Stable and efficient bargaining networks

Abstract. We examine a bargaining game among players connected by a network, where successively potential partners negotiate over terms of a project. For any network structure and ordering of selection of bargaining partners, there is a unique subgame perfect equilibrium outcome which coincides with the standard Rubinstein split. Based on this, we derive expected payoffs for players as a function of the network. We the characterize the pairwise stable and efficient bargaining networks. These two sets do not always coincide and in many situations pairwise stable networks are over-connected. Received: 10 April 2001 / Accepted: 17 January 2002 I am indebted to Matthew Jackson for suggesting this paper. I also thank Bernard Caillaud, Sanjeev Goyal, and the seminar participants at Universidad de Alicante, Universidad Carlos III de Madrid, CEMFI (Madrid), Universitat Autònoma de Barcelona, Erasmus University of Rotterdam, Universitat Pompeu Fabra (Barcelona) and CERAS (Paris) for helpful comments. This paper was written while at Universitat Pompeu Fabra and CERAS. Financial support from the Spanish Ministry of Education through research grant DGESIC PB96-0302, and from the Ecole Nationale des Ponts et Chaussées, Paris is gratefully acknowledged. All errors are of course mine.     

A note on van Damme's mechanism

We consider (two-person) bargaining games and we assume that the agents want to apply two possibly different bargaining solutions. A mechanism is a function which assigns an allocation to every bargaining game and every pair of bargaining solutions. Examining van Damme's mechanism (1986) from a cooperative point of view we see that it fails to satisfy Pareto-optimality. By modifying it we propose a new mechanism yielding the same conclusions as the original while enjoying some additional desirable properties. Received: 8 August 1998 / Accepted: 15 October 1998     

Multi-agent bilateral bargaining and the Nash bargaining solution

This paper studies a bargaining model where n   players negotiate how to share a pie through (n−1)$(n-1)$ bilateral bargaining sessions. In each session, two players bargain for a partial agreement that specifies who exits and who moves on to the next session (if there is any) via the alternating-proposal framework of Rubinstein [Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–109]. We consider two bargaining procedures under which the subgame perfect equilibrium outcomes converge to the Nash [Nash, J., 1950. The bargaining problem. Econometrica 18, 155–162] bargaining solution for the corresponding bargaining problem as the players’ discount factor goes to one. Hence, the model studied here provides a non-cooperative foundation for the Nash cooperative bargaining solution in the multilateral case.     

Achieving the first best in sequencing problems

Abstract. In a sequencing problem with linear time cost, Suijs (1996) proved that it is possible to achieve first best. By first best we mean that one can find mechanisms that satisfy efficiency of decision, dominant strategy incentive compatibility and budget balancedness. In this paper we show that among a more general and natural class of sequencing problems, sequencing problems with linear cost is the only class for which first best can be achieved. Received: 2 December 1999/Accepted: 9 May 2001     

On preferences over subsets and the lattice structure of stable matchings

This paper studies the structure of stable multipartner matchings in two-sided markets where choice functions are quotafilling in the sense that they satisfy the substitutability axiom and, in addition, fill a quota whenever possible. It is shown that (i) the set of stable matchings is a lattice under the common revealed preference orderings of all agents on the same side, (ii) the supremum (infimum) operation of the lattice for each side consists componentwise of the join (meet) operation in the revealed preference ordering of the agents on that side, and (iii) the lattice has the polarity, distributivity, complementariness and full-quota properties. Received: 5 March 1999 / Accepted: 12 May 2000     

Constitutional implementation

We consider the problem of implementing a social choice correspondence H in Nash equilibrium when the constitution of the society is given by an effectivity function E. It is assumed that the effectivity function of , is a sub-correspondence of E. We found necessary and efficient conditions for a game form to implement H (in Nash equilibria), and to satisfy, at the same time, that , the effectivity function of , is a sub-correspondence of (which guarantees that is compatible with E). We also find sufficient conditions for the coincidence of the set of winning coalitions of and , and for . All our results are sharp as is shown by suitable examples. Received: 15 December 2000 / Accepted: 3 September 2001     

Natural and double implementation of public ownership solutions in differentiable production economies

This paper examines the implementation of two public ownership solutions in convex production economies with differentiable production functions. The two public ownership solutions we focus on are the proportional and equal benefit solutions. Two “natural” mechanisms which doubly implement the proportional and equal benefit solutions respectively in Nash and strong Nash equilibria are proposed without assuming free disposal. Received: 7 May 1996 / Accepted: 1 December 1998     

Stable matchings and the small core in Nash equilibrium in the college admissions problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes in truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium in truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how different are players' equilibrium strategies from the true preferences and how many other unstable matchings are there at that Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by the direct revelation game induced by a stable mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching. Received: 30 December 1998 / Accepted: 12 October 2001 This paper is a revision of the paper “Manipulation and Stability in a College Admissions Problem” circulated since 1994. I thank Rich McLean, Abraham Neyman, Mark Satterthwaite, Sang-Chul Suh, and Tetsuji Yamada for helpful discussions. I thank the associate editor and the two anonymous referees for their helpful comments that have greatly improved the paper. I am grateful to the Kellogg G.S.M. at the Northwestern University for the hospitality for my visit. Any errors are mine.     

Buyers' and sellers' cartels on markets with indivisible goods

This paper analyzes the formation of cartels of buyers and sellers in a simple model of trade inspired by Rubinstein and Wolinsky's (1990) bargaining model. When cartels are formed only on one side of the market, there is at most one stable cartel size. When cartels are formed sequentially on the two sides of the market, there is also at most one stable cartel configuration. Under bilateral collusion, buyers and sellers form cartels of equal sizes, and the cartels formed are smaller than under unilateral collusion. Both the buyers' and sellers' cartels choose to exclude only one trader from the market. This result suggests that there are limits to bilateral collusion, and that the threat of collusion on one side of the market does not lead to increased collusion on the other side.     

Single-peakedness and coalition-proofness

We prove that multidimensional generalized median voter schemes are coalition-proof. Received: 16 July 1998 / Accepted 23 March 1999     

Multiprincipals multiagents incentive design

This paper studies a simple setting in which the contractual arrangements which determine the incentives for agents are not designed by a single central planner, but are themselves the outcome of a game among multiple noncooperatively acting principals. The notion of an Epsilon Contracting Equilibrium is introduced to predict the outcome of the contract-design game among principals. Symmetric pure strategy Epsilon Contracting Equlibria may not exist in perfectly symmetric environments. In a symmetric Epsilon Contracting Equilibrium in mixed strategies coordination failure may lead to a suboptimal institutional network in which the agents “cheat” their principals. Received: 14 June 1995 / Accepted: 09 August 1999     

A study of proportionality and robustness in economies with a commonly owned technology

Abstract. Studying one-input one-output economies, we say that an allocation is proportional if the input-output ratio is identical among agents and if each agent maximizes her welfare given this ratio. We propose three equity axioms based on this definition, and we use them to compare the main solutions to this simple equity problem. We also combine efficiency, robustness axioms and our proportionality axioms to characterize two solutions. Received: 11 June 1997 / Accepted: 26 May 2000     

Abstention and political competition

The classical literature on spatial majority voting postulates that all citizens vote. The Median Voter Theorem (MVT) then obtains when parties have perfect information on voter behavior and are either office-seekers (“Downsian”) or ideological. This paper introduces abstention, a simple yet realistic modification. We show that the main features of the MVT survive to a large extent but subject to some qualifications. First, the winning policy does not bear any necessary relation to the median voter. Second, there exist examples in which the candidates choose different positions at equilibrium. Third, equilibrium may fail to exist or be unique. Finally, the equilibria of the model with office-motivated parties may differ from the ones where parties are ideological. Received: 18 September 1999 / Accepted: 31 January 2000     

Equilibrium in a market with intermediation is Walrasian

We show that a profit maximizing monopolistic intermediary may behave approximately like a Walrasian auctioneer by setting bid and ask prices nearly equal to Walrasian equilibrium prices. In our model agents choose to trade either through the intermediary or privately. Buyers (sellers) trading through the intermediary potentially trade immediately at the ask (bid) price, but sacrifice the spread as gains. A buyer or seller who trades privately shares all the gains to trade with this trading partner, but risks costly delay in finding a partner. We show that as the cost of delay vanishes, the equilibrium bid and ask prices converge to the Walrasian equilibrium prices. Received: 2 February 1996 / Accepted: 28 March 1997     

Implementation of stable solutions in a restricted matching market

I analyze the admission mechanism used in Spanish universities. The system is open to strategic manipulation. This is because students are not allowed to express the whole list of available options. However, the mechanism implements the set of stable matchings in Nash equilibrium and the student's optimum in strong equilibrium. The mechanism also implements the students' optimum, in Nash equilibrium, under the class of “non-reverse” preferences. All these properties come from the fact that colleges do not have the opportunity to misrepresent their preferences. Received: 30 June 1995 / Accepted: 31 January 1997     

A characterization of the existenceof optimal dominant strategy mechanisms

This paper provides two theorems which characterize the domains of valuation functions for which there exist Pareto efficient and truth dominant strategy mechanisms (balanced Groves mechanisms). Theorem 1 characterizes the existence of balanced Groves mechanisms for a general class of valuation functions. Theorem 2 provides new balance-permitting domains of valuation functions by reducing the problem of solving partial differential equations to the problem of solving a polynomial function. It shows that a balanced Groves mechanism exists if and only if each valuation function in the family under consideration can be obtained by solving a polynomial function with order less than , where n is the number of individuals. Received: 5 January 1997 / Accepted: 25 May 1999     

Strategyproof multiple assignment using quotas

The allocation of heterogeneous and indivisible objects is considered where there is no medium of exchange. We characterize the set of strategyproof, nonbossy, Pareto-optimal, and neutral social choice functions when preferences are monotonic and quantity-monotonic. The characterized sets of social choice functions are sequential; agents are assigned their favorite objects among the objects not given to others before them, subject to a quota. Received: 5 January 1999 / Accepted: 14 June 1999     

The not-so-secret-agent: Professional monitors,hierarchies and implementation

Abstract. It is well-known that, when agents in an organization possess private information that is unverifiable by an outside party, games where agents simply announce their information can have multiple equilibria that may impede the successful implementation of the organization's objectives. We show that the introduction of a professional monitor (e.g. auditor, regulator, supervisor) can help to destroy the “bad' equilibria when agents have private information but have incomplete info rmation about others' information. Received: 21 May 1998 / Accepted: 26 May 2000     

On efficient trading mechanisms between one seller and n buyers

Myerson and Satterthwaite (1983) prove that if one seller and one buyer have independent private valuations for an indivisible object then no individually rational and incentive compatible trading mechanism can guarantee ex post efficiency when gains from trade are uncertain. Makowski and Mezzetti (1993) show that this is not the case when there are at least two buyers. In the latter context, if the highest possible seller's valuation is not too large, we provide an ex post efficient mechanism in which the mechanism designer and the agents are not required to know the probability distribution for the seller's valuation. Received: February 18, 1998 / Accepted: September 10, 1999     

A noncooperative approach to the Mas-Colell bargaining set

We propose an extensive form game to give a noncooperative interpretation of the Mas-Colell bargaining set for transferable utility games. Based on the objection and counterobjection in the sense of the Mas-Colell bargaining set, a bilateral negotiation is designed in our model. We show that the equilibrium outcomes of our extensive form game coincide with the Mas-Colell bargaining set in the supperadditive environments.