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"[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)

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Contents

(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).

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