Bargaining with endogenous disagreement: The extended Kalai-Smorodinsky solution

Following Vartiainen (2007) we consider bargaining problems in which no exogenous disagreement outcome is given. A bargaining solution assigns a pair of outcomes to such a problem, namely a compromise outcome and a disagreement outcome: the disagreement outcome may serve as a reference point for the compromise outcome, but other interpretations are given as well. For this framework we propose and study an extension of the classical Kalai-Smorodinsky bargaining solution. We identify the (large) domain on which this solution is single-valued, and present two axiomatic characterizations on subsets of this domain.     

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.     

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.     

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.     

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.     

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].     

A bargaining approach to the Owen value and the Nash solution with coalition structure

The mechanism by Hart and Mas-Colell (1996) for non-transferable utility (NTU) games is generalized so that a coalition structure among players is taken into account. The new mechanism yields the Owen value for transferable utility (TU) games with coalition structure as well as the consistent value (Maschler and Owen 1989, 1992) for NTU games with trivial coalition structure. Furthermore, we obtain a solution for pure bargaining problems with coalition structure which generalizes the Nash (1950) bargaining solution.     

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.     

Implementing cooperative solution concepts: a generalized bidding approach

This paper provides a framework for implementing and comparing several solution concepts for transferable utility cooperative games. We construct bidding mechanisms where players bid for the role of the proposer. The mechanisms differ in the power awarded to the proposer. The Shapley, consensus and equal surplus values are implemented in subgame perfect equilibrium outcomes as power shifts away from the proposer to the rest of the players. Moreover, an alternative informational structure where these solution concepts can be implemented without imposing any conditions of the transferable utility game is discussed as well. The authors thank Yukihiko Funaki, Andreu Mas-Colell, David Pérez-Castrillo and Jana Vyrastekova for helpful discussions. We also appreciate the comments from the seminar and conference participants at Keele University, Tilburg University, University of Haifa, University of Warwick, CORE at Louvain-la-Neuve, Catholic University Leuven, Pablo de Olavide University in Seville, University of Vigo, the 2006 Annual Conference of the Israeli Mathematical Union in Neve Ilan, Israel, and the 61st European Meeting of the Econometric Society in Vienna, Austria in 2006. In particular, we are grateful to the associate editor and an anonymous referee. Their valuable comments and constructive suggestions contributed to a significant improvement of the paper. Wettstein acknowledges the financial support of the Pinchas Sapir Center for Development in Tel Aviv University.     

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.     

Proto-coalition bargaining and the core

In the proto-coalition model of government formation, formateur F appoints a proto-coalition and asks its members whether to start negotiating a coalition contract. If all accept, then the proto-coalition forms and starts negotiating; otherwise, a caretaker government assumes office. I extend this model by allowing F to revise the chosen proto-coalition after rejections, that he states pre-conditions for the subsequent negotiations, and that F??s opponents may publicly pre-commit to accept/reject certain proposals. The set of equilibrium outcomes is identified as the core if F??s opponents can pre-commit and as the convex hull of the core if they cannot pre-commit credibly. This extended model eliminates two flaws of the standard model: it explains why F cannot always install his favored coalition (whatever the status quo) and why ??important?? coalition members may have more bargaining power in the subsequent negotiations than others.     

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.     

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.     

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.     

Demand bargaining in legislatures

We analyze a pure bargaining problem when decisions require simple majority and self interested players make unilateral demands. In contrast to the case where proposals consist of complete sharing profiles, this content of proposals prevents implicit side-payments inside the committee, and so non-discriminatory outcomes might be expected. We show that in some settings this is the case and the surplus is universally shared.     

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.     

Macroeconomic fluctuations and bargaining

I study the limit rule for bilateral bargaining when agents recognize that the aggregate economy (and thus the match surplus) follows a finite-state Poisson process. The rule derived in this paper is of special importance for decentralized exchange economies with bargaining. Two simple applications are presented to illustrate this fact. The first example is a model of wage bargaining and trade externalities. I show that in such situations sophisticated bargaining tends to increase the volatility of the wage bill. The second example is based on the Kiyotaki-Wright model of money. I explain how equilibrium prices depend in a fundamental way on the dynamic bargaining solution.     

Sharpen your bargaining skills

As the health-care revolution continues, bargaining skills have become more crucial for doctors than ever before. If you want your practice to thrive, you must regularly negotiate with colleagues, practice administrators, allied health professionals, managed-care plans, business groups, hospitals, and the government. Yet medical schools and residency programs devote little if any time to negotiating skills. The "golden days," when physicians could afford to be mavericks, are over. Doctors must develop a new collaborative outlook to be successful and happy in their careers, says Leonard J. Marcus, Ph.D., lead author of the book "Renegotiating Health Care: Resolving conflict to Build Collaboration." He and his co-authors often conduct seminars for health professionals on improving bargaining skills. The following excerpt concerns the role of conflict in negotiation, and how to resolve it.     

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).