Demand bargaining and proportional payoffs in majority games

We study a majoritarian bargaining model in which players make payoff demands in decreasing order of voting weight. The unique subgame perfect equilibrium outcome is such that the minimal winning coalition of the players that move first forms with payoffs proportional to the voting weights. This result advances previous analysis in terms of one or more of the following: a) the simplicity of the extensive form (finite horizon with a predetermined order of moves); b) the range of the majority games covered; c) the equilibrium concept (subgame perfect equilibrium is sufficient for a unique prediction).     

Demand and supply effects of bargaining power shocks

In times of crisis, social partners may consider a temporary decline in wages as a necessity to maintain employment. This paper studies the opposing demand and supply effects following declining bargaining power of workers in a New-Keynesian model with search and matching in the labour market. Lower labour income reduces aggregate demand in the presence of credit-constrained consumers. The main result is that falling bargaining power contracts output notably when monetary policy is constrained by the zero lower bound or when agents' expectations about the persistence of the shock adjust slowly.     

Walrasian bargaining

Given any two-person economy, consider an alternating-offer bargaining game with complete information where the proposers offer prices, and the responders either choose the amount of trade at the offered prices or reject the offer. We provide conditions under which the outcomes of all subgame-perfect equilibria converge to the Walrasian equilibrium (the price and the allocation) as the discount rates approach 1. Therefore, price-taking behavior can be achieved with only two agents.     

Bilateral bargaining in networks

Each connected pair of nodes in a network can jointly produce one unit of surplus. A maximum number of linked nodes is selected in every period to bargain bilaterally over the division of the surplus, according to the protocol proposed by Rubinstein and Wollinsky [Equilibrium in a market with sequential bargaining, Econometrica 53 (1985) 1133-1150]. All pairs, which reach an agreement, obtain the (discounted) payoffs and are removed from the network. This bargaining game has a unique subgame perfect equilibrium that induces the Dulmage-Mendelsohn decomposition (partition) of the bipartite network (of the set of nodes in this network).     

Nominal bargaining power, selection protocol, and discounting in legislative bargaining

The comparative static predictions of the Baron and Ferejohn [Baron, D.P., and Ferejohn, J.A., (1989). Bargaining in legislatures, American Political Science Review 83 (4), 1181-1206] model better organize behavior in legislative bargaining experiments than Gamson's Law. Regressions similar to those employed in field data produce results seemingly in support of Gamson's Law (even when using data generated by simulating agents who behave according to the Baron-Ferejohn model), but this is determined by the selection protocol which recognizes voting blocks in proportion to the number of votes controlled. Proposer power is not nearly as strong as predicted in the closed rule Baron and Ferejohn model, as coalition partners refuse to take the small shares given by the continuation value of the game. Discounting pushes behavior in the direction predicted by Baron and Ferejohn but has a much smaller effect than predicted.     

Equilibrium selection in bargaining models

This paper examines evolutionary equilibrium selection in bargaining models. We show that random best-response and continuous best-response learning dynamics give rise to (different) simple sufficient conditions for identifying outcomes as stochastically stable. This allows us to characterize the implications of these dynamics in simple bargaining games.     

Wage bargaining and employment revisited: separability and efficiency in collective bargaining*

We analyse the two-dimensional Nash bargaining solution (NBS) by deploying the standard labour market negotiations model of McDonald and Solow. We show that the two-dimensional bargaining problem can be decomposed into two one-dimensional problems, such that the two solutions together replicate the solution of the two-dimensional problem if the NBS is applied. The axiom of “independence of irrelevant alternatives” is shown to be crucial for this type of decomposability. This result has significant implications for actual negotiations because it allows for the decomposition of a multi-dimensional bargaining problem into one-dimensional problems – and thus helps to facilitate real-world negotiations.     

Competitive bargaining equilibrium

In a simple exchange economy we propose a bargaining procedure that leads to a Walrasian outcome as the agents become increasingly patient. The competitive outcome therefore obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. Our bargaining procedure involves bargaining over prices and maximum quantity constraints, and it guarantees convergence to a Walrasian outcome for any standard exchange economy. In contrast, without quantity constraints we show that equilibrium is generically inefficient.     

The separability principle in bargaining

Summary. We investigate the implications of the separability principle in the context of bargaining. For two bargaining problems with the same population, suppose that there is a subgroup of agents who receive the same payoffs in both bargaining problems. Moreover, if we imagine the departure of this subgroup with their payoffs, then the remaining agents face the same opportunities in both bargaining problems. The separability principle requires that under these hypotheses, the remaining agents should receive the same payoffs in both bargaining problems. We begin with investigating the logical relations between separability and two other axioms, contraction independence and consistency. Then, we establish characterizations of the Nash and egalitarian solutions on the basis of separability. Received: 12 August 2002, Revised: 22 March 2004, JEL Classification Numbers: C71, C78.Youngsub Chun: This work was supported by the Brain Korea 21 Project in 2003. I am grateful to William Thomson, a referee, and an associate editor for their valuable comments.     

Sociological versus strategic factors in bargaining

Most game-theoretic models of strategic interaction, indeed most economic models of any sort, specify potential outcomes entirely in terms of the preferences of the agents, as captured in their (possibly cardinal) utility functions. The underlying assumption is that the outcome of such interactions is determined entirely by these preferences, together with the strategic possibilities available to the agents. The purpose of this paper is to challenge the adequacy of this assumption in general, by investigating it in the specific context of two-person bargaining. In particular, we consider whether certain experimental results reported earlier can be accounted for strictly in terms of players' preferences and strategic possibilities, and we report a new experimental study designed to answer this question. The results strongly support the conclusion that sociological factors, unrelated to what we normally consider to be the ‘economic’ parameters of a game, can decisively influence the outcome of bargaining, in a systematic manner.     

Subgame perfection in ultimatum bargaining trees

In typical experiments on ultimatum bargaining, the game is described verbally and the majority of subjects deviate from subgame-perfect behavior. Proposers typically offer significantly more than the minimum possible and Responders reject “unfair” offers. In this work, we show that when the ultimatum bargaining game is presented as an abstract game tree, the vast majority of behavior is consistent with individualistic preferences and subgame-perfection. This finding raises doubts about theories that ignore the potential influence of social context and experiments that do not control for social context.     

Stationary equilibria in bargaining with externalities

This paper studies infinite-horizon bargaining between a seller and multiple buyers when externalities are present. We extend the analysis in Jehiel and Moldovanu by allowing for both pure and mixed equilibria [Jehiel, P., Moldovanu, B., 1995a. Cyclical delay in bargaining with externalities. Rev. Econ. Stud. 62, 619–637]. A characterization of the stationary subgame perfect equilibria in generic games is presented. Equilibria with delay exist only for strong positive externalities. Since each buyer receives a positive payoff when the seller makes an agreement with some other buyer, positive externalities induce a war of attrition between buyers.     

Randomly available outside options in bargaining

We consider an extension of the standard Rubinstein model where both players are randomly allowed to leave the negotiation after a rejection, in which case they obtain a payoff of known value. We show that, when the value of the outside opportunities is of intermediate size, there exist a continuum of subgame-perfect equilibrium outcomes, including some with delayed agreements. Considering outside opportunities of significant value, we prove that efficient delays arise caused by the bargainers' aspirations, in waiting for their outside, option rather than by threats. Moreover, if taking the outside option decreases the probability that the opponent receives an outside option in the future, then it is possible that exactly two equilibrium payoffs coexist. In this latter case, inefficiencies may be created by agreeing too early.     

Efficient performance in two agent bargaining

This paper concerns ex ante incentive efficient performance in the bargaining of one buyer and one seller over a single item. For any element in a specified class of distributions of the buyer's and the seller's valuations of the item, all of the (interim) individually rational and efficient allocation rules are characterized. It is also shown that the “buyer's bid” and the “seller's price” double auctions are efficient for all distributions in the specified class. These results are obtained by studying efficiency as a linear programming problem.     

Liquidity premia in dynamic bargaining markets

This paper develops a search-theoretic model of the cross-sectional distribution of asset returns, abstracting from risk premia and focusing exclusively on liquidity. In contrast with much of the transaction-cost literature, it is not assumed that different assets carry different exogenously specified trading costs. Instead, different expected returns, due to liquidity, are explained by the cross-sectional variation in tradeable shares. The qualitative predictions of the model are consistent with much of the empirical evidence.     

On continuous-time Markov processes in bargaining

For bilateral stochastic bargaining procedures embedded in stable homogeneous continuous-time Markov processes, we show unusual limit results when time between rounds vanish. Standard convergence results require that some states are instantaneous.     

Communication in bargaining over decision rights

This paper develops a model of bargaining over decision rights between an uninformed principal and an informed but self-interested agent. We introduce two different bargaining mechanisms: tacit and explicit bargaining. In tacit bargaining, an uninformed principal makes a take-it-or-leave-it price offer to the agent, who then decides whether to accept or reject the offer. In the equilibrium of the game, the principal inefficiently screens out some agent types so that the agent's private information cannot be fully utilized when the decision is made. In explicit bargaining in which parties can communicate explicitly via cheap talk before tacit bargaining, however, an equilibrium with no such inefficient screening exists even when the conflict of interest is arbitrarily large. We also follow a mechanism design approach, showing that under certain conditions, explicit bargaining is an optimal bargaining mechanism that maximizes the joint surplus of the parties.     

Malice and patience in Rubinstein bargaining

If two players playing a Rubinstein alternating offers game are highly malicious (getting a high utility from “malice” in every period when the other player does not obtain a share in a fixed pie), and highly patient, no equilibrium with an agreement exists and players choose perpetual disagreement. This does not change if the players are subjected to a known deadline after which the pie will be appropriated by outside agencies or disappear: perpetual disagreement is still the only outcome. If in addition players are required to pay endogenously determined fines if they fail to reach agreement, players with discount factors in a certain range do reach agreement, but only at the deadline. However, infinitely patient players would never reach agreement for any feasible level of one-time fines. The result contrasts with spiteful or envious preferences. Our results highlight a novel reason for failure to resolve property disputes.     

A bargaining model in general equilibrium

The goal of the paper is to present a simple model of rational endogenous household formation in a general equilibrium framework in which Pareto optimality at the economy level is not necessarily obtained. The simplest example of household formation is the case in which pairs of individuals engage themselves in a bargaining process on the division of some wealth: if an agreement on the distribution is (not) reached, we can say that the household is (not) formed. The vast majority of existing bargaining models predicts agreements on an efficient outcome. A seminal paper by Crawford (Econometrica 50:607–637, 1982) describes a very simple game with incomplete information in which, even with rational agents, disagreement causes welfare losses. We embed that model in a general equilibrium framework and present some results on equilibria both in the bargaining game and the associated exchange economy. Crawford’s results support Schelling’s intuition on the reasons of disagreement: it may arise if players’ commitments are reversible. Crawford shows that high probabilities of reversibility tend to favor the bargaining impasse, in fact with low probability. We prove that even if those probabilities are arbitrarily close to zero, disagreement is an equilibrium outcome, with high probability. That conclusion seems to be an even stronger support to Schelling’s original viewpoint. In the exchange economy model with that noncooperative bargaining game as a first stage, we present significant examples of economies for which equilibria exist. Because of disagreement, Pareto suboptimal exchange economy equilibria exist for all elements in the utility function and endowment spaces and they may coexist with Pareto optimal equilibria even at the same competitive prices.     

Wage bargaining and inflation

This paper investigates the issue of time inconsistency of monetary policy in a game with perfect information and infinite repetitions. The private sector is described by a wage negotiation process between a labour union and a firm. Unlike in the standard literature, the model has the feature that as the distortion of the economy gets larger, namely as the bargaining power of the union increases, the reputational equilibrium with low inflation becomes more likely to occur. This finding is consistent with the observation that higher levels of the labour cost are not typically associated with higher levels of inflation.