Core-equivalence for the Nash bargaining solution

Summary. Core equivalence and shrinking of the core results are well known for economies. The present paper establishes counterparts for bargaining economies, a specific class of production economies (finite and infinite) representing standard two-person bargaining games and their continuum counterparts as coalition production economies. Thereby we get core equivalence of the Nash solution. The results reconfirm the Walrasian approach to Nash bargaining of Trockel (1996). Moreover we establish the same speed of convergence as is known from Debreu (1975) and Grodal (1975) for replicated pure exchange economies and for regular purely competitive sequences of economies, respectively.Received: 13 June 2003, Revised: 13 January 2004, JEL Classification Numbers: C71, C78, D51.This article is dedicated to Birgit Grodal, a friend since 30 years.Financial support of the DFG under grant #444 USA 111/2/03 is gratefully acknowledged.     

On the convergence to the Nash bargaining solution for action-dependent bargaining protocols

We consider a non-cooperative multilateral bargaining game and study an action-dependent bargaining protocol, that is, the probability with which a player becomes the proposer in a round of bargaining depends on the identity of the player who previously rejected. An important example is the frequently studied rejector-becomes-proposer protocol. We focus on subgame perfect equilibria in stationary strategies which are shown to exist and to be efficient. Equilibrium proposals do not depend on the probability to propose conditional on the rejection by another player. We consider the limit, as the bargaining friction vanishes. In case no player has a positive probability to propose conditional on his rejection, each player receives his utopia payoff conditional on being recognized. Otherwise, equilibrium proposals of all players converge to a weighted Nash bargaining solution, where the weights are determined by the probability to propose conditional on one's own rejection.     

Symmetry-based compromise and the Nash solution to convex bargaining problems

A new axiom, symmetry-based compromise, is introduced to study solutions to convex bargaining problems. It is shown that any efficient solution that is also scale invariant and satisfies symmetry-based compromise must be Nash.     

Non-cooperative support for the asymmetric Nash bargaining solution

We study a model of non-cooperative multilateral unanimity bargaining on a full-dimensional payoff set. The probability distribution with which the proposing player is selected in each bargaining round follows an irreducible Markov process. If a proposal is rejected, negotiations break down with an exogenous probability and the next round starts with the complementary probability. As the risk of exogenous breakdown vanishes, stationary subgame perfect equilibrium payoffs converge to the weighted Nash bargaining solution with the stationary distribution of the Markov process as the weight vector.     

Search theory,competitive equilibrium,and the Nash bargaining solution

We investigate a canonical search-theoretic model without entry. Two agents are randomly matched with a long side being rationed. The matched agents face a pair of randomly drawn non-transferable payoffs, and then choose whether or not to form a partnership subject to a small probability of exogenous break down. As this probability and friction vanish, the Nash bargaining solution emerges as the unique undominated strategy equilibrium outcome if the mass of each party is the same. If the size of one party is larger than the other, the short side extracts the entire surplus, a sharp contrast to Rubinstein and Wolinsky (1985) [16].     

The multilinear extension and the coalition structure value

The multilinear extension has been shown to be an effective tool for computing the Shapley value of an n-person game. We modify here the method for the multilinear extension to calculate the modified coalition structure value (CS-value) for such games.     

The Nash bargaining solution in general n-person cooperative games

We present a noncooperative foundation for the Nash bargaining solution for an n-person cooperative game in strategic form. The Nash bargaining solution should be immune to any coalitional deviations. Our noncooperative approach yields a new core concept, called the Nash core, for a cooperative game based on a consistency principle. We prove that the Nash bargaining solution can be supported (in every subgame) by a stationary subgame perfect equilibrium of the bargaining game if and only if the Nash bargaining solution belongs to the Nash core.     

The endogenous objective function of a partially privatized firm: A Nash bargaining approach

We establish a model wherein a private firm competes with a partially privatized firm whose objective function is endogenously determined through bargaining between owners—the welfare-maximizing government and dividend-maximizing private shareholders. Many existing works on partial privatization have assumed that privatization increases the weight of profits in the partially privatized firm's objective, whereas it decreases the weight of welfare. However, our bargaining approach shows that this result can be reversed.     

Search, Nash bargaining and rule-of-thumb consumers

This paper analyses the effects of introducing two typical Keynesian features, namely rule-of-thumb (RoT) consumers and consumption habits, into a standard labour market search model. RoT consumers use the margin that hours and wage negotiation provides them to improve their lifetime utility, by narrowing the gap in utility with respect to Ricardian consumers. This margin for intertemporal optimisation has not been studied yet, because this class of restricted agents has been mainly used in models with no equilibrium unemployment. Our approach allows for a deeper study of the effects of shocks on vacancies, unemployment, hours, wages and how they interact. As habits increase, RoT consumers find it optimal, after a positive technology shock, to negotiate lower hours and higher wages, and this mechanism reduces the simulated correlation between the real wage (or productivity) and total hours to values closer to those obtained empirically. Thus, with the interaction of RoT consumers and consumption habits, the labour market search model improves significantly in reproducing some of the stylised facts characterising the US labour market.     

Implementing the Kalai-Smorodinsky bargaining solution

The bargaining game is proposed where the agents bid fractions of dictatorship that are used by all non-winners of the auction to threaten acceptance of the winner's proposal. That procedure non-cooperatively implements the Kalai-Smorodinsky bargaining solution.     

A theory of ethnic diversity and income distribution: A legislative bargaining approach

In this paper, I examine how the two dimensions of income inequality and ethnic diversity affect the formation of political coalitions and the reallocation of income. I construct a legislative bargaining model to analyze when coalitions form along the income dimension and when it forms along the ethnic dimension. The results of the model suggest that the agenda setter selects a partner with a lower aggregate income: a smaller population size and a lower income level. Further, I extend the model from a one-round to a two-round game. The extended model shows that forming an oversized coalition is a possible equilibrium outcome and is consistent with the empirical findings.     

Incomplete contract, bargaining and optimal divisional structure

When complete contracting is not possible, allocating control structure becomes the second-best arrangement. This paper analyzes the design of optimal divisional structure within an organization where ex post bargaining between the potential divisional managers is possible. In much the same light as Aghion and Tirole (J Political Econ 105(1):1?C29, 1997), we study the control problem in the context of search for projects. Our model shows that when the managers cannot bargain with one another, internal integration is preferred to internal separation. Where bargaining is possible, formal divisional structure defines both the ex post bargaining position of the two managers and their incentive to search ex ante. When the managers tend to arrive at a more favorable project to the principal via bargaining, the general leader of a firm may want to choose separation instead to increase the probability of bargaining, as the symmetrical incentive requires both managers to search and get informed.     

Unemployment insurance and union bargaining — an insider-outsider approach

A trade union is required to tax its own members to fund unemployment benefit paid to its unemployed members in an insider-outsider model of union bargaining over wages and employment. An increase in unemployment benefit imposed by the government increases employment overall but not necessarily the employment of insiders if the tax rate is exogenously fixed by the government.This paper has benefited from comments of Mick Common, Dipak Ghosh, Bob Hart, and Robin Ruffell. The comments of two anonymous referees and the advice of the Editor have greatly improved the presentation of the paper. Any remaining errors or omissions are the responsibility of the authors.     

Aristotle's analysis of bilateral exchange: an early formal approach to the bargaining problem

Exchange, as analysed by Aristotle in the Nichomachean Ethics, should be viewed as a bilateral relation to be approached not as a market phenomenon but in terms of cooperation between two contractors. This paper accordingly proposes a reconsideration of Aristotle's analysis in the light of modern bargaining theory. This reconsideration reconciles the two principles of distributive and corrective justice as ruling simultaneously exchange relations through the figures of geometric and arithmetic proportions, respectively. It also suggests a new reconstitution of the missing diagram supposedly illustrating Aristotle's analysis, which — contrary to the conventional square endowed with diagonals, used since Albertus Magnus' commentary — fits the function to which such a diagram was probably designed, that of exhibiting Aristotle's solution to the bargaining problem.     

A bargaining theory of the firm

Suppose that a firm has several owners and that the future is uncertain in the sense that one out of many different states of nature will realize tomorrow. An owner’s time preference and risk attitude will determine the importance he places on payoffs in the different states. It is a well-known problem in the literature that under incomplete asset markets, a conflict about the firm’s objective function tends to arise among its owners. In this paper, we take a new approach to this problem, which is based on non-cooperative bargaining. The owners of the firm play a bargaining game in order to choose the firm’s production plan and a scheme of transfers which are payable before the uncertainty about the future state of nature is resolved. We analyze the resulting firm decision in the limit of subgame-perfect equilibria in stationary strategies. Given the distribution of bargaining power, we obtain a unique prediction for a production plan and a transfer scheme. When markets are complete, the production plan chosen corresponds to the profit-maximizing production plan as in the Arrow–Debreu model. Contrary to that model, owners typically do use transfers to redistribute profits. When markets are incomplete, the production plan chosen is almost always different from the one in a transfer-free Drèze (pseudo-)equilibrium and again owners use transfers to redistribute profits. Nevertheless, our results do support the Drèze criterion as the appropriate objective function of the firm.     

A multi-agent bilateral bargaining model with endogenous protocol

People do bargain over how to bargain. We examine the role of individuals’ ability to pursue certain bargaining protocols in a multi-agent bilateral bargaining model. Bargaining protocols are not completely settled, but will emerge endogenously in equilibrium. We show that players’ ability to partially influence bargaining protocols plays a crucial role in determining equilibrium outcomes. When discounting is not too high, there are multiple subgame perfect equilibria, including inefficient ones. As the number of players increases, both the set of discount factors that support multiple equilibrium outcomes and the set of the first proposing player’s equilibrium payoffs expand. The maximum loss of efficiency increases with respect to the discount factor. We would like to thank Hongbin Cai, John Conlon, Andrew Daughety, Taiji Furusawa, Byoung Heon Jun, Akira Okada, Ping Wang, and two anonymous referees for their comments and suggestions. Both authors gratefully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada.     

A global Newton method to compute Nash equilibria

A new algorithm is presented for computing Nash equilibria of finite games. Using Kohlberg and Mertens’ structure theorem we show that a homotopy method can be represented as a dynamical system and implemented by Smale's global Newton method. The algorithm is outlined and computational experience is reported.     

A note on repeated-offers bargaining with one-sided incomplete information

Summary This note analyzes a modified version of the standard repeated-offers bargaining game with one-sided incomplete information studied in Fudenberg, Levine and Tirole (1985), Gul, Sonnenschein and Wilson (1986) and Ausubel and Deneckere (1989). The modification, which is introduced in the extensive form, is that the (uninformed) seller can choose to withdraw her offer immediately after the (informed) buyer accepts it. This modification is important because it removes the (implicit) commitment assumption built into the standard model that the seller is committed not to withdraw her price offer. A main result obtained is, that whether or not there is a gap between the seller's valuation and the lowest possible buyer's valuation, any seller payoff between zero and the static monopoly profit can be supported by sequential equilibria. Thus, even in the gap case there exist equilibria that completely reverse the Coase conjecture.I thank Ian Jewitt and an anonymous referee for their helpful advice and comments.     

Waiting to settle: Multilateral bargaining with subjective biases

We study multilateral bargaining games where agents disagree over their bargaining power. We show that if agents are extremely optimistic, there may be costly delays in an arbitrarily long finite game but if optimism is moderate, all sufficiently long games end in immediate agreement. We show that the game with extreme optimism is highly unstable in the finite-horizon, and we examine the ramifications of this instability on the infinite-horizon problem. Finally, we consider other voting rules, and show that the majority-rule may be more efficient than the unanimity rule when agents are optimistic.