Approaches to cardinal utility

In a previous article (see [3]) a system of axioms is proposed stating conditions which are necessary and sufficient to determine a cardinal utility function on any set, finite or infinite, of outcomes X. The present paper discusses and interprets the meaning of those axioms, and compares this new approach to cardinal utility with the utility differences approach proposed by Alt and Frisch, among others, and with the expected utility approach of von-Neuman and Morgenstern. The notion of repetition of the same choice situation is presented and its interpretation discussed. It is then argued that this notion leads naturally to the system of axioms presented in On Cardinal Utility. It is also argued that this notion must be used if we want to have a more clear understanding of the meaning of the axioms proposed by Alt and Frisch. Finally, it is remarked that since uncertainty is not present in the new approach, it is free of the paradoxes that have plagued the expected utility hypothesis.     

Bayesian decision theory,rule utilitarianism,and Arrow's impossibility theorem

The first part of this paper reexamines the logical foundations of Bayesian decision theory and argues that the Bayesian criterion of expected-utility maximization is the only decision criterion consistent with rationality. On the other hand, the Bayesian criterion, together with the Pareto optimality requirement, inescapably entails a utilitarian theory of morality. The next sections discuss the role both of cardinal utility and of cardinal interpersonal comparisons of utility in ethics. It is shown that the utilitarian welfare function satisfies all of Arrow's social choice postulates avoiding the celebrated impossibility theorem by making use of information which is unavailable in Arrow's original framework. Finally, rule utilitarianism is contrasted with act utilitarianism and judged to be preferable for the purposes of ethical theory.     

Expected utility without utility

This paper advances an interpretation of Von Neumann-Morgenstern's expected utility model for preferences over lotteries which does not require the notion of a cardinal utility over prizes and can be phrased entirely in the language of probability. According to it, the expected utility of a lottery can be read as the probability that this lottery outperforms another given independent lottery. The implications of this interpretation for some topics and models in decision theory are considered.     

On cardinal utility

Necessary and sufficient conditions for the existence of a cardinal utility function on a set X are presented.     

An Ethical Interpretation of the Nash Choice Rule

This paper provides an ethical intepretation of the Nash choice rule. In a setting in which (cardinal) utilities are interpersonally comparable, this procedure is characterised by an impartiality requirement and by the assumption that choices are not responsive to the agents' relative ability to convert resources into utility.     

The repetitions approach to characterize cardinal utility

Building on previous work of A. Camacho, we give necessary and sufficient conditions for the existence of a cardinal utility function to represent, through summation, a preference relation on sequences of alternatives.     

On Ordinal Utility,Cardinal Utility and Random Utility

Though the Random Utility Model (RUM) was conceived entirely in terms of ordinal utility, the apparatus through which it is widely practised exhibits properties of cardinal utility. The adoption of cardinal utility as a working operation of ordinal is perfectly valid, provided interpretations drawn from that operation remain faithful to ordinal utility. The article considers whether the latter requirement holds true for several measurements commonly derived from RUM. In particular it is found that measurements of consumer surplus change may depart from ordinal utility, and exploit the cardinality inherent in the practical apparatus.      

Estimating Risk Attitudes using Lotteries: A Large Sample Approach

Attitudes towards risk play a major role in many economic decisions. In empirical studies it is quite often assumed that attitudes towards risk do not vary across individuals. This paper questions this assumption and analyses which factors influence an individual's risk attitude. Based on questions on lotteries in a large household survey we first semiparametrically estimate an index for risk aversion. We only make weak assumptions about the underlying decision process and our estimation method allows for generalisations of expected utility. We then estimate a structural model based on Cumulative Prospect Theory. Expected utility is strongly rejected and both the value function and the probability weighting function vary significantly with (among other things) age, income, and wealth of the individual.     

Separating marginal utility and probabilistic risk aversion

This paper is motivated by the search for one cardinal utility for decisions under risk, welfare evaluations, and other contexts. This cardinal utility should have meaningprior to risk, with risk depending on cardinal utility, not the other way around. The rank-dependent utility model can reconcile such a view on utility with the position that risk attitude consists of more than marginal utility, by providing a separate risk component: a probabilistic risk attitude towards probability mixtures of lotteries, modeled through a transformation for cumulative probabilities. While this separation of risk attitude into two independent components is the characteristic feature of rank-dependent utility, it had not yet been axiomatized. Doing that is the purpose of this paper. Therefore, in the second part, the paper extends Yaari's axiomatization to nonlinear utility, and provides separate axiomatizations for increasing/decreasing marginal utility and for optimistic/pessimistic probability transformations. This is generalized to interpersonal comparability. It is also shown that two elementary and often-discussed properties — quasi-convexity (aversion) of preferences with respect to probability mixtures, and convexity (pessimism) of the probability transformation — are equivalent.     

The econometric modelling of social preferences

Experimental data on social preferences present a number of features that need to be incorporated in econometric modelling. We explore a variety of econometric modelling approaches to the analysis of such data. The approaches under consideration are: the Random Utility approach (in which it is assumed that each possible action yields a utility with a deterministic and a stochastic component, and that the individual selects the action yielding the highest utility); the Random Behavioural approach (which assumes that the individual computes the maximum of a deterministic utility function, and that computational error causes their observed behaviour to depart stochastically from this optimum); and the Random Preference approach (in which all variation in behaviour is attributed to stochastic variation in the parameters of the deterministic component of utility). These approaches are applied in various ways to an experiment on fairness conducted by Cappelen et al. (Am Econ Rev 97(3):818–827, 2007). Various models that we estimate succeed in capturing the key features of the dataset. Conclusions concerning fairness-related behaviour depend crucially on the choice of econometric model.     

An integrated axiomatic approach to the existence of ordinal and cardinal utility functions

Summary This paper presents a unified axiomatic treatment of the existence of ordinal and cardinal utility functions. The contribution of the paper is mainly pedagogical.     

Ordered sum and tensor product of linear utility structures

Measurement-theoretic behavior of the category of social utility models is studied in terms of ordered sums and tensor products of mixture spaces. Necessary and sufficient conditions are given for the existence of individual utility functions and a social utility function (being a weighted average, and in another case a product, of the individual utilities) in terms of individual and social preference rankings.Part of the research presented here was supported by the NSF Grant GS-2936.I am deeply indebted to Richard C. Jeffrey for his stimulation and advice, and to Duncan Luce for many valuable discussions on related topics of measurement theory.     

A Method for Eliciting Utilities and its Application to Collective Choice

Designing a mechanism that provides a direct incentive for an individual to report her utility function over several alternatives is a difficult task. A framework for such mechanism design is the following: an individual (a decision maker) is faced with an optimization problem (e.g., maximization of expected utility), and a mechanism designer observes the decision maker’s action. The mechanism does reveal the individual’s utility truthfully if the mechanism designer, having observed the decision maker’s action, infers the decision maker’s utilities over several alternatives. This paper studies an example of such a mechanism and discusses its application to the problem of optimal social choice. Under certain simplifying assumptions about individuals’ utility functions and about how voters choose their voting strategies, this mechanism selects the alternative that maximizes Harsanyi’s social utility function and is Pareto-efficient.     

Some quantitative concepts of value and utility from a utilitarian point of view

This paper clarifies and interprets some basic quantitative concepts of value, utility and utility function from a utilitarian point of view. First, I discuss the question as to whether value is objective or subjective. I hold that value is subjective and statistical in nature (although from the various subjective values of a certain object a norm can usually be obtained). Second, I emphasize the distinction between use value and exchange value in relation to utility. Third, I propose a law of diminishing incremental interest, which refers to the incremental (marginal) utility of money. Fourth, I identify the utility of money with the von Neumann-Morgenstern utility. Fifth, I question the necessity of the usual normalization of utility functions and the restricted linear transformation (and the consequent concept of strategic equivalence). Sixth, I discuss in detail the terminal values and utilities of a utility function from a philosophical rather than mathematical point of view, particularly the boundedness of a utility function and the magnitudes of V ₀ and U ₀. Finally, I conclude that, in order to be able to have interpersonal comparisons of utility, utility should have the same dimension as value rather than no dimension, and the normalization problem should be reconsidered in the light of terminal values and utilities.     

The Generalized Means Model (GMM) for non-deterministic decision making: Its normative and descriptive power,including sketch of the representation theorem

Two schools of thought have been arguing during the last thirty years about the foundations of the theory of choice under uncertainty, namely: The neo-Bernoullian or American school defending an Expected Utility Model (EUM); and the Allais or French school proposing a model based on the moments of the probability distribution over psychological values (MM). In this paper we present a unified theory: the Generalized Means Model (GMM). By using the well known concept of the generalized mean it is possible to derive both contesting models from the same core of axioms, that — surprisingly enough — includes an extended form of the hotly debated Substitution Principle. It can be seen that the differences between the two models occur at the beginning and at the end of their axiomatic derivation, as follows: the EUM starts from a probability distribution function over consequences whereas the MM begins with a probability distribution over psychological values. The EUM finishes with an early introduction of a behavioural axiom on the existance of utility, whereas the MM uses first the properties of the distribution function, and then introduces the behavioural assumptions. The simplest MM consistent with all axioms of the GMM is the model proposed by the author some years ago. It is suggested that a reduced version, the Three Moments Model (TMM), is sufficient for practical applications. The second part of the paper demonstrates how the TMM solves in a very natural way the Allais Paradox, the certainty effect, the reflection effect, and several other behavioural observations.     

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaari's (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaari's language, when subjective odds differ from the market odds).     

Avoiding the curves: Direct elicitation of time preferences

We propose and test a new method for eliciting curvature-controlled discount rates that are invariant to the form of the utility function. Our method uses a single elicitation task and obtains individual discount rates without knowledge of risk attitude or parametric assumptions about the form of the utility function. We compare our method to a double elicitation technique in which the utility function and discount rate are jointly estimated. Our experiment shows that these methods yield consistent estimates of the discount rate, which is reassuring given the wide range of estimates in the literature. We find little evidence of probability weighting, but in a second experiment, we observe that discount rates are sensitive to the length of the front-end delay, suggesting present bias. When the front-end delay is at least two weeks, we estimate average discount rates to be 11.3 and 12.2% in the two experiments.     

Social choice in a sequential environment

In a recent paper [2] we presented a model of societies. In the context of that model, we argued that in the field of social choice it is necessary to consider some type of cardinal utility indices if we want to develop a sensible analysis. The main purpose of the present article is to complete and extend an argument initiated in Section 5 of [2] by giving a rigorous formulation and proof of a theorem informally discussed there.To formulate rigorously our theorem a concept of impersonality is introduced that appears to be more general than those which have been used in the economic literature.Our arguments concerning the need to use cardinal utility indices in the field of social choice are shown to be similar to those used by the proponents of decentralization in the controversy regarding centralization and decentralization as alternative ways of organizing the economic activity of a society.An earlier version of this paper was presented at the Econometric Society Winter Meeting held in Toronto, Canada, December 1972.