"[Under uncertainty] there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability waiting to be summed."
(John Maynard Keynes, "General Theory of Employment", 1937, Quarterly Journal of Economics)
"Many idle controversies involving the nature of expectation could be avoided by recognizing at the outset that man's conscious actions are the reflection of his beliefs and of nothing else."
(Nicholas Georgescu-Roegen, 1958)
(A) The Concept of Subjective Probability
(B) Savage's Axiomatization
(C) The Anscombe-Aumann Approach
(D) The Ellsberg Paradox and State-Dependent Preferences
(A) The Concept of Subjective Probability
In the von Neumann-Morgenstern theory, probabilities were assumed to be "objective". In this respect, they followed the "classical" view that randomness and probabilities, in a sense, "exist" inherently in Nature. There are roughly three versions of the objectivist position. The oldest is the "classical" view perhaps stated most fully by Pierre Simon de Laplace (1795). Effectively, the classical view argues that the probability of an event in a particular random trial is the number of equally likely outcomes that lead to that event divided by the total number of equally likely outcomes. Underlying this notion is the "principle of cogent reason" (i.e physical symmetry implies equal probability) and the "principle of insufficient reason" (i.e. if we cannot tell which outcome is more likely, we ought to assign equal probability).
There are great deficiencies in the classical approach - particularly the meaning of symmetry and the possibly non-additive and often counterintuitive consequences of the principle of insufficient reason. As a result, it has been challenged in the twentieth century by a variety of competing conceptions Its most prominent successor was the "relative frequentist" view famously set out by Richard von Mises (1928) and popularized by Hans Reichenbach (1949). The relative frequency view argues that the probability of a particular event in a particular trial is the relative frequency of occurrence of that event in an infinite sequence of "similar" trials.
In a sense, the relative frequentist view is related to Jacob Bernoulli's (1713) "law of large numbers". This claims, in effect, that if an event occurs a particular set of times (k) in n identical and independent trials, then if the number of trials is arbitrarily large, k/n should be arbitrarily close to the "objective" probability of that event. What the relative frequentists added (or rather subtracted) is that instead of positing the independent existence of an "objective" probability for that event, they defined that probability precisely as the limiting outcome of such an experiment.
The relative frequentist idea of infinite repetition, of course, is merely an idealization. Nonetheless, this notion caused a good amount of discomfort even to partisans of the objectivist approach: how is one to discuss the probability of events that are inherently "unique" (e.g. the outcome of the U.S. presidential elections in the year 2000). As a consequence, some frequentists have accepted the limitations of probability reasoning merely to controllable "mechanical" situations and allow unique random situations to fall outside their realm of applicability.
However, many thinkers remained unhappy with this practical compromise on the applicability of probability reasoning. As an alternative, some have appealed to a "propensity" view of objective probabilities, initially suggested by Charles S. Peirce (1910), but most famously associated with Karl Popper (1959). The "propensity" view of objective probabilities argues that probability represents the disposition or tendency of Nature to yield a particular event on a single trial, without it necessarily being associated with long-run frequency. It is important to note that these "propensities" are assumed to objectively exist, even if only in a metaphysical realm. Given the degree of looseness of the concept, one should expect its formalization to be somewhat more difficult. For a noble attempt, see Patrick Suppes (1973).
However, many statisticians and philosophers have long objected to this view of probability, arguing that randomness is not an objectively measurable phenomenon but rather a "knowledge" phenomena, thus probabilities are an epistemological and not an ontological issue. In this view, a coin toss is not necessarily characterized by randomness: if we knew the shape and weight of the coin, the strength of the tosser, the atmospheric conditions of the room in which the coin is tossed, the distance of the coin-tosser's hand from the ground, etc., we could predict with certainty whether it would be heads or tails. However, as this information is commonly missing, it is convenient to assume it is a random event and ascribe probabilities to heads or tails. In short, in this view, probabilities are really a measure of the lack of knowledge about the conditions which might affect the coin toss and thus merely represent our beliefs about the experiment. As Knight expressed it, "if the real probability reasoning is followed out to its conclusion, it seems that there is `really' no probability at all, but certainty, if knowledge is complete." (Knight, 1921: 219).
This epistemic or knowledge view of probability can be traced back to arguments in the work of Thomas Bayes (1763) and Pierre Simon de Laplace (1795). The epistemic camp can also be roughly divided into two groups: the "logical relationists" and the "subjectivists".
The logical relationist position was perhaps best set out in John Maynard Keynes's Treatise on Probability (1921) and, later on, Rudolf Carnap (1950). In effect, Keynes (1921) had insisted that there was less "subjectivity" in epistemic probabilities than was commonly assumed as there is, in a sense, an "objective" (albeit not necessarily measurable) relation between knowledge and the probabilities that are deduced from them. It is important to note that, for Keynes and logical relationists, knowledge is disembodied and not personal. As he writes:
"In the sense important to logic, probability is not subjective. A proposition is not probable because we think it so. When once the facts are given which determine our knowledge, what is probable or improbable in those circumstances has been fixed objectively, and is independent of our opinion." (Keynes, 1921: p.4)
Frank P. Ramsey (1926) disagreed with Keynes's assertion. Rather than relating probability to "knowledge" in and of itself, Ramsey asserted instead that it is related to the knowledge possessed by a particular individual alone. In Ramsey's account, it is personal belief that governs probabilities and not disembodied knowledge. Probability is thus subjective.
This "subjectivist" viewpoint had been around for a while - even economists such as Irving Fisher (1906: Ch.16; 1930: Ch.9) had expressed it. However, the difficulty with the subjectivist viewpoint is that it seemed impossible to derive mathematical expressions for probabilities from personal beliefs. If assigned probabilities are subjective, which almost implies that randomness itself is a subjective phenomenon, how is one to construct a consistent and predictive theory of choice under uncertainty? After von Neumann and Morgenstern (1944) achieved this with objective probabilities, the task was at least manageable. But with subjective probability, far closer in meaning to Knightian uncertainty, the task seemed impossible.
However, Frank Ramsey's great contribution in his 1926 paper was to suggest a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities. In so doing, Ramsey provided the first attempt at an axiomatization of choice under uncertainty - more than a decade before von Neumann-Morgenstern's attempt (note that Ramsey's paper was published posthumously in 1931). Independently of Ramsey, Bruno de Finetti (1931, 1937) had also suggested a similar derivation of subjective probability.
The subjective nature of probability assignments is can be made clearer by thinking of situations like a horse race. In this case, most spectators face more or less the same lack of knowledge about the horses, the track, the jockeys, etc. Yet, while sharing the same "knowledge" (or lack thereof), different people place different bets on the winning horse. The basic idea behind the Ramsey-de Finetti derivation is that by observing the bets people make, one can presume this reflects their personal beliefs on the outcome of the race. Thus, Ramsey and de Finetti argued, subjective probabilities can be inferred from observation of people's actions.
To drive this point further, suppose a person faces a random venture with two possible outcomes, x and y, where the first outcome is more desirable than the second. Suppose that our agent faces a choice between two lotteries, p and q defined over these two outcomes. We do not know what p and q are composed of. However, if an agent chooses lottery p over lottery q, we can deduce that he must believe that lottery p assigns a greater probability to state x relative to y and lottery q assigns a lower probability to x relative to y. The fact that x is more desirable than y, then, implies that his behavior would be inconsistent with his tastes and/or his beliefs had he chosen otherwise. In essence, then, the Ramsey-de Finetti approach can be conceived of as a "revealed belief" approach akin to the "revealed preference" approach of conventional consumer theory.
We should perhaps note, at this point, that another group of subjective probability theorists, most closely associated with B.O. Koopman (1940) and Irving J. Good (1950, 1962), holds a more "intuitionist" view of subjective probabilities which disputes this conclusion. In their view, the Ramsey-de Finetti "revealed belief" approach is too dogmatic in its empiricism as, in effect, it implies that a belief is not a belief unless it is expressed in choice behavior. In contrast, "the intuitive thesis holds that...probability derives directly from intuition, and is prior to objective experience" (Koopman, 1940: p.269). Thus, subjective probability assignments need not necessarily always reveal themselves through choice - and even then, usually through intervals of upper and lower probabilities rather than single numerical measures, and therefore, only partially ordered - a concept that stretches back to John Maynard Keynes (1921, 1937) and finds its most prominent economic voice in the work of George L.S. Shackle (e.g. Shackle, 1949, 1955, 1961) (although one can argue, quite reasonably, that the Arrow-Debreu "state-preference" approach expresses precisely this intuitionist view).
More importantly, the intuitionists hold that not all choices reveal probabilities. If the Ramsey-de Finetti analysis is taken to the extreme, choice behavior may reveal "probability" assignments that the agent had no idea he possessed. For instance, an agent may bet on a horse simply because he likes its name and not necessarily because he believes it will win. A Ramsey-de Finetti analyst would conclude, nonetheless, that his choice behavior would reveal a "subjective" probability assignment - even though the agent had actually made no such assignment or had no idea that he made one. One can consequently argue, the hidden assumption behind the Ramsey-de Finetti view is the existence of state-independent utility, which we shall touch upon later (cf. Karni, 1996).
Finally we should mention that one aspect of Keynes's (1921) propositions has re-emerged in modern economics via the so-called "Harsanyi Doctrine" - also known as the "common prior" assumption (e.g. Harsanyi, 1968). Effectively, this states that if agents all have the same knowledge, then they ought to have the same subjective probability assignments. This assertion, of course, is nowhere implied in subjective probability theory of either the Ramsey-de Finetti or intuitionist camps. The Harsanyi doctrine is largely an outcome of information theory and lies in the background of rational expectations theory - both of which have a rather ambiguous relationship with uncertainty theory anyway. For obvious reasons, information theory cannot embrace subjective probability too closely: its entire purpose is, after all, to set out a objective, deterministic relationship between "information" or "knowledge" and agents' choices. This makes it necessary to filter out the personal peculiarities which are permitted in subjective probability theory.
The Ramsey-de Finetti view was famously axiomatized and developed into a full theory by Leonard J. Savage in his revolutionary Foundations of Statistics (1954). Savage's subjective expected utility theory has been regarded by some observers as "the most brilliant axiomatic theory of utility ever developed" (Fishburn, 1970: p.191) and "the crowning glory of choice theory" (Kreps, 1988: p.120). Savage's brilliant performance was followed up by F.J. Anscombe and R.J. Aumann's (1963) simpler axiomatization which incorporated both objective and subjective probabilities into a single theory, but lost a degree of generality in the process. We will first go through Savage's axiomatization rather heuristically and save a more formal account for our review of Anscombe and Aumann's theorem. (note, it might be useful to go through Anscombe and Aumann before Savage).