of Economic LiteratureClassification Numbers: C78, C73, D44.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

Coalitional Bargaining with Competition to Make Offers

The paper analyzes a simple discrete-time noncooperative coalitional bargaining game in which, at each stage, there is a contest for the right to make a proposal. The pure stationary subgame-perfect equilibrium payoff set is equal to the core.Journal of Economic LiteratureClassification Numbers: 026.     

Rubinstein Auctions: On Competition for Bargaining Partners

This paper analyzes the effect of competition for bargaining partners on the prices that prevail in thin markets, as well as how the matches are simultaneously determined. Three trading processes or bargaining procedures are described. In all the variants that we consider, except for one case of public offers, either there is no pure strategy subgame perfect equilibrium or such equilibria exhibit delay in reaching agreement.Journal of Economic LiteratureClassification Numbers: C72, D43.     

Folk Theorem For Bilateral Bargaining with Vector Endowments

We analyse a bargaining game of two players on the division of the sum of their vector endowments, with alternating proposals and discounting of single period utilities. The pair of endowments is not weakly Pareto‐efficient. Until they reach an agreement, each of the parties can withdraw from bargaining and keep their endowments. Any strictly individually rational division of the sum of endowments can emerge in some subgame perfect equilibrium if discount factors are close enough to one. Each subgame perfect equilibrium, in which the parties’ decisions do not depend on past rejected proposals, leads to a weakly Pareto‐efficient agreement in the first period.     

Wage Bargaining Versus Efficiency Wages: A Synthesis

We construct a model integrating the efficiency wage model of Shapiro–Stiglitz (1984) (SS), with an individual wage bargaining model in the Diamond–Mortensen–Pissarides (DMP) tradition where firms and workers form pairwise matches. We show that when workers may threaten to shirk on the job and there is individual wage bargaining, the wage is always higher and employment lower than in either the SS model, or the (appropriately modified) DMP model. When firms determine workers' efforts unilaterally, efforts are set inefficiently low in the SS model. In the bargaining model, effort is higher, and is first best when the worker non–shirking constraint does not bind. The overall equilibrium allocation may then be more or less efficient than in the SS model, but is always less efficient than in a pure bargaining model with no moral hazard.     

Bargaining under Uncertainty and the Monotone Path Solutions

Uncertainty with respect to the feasible set of utility vectors is introduced in an axiomatic bargaining model. Given a criterion for nonprobabilistic decision-making under uncertainty, a natural efficiency requirement can be imposed on a bargaining solution. Using the maximin ordering, thestrictly monotone pathsolutions (generalizations of theegalitariansolution) to the bargaining problem are characterized as the only continuous solutions that satisfy this efficiency axiom. If the maximin criterion is replaced by the maximax ranking or a strict convex combination of the maximin and the maximax criterion, imposing our efficiency axiom and continuity leads to thedictatorialsolutions.Journal of Economic LiteratureClassification Number: C78.     

Bargaining Economies with Patient and Impatient Agents: Equilibria and Intermediation

This paper investigates the characteristics of steady-state equilibria in a bargaining economy in which there are both patient and impatient agents and the possibility of intermediation arising in such an economy. The equilibrium in the unmediated economy is inefficient if the proportion of impatient agents is sufficiently high. There is a simple fee mechanism, which induces patient and impatient agents to trade in separate groups, under which there is always an efficient equilibrium. Such separation is also induced by the option to trade at fixed prices, and profit-seeking intermediaries have an incentive to facilitate such trade.Journal of Economic LiteratureClassification Number: C78.     

Bargaining foundations of the median voter theorem

We give a game-theoretic foundation for the median voter theorem in a one-dimensional bargaining model based on Baron and Ferejohn's [D. Baron, J. Ferejohn, Bargaining in legislatures, Amer. Polit. Sci. Rev. 83 (1989) 1181-1206] model of distributive politics. We prove that as the agents become arbitrarily patient, the set of proposals that can be passed in any pure strategy, subgame perfect equilibrium collapses to the median voter's ideal point. While we leave the possibility of some delay, we prove that the agents' equilibrium continuation payoffs converge to the utility from the median, so that delay, if it occurs, is inconsequential. We do not impose stationarity or any other refinements. Our result counters intuition based on the folk theorem for repeated games, and it contrasts with the known result for the distributive bargaining model that as agents become patient, any division of the dollar can be supported as a subgame perfect equilibrium outcome.