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 Bernoulli's "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 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"? 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.

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."

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). In effect, Keynes 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."

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, 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 achieved this with objective probabilities, the task was at least manageable. But with subjective probability 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.

The subjective nature of probability assignments 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, holds a more "intuitionist" view of subjective probabilities which disputes this conclusion. In their view, the "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,). 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 Keynes and finds its most prominent economic voice in the work of George L.S. Shackle (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.

Finally we should mention that one aspect of Keynes' propositions has re-emerged in modern economics via the so-called "Harsanyi Doctrine" - also known as the "common prior" assumption. 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. 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 axiomatized and developed into a full theory by Leonard J. Savage0 in his Foundations of Statistics (1954). Savage's work 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.

 Savage's axiomatization of subjective expected utility theory is a rather involved affair. A simpler derivation of subjective expected utility theory was provided by F.J. Anscombe and Robert J. Aumann. However, Anscombe and Aumann's derivation can be regarded as an intermediate theory as it requires the presence of lotteries with objective probabilities. What they assume is that an action ¦ is no longer merely a function from states S to outcomes X, but rather ¦ : S ® D (X), where D (X) is the set of simple probability distributions on the set X. Thus, a consequence is no longer a particular x, but rather a distribution p Î D (X). The set of consequences D (X) are themselves lotteries - but now lotteries with "objective" probabilities.

As a result the components of the Anscombe and Aumann theory are the following:

S is a set of states

D (X) a set of consequences (objective lotteries on outcomes)

¦ : S ® D (X) is an action (horse/roulette lottery combination)

F = {¦ | ¦ : S ® D (X)} a set of actions

³ _h Ì F ´ F are preferences on actions

Thus, an agent's preferences ³ _h is a binary relation on actions F that fulfills the following axioms:

(A.1) ³ _h is complete, i.e. either ¦ ³ _h g or g ³ _h ¦ for all ¦ , g Î F.

(A.2) ³ _h is transitive, i.e. if ¦ ³ _h g and g ³ _h h then ¦ ³ _h h for all ¦ , g, h Î F.

(A.3) Archimedean Axiom: if ¦ , g, h Î F such that ¦ >_h g >_h h, then there is an a , b Î (0, 1) such that a ¦ + (1-a )h >_h g and g >_h b ¦ + (1-b )h.

(A.4) Independence Axiom, i.e. for all ¦ , g, h Î F and any a Î [0, 1], then ¦ ³ _h g if and only if a ¦ + (1-a )h ³ _h a g + (1-a )h.

which, of course, are merely analogues of axioms (A.1)-(A.4) set out earlier in the V-M structure. As before, F is a "mixture set", i.e. for any ¦ , g Î F and for any a Î [0, 1], we can associate another element a ¦ + (1-a )g Î F defined pointwise as (a ¦ + (1-a )g)(s) = a ¦ (s) + (1-a )g(s) for all s Î S.

We can think of this as a combination of "horse race lotteries" (i.e. with subjective probabilities) and "roulette lotteries" (i.e. with objective probabilities). Or, more simply, ¦ : S ® D (X) is a horse race but the bettor, instead of receiving the winnings on his bet in cold cash, is actually given a voucher for a roulette bet, or a ticket for a lottery with objective probabilities. This can be visualized in Figure 1, where we have a tree diagram for a particular action ¦ where S = {1, 2} and X = {x₁, x₂, x₃}, so ¦ : S ® D (X) is a particular action. As Nature chooses states, then depending on which s Î S occurs, we will obtain ¦ ₁ or ¦ ₂. However, recall ¦ _s is lottery ticket, thus ¦ _s Î D (X) is a probability distribution over X, or ¦ _s = [¦ _s(x₁), ¦ _s(x₂), ¦ _s(x₃)].

This helps our analysis as, immediately, we know that we can evaluate different (objective) lotteries with the old V-M expected utility function. However, as the lottery is only played after a particular state s Î S occurs, then the von Neumann-Morgenstern expected utility function will be dependent on the state, i.e. U_s: D (X) ® R. We also know that U_s(¦ _s) has an expected utility form:

U_s(¦ _s) = å _{xÎ X} ¦ _s(x_i)u_s(x_i)

where u_s: X ® R is the elementary utility function which corresponds to the particular V-M expected utility function U_s: D (X) ® R that obtains when state s Î S. Thus, note that u_s: X ® R is a state-dependent elementary utility function. Thus, if state s = 1 obtains, then the expected utility of ¦ ₁ is U₁(¦ ₁) = ¦ ₁(x₁)u₁(x₁) + ¦ ₁(x₂)u₁(x₂) + ¦ ₁(x₃)u₁(x₃) and if state s = 2 obtains, then the expected utility of ¦ ₂ is U₂(¦ ₂) = ¦ ₂(x₁)u₂(x₁) + ¦ ₂(x₂)u₂(x₂) + ¦ ₂(x₃)u₂(x₃).

As we can see immediately, U_s(¦ _s) can be thought of as the expected utility of state s Î S given that a particular action ¦ : S ® D (X) is chosen. If S is finite, then obviously the utility of the action ¦ is:

U(¦ ) = å _{sÎ S} U_s(¦ _s)

where, notice, we are not multiplying U_s(¦ _s) by the probability that state s occurs - because we do not know what those probabilities are. That is, after all, the purpose of this subjective expected utility theory - otherwise it would be merely a case of compound lotteries and we would simply apply V-M. However, as we do have expressions for U_s(¦ _s), then we can write out the utility from the act ¦ as:

U(¦ ) = å _{sÎ Så xÎ X} ¦ _s(x_i)u_s(x_i).

We can thus call this a state-dependent expected utility representation of the utility of act ¦ . The next question should be obvious: does this represent preferences over actions? To formalize all this intuition and prove this last result, let us state the first theorem:

Theorem: (State-Dependent Expected Utility) Let S = [s₁, .., s_n] and let D (X) be a set of simple probability distributions on X. Let ³ _h be a preference relation on the set F = {¦ | ¦ :S ® D (X)}. Then ³ _h fulfills axioms (A.1)-(A.4) if and only if there is a collection of functions {u_s: X ® R}_{sÎ S} such that for every ¦ , g Î F:

¦ >_h g if and only if å _{sÎ Så xÎ X} ¦ _s(x)u_s(x).³ å _{sÎ Så xÎ X} g_s(x)u_s(x).

Moreover, if {v_s: X ® R}_{sÎ S} is another collection of state-dependent utility functions which represent preferences, then there is b ³ 0 and a_s such that v_s = bu_s + a_s.

Proof: This is an if and only if statement thus we must prove from axioms to representation and representation to axioms. We omit the latter, and concentrate on the former. Now, by the von Neumann-Morgenstern theorem, we know that if (A.1)-(A.4) are fulfilled over a linear convex set F, then there exists a function U: F ® R such that for every ¦ , g Î H, ¦ ³ _h g iff U(¦ ) ³ U(g) and U is affine, i.e. U(a ¦ + (1-a )g) = a U(¦ ) + (1-a )U(g). Now, let us fix ¦ * Î F, thus ¦ * = (¦ ₁*, ..., ¦ _n*). Consider now another function ¦ and define ¦ ^s = [¦ ₁*, ..., ¦ _s-1*, ¦ _s, ¦ _s+1*, .., ¦ _n*], thus ¦ ^s is identical to ¦ except for the sth position. Doing so for all s Î S, then we obtain a collection of n functions, {¦ ^s}_{sÎ S}. Now, observe that:

å _{sÎ S} ¦ ^s = ¦ + (n-1)¦ *

where ¦ = [¦ ₁, ¦ ₂, .., ¦ _n]. To see this heuristically, let n = 3. Then:

or rearranging:

Thus, in general, for any n, we see å _{sÎ S} ¦ ^s = ¦ + (n-1)¦ *. Now, dividing through by n:

(1/n)å _{sÎ S} ¦ ^s = (1/n)¦ + ((n-1)/n)¦ *

Now, by affinity of U: F ® R:

(1/n)å _{sÎ S} U(¦ ^s) = (1/n)U(¦ ) + ((n-1)/n)U(¦ *)

or:

(1/n)U(¦ ) = (1/n)å _{sÎ S} U(¦ ^s) - ((n-1)/n)U(¦ *)

Now, let us turn to the following. For any p Î D (X), let us define state-dependent U_s(p) as:

U_s(p) = U(¦ ₁*, .., ¦ _s-1*, p, ¦ _s+1*, .., ¦ _n*) - ((n-1)/n)U(¦ *)

Letting p = ¦ _s Î D (X) then obviously:

U_s(¦ _s) = U(¦ ^s) - ((n-1)/n)U(¦ *) - ((n-1)/n))U(¦ *)

by the definition of ¦ ^s. Thus, summing up over s Î S and dividing through by n:

(1/n)å _{sÎ S}U_s(¦ _s) = (1/n)å _{sÎ S}U(¦ ^s) - ((n-1)/n)U(¦ *)

But recall from before that the entire right hand side is merely (1/n)U(¦ ), thus:

(1/n)U(¦ ) = (1/n)å _{sÎ S}U_s(¦ _s)

or simply:

U(¦ ) = å _{sÎ S}U_s(¦ _s)

thus we have a representation of the utility of the action ¦ U(¦ ) expressed as the sum of state-dependent utility function over lotteries, U_s(¦ _s), as we had intimated before. Thus, we know that as U represents preferences, then:

¦ ³ _h g Û U(¦ ) ³ U(g) Û å _{sÎ S}U_s(¦ _s) ³ å _{sÎ S}U_s(¦ _s)

We are half-way there. Define u_s(x) = U_s(d _x) where d _x(y) = 1 if y = x and = 0 otherwise. Now, recalling the definition of U_s(p) = U(¦ ₁*, .., ¦ _s-1*, p, ¦ _s+1*, .., ¦ _n*) - ((n-1)/n)U(¦ *), then for any p, q Î D (X), then:

U_s(a p + (1-a )q) = U(¦ ₁* .., ¦ _s-1*, a p + (1-a )q, ¦ _s+1*, .., ¦ _n*) - ((n-1)/n)U(¦ *)

= U(a ¦ ₁* + (1-a )¦ ₁* .., a p + (1-a )q, .., a ¦ _n* + (1-a )¦ _n*) - ((n-1)/n)U(a ¦ * + (1-a )¦ *)

or as U is affine, then we obtain:

U_s(a p + (1-a )q) = a [U(¦ ₁*, .., p, .., ¦ _n*) - ((n-1)/n)U(¦ *)] + (1-a )[U(¦ ₁*, .., q, .., ¦ _n*) - ((n-1)/n)U(¦ *)]

thus:

U_s(a p + (1-a )q) = a U_s(p) + (1-a )U_s(q)

so U_s is also affine.

Now, by the corollary to the V-M theorem, since D (X) is a set of simple lotteries and d _x Î D (X), then there is a function u_s: X ® R such that:

U_s(¦ _s) = å _{xÎ X} ¦ _s(x)u_s(x)

As this is true for any s Î S, then:

U(¦ ) = å _{sÎ S}U_s(¦ _s) = å _{sÎ Så xÎ X} ¦ _s(x)u_s(x)

thus we conclude that for any ¦ , g Î F, then:

¦ ³ _h g Û U(¦ ) ³ U(g) Û å _{sÎ S}U_s(¦ _s) ³ å _{sÎ S}U_s(g_s)

Û å _{sÎ Så xÎ X} ¦ _s(x)u_s(x) ³ å _{sÎ Så xÎ X} g_s(x)u_s(x)

which is what we sought. Finally, we shall not prove the "moreover" remark as it follows directly from the uniqueness of U. All we wish to note from this uniqueness statement, v_s = bu_s + a_s, is that b ³ 0 is state-independent.§

Now, so far we have obtained an additive representation of U(¦ ) with state-dependent elementary utility functions on outcomes, u_s: X ® R. Our aim, however, is to derive an additive representation with a state-independent elementary utilities on outcome, u:X ® R. This is the important task and requires some additional structure. Before we do this, let us provide a definition:

Null States: a state s Î S is a null state if (¦ ₁, .., ¦ _s-1, p, ¦ _s+1, .., ¦ _n) ~_h (¦ ₁, .., ¦ _s-1, q, ¦ _s+1, .., ¦ _n) for all p, q Î D (X).

Notice that the action on the left is the same as the action on the right except for the component at position s, where that on the left yields p and the right has q. If one is nonetheless indifferent between the two acts, then effectively state s does not matter, it i.e. it is equivalent to stating that the agent believes s will never happen. We do not want to rule this out, but we do want to prove that there are at least some states that are non-null states. To establish this, we need the following axiom:

(A.5) Non-degeneracy Axiom: there is an ¦ , g Î F such that ¦ >_h g. (i.e. >_h is non-empty).

We can see that non-degeneracy guarantees the existence of non-null states. To see this, suppose not. Suppose all states are null. Then, (¦ ₁, ¦ _{2 ..,¦ n}) ~_h (¦ _1¢ , ¦ ₂, .., ¦ _n) ~_h (¦ _1¢ , ¦ _2¢ , .., ¦ _n) ~_h .... ~_h (¦ _1¢ , ¦ _2¢ , .., ¦ _n¢ ). But, (¦ _1¢ , ¦ _2¢ , .., ¦ _n¢ ) can be any g Î F. Thus, ¦ ~_h g for all g Î F, or there is no g Î F such that ¦ >_h g. Thus, (A.5) is contradicted.

Let us now turn to a rather important axiom:

(A.6) State-Independence Axiom: Let s Î S be a non-null state and p, q Î D (X). Then if:

(¦ ₁, ..., ¦ _s-1, p, ¦ _s+1, .., ¦ _n) >_h (¦ ₁, ..., ¦ _s-1, q, ¦ _s+1, .., ¦ _n)

then, for every non-null state t Î S:

(¦ ₁, ..., ¦ _t-1, p, ¦ _t+1, .., ¦ _n) >_h (¦ ₁, ..., ¦ _t-1, q, ¦ _t+1, .., ¦ _n)

The state-independent axiom is quite important so let us be clear as to what is says. Effectively, it claims that if p >_h q at non-null state s Î S, then p ³ _h q at any non-null state t Î S. Thus, the preference ranking between lotteries p and q is state independent.

With these two axioms, we can now turn to the main theorem we seek from Anscombe and Aumann (1963) to derive the state-independent expected utility representation:

Theorem: (Anscombe-Aumann) Let S = [s₁, .., s_n] and let D (X) be a set of simple probability distributions on X. Let ³ _h be a preference relation on the set F = {¦ | ¦ :S ® D (X)}. Then ³ _h fulfills axioms (A.1)-(A.4), (A.5), (A.6) if and only if there is a unique probability measure p on S and a non-constant function u: X ® R such that for every ¦ , g Î H:

¦ ³ _h g if and only if å _{sÎ Sp} (s)å _{xÎ X} ¦ _s(x)u(x).³ å _{sÎ Sp} (s)å _{xÎ X} g_s(x)u(x).

Moreover, (p , u) is unique (p ¢ , v) is another probability measure on S and if v: X ® R represents ³ _h in the sense above, then there is b > 0 and a such that v = bu + a and p = p ¢ .

Proof: We shall go from axioms to representations first. Notice that from the previous theorem, (A.1)-(A.4) there is a collection of functions {u_s: X ® R}_{sÎ S} such that for every ¦ , g Î F:

¦ ³ _h g if and only if å _{sÎ S}å _{xÎ X} ¦ _s(x)u_s(x).³ å _{sÎ S}å _{xÎ X} g_s(x)u_s(x).

Now, let s Î S be a non-null state (which we know exists by non-degeneracy axiom (A.5)). Consider now two actions, ¦ ^s = (¦ ₁, .., ¦ _s-1, p, ¦ _s+1, .., ¦ _n) and g^s = (¦ ₁, .., ¦ _s-1, q, ¦ _s+1, .., ¦ _n) where p, q Î D (X). Then by the above representation, notice that ¦ ^s ³ _h g^s if and only if å _{så x} ¦ _s^s(x)u_s(x).³ å _{så x} g_s^s(x)u_s(x).which reduces to ¦ ^s ³ _h g^s Û å _x p(x)u_s(x).³ å _x q(x)u_s(x). But we know by state-independence axiom (A.6) that if ¦ ^s ³ _h g^s for non-null s Î S, then ¦ ^t ³ _h g^t for all non-null t Î S. Thus, it is also true that if t is non-null, then ¦ ^t ³ _h g^t Û å _x p(x)u_t(x).³ å _x q(x)u_t(x). But recall that the von Neumann-Morgenstern representation argued that if U(p) ³ U(q), then there is a real-valued function u: X ® R such that å _x p(x)u(x).³ å _x q(x)u(x) and if any v: X ® R also represented preferences over D (X), then there is a b > 0 such that v = bu + a. Well, in our case, we have u_s and u_t representing preferences over D (X). Thus, there is a b_s, b_t > 0 and a_s, a_t such that u_s = b_su + a_s and u_t = b_tu + a_t. This will be true for any non-null s, t Î S. If, however, s is null, then b = 0. Thus, substituting into our earlier expression:

¦ ³ _h g Û å _{sÎ S} å _{xÎ X} ¦ _s(x)(b_su + a_s)(x).³ å _{sÎ S} å _{xÎ X} g_s(x)(b_su + a_s)(x).

or:

¦ ³ _h g Û å _{sÎ S} b_s å _{xÎ X} ¦ _s(x)u(x).³ å _{sÎ S}b_s å _{xÎ X} g_s(x)u(x).

so, defining B = å _{sÎ S}b_s > 0 (by non-degeneracy (A.5), there is at least one such S), then dividing through by B:

¦ ³ _h g Û å _{sÎ S} (b_s/B) å _{xÎ X} ¦ _s(x)u(x).³ å _{sÎ S} (b_s/B) å _{xÎ X} g_s(x)u(x).

so, finally, defining p (s) = b_s/B and it will be noted that å _{sÎ S} b_s/B = å _{sÎ S} p (s) = 1, then:

¦ ³ _h g Û å _{sÎ S p} (s) å _{xÎ X} ¦ _s(x)u(x).³ å _{sÎ S p} (s) å _{xÎ X} g_s(x)u(x).

and thus we have it. We leave the uniqueness and the converse proof undone.§

We have now obtained the state-independent utility function u: X ® R and expressed preferences over actions via this expected utility decomposition. To see the expected utility composition more clearly, recall that U_s(¦ _s) = ¦ _s(x)u_s(x) = ¦ _s(x)u(x) = U(¦ _s) by our last result. Thus this becomes:

¦ ³ _h g Û å _{sÎ S p} (s)u(¦ _s).³ å _{sÎ S p} (s)u(g_s).

Thus, we have obtained an expected utility representation of preferences over actions, ¦ : X ® D (X). Thus, a particular action ¦ is preferred to another g if the expected utility of action ¦ is greater than the expected utility of action g. Note the terms we use. The term å _{sÎ S} p (s)u(¦ (s)) is the expected utility of action ¦ because it sums up the utility of the consequences of this action (u(¦ (s)) over states weighted by the probability of a state happening, p (s). The crucial thing to recall here is that the probabilities p (s) were derived from preferences over actions and not imposed externally! Thus, these are subjective probabilities and, hopefully, they represent individual belief.

This last part is something of a leap here, but the basic notion is that a rational agent would not choose an action ¦ over an action g if they did not correspond rationally to his beliefs on the probabilities of the occurrences of states. In horse-racing language, then p (s) corresponds to the beliefs on the outcome of the horse race (the different states) because a bettor would not rationally prefer a betting strategy that yields contradicts his beliefs.