Weak monotonicity and Bayes–Nash incentive compatibility

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.     

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.     

Implementations of the Nash solution based on its Walrasian characterization

Summary. The present paper provides three different support results for the Nash bargaining solution of -person bargaining games. First, for any bargaining game there is defined a non-cooperative game in strategic form, whose unique Nash equilibrium induces a payoff vector that coincides with the Nash solution of the bargaining game. Next this game is modified in such a way that the unique Nash equilibrium that supports the Nash solution is even in dominant strategies. After that an -stage game in extensive form is presented whose unique subgame perfect equilibrium supports the Nash solution of the bargaining game. Finally, the support results are shown to induce implementation results in the sense of mechanism theory. Received: October 3, 1999; revised version: October 26, 1999     

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 Cournot–Nash Model of Family Decision Making

This paper models a two-person family. Each family member is utility maximising, yet family members are interdependent because of caring and public goods within the family. The two family members' interdependent utility maximisation problems are first solved using a non-cooperative, or Cournot–Nash, game theoretic framework. The Cournot–Nash equilibrium is then used as a threat point in a bargaining game. The paper provides a rigorous derivation of the properties of household demands, a full analysis of the determinants of intra-household resource allocation, including the effect of varying household bargaining power, and consideration of policy implications.     

Nash equilibrium when players account for the complexity of their forecasts

Nash equilibrium is often interpreted as a steady state in which each player holds the correct expectations about the other players' behavior and acts rationally. This paper investigates the robustness of this interpretation when there are small costs associated with complicated forecasts. The model consists of a two-person strategic game in which each player chooses a finite machine to implement a strategy in an infinitely repeated 2×2 game with discounting. I analyze the model using a solution concept called Nash Equilibrium with Stable Forecasts (ESF). My main results concern the structure of equilibrium machine pairs. They provide necessary and sufficient conditions on the form of equilibrium strategies and plays. In contrast to the “folk theorem,” these structural properties place severe restrictions on the set of equilibrium paths and payoffs. For example, only sequences of the one-shot Nash equilibrium can be generated by any ESF of the repeated game of chicken.     

Management contribution and the allocation of time in a labor-managed firm

The problem of time allocation in a labor-managed firm is considered under a technology specification in which the role of management in the production process is different from that of “ordinary” members. Two allocation schemes are discussed, the egalitarian and the Nash equilibrium. It is shown that in the model discussed the Nash solution brings about an output level that is lower than the Pareto-optimal output level. If the manager seeks employment opportunities outside the cooperative that reflect his success as manager, he increases his time input so that the output of the cooperative and the “ordinary” members' utility levels increase.     

Symmetrically pairwise-bargained allocations in an assignment market

Consider an assignment game with transferable utility where the optimally assigned partners engage in bargaining of the sort modelled by Nash, using as their threats the maximum they could receive in an alternative match. A symmetrically pairwise-bargained (SPB) allocation is a core allocation such that all partners are in bargained equilibrium. It is shown that an SPB allocation always exists, that the set of SPB allocations coincides with the intersection of the kernel and the core, and that there is a rebargaining process which converges to an SPB allocation if it begins at a “distinguished point” in the core.     

Existence and Uniqueness of Ordinal Nash Outcomes

In this paper we present necessary and sufficient conditions for existence and uniqueness of ordinal Nash outcomes. These outcomes are derived from the ordinal Nash solution—a reinterpretation and an extension of the Nash bargaining solution that allows bargainers to have preference relations that are more general than expected utility. Our task is undertaken by the construction of a new notion called “induced utilities”. Journal of Economic Literature Classification Number: C78.     

Coalitional bargaining games with random proposers: Theory and application

We consider a non-cooperative coalitional bargaining game with random proposers in a general situation for which players differ in recognition probability and time preference. We characterize an efficient equilibrium as the generalized Nash bargaining solution that belongs to the core. The model is applied to wage bargaining between an employer and multiple workers. Although involuntary unemployment may occur in equilibrium, full employment emerges as players become sufficiently patient.     

Codetermination,Price Competition and the Network Industry

This research develops a tractable two‐stage non‐cooperative game with complete information describing the behaviour of price‐setting firms that must choose to be profit maximisers or bargainers under codetermination in a network industry with horizontal product differentiation. The existing theoretical literature has already shown that codetermination might arise as the endogenous market outcome in a strategic competitive quantity‐setting duopoly. In sharp contrast with this result, the present article shows that codetermination does never emerge as a Nash equilibrium in a price‐setting non‐network duopoly. Then, it aims at highlighting the role of network externalities in determining changes of paradigm of the game and letting codetermination become a sub‐game perfect Nash equilibrium when prices are strategic substitutes or strategic complements. This equilibrium may be Pareto efficient. Results allow distinguishing between mandatory codetermination and voluntary codetermination. The article also proposes a model of endogenous codetermination according to which every firm may choose to bargain with its own corresponding union bargaining unit only whether the firm's bargaining strength is exactly the profit‐maximising one. The equilibrium outcomes emerging in this case range from a uniform Nash equilibrium, in which both firms are codetermined, to mixed Nash equilibria, in which only one of them chooses to be codetermined. These results are ‘network depending’ and do not hold in a non‐network duopoly.     

Nash consistent representation of effectivity functions through lottery models

Effectivity functions for finitely many players and alternatives are considered. It is shown that every monotonic and superadditive effectivity function can be augmented with equal chance lotteries to a finite lottery model—i.e., an effectivity function that preserves the original effectivity in terms of supports of lotteries—which has a Nash consistent representation. The latter means that there exists a finite game form which represents the lottery model and which has a Nash equilibrium for any profile of utility functions satisfying the minimal requirement of respecting first order stochastic dominance among lotteries. No additional condition on the original effectivity function is needed.     

Consistency of beliefs and epistemic conditions for Nash and correlated equilibria

A consistency condition (action-consistency) on the interim beliefs of players in a game is introduced. Action-consistency is weaker than common priors and, unlike common priors, is characterized by a “no-bets” condition on verifiable events. Using action-consistency, we provide epistemic conditions to Nash and correlated equilibria weakening the common knowledge restrictions in Aumann and Brandenburger [Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180] and Aumann [Aumann, R., 1987. Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1–18].     

A Pie Allocation Among Sharing Groups

This paper deals with a pie allocation problem to users, who are divided intosharing groups, that are not necessarily disjoint. The pie is allocated among the groups, rather than the users. The portion of a sharing group is a public good for its members, that is, all of them can enjoy it. A non-cooperative game for allocating the pie and a special equilibrium outcome,strongly disappointment-free allocation, are defined. We prove the existence of a unique such allocation and show that it coincides with the Nash solution of the related bargaining problem.Journal of Economic LiteratureClassification Numbers: C70, C71, C72 and C78.     

Sociological versus strategic factors in bargaining

Most game-theoretic models of strategic interaction, indeed most economic models of any sort, specify potential outcomes entirely in terms of the preferences of the agents, as captured in their (possibly cardinal) utility functions. The underlying assumption is that the outcome of such interactions is determined entirely by these preferences, together with the strategic possibilities available to the agents. The purpose of this paper is to challenge the adequacy of this assumption in general, by investigating it in the specific context of two-person bargaining. In particular, we consider whether certain experimental results reported earlier can be accounted for strictly in terms of players' preferences and strategic possibilities, and we report a new experimental study designed to answer this question. The results strongly support the conclusion that sociological factors, unrelated to what we normally consider to be the ‘economic’ parameters of a game, can decisively influence the outcome of bargaining, in a systematic manner.     

Run equilibria in the Green–Lin model of financial intermediation

We study the Green–Lin model of financial intermediation [E.J. Green, P. Lin, Implementing efficient allocations in a model of financial intermediation, J. Econ. Theory 109 (2003) 1–23] under a more general specification of the distribution of types across agents. We derive the efficient allocation in closed form. We show that, in some cases, the intermediary cannot uniquely implement the efficient allocation using a direct revelation mechanism. In these cases, the mechanism also admits an equilibrium in which some (but not all) agents “run” on the intermediary and withdraw their funds regardless of their true liquidity needs. In other words, self-fulfilling runs can arise in a generalized Green–Lin model and these runs are necessarily partial, with only some agents participating.     

Trading networks with price-setting agents

In a wide range of markets, individual buyers and sellers trade through intermediaries, who determine prices via strategic considerations. Typically, not all buyers and sellers have access to the same intermediaries, and they trade at correspondingly different prices that reflect their relative amounts of power in the market. We model this phenomenon using a game in which buyers, sellers, and traders engage in trade on a graph that represents the access each buyer and seller has to the traders. We show that the resulting game always has a subgame perfect Nash equilibrium, and that all equilibria lead to an efficient allocation of goods. Finally, we analyze trader profits in terms of the graph structure — roughly, a trader can command a positive profit if and only if it has an “essential” connection in the network, thus providing a graph-theoretic basis for quantifying the amount of competition among traders.     

Weak monotonicity and Bayes–Nash incentive compatibility

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.     

An efficient solution to the informed principal problem

In this paper I study mechanism design by an informed principal. I show that generically this problem has an ex-post efficient solution. In the equilibrium mechanism, the informed principal appropriates all expected social surplus, with each type of her getting all expected social surplus conditional on that type. This outcome is supported as a perfect sequential equilibrium of the informed principal game when the joint probability distribution from which the agents’ types are drawn satisfies two conditions: the well-known condition of Cremer and McLean and Identifiability condition introduced by Kosenok and Severinov [Individually rational, budget-balanced mechanisms and allocation of surplus, J. Econ. Theory (2002), forthcoming]. Conversely, these conditions are necessary for an ex-post efficient outcome to be attainable in an equilibrium of the informed principal game. Under these conditions only our equilibrium outcome constitutes a neutral optimum, i.e. cannot be eliminated by any reasonable concept of blocking [R. Myerson, Mechanism design by an informed principal, Econometrica 51 (1983) 1767-1797]. Identifiability and Cremer-McLean conditions are generic when there are at least three agents, and none of them has more types than the number of type profiles of the other agents.