Changes in risk and strategic interaction

In this paper we study the comparative statics of Nth degree stochastic dominance shifts in a large class of non-cooperative games. We consider symmetric equilibria as well as asymmetric equilibria in which the risk changes are idiosyncratic and not necessarily of the same stochastic order. Furthermore, we establish conditions for risk changes to produce multiplier effects on equilibrium strategies. Finally, we evaluate the comparative statics of stochastic dominance shifts in supermodular games, which may feature multiple equilibria and non-convex strategy sets.     

Is having a unique equilibrium robust?

We investigate whether having a unique equilibrium (or a given number of equilibria) is robust to perturbation of the payoffs, both for Nash equilibrium and correlated equilibrium. We show that the set of n  -player finite games with a unique correlated equilibrium is open, while this is not true of Nash equilibrium for n>2$n>2$. The crucial lemma is that a unique correlated equilibrium is a quasi-strict Nash equilibrium. Related results are studied. For instance, we show that generic two-person zero-sum games have a unique correlated equilibrium and that, while the set of symmetric bimatrix games with a unique symmetric Nash equilibrium is not open, the set of symmetric bimatrix games with a unique and quasi-strict symmetric Nash equilibrium is.     

Pre-play communication in procurement auctions: Silence is not golden

I study the effect of cheap talk between bidders on the outcome of a first-price procurement auction in which participation is costly. Although no side payments or commitments are allowed, there exists a family of equilibria in which sellers use communication to collude on a subset of participants and/or reveal information about their cost. Cheap talk matters in the sense that it strictly enlarges the set of Nash equilibria (symmetric and asymmetric) and the set of public correlated equilibria of the game. I show that the buyer may benefit from cheap talk between sellers and that the surplus increases in the amount of information revealed in equilibrium under one fairly general condition. This is because when communication is cheap, sellers cannot directly collude on higher prices. Rather, communication leads to competition between fewer, but more aggressive bidders, which entails greater allocative efficiency and a decrease in the total wasteful entry cost.     

Stable risk-sharing

We analyze the evolution of contract participation and evaluate the selection of risk-sharing contracts in the presence of moral hazard. Organizations specify rules for sharing output among producers, and so affect the extent of private investment in production. Organizations are rigid, as some details of the contract are fixed, but people are free to move around. In the presence of rigidity, equilibrium displays coordination failure because potentially efficient contracts can fail to attract participants. Methods of evolutionary stability are used to select equilibria when organizations compete for members. We identify stable contracts which survive competition against any other. Stable contracts need not be efficient, but for large groups the loss becomes small.     

Equilibrium selection in games: the mollifier method

This paper introduces an embedding of a Nash equilibrium into a sequence of perturbed games, which achieves continuous differentiability of best responses by mollifying them over a continuously differentiable density with compact support (window size). Along any sequence with shrinking window size, the equilibria are single-valued whenever the function has a regular Jacobian and the set of equilibria where it is singular has measure zero. We achieve a further reduction of the equilibrium set by inserting within the embedding a procedure that eliminates the strict interior of equilibrium sets. The mollifier can be approximated consistently using kernel density regression, and we sketch a non-stationary stochastic optimization algorithm that uses this approximation and converges with probability one to an equilibrium of the original game.     

Taxation in an economy with private provision of public goods

This paper analyses the effects of taxation and subsidies in an economy with private provision of a public good. It is shown that in a situation where all individuals contribute, taxation affects the equilibrium allocation if and only if at least one individual's voluntary contribution to the public good has an impact on the aggregate tax payments of the others. We then consider linear nonneutral tax-subsidy schemes and analyse efficiency and uniqueness of the resulting Nash equilibria. We show that an efficient Nash equilibrium, where all individuals contribute, will in general not be unique, and establish a non-uniformity property which a tax-subsidy scheme must fulfil in order to induce a unique interior equilibrium that is efficient. Throughout the paper it is assumed that individuals fully understand and take into account the government's budget constraint. Received: 3 November 1997 / Accepted: 23 March 1999     

Consistent collusion-proofness and correlation in exchange economies

We present a feasible strategic market mechanism with finitely many agents whose Nash, semi-strong Nash and coalition-proof Nash equilibria fully implement the Walrasian equilibria. We define a strategic equilibrium concept, called correlated semi-strong equilibrium, and show that the Walrasian equilibria can be implemented by these equilibria, and also by the coalition-proof correlated equilibria of our mechanism. We show that these two concepts, suitably modified with transfers, fully implement the Pareto optimal allocations.     

On the existence,uniqueness and stability of spatial equilibrium in an open city with externalities

This paper deals with the existence, uniqueness and stability of a spatial equilibrium in an open city with external diseconomies like air pollution generated by manufacturing activities. First, assuming that both production functions and utility functions are Cobb-Douglas, we prove the existence of a spatial equilibrium under some reasonable assumptions. It is shown, however, that the uniqueness and stability of the equilibrium may not be obtained, unless the degree of externality is sufficiently small. In fact, none of the equilibria will be stable for a certain set of speeds of adjustment in the labor market and the land market, if the degree of externality is relatively large. Finally, some implications of our results are derived in regard to the application of the hedonic price concept within the context of a spatial equilibrium model.     

Symmetry and continuity properties of Lindahl equilibria

In this paper it is shown that: (1) some non-symmetric Lindahl equilibria may exist; (2) under usual assumptions, the set of symmetric Lindahl equilibria is non-empty; (3) however, the correspondence which associates with every economy the set of symmetric Lindahl equilibria is not a closed correspondence; (4) there are economies with a measure space of economic agents in which every Lindahl equilibrium is discontinuous.     

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.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Double implementation of the Lindahl equilibrium by a continuous mechanism

We consider a class of economies with public goods that have the following properties: (i) The preferences of the agents are convex, interior, and strictly increasing. (ii) The technology for production of public goods is a closed convex cone that satisfies free disposal and an additional mild assumption. No assumptions are made on continuity, completeness or transitivity of preferences. We provide a continuous and feasible mechanism that implements the Lindahl equilibrium by Nash equilibria, and has the following property: For every economy in our class every Nash equilibrium of the game induced by the mechanism is a strong Nash equilibrium.     

The revelation of information in strategic market games: A critique of rational expectations equilibrium

We criticize the R.E.E. approach to asymmetric information general equilibrium because it does not explain how information gets ‘into’ the prices. This leads to well-known paradoxes. We suggest a multiperiod game instead, where the flow of information into and out of prices is explicitly modeled. In our game Nash equilibria (N.E.) (1) generalize Walrasian equilibria to asymmetric information, (2) exist generically, (3) eliminate pure speculation, (4) allow prices to reveal information and markets to become more efficient over time, (5) are consistent with the weak efficient markets hypothesis that tracking past prices is not profitable, (6) yet always lead to higher utility for better informed agents (such as experts). Throughout the paper we use one concrete game. In the last section we prove that there is a broad range of games that would have the same properties.     

Counting Groves-Ledyard equilibria via degree theory

We study the Groves-Ledyard mechanism for determining optimal amounts of public goods in economies whose agents have the most general class of preferences for which a Pareto amount of public goods can be computed independently of income distribution. We use degree theory on affine spaces to show that the number of equilibria in such economies grows exponentially as the number of agents in the economy increases. The large number of equilibria in such simple economic models raises doubts as to whether the Groves-Ledyard mechanism is a workable solution to the Free Rider Problem since individuals may have incentives to falsify their preferences in order to drive the adjustment process to a preferred Nash equilibrium.     

Coalitional stability in the location problem with single-dipped preferences: An application of the minimax theorem

This study considers a situation in which agents choose the location of a public facility from a street according to a given mechanism. Agents have single-dipped preferences over a set of feasible locations. We analyze coalitional behavior for any given mechanism for this situation. We identify a necessary and sufficient condition for a mechanism to possess a strong Nash equilibrium by applying the minimax theorem of von Neumann and Morgenstern (1944). We introduce a class of core solutions and show that these solutions are characterized by strong Nash implementability. As a byproduct of these results, we propose a simple mechanism that implements any core solution in strong Nash equilibria.     

A decentralized and informationally efficient mechanism realizing fair outcomes in economies with public goods

We present a decentralized mechanism (called Lindahl Egalitarian), which yields Pareto efficient and envy free allocations (i.e. fair outcomes). We show that the mechanism is informationally efficient in general production economies with an arbitrary, but finite, number of private and public goods, and a finite number of agents. The mechanism reduces to the Walrasian mechanism starting from equal wealth when no agent cares about public goods. We also prove that the set of Public Competitive equilibrium allocations (from equal endowments and proportional taxation), and the set of the Lindahl Egalitarian equilibrium allocations are the same. We are grateful to Xavier Calsamiglia and Albert Marcet for helpful conversations, and to A. de la Fuente, I. Macho, and an anonimous referee for useful suggestions. A. Manresa’s research has been supported by the grant CICYT PB90-0172. J. Aizpurua acknowledges financial support from the Government of Navarra.     

Asymmetric‐information allocation to avoid coordination failure

In the context of team production, this paper studies the optimal (deterministic and stochastic) information allocation that implements desired effort levels as the unique Bayesian equilibrium. We show that under certain conditions, it is optimal to asymmetrically inform agents even though they may be ex ante symmetric. The main intuition is that informing the agents asymmetrically can be effective in avoiding “bad” equilibria, that is, equilibria with coordination failure.     

A bargaining set for roommate problems

Since stable matchings may not exist, we propose a weaker notion of stability based on the credibility of blocking pairs. We adopt the weak stability notion of Klijn and Massó (2003) for the marriage problem and we extend it to the roommate problem. We first show that although stable matchings may not exist, a weakly stable matching always exists in a roommate problem. Then, we adopt a solution concept based on the credibility of the deviations for the roommate problem: the bargaining set. We show that weak stability is not sufficient for a matching to be in the bargaining set. We generalize the coincidence result for marriage problems of Klijn and Massó (2003) between the bargaining set and the set of weakly stable and weakly efficient matchings to roommate problems. Finally, we prove that the bargaining set for roommate problems is always non-empty by making use of the coincidence result.     

Cournot outcomes under Bertrand–Edgeworth competition with demand uncertainty

We provide new results for two-stage games in which firms make capacity investments when demand is uncertain, then, when demand is realized, compete in prices. We consider games with demand rationing schemes ranging from efficient to proportional rationing. In all cases, there is a subgame perfect equilibrium outcome coinciding with the outcome of the Cournot game with demand uncertainty if and only if (i) the fluctuation in absolute market size is small relative to the cost of capacity, or (ii) uncertainty is such that with high probability the market demand is very large and with the remaining probability the market demand is extremely small. Otherwise, equilibria involve mixed strategies. Further, we show under efficient rationing that condition (i) is sufficient for the unique equilibrium outcome to be an equilibrium outcome of the Cournot game with demand uncertainty.     

Gross substitutability of point-to-set correspondences

In this paper the notion of gross substitutability for the multi-valued case is studied. It is proved that, if in a pure exchange equilibrium model gross substitutability and some auxiliary conditions prevail, then (a) the set of equilibria is a Cartesian product of a convex set of equilibrium resource allocations and a convex cone of equilibrium prices; hence all equilibria are equiadvantageous for every trader; (b) the weak axiom of revealed preference holds in any equilibrium; (c) any equilibrium is stable with respect to reallocations of initial resources. Some situations in which Walras' law does not hold are considered as well.