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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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