Utility and Probability

Acyclicity is a consistency property of preferences and other binary relations. It requires that the asymmetric part P of the relation (e.g. the subrelation of strict preference) contain no cycles; that is, for no sequence of alternatives x₁, x₂,…, x_n is it true that x₁Px₂, x₂Px₃, …, x_n−1Px_n, and x_nPx₁. The study of cyclic preferences dates at least to Condorcet’s (1785) treatment of the paradox of voting, in which transitive individual voters generate cyclic majority preferences.

THE ST PETERSBURG PARADOX AND THE BERNOULLIAN FORMULATION. Let there be a random prospect g₁, …, g_i, …, g_n, …, p₁, …, p_i, …, p_n (Σ_ip_i = 1) giving the probability p_i of positive or negative gains g_i.

The Rev. Thomas Bayes (1702–1761) was the eldest son of Joshua Bayes, a minister in the Nonconformist church. He was probably educated at Coward’s Academy. After assisting his father as pastor in Hatton Garden, London, he became, in 1731, Presbyterian minister at Mount Sion, Tunbridge Wells where he remained until his death on 17 April 1761. His fame today rests entirely on one paper, found by his friend Richard Price amongst Bayes’ effects after his death and presented to the Royal Society: Bayes (1763). (A convenient recent reference is Bayes, 1958.) The paper appears to have aroused little interest at the time and a proper appreciation was left to Laplace. Even today there is much discussion over just what Bayes meant, but the fact that so much interest is taken in a paper over 200 years old testifies to the importance of the problem and the brilliance of Bayes’ argument.

Swiss mathematician and theoretical physicist; born at Groningen, 8 February 1700; died at Basel, 17 March 1782.

The term ‘bounded rationality’ is used to designate rational choice that takes into account the cognitive limitations of the decision-maker — limitations of both knowledge and computational capacity. Bounded rationality is a central theme in the behavioural approach to economics, which is deeply concerned with the ways in which the actual decision–making process influences the decisions that are reached.

In order to take a decision in an uncertainty context, it is necessary, from a theoretical point of view, to build a model and specify all the consequences in every possible state of the world. In applied work this method is much too involved. Consequently, for applied purposes, it would be interesting to have a model where uncertainty is treated in such a way that the decision problems are as simple as the equivalent ones in a certainty framework. The identification of the conditions under which such an isomorphism between the optimal decisions under uncertainty and the optimal decisions in an equivalent certainty context holds is called the certainty equivalent problem.

The theory of general competitive equilibrium was originally developed for environments where no uncertainty prevailed. Everything was certain and phrases like ‘it might rain’ or ‘the weather might be hot’ were outside the scope of the theory. The idea of contingent commodity, that was introduced by Arrow (1953) and further developed by Debreu (1953), was an ingenious device that enabled the theory to be interpreted to cover the case of uncertainty about the availability of resources and about consumption and production possibilities. Basically, the idea of contingent commodity is to add the environmental event in which the commodity is made available to the other specifications of the commodity. With no uncertainty every commodity is specified by its physical characteristics and by the location and date of its availability. It is fairly clear, however, that such a commodity can be considered to be quite different where two different environmental events have been realised. The following examples clarify this: an umbrella at a particular location and at a given date in case of rain is clearly different from the same umbrella at the same location and date when there is no rain; some ice cream when the weather is hot is clearly different from the same ice cream (and at the same location and date) when the weather is cold; finally, the economic role of wheat with specified physical characteristics available at some location and date clearly depends on the precipitation during its growing season.

In this paper, I want to disentangle some of the senses in which the hypothesis of rationality is used in economic theory. In particular, I want to stress that rationality is not a property of the individual alone, although it is usually presented that way. Rather, it gathers not only its force but also its very meaning from the social context in which it is embedded. It is most plausible under very ideal conditions. When these conditions cease to hold, the rationality assumptions become strained and possibly even self–contradictory. They certainly imply an ability at information processing and calculation that is far beyond the feasible and that cannot well be justified as the result of learning and adaptation.

Giving the Sidney Ball Lecture in May 1929, A.C. Pigou remarked that ‘During some thirty years until their recent deaths in honoured age, the two outstanding names in English economics were Marshall at Cambridge and Edgeworth here in Oxford’ (Pigou and Robertson, 1931, p. 3). That the names were presented in that non-alphabetical order was not just Cambridge insularity but a universal perception. In a letter to Edgeworth from Bari in November 1890, Pantaleoni wrote that ‘… you are the closest approximation of a match for Marshall in England. You know that to my mind, Marshall is simply a new Ricardo who has appeared in the field — and to be second to him is as great an honour as a scientific man can wish for, in our time.’ The modest Edgeworth would have concurred: ‘Marshall was at the Council to-day; it was as if Achilles had come back,’ (reported by Bonar, 1926, p. 650).

1. Expected utility theory deals with choosing among acts where the decisionmaker does not know for sure which consequence will result from a chosen act. When faced with several acts, the decision–maker will choose the one with the highest ‘expected utility’, where the expected utility of an act is the sum of the products of probability and utility over all possible consequences.

The expected utility hypothesis of behaviour towards risk is essentially the hypothesis that the individual decision–maker possesses (or acts as if possessing) a ‘von Neumann-Morgenstern utility function’ U(·) or ‘von Neumann-Morgenstern utility index’ {U¡} defined over some set of outcomes, and when faced with alternative risky prospects or ‘lotteries’ over these outcomes, will choose that prospect which maximizes the expected value of U(·) or {U¡}. Since the outcomes could represent alternative wealth levels, multidimensional commodity bundles, time streams of consumption, or even non–numerical consequences (e.g. a trip to Paris), this approach can be applied to a tremendous variety of situations, and most theoretical research in the economics of uncertainty, as well as virtually all applied work in the field (e.g. optimal trade, investment or search under uncertainty) is undertaken in the expected utility framework.

De Finetti was born in Innsbruck, Austria in 1906, and died in Rome in 1985. After a degree in mathematics at Milan University, he chose practical activities rather than an academic career, and worked at the Istituto Centrale di Statistica (1927–31) and then at the Assicurazioni Generali (1931–46). Only later did he turn to an academic career and win a chair in Financial Mathematics at Trieste University (1939); from 1954 to 1961 he held the chair in the same subject at the University of Rome and from 1961 to 1976 the chair of Calculus of Probabilities at the same university. He was a member of the Accademia Nazionale dei Lincei and Fellow of the International Institute of Mathematical Statistics.

Gossen was born in Düren (between Aachen and Cologne) on 7 September 1810; he died in Cologne on 13 February 1858. Little is known about his life, partly because the inconspicuous bachelor did not attract attention, partly because most of those who had known him were dead by the time he became famous, partly also because his literary remains, scant as they must have been, are lost. The principal biographical source is the essay by Walras (1885). The available facts are admirably surveyed by Georgescu–Roegen (1983), on whose masterly introduction to the English translation of Gossen’s book the following life sketch is mostly based.

Impatience refers to the preference for earlier rather than later consumption, an idea which stems from Böhm–Bawerk (1912) and Fisher (1930), among others. Preference orderings that exhibit impatience are also described as being myopic or as embodying discounting. Because in many contexts the future has no natural termination date, an infinite horizon framework is most appropriate and convenient for the analysis of many problems in intertemporal economics. The open-endedness of the future raises several issues surrounding impatience (its presence, degree, and the precise form it takes) which do not arise in finite horizon models.

Induction, in its most general form, is the making of inferences from the observed to the unobserved. Thus, inferences from the past to the future, from a sample to the population, from data to an hypothesis, and from observed effects to unobserved causes are all aspects of induction, as are arguments from analogy. A successful account of induction is required for a satisfactory theory of causality, scientific laws and predictive applications of economic theory. But induction is a dangerous thing, and especially so for those who lean towards empiricism, the view that only experience can serve as the grounds for genuine knowledge. Because induction, by its very nature, goes beyond the observed, its use is inevitably difficult to justify for the empiricist. In addition, inductive inferences differ from deductive inferences in three crucial respects. First, the conclusion of an inductive inference does not follow with certainty from the premises, but only with some degree of probability. Second, whereas valid deductive inferences retain their validity when extra information is added to the premises, inductive inferences may be seriously weakened. Third, whereas there is widespread agreement upon the correct characterization of deductive validity. there is widespread disagreement about what constitutes a correct inductive argument, and indeed whether induction is a legitimate part of science at all.

Interdependent preferences arise in economic theory in the study of both individual decisions and group decisions. We imagine that a decision is required among alternatives in a set X and that the decision will depend on preferences between the elements in X. If the preferences represent different points of view about the relative desirability of the alternatives, or if they are based on multiple criteria that impinge on the decision, then we encounter the possibility of interdependent preferences.

Suppose I am left with a ticket to a Mozart concert I am unable to attend and decide to give it to one of my closest friends. Which friend should I actually give it to? One thing I will surely consider in deciding this is which friend of mine would enjoy the concert most. More generally, when we decide as private individuals whom to help, or decide as voters or as public officials who are to receive government help, one natural criterion we use is who would derive the greatest benefit, that is, who would derive the highest utility, from this help. But to answer this last question we must make, or at least attempt to make, interpersonal utility comparisons.

Lexicographic orderings are orderings in which certain elements of the space being ordered have been selected for special treatment. I begin with an example. Suppose an agent has an ordering over commodities a and b. Although he or she likes both a and b, any bundle which has more of a is preferred to any bundle which has less of a. Of course among bundles which have the same amount of a, bundles with more b are preferred to those with less. Thus, there are no trade-offs between a and b and each indifference set is a single point. The name ‘lexicographic’ comes from the way words are ordered in a dictionary, alphabetically by the first letter and then the second and so on.

In a dynamic context a decision maker at any instant t has information about his exogenous economic environment both at time t and at later dates. We represent the environment at t by a vector x(t) of exogenous variables, and their future values by (x(t + l),c + 2),…,ü + T)). The horizon T is determined by such considerations as length of life, technology, resource limitations etc.; it might be infinite. A decision rule at time t is a map ψ_t associating with a vector of a variables z, the variable d representing the choice of the decision maker. We write d = ψ_t(z). Myopic decision rules refer to those maps of the form d(t) = ψ_t(x(t)) in which d(t) depends only upon the values of the exogenous variables at time t, disregarding any information about future conditions of the economic environment. A decision rule is said to be non–myopic if it is of the form d(t) = ψ_t(x(t), x(t=l),…,x(t+T)).

An ordering (also called a complete preordering or a weak ordering) is a binary relation which is reflexive, transitive and complete, that is, it is a preordering that is complete.

Perfect foresight is an occasionally convenient theoretical assumption whose total lack of realism is undisputed, and perhaps unrivalled. There are two elements to perfect foresight; firstly that people have definite point expectations, allowing no uncertainty, of future variables, and secondly that these expectations are correct. In practice, as these fortunate perfectly foresightful individuals generally inhabit models with instantaneously clearing perfectly competitive markets, they only need to forecast prices. The pioneering work by Hicks (1939) on intertemporal general equilibrium theory provides a framework in which the issues associated with perfect foresight can be explored. Writing prior to the development of the expected utility theory of choice under uncertainty (von Neumann and Morgenstern, 1944), Hicks had no alternative to a deterministic model in his discussion. He acknowledges the existence and importance of uncertainty in expectation formation, but argues in a somewhat unsatisfactory fashion that point predictions can be interpreted as risk adjusted summaries of underlying probability distributions. Hicks divides time into weeks. Trade takes place weekly. Supply and demand in each week depend upon decisions made in the past, expectations of spot prices in future weeks and current spot prices. In temporary equilibrium these spot prices adjust to clear markets, but expectations may be wrong. In the situation which Hicks terms ‘Equilibrium over Time’, markets clear at each date, and, crucially, everyone has perfect foresight; price expectations are fulfilled.

1. The concept of preference holds a pivotal position in value theory. It may even be considered a ‘value radical’ or common conceptual root of the three main types of evaluative discourse, namely, aesthetic, economic and moral.

Preference reversal is an experimentally observed phenomenon in which subjects, when asked to choose between suitably matched pairs of lotteries and then to state the lowest amount of money they would be willing to accept in exchange for the right to participate in each of these lotteries, announce the lowest amount for the chosen lottery. Preference reversals were first reported by Lichtenstein and Slovic (1971) and have since been replicated in numerous studies, for example Lindman (1971), Grether and Plott (1979), Pommerehne, Schneider and Zweifel (1982), Reilly (1982). The latter studies introduce variations in the experimental design to increase the motivation and reduce the possibility of confusion and errors on the part of the subjects. This experimental evidence seems inconsistent with transitive preferences and, consequently, with any theory of decision making under risk based on such preferences. Recently, however, Karni and Safra (1986) and Holt (1986) demonstrated that preference reversals may occur even when preferences are transitive, and that the preference reversal phenomenon may be the result of the interaction between the subjects’ preferences and the experimental design. According to this interpretation, preference reversals constitute a violation of the independence axiom of expected utility theory.

A preordering (also called a weak ordering or a quasi-ordering) is a reflexive and transitive binary relation which is not necessarily complete.

Probability denotes a family of ideas that originally centred on the notion of credibility, or reasonable belief falling short of certainty. There have arisen two quite distinct uses of this group of ideas, namely in the modelling of physical or social processes, and in drawing inferences from, or making decisions on the basis of, inconclusive data.

Several developments have joined to stimulate economists to think about issues that have been on the forefront of psychological research. Firstly, the information revolution in economics has focused economists on the subtle nature of individual information processing. Secondly, developments in game theory have so successfully identified new solution concepts that for almost any pattern of market behaviour there exists a reasonable theory consistent with that pattern. Introspection, a few principles of decision making, internal consistency and a few stylized facts do not constrain possibilities enough to be sufficient guides to theory. Theorists are being forced to seek more systematic sources of data and additional principles to reduce the number of competing theories. Thirdly, the rapid development of experimental methods applicable to economics has brought the testing of psychologically based economic theories within the realm of reality. Economists can accurately measure behaviour in economically relevant settings. As behavioural patterns become established that are difficult to reconcile with economic models alone, the profession has begun to look to psychology for answers. The data thus force the attention of economists to a broader class of models.

There are interesting parallels in the careers of Frank Ramsey and John von Neumann. Each was born in 1903, one the product of the ‘High Intelligentsia of England’ (Keynes, 1933, p. vii) and the other the son of a wealthy banker in Budapest (Ulam, 1976, p. 79). Each was a creative mathematician of high order but each also made major contributions to at least two other disciplines. Each wrote just three papers in economic theory, all six of which were of fundamental importance. Moreover, with one exception every one of these seminal papers had to wait many years for its proper recognition; even the exception — the utility theory set out in the Appendix to von Neumann and Morgenstern (1947) — at first encountered serious misunderstanding within the profession. Indeed, considering them purely as economists, one wonders how these two geniuses would fare today, when promotion and tenure so often depend on a good immediate showing in citation indexes and the like.

The concept of rational behaviour is frequently used in economic theory. The interest in this concept springs from two quite distinct motivations. First, insofar as economic exercises often take a perspective form, it is interesting to know how one could behave rationally in a given situation. This may be called the ‘prescriptive motivation’. It should be warned that the prescription need not be necessarily of an ethical kind. Indeed, the prescriptive motivation is sometimes described in clearly non-ethical terms, involving the pursuit of self-interest only. In a classic presentation of this position, Harsanyi (1977) describes ‘perfectly rational behaviour’ in the context of game theory in the following terms:

…our theory is a

normative

(prescriptive) theory rather than a

positive

(descriptive) theory. At least formally and explicitly it deals with the question of how each player

should

act in order to promote his own interests most effectively in the game and not with the question of how he (or persons like him)

will

actually act in a game of this particular type (Harsanyi, 1977, p. 16).

Three facets of subjective preferences have played central roles in economics. They are the qualitative structure of an agent’s preferences, numerical representations of preferences, and the use of numerical representations or utility functions in economic analysis. We consider various representations and their ties to qualitative preference structures.

The phenomenon of risk (or alternatively, uncertainty or incomplete information) plays a pervasive role in economic life. Without it, financial and capital markets would consist of the exchange of a single instrument each period, the communications industry would cease to exist, and the profession of investment banking would reduce to that of accounting. One need only consult the contents of any recent economics journal to see how the recognition of risk has influenced current research in economics. In this essay we present an overview of the modern economic theory of the characterization of risk and the modelling of economic agents’ responses to it.

L.J. (Jimmie) Savage, né Leonard Ogashevitz, was born in Detroit on 20 November 1917 and died in New Haven on 1 November 1971. His interests were encyclopaedic: as a youth he immersed himself in the Book of Knowledge, and at the time of his death he was preparing for the Peabody Museum a demonstration-exhibit on animal odorants. The dominant theme of Savage’s professional work was the mathematical analysis of normative behaviour.

It is commonplace to formulate theories of individual decision-making under uncertainty using three sets: the set of states of nature, S, the set of consequences, C, and the set of acts, L. Following Savage (1954) we define nature as the object of concern to the decision–maker and a state of nature as a portrayal of nature leaving no relevant aspect undescribed. A consequence is anything that may happen to a person. An act is a course of action. Each combination of an act fϵL and a state of nature sϵS determines a unique consequence denoted c(f, s) in C. In addition, the theory postulates the existence of a preference relation, ≽, on acts. For our purposes preference relations are taken to be complete and transitive binary relations on L, with the symbol ≽ being interpreted as ‘preferred or indifferent’.

Under certainty, with commodities iϵI, individual preferences are defined over commodity bundles c = (c_i:iϵI), which are the objects of choice of individuals. Under uncertainty, production possibilities and individual and aggregate endowments, for instance, may vary with the realization of random states of nature sϵS.

The notion of stochastic dominance is quite old (see, for example, Blackwell, 1953). Although it was (in various forms) used in statistical or economic theory, it was for some reason not developed until 1969–70, when four papers were published by Hadar and Russell (1969), Hanoch and Levy (1969), Rothschild and Stiglitz (1970) and Whitmore (1970). Since then almost 400 papers have been written on this topic; for a good survey article see Kroll and Levy (1980) and for a good bibliography see Bawa (1982).

1. KINDS OF PROBABILITY. The usual meaning of ‘probable’ in ordinary conversation is closely related to its derivation from a Latin word meaning provable or capable of being made convincing. The concept is even clearer in the derivation of the German word Wahrscheinlichkeit, ‘having the appearance of truth’. In fact, when we say an event is probably we usually mean that we would not be surprised (or we ought not to be) if it occurred, or that we would be somewhat surprised (or ought to be) if it did not occur. Since ‘surprise’ refers to a personal or subjective experience it seems clear that the ordinary concept of probability is subjectivistic (or else in some sense logical). Also a probability, in this subjective or logical sense, can be more or less large so it can be interpreted as a degree of belief or intensity of conviction. A subjective probability is usually regarded as somewhat more than just a degree of belief — it is a degree of belief that belongs to a body of beliefs from which the worst inconsistencies have been removed by means of detached judgements. In short, the degree of belief should be more or less rational.

Time preference is the insight that people prefer ‘present goods’ (goods available for use at present) to ‘future goods’ (present expectations of goods becoming available at some date in the future), and that the social rate of time preference, the result of the interactions of individual time preference schedules, will determine and be equal to the pure rate of interest in a society. The economy is pervaded by a time market for present as against future goods, not only in the market for loans (in which creditors trade present money for the right to receive money in the future), but also as a ‘natural rate’ in all processes of production. For capitalists pay out present money to buy or rent land, capital goods, and raw materials, and to hire labour (as well as buying labour outright in a system of slavery), thereby purchasing expectations of future revenue from the eventual sales of product. Long–run profit rates and rates of return on capital are therefore forms of interest rate. As businessmen seek to gain profits and avoid losses, the economy will tend toward a general equilibrium, in which all interest rates and rates of return will be equal, and hence there will be no pure entrepreneurial profits or losses.

Transitivity is formally just a property that a binary relation might possess, and thus one could discuss the concept in any context in economics in which an ordering relation is used. Here, however, the discussion of transitivity will be limited to its role in describing an individual agent’s choice behaviour. In this context transitivity means roughly that if an agent choses A over B, and B over C, that agent ought to choose A over C, or at least be indifferent. On the surface this seems reasonable, even ‘rational’, but this ignores how complicated an agent’s decision making process can be. For an excellent discussion of this issue see May (1954). Given a model of agent behaviour, transitivity can be imposed as a direct assumption, or can be an implication of the model for choice behaviour. The standard model of agent behaviour in economics is that the agent orders prospects by means of a utility function, which in effect assumes transitivity. With appropriate continuity and convexity restrictions on utility functions, the model allows one to demonstrate that: (1) Individual demand functions are well defined, continuous, and satisfy the comparative static restriction, the Strong Axiom of Revealed Preference (SARP). (In the smooth case, this corresponds to the negative semidefiniteness and symmetry of the Slutsky matrix.) (2) Given a finite collection of such agents with initial endowments of goods, a competitive equilibrium exists. What will be discussed in the remaining part of this essay is to what extent one can obtain results analogous to (1) and (2) above while using a model of agent behaviour which does not assume or implv transitive behaviour.

Nothing is more certain than the prevalence of uncertainty about the consequences of any economic decision. It is therefore entirely appropriate that uncertainty has been the subject of a large literature that grew out of important work in the early 1950s, and is still flourishing, as testified by the number of recent surveys and books such as Balch, McFadden and Wu (1974), Diamond and Rothschild (1978), Hirshleifer and Riley (1979), Lippman and McCall (1981), Fishburn (1982), Schoemaker (1982), Sinn ( 1983). No attempt will be made here to provide a comprehensive new survey. Rather, the devices of state contingent consequence functions and state preferences will be explained, and various types of uncertainty categorized. Following the pioneering work of Ramsey (1926), Savage (1954), and Anscombe and Aumann (1963) in particular, I shall discuss decision theory and when uncertainty can be described by subjective probabilities, based on an analysis of decision trees in the spirit of Raiffa (1968). While uncertainty per se can be largely treated through devices such as Debreu (1959) state contingent commodity contracts, the problems posed by asymmetric information and the lack of common knowledge are much more fundamental and intractable.

Utility is a term which has a long history in connection with the attempts of philosophers and political economists to explain the phenomenon of value. It has most frequently been given the connotation of ‘desiredness’, or the capacity of a good or service to satisfy a want, of whatever kind. Its use with that meaning can be traced back at least to Gershom Carmichael’s 1724 edition of Pufendorf s De Officio Hominis et Civis Iuxta Legem Naturalem, and arguably came down to him through the medieval schoolmen from Aristotle’s Politics.

The conjunction of utility theory and decision theory involves formulations of decision making in which the criteria for choice among competing alternatives are based on numerical representations of the decision agent’s preferences and values. Utility theory as such refers to those representations and to assumptions about preferences that correspond to various numerical representations. Although it is a child of decision theory, utility theory has emerged as a subject in its own right as seen, for example, in the contemporary review by Fishburn (see REPRESENTATION OF PREFERENCES). Readers interested in more detail on representations of preferences should consult that essay.