Finite allocation mechanisms: approximate Walrasian versus approximate Direct Revelation

Summary. This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in exchange economies while using a continuum of possible messages. In particular, we develop finite counterparts of the superiority, with respect to message-space dimension, of the Walrasian mechanism over Direct Revelation (DR). We measure a finite mechanism's cost by the number of its (equilibrium) messages. Our two main results are as follows: (1) For exchange economies we find that the overall (maximum) error of a (sufficiently fine) approximate Walrasian mechanism is less than the overall error of a not-more-costly approximate DR mechanism whose equilibrium outcomes are trades that are (approximately) Pareto optimal and individually rational; more generally, approximate Walrasian mechanisms are superior, in the same sense, to approximations of any continuum mechanism whose outcomes are Pareto optimal individ ually rational trades and whose message space has higher dimension than that of the Walrasian mechanism. (2) As we increase without limit the dimension of the set of environments (characteristics) defining our class of exchange economies, the extra cost of DR approximations relative to Walrasian approximations, when both achieve the same overall error, also grows without limit. Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages in the approximation as our cost measure. Received: June 16, 2002; revised version: July 22, 2002 RID="*" ID="*" The second author is grateful for support from National Science Foundation grant #IIS-0118600. Correspondence to: T. Marschak     

Mechanism design with partial state verifiability

We study implementation in environments where agents have limited ability to imitate others. Agents are randomly and privately endowed with type-dependent sets of messages. So sending a message becomes a partial proof regarding type. For environments where agents can send any combination of available messages, we develop an Extended Revelation Principle and characterize the incentive constraints which implementable allocations must satisfy. When not all message combinations are feasible, static mechanisms no longer suffice. If a ‘punishment’ allocation exists for each agent, then implementable allocations can be characterized as equilibria of a “Revelation Game,” in which agents first select from the menus of allocation rules, then the mediator requests each agent to send some verifying messages. When a punishment allocation fails to exist for some agent, dynamic games in which agents gradually reveal their evidence implement a larger set of outcomes. The latter result provides a foundation for a theory of debate.     

Mechanism design with partial state verifiability

We study implementation in environments where agents have limited ability to imitate others. Agents are randomly and privately endowed with type-dependent sets of messages. So sending a message becomes a partial proof regarding type. For environments where agents can send any combination of available messages, we develop an Extended Revelation Principle and characterize the incentive constraints which implementable allocations must satisfy. When not all message combinations are feasible, static mechanisms no longer suffice. If a ‘punishment’ allocation exists for each agent, then implementable allocations can be characterized as equilibria of a “Revelation Game,” in which agents first select from the menus of allocation rules, then the mediator requests each agent to send some verifying messages. When a punishment allocation fails to exist for some agent, dynamic games in which agents gradually reveal their evidence implement a larger set of outcomes. The latter result provides a foundation for a theory of debate.     

An ascending double auction

We show why the failure of the affiliation assumption prevents the double auction from achieving efficient outcomes when values are interdependent. This motivates the study of an ascending price version of the double auction. It is shown that when there is a sufficiently large, but still finite, number of sellers, this mechanism has an approximate perfect Bayesian equilibrium in which traders continue bidding if and only if their true estimates of the ‘value’ of the object being traded exceed the current price. This equilibrium is ex post efficient and has a rational expectations property in the sense that along the equilibrium path traders appear to have made the best possible trades conditional on information revealed by the trading process. We thank two anonymous referees and Dan Kovenock, the Editor, whose detailed comments and suggestions have allowed us to substantially improve the paper. We also thank seminar participants at University of Toronto, University of Wisconsin-Madison, Summer 2003 North American Meetings of the Econometric Society, 2004 NSF Decentralization Conference for their comments.     

Broadcasting Opinions with an Overconfident Sender*

We analyze a model where an altruistic, but possibly overconfident sender broadcasts one of a finite set of messages to rational receivers. If broadcasting is costless and the sender is rational, there is an informationally efficient equilibrium, but multiple equilibria may arise, and asymmetric equilibria might be more informative than the symmetric equilibrium even if the prior is symmetric. Although overconfidence on the part of the sender reduces informativeness in some cases, it may also eliminate less informative equilibria and lead to better information transmission. Overconfidence can also improve the informativeness of the message when broadcasting is costly.     

Review article. Growth and stagnation in a two-sector model: Kaldor's Mattioli Lectures

Kaldor's Mattioli Lectures analyse a two-sector model with increasingreturns to scale (IRS) in industry and diminishing returns inagriculture (DR). This review article shows that (i) with IRSin industry, a long-run equilibrium growth path with strictlypositive growth rates may exist even if agriculture is subjectto DR; (ii) the industrial sector is the 'engine of growth'if agricultural investment is determined passively by availablesaving; and (iii) if one introduces a separate agriculturalinvestment function, both positive and negative agriculturalsupply shocks may lead to stagnation, thus vindicating Kaldor'semphasis on commodity price stabilisation.     

Linear exchange economies with a continuum of agents

The purpose of this paper is to study how the equilibrium prices vary with respect to the initial endowments in a linear exchange economy with a continuum of agents. We first state the model and give conditions of an increasing strength for existence, uniqueness and continuity of equilibrium prices. Then, if we restrict ourselves to economies with essentially bounded initial endowments and if we assume that there is, from the point of view of preferences, only a finite number of types of agents, we show that, on an open dense subset of the space of initial endowments, the equilibrium price vector is an infinitely differentiable function of the initial endowments. The proof of this claim is based on a formula allowing to compute the equilibrium price vector around a so-called “regular” endowment where it is known.     

Efficiency properties of strategies market games: An axiomatic approach

The paper investigates the conditions under which an abstractly given market game will have the property that if there is a continuum of traders then every noncooperative equilibrium is Walrasian. In orther words, we look for a general axiomatization of Cournot's well-known result. Besides some convexity, continuity, and nondegeneracy hypotheses, the crucial axioms are: anonymity (i.e., the names of traders are irrelevant to the market) and aggregation (i.e., the net trade received by a trader depends only on his own action and the mean action of all traders). It is also shown that the same axioms do not guarantee efficiency if there is only a finite number of traders. Some examples are discussed and a notion of strict noncooperative equilibrium for anonymous games is introduced.     

Equilibria in incomplete assets economies with infinite dimensional spot markets

The paper studies the two period incomplete markets model where assets are claims on state contingent commodity bundles and there are no bounds on portfolio trading. The important results on the existence of equilibrium in this model assume that there is a finite number of commodities traded in each spot market and that preferences are given by smooth utility functions. With these assumptions an equilibrium exists outside an “exceptional” set of assets structures and initial endowments. The present paper extends these results by allowing for general infinite dimensional commodity spaces in each spot market. These include all the important commodity spaces studied in the literature on the existence of Walrasian equilibrium—in each spot market the consumption sets are the positive cone of an arbitrary locally solid Riesz space or of an ordered topological vector space with order unit or of a locally solid Riesz space with quasi-interior point. The paper establishes that even with our very general commodity spaces there exists an equilibrium for a “very” dense set of assets structures. Our approach is in the main convex analytic and the results do not require that preferences be smooth or complete or transitive. The concepts and techniques studied in this paper have important finite as well as infinite dimensional applications. This paper has benefited from the comments of Martine Quinzii, Wayne Shafer, Manuel Santos and Yeneng Sun. The research of C. D. Aliprantis is supported by the NSF Grants SES-0128039, DMS-0437210, and ACI-0325846. The research of R. Tourky is funded by the Australian Research Council Grant A00103450.     

Perturbed communication games with honest senders and naive receivers

This paper studies communication games in which the sender is possibly honest (tells the truth) and the receiver is possibly naive (follows messages as if truthful). The characterization of message-monotone equilibria in the perturbed games explain several important aspects of strategic communication including sender exaggeration, receiver skepticism and message clustering. Surprisingly, the strategic receiver may respond to more aggressive claims with more moderate actions. In the limit as the probabilities of the non-strategic players approach zero, (i) the limit equilibrium corresponds to a most-informative equilibrium of the limit (Crawford-Sobel) game; (ii) only the top messages are sent.     

Approximate tâtonnement processes

Summary We introduce a probabilistic model for price adjustment in an exchange economy which approximates the classical Walras tâtonnement process while avoiding many of its unrealistic features. The model is decentralized in that the trades permitted to an agent and the resulting price changes depend only on the commodity vector currently held by that agent, and not on the commodity vectors held by the other agents in the economy. Our results will show that the Walras tâtonnement process can be decentralized without changing its behavior on the macroeconomic scale. Our model has a finite set of commodities, a market maker who adjusts prices, and a large finite set of agents who trade only with the market maker. Each agent has a demand function depending on his commodity vector and the price vector. At each discrete time, one agent is chosen at random and exchanges his current commodity vector for his demand vector. Then the market maker adjusts the price vector by an amount which depends on the selected agent's commodity vector and the current price. Prices are adjusted rapidly enough to avoid prolonged trading at the wrong price, but slowly enough so that a substantial price change will depend on a significant simple of agents. The main result shows that with probability arbitrarily close to one the price will rapidly approach and then remain close to an equilibrium value, following a path which is close to the price path of the corresponding tâtonnement process.     

A Revelation Mechanism for Soft Information under Moral Hazard

This paper constructs a revelation mechanism to address a problem of moral hazard under soft information. The agent alone observes the stochastic outcome of her action, which she reports to the principal. Therefore the principal also faces a problem of ex post adverse selection. Economically relevant restrictions induce constraints on the principal's choice of mechanism and the Revelation Principle fails to apply. Specifically, a direct mechanism induces some pooling, which does not replicate the allocation obtained using a larger message space. Pooling also weakens the ex ante incentives. The Revelation Principle is extended to obtain type separation. A better audit relaxes frictions.     

An experimental study of strategic information transmission

Summary We examine strategic information transmission in an experiment. Senders are privately informed about a state. They send messages to Receivers, who choose actions resulting in payoffs to Senders and Receivers. The payoffs depend on the action and the state. We vary the degree to which the Receivers' and the Senders' preferences diverge. We examine the relationship between the Senders' messages and the true state as well as that between actions and the true state and contrast the ability of different equilibrium message sets to explain the data.When preferences are closely aligned Senders disclose more. We assess two comparative statics: (i) as preferences diverge, state and action are less frequently matched, and (ii) messages tend to become less informative as preferences diverge. The first result is weakly confirmed for adjacent treatments but is considerably stronger when non-adjacent treatments are compared. We find that as preferences diverge messages become less informative. While the ex-ante Pareto-optimal Bayesian Nash Equilibrium does not explain our conditions, the equilibrium message sets supported by the data are similar to the ex-ante Pareto Optimal message sets.We would like to thank seminar participants at the Economic Science Association meetings, the University of Iowa, the University of Minnesota, Northwestern University and the Society for the Advancement of Economic Theory meetings for their comments. We would like to thank Beth Allen, Joyce Berg, Robert Forsythe, Yong-Gwan Kim, Antonio Merlo, Leonard Mirman, In-Uck Park, Charles Plott, Jennifer Reinganum and an anonymous referee for their suggestions. Financial support from the Accounting Research Center at the University of Minnesota is also acknowledge.     

Private provision of a discrete public good: efficient equilibria in the private-information contribution game

We study the voluntary provision of a discrete public good via the contribution game. Players independently and simultaneously make nonrefundable contributions to fund a discrete public good, which is provided if and only if contributions cover the cost of production. We characterize nonconstant continuous symmetric equilibria, giving sufficient conditions for their existence. We show the common normalization by which players’ values are distributed over [0, 1] is not without loss of generality: if the distribution over this interval has continuous density f with f(0) >  0, then no (nonconstant) continuous symmetric equilibrium exists. We study in detail the case in which players’ private values are uniformly distributed, showing that, generically, when one continuous equilibrium exists, a continuum of continuous equilibria exists. For any given cost of the good, multiple continuous equilibria cannot be Pareto ranked. Nevertheless, not all continuous equilibria are interim incentive efficient. The set of interim incentive efficient equilibria is exactly determined. The authors thank Manfred Dix, George Mailath, Andrew Postlewaite, and an anonymous referee for their comments.     

A discrete approach to continuum economies

Summary We consider a perfectly competitive economy in which only a finite number of different agents' characteristics can be distinguished. We associate this economy with ann-agents' economy with an ordered Banach commodity space, and we prove that the continuum and the discrete approach to the equilibrium problem can be considered equivalent.The authors are indebted to Javier Ruiz-Castillo for his helpful suggestions to an earlier version of this paper. We would like to thank Nicholas C. Yannelis and an anonymous referee for helpful suggestions and ecomments.     

The effects of costless pre-play communication: Experimental evidence from games with Pareto-ranked equilibria

Cheap talk is shown to facilitate coordination on the unique efficient equilibrium in experimental order-statistic games. This result is roughly consistent with theoretical predictions according to which cheap talk promotes efficient Nash play. The evidence concerning the mechanisms that theory appeals to is mixed: Frequent agreement of messages and actions is consistent with messages being viewed as self-committing. Risk in the underlying game and the absence of self-signaling messages may explain why message profiles are not unanimous. Time-varying message profiles can be interpreted as evidence for players trying to negotiate equilibria and/or trying to rely on secret handshakes.     

Strategyproof sharing of submodular costs:budget balance versus efficiency

Summary. A service is produced for a set of agents. The service is binary, each agent either receives service or not, and the total cost of service is a submodular function of the set receiving service. We investigate strategyproof mechanisms that elicit individual willingness to pay, decide who is served, and then share the cost among them. If such a mechanism is budget balanced (covers cost exactly), it cannot be efficient (serve the surplus maximizing set of users) and vice-versa. We characterize the rich family of budget balanced and group strategyproof mechanisms and find that the mechanism associated with the Shapley value cost sharing formula is characterized by the property that its worst welfare loss is minimal. When we require efficiency rather than budget balance – the more common route in the literature – we find that there is a single Clarke-Groves mechanism that satisfies certain reasonable conditions: we call this the marginal cost pricing mechanism. We compare the size of the marginal cost pricing mechanism's worst budget surplus with the worst welfare loss of the Shapley value mechanism. Received: October 26, 1998; revised version: December 3, 1999     

Frontrunners and Laggards: The Strategy of Environmental Regulation under Uncertainty

The extant literature on the political economy of environmental regulation does not provide a unified theoretical explanation for three salient stylized facts. First, companies voluntarily invest to reduce the environmental burden that they cause under threat of regulation. Second, ex ante estimates of the compliance cost tend to be systematically higher than ex post estimates. Finally, regulators use limited information provided by the industry. I construct a game-theoretic model of environmental regulation under uncertainty with a benevolent regulator. In equilibrium, companies undertake voluntary action to induce regulation that raises barriers to entry. This profit-driven behavior is not always socially detrimental, however, as the regulator obtains a credible commitment to production and a more accurate estimate of the compliance cost. Additionally, the results provide a selection explanation for the mismatch between ex ante and ex post cost estimates: if companies condition compliance on the installation cost, only low-cost companies install in equilibrium. The analysis combines “regulatory capture” with social welfare maximization to explain the curious combination of voluntary action and low ex post compliance cost without serious information collection by the regulator.     

Does commodity money eliminate the indeterminacy of equilibrium?

Previous studies have shown that a random-matching model with divisible fiat money and without constraint on agents’ money inventories possesses a continuum of stationary single-price equilibria. Wallace (J. Econom. Theory 81 (1998) 223) conjectures that the indeterminacy can be eliminated by the use of commodity money. Instead, I find that in a similar random-matching model with dividend-yielding commodity money, a continuum of stationary single-price equilibria exists when the utility of dividend is not too high. This result casts doubt on the conventional belief that the indeterminacy of monetary equilibrium is be caused only by the nominal nature of money.     

Cournot-Nash and Lindahl equilibria in pure public “bad” economies

Summary. Using a general equilibrium framework, this paper analyzes the equilibrium provision of a pure public bad commodity (for example pollution). Considering a finite economy with one desired private good and one pure public “bad” we explicitly introduce the concept of Lindahl equilibrium and the Lindahl prices into a pure public bad economy. Then, the Lindahl provision is analyzed and compared with the Cournot-Nash provision. The main results for economies with heterogeneous agents state that the asymptotic Lindahl allocation of the pure public bad is the null allocation. In contrast, the asymptotic Cournot-Nash provision of the public bad might approach infinity. Other results were obtained in concert with the broad analysis of the large finite economies with pure public bad commodities. Received: July 26, 2001; revised version: March 12, 2002 RID="*" ID="*" We are indebt to Nicholas Yannelis and anonymous referee for their valuable comments and suggestions. Correspondence to: B. Shitovitz