On the Nash bargaining solution with noise

Suppose two parties have to share a surplus of random size. Each of the two can either commit to a demand prior to the realization of the surplus - as in the Nash demand game - or remain silent and wait until the surplus was observed. Adding the strategy to wait results in two strict equilibria, in each of which one player takes almost the whole surplus, provided uncertainty is small. If commitments concern only who makes the first offer, the more balanced Nash bargaining solution is approximately restored. In all cases, commitment occurs in equilibrium - despite the risk of breakdown of negotiations.     

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.     

From ultimatum to Nash bargaining: Theory and experimental evidence

We consider a sequential two-party bargaining game with uncertain information transmission. When the first mover states her demand she does only know the probability with which the second mover will be informed about it. The informed second mover can either accept or reject the offer and payoffs are determined as in the ultimatum game. Otherwise the uninformed second mover states his own demand and payoffs are determined as in the Nash demand game. In the experiment we vary the commonly known probability of information transmission. Our main finding is that first movers’ and uninformed second movers’ demands adjust to this probability as qualitatively predicted, that is, first movers’ (uninformed second movers’) demands are lower (higher) the lower the probability of information transmission. JEL Classification C72 · C78 · C92     

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.     

Distortion of preferences and the Nash theory of bargaining

It is shown that in Nash bargaining over division of a single good, when agents are allowed to distort their von Neumann-Morgenstern utility functions into any (weakly) concave form, reporting linear utility functions constitutes a unique dominant-strategy Nash equilibrium.     

A general equilibrium approach to wage bargaining and employment

This paper attempts to capture the effects of cyclical fluctuations in business activity on factor prices and employment levels in a two-sector model with wage bargaining.     

Extending the Nash solution to choice problems with reference points

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by “unanimity for the grand coalition” and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms – including independence of irrelevant alternatives – may be logically independent of the remaining axioms.     

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.     

An implementation of the Owen value

We present an implementation of the Owen value (Owen, 1977, Essays in Mathematical Economics and Game Theory, pp. 76–88), inspired by the bidding mechanism introduced by Pérez-Castrillo and Wettstein (2001, J. Econ. Theory 100, 274–294). The idea is the following: first players of each coalition play the bidding mechanism trying to obtain the resources of the coalition. Later, players who obtained the resources play the bidding mechanism in order to share the surplus.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

Modeling longevity risk transfers as Nash bargaining problems: Methodology and insights

The problem of longevity risk has recently received considerable attention. In this paper, we apply economic modeling methods to longevity risk securitization, which is now regarded by pension and insurance industries as a solution to the problem. Specifically, we model the trade of a longevity security as a two-player bargaining game, and use Nash's bargaining solution to determine the outcome of it. Our work not only offers an alternative method for pricing longevity securities, but also reveals several properties about the market for longevity securities. First, a trade would occur if the longevity security is an effective hedging instrument, and the trade would benefit all agents involved. Second, a trade of longevity risk can reduce pension plans' bankruptcy risk, safeguarding the financial security of pension plan members. Finally, compared to the competitive equilibrium, Nash's bargaining solution yields higher trading prices. Therefore, as the market becomes more competitive, pension plans may hedge longevity risk at a lower cost.