Implementation of college admission rules

Summary We consider both Nash and strong Nash implementation of various matching rules for college admissions problems. We show that all such rules are supersolutions of the stable rule. Among these rules the lower bound stable rule is implementable in both senses. The upper bound Pareto and individually rational rule is strong Nash implementable yet it is not Nash implementable. Two corollaries of interest are the stable rule is the minimal (Nash or strong Nash) implementable solution that is Pareto optimal and individually rational, and the stable rule is the minimal (Nash or strong Nash) implementable extension of any of its subsolutions.We wish to thank Professor William Thomson for his efforts in supervision as well as his useful suggestions. We are grateful to the participants in his reading class, workshops at Bilkent University, University of Rochester, and in particular Jeffrey Banks, Stephen Ching, Bhaskar Dutta, Rangarajan Sundaram and an anonymous referee for their helpful comments.     

Exchange in a general market with indivisible goods

We show how to restrict trades in exchange markets with heterogeneous indivisible goods so that the resulting restricted exchange markets, the fixed deal exchange markets, have a unique core allocation. Our results on fixed deal exchange markets generalize classical results on the Shapley-Scarf housing market, in which each agent owns one good only. Furthermore, we define the class of fixed deal exchange rules for general exchange markets, and prove that these are the only exchange rules that satisfy strategyproofness, individual rationality, and a weak form of efficiency.     

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.     

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).     

Multiple equilibria in bargaining models of decentralized trade

Summary A market in which population size is endogenously determined is modeled. Buyers and sellers are randomly matched upon entering the market, whereupon they engage in non-cooperative bilateral bargaining. It is shown that general matching technologies and heterogeneity of buyer or seller populations often result in multiple market equilibria. Up to four equilibria can occur and the equilibria can always be ranked according to population size. Under some conditions all equilibria can also be Pareto ranked. The set of equilibria depends on the relative population sizes of different types of potential entrant, market entry costs, and the degree of differentiation between agents on the same side of the market.This paper evolved from a chapter in my dissertation. I am indebted to Leo Simon for many helpful conversations. Michael Hanemann, Jeff Perloff, Stefan Reichelstein, Jim Vercammen, Brian Wright, and participants in seminars at U.C. Berkeley, the University of Maryland, the University of British Columbia and the Australian National University also provided useful comments. Lastly, I benefited greatly from the thorough comments and corrections provided by an anonymous referee.     

On the asymptotic uniqueness of bargaining equilibria

This note reexamines the connection between the asymmetric Nash bargaining solution and the equilibria of strategic bargaining games. A first example shows non-convergence to the asymmetric Nash bargaining solution. A second example demonstrates the possibility of multiple limits.     

RISKY VERSUS RISKLESS BARGAINING PROCEDURES: THE AUMANN‐ROTH CONTROVERSY REVISITED*

In a series of papers, Aumann and Roth discussed a game in which players can cooperate in pairs and two of them prefer to form a coalition with each other. Roth argued that the only rational outcome is that the players who prefer each other form a coalition; Aumann argued that all three coalitions are possible because the players have a problem of expectation coordination. A non‐cooperative analysis provides additional support for Aumann's arguments and shows that the difference between Aumann's and Roth's views can be traced back to a difference (risky versus/riskless) in the bargaining procedure.     

Proportional concessions and the leximin solution

In this note we introduce a new axiom for bargaining solutions, named Proportional Concession Monotonicity (PCM), which imposes that no player benefits when all players collectively make proportional concessions with respect to their respective utopia values. We reconsider the leximin solution (Imai, 1983), and obtain an alternative characterization on the basis of PCM.     

Random matching in the college admissions problem

In the college admissions problem, we consider the incentives confronting agents who face the prospect of being matched by a random stable mechanism. We provide a fairly complete characterization of ordinal equilibria. Namely, every ordinal equilibrium yields a degenerate probability distribution. Furthermore, individual rationality is a necessary and sufficient condition for an equilibrium outcome, while stability is guaranteed in ordinal equilibria where firms act straightforwardly. Finally, we relate equilibrium behavior in random and in deterministic mechanisms.     

Alliances and negotiations

Negotiations often occur not between individuals but among groups (companies, trade unions, political parties, etc.) each representing composite interests, whose bargaining behaviour is regulated by some collective decision mechanism. In this paper we provide a natural model of such circumstances. We formalise the notion of ‘alliance’ and show how different preference aggregation procedures within the alliance affect the bargaining outcome. In particular, we find that unanimity procedures lead to ‘more aggressive’ negotiating tactics than majority procedures.     

A decentralized model of information pricing in networks

We propose a recursive method of pricing an information good in a network of holders and demanders of this good. The prices are determined via a unique equilibrium outcome in a sequence of bilateral bargaining games that are played by connected agents. If the information is a homogenous, non-depreciating good without network effects, we derive explicit formulae which elucidate the role of the link pattern among the players. Particularly, we find out that the equilibrium price is intimately related to the existence of cycles in the network: it is zero if a cycle covers the trading pair and it is proportional to the direct and indirect utility that the good generates otherwise.     

Representing roommates’ preferences with symmetric utilities

In the context of the stable roommates problem, it is shown that acyclicity of preferences is equivalent to the existence of symmetric utility functions, i.e. the utility of agent i when matched with j is the same as j's utility when matched with i.     

One-dimensional bargaining with Markov recognition probabilities

We study a process of bargaining over alternatives represented by points in the unit interval. The paper focuses on the asymptotic behavior of the subgame perfect equilibrium in stationary strategies as the continuation probability approaches one. We give a complete characterization of the limit of the equilibrium proposals as the generalized fixed point of the decumulative distribution of the players' ideal points as induced by the recognition probabilities. In contrast to the existing literature, we find no general relationship between the limit equilibrium proposals and either the Nash bargaining solution or the median voter outcome.     

Coase theorem, complexity and transaction costs

This paper, by introducing complexity considerations, provides a dynamic foundation for the Coase theorem and highlights the role of transaction costs in generating inefficient bargaining/negotiation outcomes. We show, when the players have a preference for less complex strategies, the Coase theorem holds in negotiation models with repeated surplus and endogenous disagreement payoffs if and only if there are no transaction costs. Specifically, complexity considerations select only efficient equilibria in these models without transaction costs while every equilibrium induces perpetual disagreement and inefficiency with transaction costs. We also show the latter is true in the Rubinstein bargaining model with transaction costs.