A cardinal approach to straightforward probabilistic mechanisms

In every probabilistic mechanism, society selects an alternative, through a random device, out of a subset of indifferent alternatives. Consequently, in this context individuals face uncertainty and value the different lotteries on alternatives by their expected utility, so that they make use of a Von Neumann-Morgenstern cardinal utility function. Surprisingly, the social choice approach to probabilistic mechanisms assumes the use of ballots which preclude the complete expression of behaviour towards risk: individuals can only announce their ordinal preferences, or an approximation of their cardinal preferences, since in any case only a finite number of representations of preferences is available. This paper attempts to study voting systems in which individuals can express the cardinality of their preferences by assigning weights to the alternatives. It is shown that by voting with ballots which reflect weighting a new class of straightforward probabilistic mechanisms is defined, and that this class strictly contains the class of probabilistic straightforward mechanism designed by Gibbard.     

A contribution to duality theory,applied to the measurement of risk aversion

This paper determines the precise connection between the curvature properties of an objective function and the ray-curvature properties of its dual. When the objective function is interpreted as a Bernoulli or cardinal utility function, our results characterize the relationship between an agent’s attitude towards income risks and her attitude towards risks in the underlying consumption space. We obtain these results by developing and applying a number of representation theorems for concave functions.The work of Juan E. Martínez-Legaz has been supported by the Spanish Ministry of Science and Technology and the FEDER, project BEC2002-00642, and by the Departament d’Universitats, Recerca i Societat de la Informació, Direcció General de Recerca de la Generalitat de Catalunya, project 2001SGR-00162. He also thanks the Barcelona Economics Program of CREA for its support. John Quah would like to acknowledge with gratitude the financial support of the ESRC (grant number R000271171). He would also like to thank the Department of Economics at UC Berkeley, whose hospitality he enjoyed while completing this project. Both authors would like to thank Simon Cowan for pointing the way to some important references. They are also very grateful to the referee whose insightful suggestions led to a much improved paper     

Expected utility theory and experimental utility measurement, 1950–1985. From confidence to scepticism

Abstract

The paper reconstructs the history of the experimental attempts to measure the cardinal utility of money between 1950 and 1985 within the framework provided by expected utility theory (EUT). It is shown that this history displays a definite trajectory: from the confidence in EUT and the EUT-based measurement of utility of the 1950s to the scepticism that, from the mid-1970s, haunted the validity of EUT as well as the significance of the utility measures obtained through it. By exploring the diverse aspects and causes of this trajectory, the paper covers new ground in the history of both decision theory and utility measurement.     

Another proof that assignment games have singleton cores only if multiple optimal matchings exist

Complementary slackness of a pair of dual linear programming problems implies the remark of Sotomayor (2003) that assignment games have singleton cores only when multiple optimal matchings exist.The author thanks Marilda Sotomayor, William Thomson, and an anonymous referee for their helpful comments. This paper was finished when the author was visiting University of Rochester. JEL Classification Numbers: C71, C78, D58     

An ordinal characterization of the linear opinion pool

Given a collection of individual ordinal probabilities on a finite state space, we discuss an ordinal condition that is necessary and sufficient for an ordinal probability to be represented as a weighted average of probability representations of the individual probabilities. We also give necessary and sufficient conditions for when such an ordinal probability can be represented as an unweighted average of probability representations. Many thanks are due to an anonymous referee, who provided detailed comments and corrections which greatly improved this paper.     

Duality between direct and indirect preferences

Summary This paper develops a duality theory for preference orders, and highlights important differences with respect to the cardinal utility approach. Our main result is a symmetric duality theorem, under the minimal hypotheses, between direct and indirect preferences. In contrast to the cardinal theory, we also find that these conditions do not completely characterize the class of indirect preference orders.We are grateful to an editor for helpful remarks. This work was partially supported by DGICYT (Spain), Project PS89-OO58.     

The role of diminishing marginal utility in the ordinal and cardinal utility theories

We illustrate the role of the law of diminishing marginal utility in the two main modern utility theories, the ordinal and cardinal utility theories, using a generalised total utility function. In short, the ordinal utility theory, in which utility is immeasurable, must abandon the law of diminishing marginal utility; the cardinal utility theory, although able to retain this law, suffers from keeping the unrealistic view of utility measurability, which Samuelson criticises as “infinitely improbable.” A new utility theory with the advantages of the two mentioned theories (i.e. the notions of both diminishing marginal utility and utility immeasurability) but without the disadvantages (i.e. the law of diminishing marginal utility is excluded and utility is measurable) therefore still seems to be a Holy Grail deserving search and development by economists.     

Additive utility

Summary We put the theory of cardinal and additive utilities on the same kind of simple foundation as the theory of ordinal utility. We give necessary and sufficient conditions for preferences to have continuous cardinal or additive utility functions, on connected topological spaces. Basing our proofs on fundamental algebraic theorems yields new techniques that allow us to give simple proofs of earlier results (cf. [4, 14] and to provide a basis for new results [10].The assistance of the National Science Foundation, Grant SES-8510620, is gratefully acknowledged.     

A note on small income effects

This note provides an alternative sufficient condition for the small income effect result that is first shown by Vives [Small income effects: a Marshallian theory of consumer surplus and downward sloping demand, Rev. Econ. Stud. 54(1) (1987) 87-103]. The condition is stated by ordinal terms only, whereas Vives assumes cardinal properties of utility representation. Second, as its application, we provide a sufficient condition for the preference being asymptotically quasi-linear, in a two good economy where the second good is a composition of a large number of goods.     

Subjective expected utility theory with costly actions

This paper explores alternative axiomatizations of subjective expected utility theory for decision makers with direct preferences over actions; including a general subjective expected utility representation with action-dependent utility, and separately additive representations. In the context of the state-space formulation of agency theory, the results of this paper constitute axiomatic foundations of the agent's behavior.     

Undecidability of the existence of pure Nash equilibria

Summary. The purpose of this paper is to show that for any positive integer n, there exists no algorithm which decides for each non-cooperative n-person game in strategic form with partially computable payoff functions whether it has a pure Nash equilibrium or not.Received: 20 March 2002, Revised: 23 April 2003, JEL Classification Numbers: C72.The author thanks the Referee and the Associate Editor for offering critical comments on this paper, which is dedicated to his brother Konstantinos Efstathiou Sofronidis.     

The assumption of equal marginal utility of income: how much does it matter?

In most applied cost–benefit analyses, individual willingness to pay (WTP) is aggregated without using explicit welfare weights. This can be justified by postulating a utilitarian social welfare function along with the assumption of equal marginal utility of income for all individuals. However, since marginal utility is a cardinal concept, there is no generally accepted way to verify the plausibility of this latter assumption, nor its empirical importance. In this paper, we use data from seven contingent valuation studies to illustrate that if one instead assumes equal marginal utility of the public good for all individuals, aggregate monetary benefit estimates change dramatically.     

Consumers' Surplus As an Exact and Superlative Cardinal Welfare Indicator

This article shows that the Bennet-Bowley consumer surplus measure is an exact measure of Allais' disposable surplus if the consumer's utility function is of the translation-homothetic generalized quadratic form. The Bennet-Bowley consumer surplus measure, therefore, is a superlative cardinal welfare measure for the entire class of translation homothetic preferences. Because the exactness results in this article apply for cardinal welfare measures, they can be meaningfully aggregated across consumers to make aggregate welfare comparisons.     

Outliers - robust ECM cointegration tests based on the trend components

The main goal of this paper is to analyze the behavior of the ECM non-cointegration test when there are additive outliers in the time series under different co-breaking situations. We show that the critical values of the usual ECM test are not robust to the presence of transitory shocks and we suggest a procedure based on signal extraction to bypass this problem. These procedure renders ECM tests with a left tail of distribution under the null that is robust to the presence of additive outliers in the series. The small sample critical values and the empirical power of the test are analyzed by Monte Carlo simulations for several low frequency filters. The proposed empirical methodology is applied to the CPI-based US/Finland real exchange rate.JEL Classification: C22, C12, C15, C52, C51 Corresponding author: Alvaro EscribanoWe thank A. Lucas for kindly providing us with the data for the empirical example. Previous drafts of this paper have greatly improved thanks to the comments of two anonymous referees and the Associate Editor. The first author wishes to thank members of Department of Statistics and Econometrics, Universidad Carlos III de Madrid, and Department of Economics, University of Maryland at College Park. The second author acknowledeges support from the following grants: Spanish MCyT BEC2002-00279, The European TMR-ERB-40618C97-0994 and the Secretaría de Estado de Universidades PR2003-0305. This paper was finished while Alvaro Escribano was visiting the Department of Economics, Georgetwon University, Washington DC, USA.     

Subjective Expected Utility Theory with State-Dependent Preferences

This paper extends Savage′s subjective expected utility theory to include state-dependent preferences. The dependence of the decision maker′s preferences over consequences on the states of nature is represented by state-specific mappings of the set of consequences onto itself. Within this framework Savage′s postulates are reformulated and it is shown that there exist subjective expected utility representations of the preference relation over acts with unique, nonatomic, probability measure on the algebra of all events, and a state-dependent utility function over the set of consequences. Journal of Economic Literature Classification Number: D81.     

On Risk and Uncertainty,and Objective versus Subjective Probability

In this paper, we use the algebra to characterise decision‐makers’ representations of risk and uncertainty. We show that risk can be represented by objective probabilities on one part of the algebra, and that uncertainty can be represented by subjective probabilities on the other part. Decision‐makers are shown to maximise a generalised form of rank‐dependent expected utility. Their occasionally anomalous behaviour is discussed.     

Bargaining with subjective mixtures

This paper reconsiders the Bargaining Problem of Nash (Econometrica 28:155–162, 1950). I develop a new approach, Conditional Bargaining Problems, as a framework for measuring cardinal utility. A Conditional Bargaining Problem is the conjoint extension of a Bargaining Problem, conditional on the fact that the individuals have agreed on a “measurement event”. Within this context, Subjective Mixture methods are especially powerful. These techniques are used to characterise versions of the Nash and the Kalai–Smorodinsky solutions. This approach identifies solutions based only on the individuals’ tastes for the outcomes. It is therefore possible to do Bargaining theory in almost complete generality. The results apply to Biseparable preferences, so are valid for almost all non-expected utility models currently used in economics.     

A theory of anticipated utility

A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.     

Equilibrium dumping with endogenous anti-dumping penalties

The author develops a quantity setting model of firm behavior that captures the twofold effects of a dumping margin between market periods. The dumping margin determines the probability of getting charged with dumping and the size of the fine imposed, if caught. Both the probability of being charged with dumping and the size of the fine associated with dumping are endogenized. Profits are inter-temporally linked. The solution concept is the pure strategy Markov Perfect Equilibrium. The results are appealing: a margin of dumping that recurs is relatively easy to find under uncertain enforcement. The author finds reason to believe that the current anti-dumping legal structure is not likely to cause either an end to dumping or price equalization.The author thanks Marie Thursby, Jim Moore, Dan Kovenock, and Yukiko Hirao for their work. Appreciation also goes to Jon Haveman for helpful suggestions and Usha Nair for noisy discussions. Any errors found in this paper are the responsibility of the author.     

What do uncertainty-averse decision-makers believe?

Summary. This paper introduces the concept of firm belief, which is proposed as a new epistemic model for a wide class of preferences. In particular, firm beliefs are shown to have the following desirable properties: (i) they are derived from preferences according to a plausible rule of epistemic inference; (ii) they satisfy standard logical properties; and (iii) tractable representations of firm belief are available for all (suitably continuous) biseparable preferences [13, 14], including the Choquet expected utility [30] and maxmin expected utility [16] classes. We also use firm belief to construct a generalization of Nash equilibrium for (two-player) normal form games. Received: December 14, 1999; revised version: February 26, 2001