Choice under Risk and Uncertainty
- General Introduction -

Magister Ludi


(A) Randomness in Economic Theory
(B) Risk, Uncertainty and Expected Utility


(A) Randomness in Economic Theory

Surprisingly, risk and uncertainty have a rather short history in economics. The formal incorporation of risk and uncertainty into economic theory was only accomplished in 1944, when John von Neumann and Oskar Morgenstern published their Theory of Games and Economic Behavior - although the exceptional effort of Frank P. Ramsey (1926) must be mentioned as an antecedent. Indeed, the very idea that risk and uncertainty might be relevant for economic analysis was only really suggested in 1921, by Frank H. Knight in his formidable treatise, Risk, Uncertainty and Profit. What makes this lateness even more surprising is that not only could early economists count several prominent statistical theorists among their ranks (notably Francis Y. Edgeworth and John Maynard Keynes), but that the very concept of marginal utility, the foundation stone of Neoclassical economics, was introduced by Daniel Bernoulli (1738) in the context of choice under risk.

Previous to Frank H. Knight's 1921 treatise, only a handful of economists, notably Carl Menger (1871), Irving Fisher (1906) and Francis Y. Edgeworth (1908), even deigned to acknowledge the potential modifications risk and uncertainty might make to economic theory. It was in Knight's treatise that for effectively the first time the case was made for the economic importance of these concepts. Indeed, he linked profits, entrepreneurship and the very existence of the free enterprise system to risk and uncertainty. After Knight, economists finally began to take it into account: John Hicks (1931), John Maynard Keynes (1936, 1937), Michal Kalecki (1937), Helen Makower and Jacob Marschak (1938), George J. Stigler (1939), Gerhard Tintner (1941), A.G. Hart (1942) and Oskar Lange (1944), appealed to risk or uncertainty to explain things like profits, investment decisions, demand for liquid assets, the financing, size and structure of firms, production flexibility, inventory holdings, etc.

As Arrow's (1951) survey of the state of affairs illustrates, it was a growing field with severe growing pains and much confusion. The great barrier in a lot of this early work was in making precise what it means for "uncertainty" or "risk" to affect economic decisions. How do agents evaluate ventures whose payoffs are random? How exactly does increasing or decreasing uncertainty consequently lead to changes in behavior? These questions were crucial, but with several fundamental concepts left formally undefined, appeals risk and uncertainty were largely of a heuristic and unsystematic nature.

The great missing ingredient was the formalization of the notion of "choice" in risky or uncertain situations. Already Hicks (1931), Marschak (1938) and Tintner (1941) had a sense that people should form preferences over distributions, but how does one separate the element of attitudes towards risk or uncertainty from pure preferences over outcomes? Alternative hypotheses included ordering random ventures via their means, variances, etc., but no precise or satisfactory means were offered up. Ocassionally, they took some quite bizarre turns: for instance, Arthur C. Pigou attempted to measure a "fundamental unit of uncertainty-bearing" by defining it as "the exposure of a 」 to a given scheme of uncertainty, or ... to a succession of like schemes of uncertainty during a year ... by a man of representative temperament and with representative knowledge." (Pigou, 1912: p.772).

Surprisingly, Daniel Bernoulli's (1738) notion of expected utility which decomposed the valuation of a risky venture as the sum of utilities from outcomes weighted by the probabilities of outcomes, was generally not appealed to by these early economists. Part of the problem was that it did not seem sensible for rational agents to maximize expected utility and not something else. Specifically, Bernoulli's assumption of diminishing marginal utility seemed to imply that, in a gamble, a gain would increase utility less than a decline would reduce it. Consequently, many concluded, the willingness to take on risk must be "irrational", and thus the issue of choice under risk or uncertainty was viewed suspiciously, or at least considered to be outside the realm of an economic theory which assumed rational actors.

The great task of John von Neumann and Oskar Morgenstern (1944) was to lay a rational foundation for decision-making under risk according to expected utility rules. Once this was done, the floodgates opened - albeit, even then, only slowly. The novelty of using the axiomatic method - combining sparse explanation with often obtuse axioms - ensured that most economists of the time would find their contribution inaccessible and bewildering. Indeed, there was substantial confusion regarding the structure and meaning of the von Neumann-Morgenstern expected utility hypothesis itself. Restatements and re-axiomatizations by Jacob Marschak (1950), Paul Samuelson (1952) and I.N. Herstein and J. Milnor (1953) did much to improve the situation.

A second revolution occurred soon afterwards. The expected utility hypothesis was given a celebrated subjectivist twist by Leonard J. Savage in his classic Foundations of Statistics (1954). Inspired by the work of Frank P. Ramsey (1926) and Bruno de Finetti (1931, 1937), Savage derived the expected utility hypothesis without imposing objective probabilities but rather by allowing subjective probabilities to be determined jointly. Savage's brilliant performance was followed up by F.J. Anscombe and R.J. Aumann (1963). In some regards, the Savage-Anscome-Aumann "subjective" approach to expected utility has been considered more "general" than the older von Neumann-Morgenstern concept.

Another "subjectivist" revolution was initiated with the "state-preference" approach to uncertainty of Kenneth J. Arrow (1953) and Gerard Debreu (1959). Although not necessarily "opposed" to the expected utility hypothesis, the state-preference approach does not involve the assignment of mathematical probabilities, whether objective or subjective, although it often might be useful to do so. The structure of the state-preference approach is more amenable to Walrasian general equilibrium theory where "payoffs" are not merely money amounts but actual bundles of goods. It became particularly popular after useful applications were pursued by Jack Hirshleifer (1965, 1966), Peter Diamond (1967) and Roy Radner (1968, 1972) and has since become the dominant method of incorporating uncertainty in general equilibrium contexts.

The comparative properties of the expected utility hypothesis when payoffs are univariate (i.e. "money") were further examined and developed in the post-war period. The concept of "risk aversion" was analyzed by Milton Friedman and Leonard J. Savage (1948) and Harry Markowitz (1952) and measurements of risk aversion developed by John W. Pratt (1964) and Kenneth J. Arrow (1965) and later refined by Stephen Ross (1981). Menachem Yaari (1968) and Richard Kihlstrom and L. Mirman (1974) pursued definitions of risk-aversion in multi-variate contexts. Measurements of "riskiness" were suggested by Michael Rothschild and Joseph E. Stiglitz (1970, 1971), Peter Diamond and J.E. Stiglitz (1974) and others. These have been particularly useful in many economic applications (for a survey, see Lippmann and McCall (1981)).

There have also always been disputants. George L.S. Shackle (1949), Maurice Allais (1953) and Daniel Ellsberg (1961) were among the first to challenge the expected utility decomposition of choice under risk or uncertainty and to suggest substantial modifications. Influential experimental studies, such as those by Daniel Kahneman and Amos Tversky (e.g. 1979), have reinforced the need to rethink much of the theory. Towards this end, in recent years, many attempts have been made to reaxiomatize the theory of choice under uncertainty, with weighted expected utility (e.g. Allais, 1979; Chew and McCrimmon, 1979; Fishburn, 1983), rank-dependent expected utility (Quiggin, 1982; Yaari, 1987), non-linear expected utility (e.g. Machina, 1982), regret theory (Loomes and Sugden, 1982), non-additive expected utility (Shackle , 1949; Schmeidler, 1989) and state-dependent preferences (Karni, 1985).

(B) Risk, Uncertainty and Expected Utility

Much has been made of Frank H. Knight's (1921: p.20, Ch.7) famous distinction between "risk" and "uncertainty". In Knight's interpretation, "risk" refers to situations where the decision-maker can assign mathematical probabilities to the randomness which he is faced with. In contrast, Knight's "uncertainty" refers to situations when this randomness "cannot" be expressed in terms of specific mathematical probabilities. As John Maynard Keynes was later to express it:

"By `uncertain' knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty...The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence...About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know." (J.M. Keynes, 1937)

Nonetheless, many economists dispute this distinction, arguing that Knightian risk and uncertainty are one and the same thing. For instance, they argue that in Knightian uncertainty, the problem is that the agent does not assign probabilities, and not that she actually cannot, i.e. that uncertainty is really an epistemological and not an ontological problem, a problem of "knowledge" of the relevant probabilities, not of their "existence". Going in the other direction, some economists argue that there are actually no probabilities out there to be "known" because probabilities are really only "beliefs". In other words, probabilities are merely subjectively-assigned expressions of beliefs and have no necessary connection to the true randomness of the world (if it is random at all!).

Nonetheless, some economists, particularly Post Keynesians such as G.L.S. Shackle (1949, 1961, 1979) and Paul Davidson (1982, 1991) have argued that Knight's distinction is crucial. In particular, they argue that Knightian "uncertainty" may be the only relevant form of randomness for economics - especially when that is tied up with the issue of time and information. In contrast, situations of Knightian "risk" are only possible in some very contrived and controlled scenarios when the alternatives are clear and experiments can conceivably be repeated -- such as in established gambling halls. Knightian risk, they argue, has no connection to the murkier randomness of the "real world" that economic decision-makers usually face: where the situation is usually a unique and unprecedented one and the alternatives are not really all known or understood. In these situations, mathematical probability assignments usually cannot be made. Thus, decision rules in the face of uncertainty ought to be considered different from conventional expected utility.

The "risk versus uncertainty" debate is long-running and far from resolved at present. As a result, we shall attempt to avoid considering it with any degree of depth here. What we shall refer throughout as "uncertainty" does not correspond to its Knightian definition. Instead, we will use the term with more fluidity in analyzing modern theories of choice in random situations. However, some form of the Knightian distinction may still be useful, in that it permits us to roughly divide theories between those which use the assignment of mathematical probabilities and those which do not make such assignments. In this manner, the expected utility theory with objective probabilities of von Neumann and Morgenstern (1944) is clearly one of "risk", whereas the state-preference approach of Arrow (1953) and Debreu (1959). in which there are no assignments of probabilities whatsoever is (perhaps less obviously) one of "uncertainty". However, the intermediate theory of Savage (1954), which yields expected utility with subjective probabilities, is not clearly in one camp or another: on the one hand, the very assignment of numerical probabilities - even if subjective - implies that it represents choice under "risk"; on the other hand, these probabilities are merely expressions of what is ultimately amorphous belief and thus may seem more like "uncertainty". [A more extensive discussion of the various categories of "objective" and "subjective" probabilities are contained in the introduction to our exposition of Savage's theory].

In the first section, we shall concentrate on the expected utility hypothesis with objective probabilities of von Neumann and Morgenstern (1944). We shall consider Savage's theory after that and the Arrow-Debreu approach later on. As we have noted, there are other approaches to choice under uncertainty, but these three are the most developed and prominent ones and thus we shall concentrate on them.

Excellent surveys of uncertainty theory include Peter C. Fishburn (1970, 1982, 1988, 1994) and Edi Karni and David Schmeidler (1991) at a relatively advanced level and Jack Hirshleifer and John G. Riley (1979, 1992), Jean-Jacques Laffont (1989) and Mark Machina (1987) at a more accessible level. The remarkable little classic of David M. Kreps (1988) is especially recommended for its excellent exposition and intuition. The older text by R.D. Luce and H. Raiffa (1957) may also be still worth consulting. The volume edited byPeter Diamond and Michel Rothschild (1978) reproduces several classical articles. Finally, the relevant entries in The New Palgrave, several of them conveniently collected and reprinted in a distinct volume (Eatwell, Milgate and Newman, 1990), are highly recommended. For surveys focused more on applications of uncertainty theory, see J.D. Hey (1979) and S.M. Lippmann and J.J. McCall (1981).

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