Exactly what happens after the Anscombe–Aumann race?

This paper derives a representation of preferences for a choice theory with vague environments; vague in the sense that the agent does not know the precise lotteries over outcomes conditional on states. Instead, he knows only a possible set of these lotteries for each state. Thus, this paper’s main departure from the standard subjective expected utility model is to relax an assumption about the environment, rather than weakening the axiomatic structure. My model is consistent with the behavior observed in the Ellsberg experiment. It can capture the same type of behavior as the multiple priors models, but can also result in behavior that is different from both the behavior implied by standard subjective expected utility models and the behavior implied by the multiple priors models. This paper is a revised chapter of my Ph.D. dissertation at Cornell University. I am very grateful to David Easley for extensive comments and suggestions. I also thank Larry Blume, Ani Guerdjikova, Edi Karni, Ted O’Donoghue, Maureen O’Hara, Jacob Sagi, seminar participants at the University of Copenhagen, Cornell University, ESEM 2006, Fuqua—Duke University, FUR 2006, Johns Hopkins University, Purdue University, Queen’s University, the University of Warwick, the associate editor, and an anonymous referee for comments and suggestions.     

Indivisibilities,lotteries, and sunspot equilibria

Summary We analyze economies with indivisible commodities. There are two reasons for doing so. First, we extend and provide some new insights into sunspot equilibrium theory. Finite competitive economies with perfect markets and convex consumption sets do not allow sunspot equilibria; these same economies with nonconvex consumption sets do, and they have several properties that can never arise in convex environments. Second, we provide a reinterpretation of the employment lotteries used in contract theory and in macroeconomic models with indivisible labor. We show how socially optimal employment lotteries can be decentralized as competitive equilibria without lotteries once sunspots are introduced.We thank Kenneth Arrow, Aditya Goenka, Ed Green, Jeremy Greenwood, Walter Heller, Steve Matthews, Herve Moulin, Roger Meyerson, Jim Peck, Patrick Kehoe, Ramon Marimon, Ed Prescott, Richard Rogerson, Nancy Stokey and Raghu Sundaram for their comments. We also thank participants in seminars at Northwestern, Yale, USC, Cornell, Barcelona, Madrid, Santander, and the Canadian Economics Association annual meetings in Victoria. We are grateful to the National Science Foundation (through grants SES-8606944 and SES-8821225), the Center for Analytic Economics, the Thorne Fund, and the University of Pennsylvania Research Foundation for research support. The views expressed here are those of the authors, and not necessarily those of the Federal Reserve System or the Federal Reserve Bank of Minneapolis.     

Group contest success functions

This paper extends the axiomatic characterization of contest success functions of Skaperdas (Econ Theory 7:283–290, 1996) and Clark and Riis (Econ Theory 11:201–204, 1998) to contests between groups. Comments by Dan Kovenock significantly improved the paper and are gratefully acknowledged. I would also like to thank Pavlo Blavatskyy, Aron Kiss, Kai Konrad, Florian Morath, Dana Sisak, participants of the SFB/TR 15 meeting in Gummersbach 2004, the 2004 meeting of the Verein für Socialpolitik in Dresden, and two anonymous referees. Any errors are mine. Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged.     

Subjective recursive expected utility

We axiomatize a subjective version of the recursive expected utility model. This development extends the seminal results of Kreps and Porteus (Econometrica 46:185–200 (1978)) to a subjective framework and provides foundations that are easy to relate to axioms familiar from timeless models of decision making under uncertainty. Our analysis also clarifies what is needed in going from a represention that applies within a single filtration to an across filtration representation.Part of this research was conducted when Ozdenoren visited MEDS in Fall 2003. We thank Tapas Kundu, Costis Skiadas, Jean-Marc Tallon and Tan Wang for helpful discussions and also thank audiences at Koc University, Northwestern University, the CERMSEM conference “ Mathematical Models in Decision Theory” at Universite Paris I, and the FUR XI conference on foundations and applications of utility, risk and decision theory     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

Completing the State Space with Subjective States

We define an opportunity act as a mapping from an exogenously given objective state space to a set of lotteries over prizes, and consider preferences over opportunity acts. We allow the preferences to be possibly uncertainty averse. Our main theorem provides an axiomatization of the maxmin expected utility model. In the theorem we construct subjective states to complete the objective state space. As in E. Dekel et al. (Econometrica, in press), we obtain a unique subjective state space. We also allow for preference for flexibility in some of the subjective states and commitment in others. Journal of Economic Literature Classification Number: D81.     

Preference for Flexibility and Dynamic Consistency

Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency, is linked to a comparative theory of preference for flexibility. When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.     

Edgeworth and Walras equilibria of an arbitrage-free exchange economy

Summary. In this paper, we first give a direct proof of the existence of Edgeworth equilibria for exchange economies with consumption sets which are (possibly) unbounded below. The key assumption is that the individually rational utility set is compact. It is worth noticing that the statement of this result and its proof do not depend on the dimension or the particular structure of the commodity space. In a second part of the paper, we give conditions under which Edgeworth allocations can be decentralized by continuous prices in a finite dimensional and in an infinite dimensional setting. We then show how these results apply to some finance models.Received: 15 April 2002, Revised: 25 February 2003, JEL Classification Numbers: C62, D58, G12. Correspondence to: Monique FlorenzanoThis paper was presented at seminars of the Economic Departments of Brown University, The Johns Hopkins University, The University of Alabama, Purdue University and at the Centro de Modelamiento Matematico de la Universidad de Chile at Santiago. The paper has benefitted of the comments of these diverse audiences. Part of this work was done while the authors were visiting respectively Brown University and the Centro de Modelamiento Matematico. We thank them for their hospitality.     

Strongly stable networks

We analyze the formation of networks among individuals. In particular, we examine the existence of networks that are stable against changes in links by any coalition of individuals. We show that to investigate the existence of such strongly stable networks one can restrict focus on a component-wise egalitarian allocation of value. We show that when such strongly stable networks exist they coincide with the set of efficient networks (those maximizing the total productive value). We show that the existence of strongly stable networks is equivalent to core existence in a derived cooperative game and use that result to characterize the class of value functions for which there exist strongly stable networks via a “top convexity” condition on the value function on networks. We also consider a variation on strong stability where players can make side payments, and examine situations where value functions may be non-anonymous—depending on player labels.     

A new solution to the problem of adjudicating conflicting claims

Summary. For the problem of adjudicating conflicting claims, we consider the requirement that each agent should receive at least 1/n his claim truncated at the amount to divide, where n is the number of claimants (Moreno-Ternero and Villar, 2004a). We identify two families of rules satisfying this bound. We then formulate the requirement that for each problem, the awards vector should be obtainable in two equivalent ways, (i) directly or (ii) in two steps, first assigning to each claimant his lower bound and then applying the rule to the appropriately revised problem. We show that there is only one rule satisfying this requirement. We name it the “ rule”, as it is obtained by a recursion. We then undertake a systematic investigation of the properties of the rule.Received: 7 September 2004, Revised: 21 April 2005, JEL Classification Numbers: C79, D63, D74.William Thomson: Correspondence toWe thank Christopher Chambers, Youngsub Chun, Tarík Kara, Çağatay Kayí, Juan Moreno-Ternero, Toyotaka Sakai, and Michel Truchon for their comments. We also thank a referee for useful comments. This paper was presented at Columbia University, CORE, Bilkent University, Sabancí University, Université Laval, and Caltech, and at the Mallorca Review of Economic Design meeting. Thomson acknowledges support from the NSF under grant SES-0214691.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

On the Evaluation of Options on Lotteries: An Experimental Study

We present the results of a comparative experimental study of the evaluation of simple lotteries and call/put/insurance options on these lotteries. The main findings and conclusions are:

(a) The observed bidding patterns depend on the type of asset under evaluation. In particular, subject behavior when buying or selling a basic lottery seems much more cautious than their behavior when buying or selling options on that lottery.

(b) The observed bidding patterns also depend on subject positions with respect to the underlying asset. In particular, the bids for buying lotteries and options long are statistically uncorrelated with the bids for selling the same lotteries and options short.

(c) Subjects with extreme risk attitudes are more inclined to violate basic no-arbitrage conditions (like the call-put parity) when bidding for the different lotteries.

We demonstrate that it is difficult to reconcile the experimental evidence with mainstream theories on individual decision and choice (although we find strong support for prospect theory in some parts of the data). We conclude that the evaluation of options on lotteries is context-dependent and subtler than perceived by existing theories.     

Campaign spending regulation in a model of redistributive politics

Summary. We consider a model in which parties that differ in perceived valence choose how to allocate electoral promises (money, pork-barrel projects) among voters. The party perceived to be less valent has a greater incentive to “sell out” to a favored minority and completely expropriate a fraction of the electorate. By reducing the difference in perceived valence, campaign-finance regulations may reduce the extent of the expropriation and achieve a more equitable political outcome. We analyze various instruments of campaign-finance regulation from this perspective.Received: 20 Februay 2003, Revised: 25 January 2005, JEL Classification Numbers: D72, H2.Nicolas Sahuguet: Correspondence toWe thank Alessandro Lizzeri, George Mailath, and Andrew Postlewaite for their comments. We also thank the editor Dan Kovenock and an anonymous referee. The second author is grateful to the National Science Foundation for financial support under grant SES-0078870.     

Finding a Nash equilibrium in spatial games is an NP-complete problem

Summary. We consider the class of (finite) spatial games. We show that the problem of determining whether there exists a Nash equilibrium in which each player has a payoff of at least k is NP-complete as a function of the number of players.Received: 15 September 2002, Revised: 9 March 2003, JEL Classification Numbers: C72.Correspondence to: H. HallerWe thank a referee for helpful comments. The hospitality of the Institute of Economics, University of Copenhagen, and the Institute for Advanced Studies, Vienna, is gratefully acknowledged by the third author.     

Decentralizing lottery allocations in markets with indivisible commodities

Summary In economies with indivisible commodities, consumers tend to prefer lotteries in commodities. A potential mechanism for satisying these preferences is unrestricted purchasing and selling of lotteries in decentralized markets, as suggested in Prescott and Townsend [Int. Econ. Rev.25, 1–20]. However, this paper shows in several examples that such lottery equilibria do not always exist for economies with finitely many consumers. Other conditions are needed. In the examples, equilibrium and the associated welfare gains are realized if consumptions are bounded or if lotteries are based upon a common sunspot device as defined by Shell [mimeo, 1977] and Cass and Shell [J. Pol. Econ.91, 193–227]. The paper shows that any lottery equilibrium is either a Walrasian equilibrium or a sunspot equilibrium, but there are Walrasian and sunspot equilibria that are not lottery equilibria.This paper is based on Chapter 3 of my doctoral dissertation, written while I was a student at Cornell University. I thank Larry Blume, Yue Yun Chen, David Easley, Aditya Goenka, John Marshall, Bruce Smith, John Wooders and an anonymous referee. I am particularly grateful to Karl Shell and Cheng-Zhong Qin. I thank the Academic Senate at UCSB for financial support.     

Lotteries, Sunspots, and Incentive Constraints

We study a prototypical class of exchange economies with private information and indivisibilities. We establish an equivalence between lottery equilibria and sunspot equilibria and show that the welfare and existence theorems hold. To establish these results, we introduce the concept of the stand-in consumer economy, which is a standard, convex, finite consumer, finite good, pure exchange economy. With decreasing absolute risk aversion and no indivisibilities, we prove that no lotteries are actually used in equilibrium. We provide a simple numerical example with increasing absolute risk aversion in which lotteries are necessarily used in equilibrium. We also show how the equilibrium allocation in this example can be implemented in a sunspot equilibrium. Journal of Economic Literature Classification Numbers: D11, D50, D82.     

When and why not to auction

Summary. Standard auctions are known to be a revenue-maximizing way to sell an object under broad conditions when buyers are symmetric and have independent private valuations. We show that when buyers have interdependent valuations, auctions may lose their advantage, even if symmetry and independence of information are maintained. In particular, simple alternative selling mechanisms that sometimes allow a buyer who does not have the highest valuation to win the object will in general increase all buyers willingness to pay, possibly enough to offset the loss to the seller of not always selling to the buyer with the greatest willingness to pay.Received: 18 June 2003, Revised: 7 September 2004, JEL Classification Numbers: D44, D82. Correspondence to: Colin M. CampbellWe thank Richard McLean, Dan Kovenock, two referees, and seminar participants at the University of Pittsburgh, Johns Hopkins University, Washington University, the Federal Communications Commission, the Institute for Advanced Studies at Princeton, UC-Santa Barbara, UC-Santa Cruz, and the Ohio State University Department of Finance for helpful suggestions. We thank the NSF for funding.     

Creditworthiness and thresholds in a credit market model with multiple equilibria

Summary. The paper studies creditworthiness in a model with endogenous credit cost and debt constraints. Such a model can give rise to multiple candidates for steady state equilibria. We use new analytical techniques such as dynamic programming (DP) with flexible grid size to find solutions and to locate thresholds that separate different domains of attraction. More specifically, we (1) compute present value borrowing constraints and thus creditworthiness, (2) locate thresholds where the dynamics separate to different domains of attraction, (3) show jumps in the decision variable, (4) distinguish between optimal and non-optimal steady states, (5) demonstrate how creditworthiness and thresholds change with change of the credit cost function of the debtor and (6) explore the impact of debt ceilings and consumption paths on creditworthiness.JEL Classification Numbers: C61, C63, D91, D92, E51, G12, G32.An earlier version of this paper has been prepared for the 1998 North American Winter Meeting of the Econometric Society, January 1998, Chicago. We want to thank Jess Benhabib, Buz Brock, Gustav Feichtinger, Franz Wirl, Michael Woodford, Wolf-Jürgen Beyn and Thorsten Pampel for helpful discussions and comments on various versions of the paper. We also want to thank participants in a workshop at the University of Technology, Vienna, the Macroeconomic Workshop at Columbia University, and the SCE conference, at Yale University, June 2001. We are also grateful for comments from a referee of the journal.     

Maxmin under risk

Summary. Let be a continuous and convex weak order on the set of lotteries defined over a set Z of outcomes. Necessary and sufficient conditions are given to guarantee the existence of a set of utility functions defined on Z such that, for any lotteries p and q, The interpretation is simple: a conservative decision maker has an unclear evaluation of the different outcomes when facing lotteries. She then acts as if she were considering many expected utility evaluations and taking the worst one. Received: January 19, 2000; revised version: December 20, 2000     

On the welfare effects of allowing unlimited renegotiation in agency relationships

We study renegotiation in an agency setting where the number of offers and accept/reject decisions parties can make is potentially unlimited. Thus any contract, either on or off the equilibrium path, may be subject to possible renegotiation. We first show that the principal will not be able to gain complete access to the agent’s private information with unlimited renegotiation, unlike when the potential number of renegotiations is finite. Rather the agent either employ a randomized reporting strategy or do not to report at all. We then identify conditions under which expected allocations are most efficient with the contract that induces no agent communication. More significantly, by doing so we also identify conditions under which the parties are made strictly worse off by committing to end renegotiation after a fixed number of rounds. We thank workshop participants at University of California, Irvine, University of Chicago, Duke University, Hong Kong University of Science and Technology, University of Houston, London School of Economics, University of Minnesota, Ohio State University, the Carnegie Mellon Accounting Research Conference, the editor, and an anonymous referee for their helpful comments and suggestions.