As we have seen, Savage's axiomatization of subjective expected utility theory is a rather involved affair.
A simpler derivation of subjective expected utility theory was famously provided by F.J.
Anscombe and Robert J. Aumann (1963). 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
ｦ is no longer merely a function from states S to outcomes X,
but rather ｦ : S ｮ 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 ﾎ 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 (1963) theory are the following:

S is a set of states

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

ｦ : S ｮ D (X) is an action (horse/roulette lottery combination)

F = {ｦ | ｦ : S ｮ D (X)} a set of actions

ｳ

_{h}ﾌ F ｴ F are preferences on actions

Thus, an agent's preferences ｳ _{h} is
a binary relation on actions F that fulfills the following axioms:

(A.1) ｳ

_{h}iscomplete, i.e. either ｦ ｳ_{h }g or g ｳ_{h}ｦ for all ｦ , g ﾎ F.(A.2) ｳ

_{h}istransitive, i.e. if ｦ ｳ_{h}g and g ｳ_{h}h then ｦ ｳ_{h}h for all ｦ , g, h ﾎ F.(A.3)

Archimedean Axiom: if ｦ , g, h ﾎ F such that ｦ >_{h}g >_{h}h, then there is an a , b ﾎ (0, 1) such that a ｦ + (1-a )h >_{h}g and g >_{h}b ｦ + (1-b )h.(A.4)

Independence Axiom, i.e. for all ｦ , g, h ﾎ F and any a ﾎ [0, 1], then ｦ ｳ_{h}g if and only if a ｦ + (1-a )h ｳ_{h}a g + (1-a )h.

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

Heuristically, as Aumann and Anscombe (1963) indicate, 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, ｦ : S ｮ 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 ｦ where S
= {1, 2} and X = {x_{1}, x_{2}, x_{3}}, so ｦ
: S ｮ D (X) is a particular action.
As Nature chooses states, then depending on which s ﾎ S
occurs, we will obtain ｦ _{1} or ｦ
_{2}. However, recall ｦ _{s} is lottery
ticket, thus ｦ _{s} ﾎ D (X) is a probability distribution over X, or ｦ
_{s} = [ｦ _{s}(x_{1}), ｦ _{s}(x_{2}), ｦ _{s}(x_{3})].

** **

Figure 1 -An Anscombe-Aumann action ｦ : S ｮ D (X)

This helps our analysis as, immediately, we know that we can evaluate
different (objective) lotteries with the old von
Neumann-Morgenstern expected utility function. However, as the lottery is only played *after*
a particular state s ﾎ S occurs, then the von
Neumann-Morgenstern expected utility function will be dependent on the state, i.e. U_{s}:
D (X) ｮ R. We also know that U_{s}(ｦ _{s}) has an expected utility form:

U

_{s}(ｦ_{s}) = ・/font>_{xﾎ X}ｦ_{s}(x_{i})u_{s}(x_{i})

where u_{s}: X ｮ R is the elementary
utility function which corresponds to the particular von Neumann-Morgenstern expected
utility function U_{s}: D (X) ｮ
R that obtains when state s ﾎ S. Thus, note that u_{s}:
X ｮ R is a *state-dependent* elementary utility function.
Thus, in terms of Figure 1, *if* state s = 1 obtains, then the expected utility of ｦ _{1} is U_{1}(ｦ _{1})
= ｦ _{1}(x_{1})u_{1}(x_{1}) + ｦ _{1}(x_{2})u_{1}(x_{2}) + ｦ _{1}(x_{3})u_{1}(x_{3}) and if
state s = 2 obtains, then the expected utility of ｦ _{2}
is U_{2}(ｦ _{2}) = ｦ
_{2}(x_{1})u_{2}(x_{1}) + ｦ _{2}(x_{2})u_{2}(x_{2})
+ ｦ _{2}(x_{3})u_{2}(x_{3}).

As we can see immediately, U_{s}(ｦ _{s})
can be thought of as the expected utility of *state* s ﾎ
S *given* that a *particular* action ｦ : S ｮ D (X) is chosen. If S is finite, then
obviously the utility of the *action* ｦ is:

U(ｦ ) = ・/font>

_{sﾎ S}U_{s}(ｦ_{s})

where, notice, we are *not* multiplying U_{s}(ｦ _{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 von Neumann-Morgenstern.
However, as we do have expressions for U_{s}(ｦ _{s}),
then we can write out the utility from the act ｦ as:

U(ｦ ) = ・/font>

_{sﾎ S・/font> xﾎ X}ｦ_{s}(x_{i})u_{s}(x_{i}).

We can thus call this a state-dependent expected utility representation of the utility of act ｦ . 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 = [s_{1}, .., s_{n}] and let D (X) be a set of simple probability distributions on X. Let ｳ_{h}be a preference relation on the set F = {ｦ | ｦ :S ｮ D (X)}. Then ｳ_{h}fulfills axioms (A.1)-(A.4) if and only if there is a collection of functions {u_{s}: X ｮ R}_{sﾎ S}such that for every ｦ , g ﾎ F:ｦ >

_{h}g if and only if ・/font>_{sﾎ S・/font> xﾎ X}ｦ_{s}(x)u_{s}(x).ｳ ・/font>_{sﾎ S・/font> xﾎ X }g_{s}(x)u_{s}(x).Moreover, if {v

_{s}: X ｮ R}_{sﾎ S}is another collection of state-dependent utility functions which represent preferences, then there is b ｳ 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 ｮ R such that for every ｦ , g ﾎ H, ｦ ｳ _{h}
g iff U(ｦ ) ｳ U(g) and U is
affine, i.e. U(a ｦ + (1-a )g) = a U(ｦ
) + (1-a )U(g). Now, let us fix ｦ *
ﾎ F, thus ｦ * = (ｦ _{1}*, ..., ｦ _{n}*).
Consider now another function ｦ and define ｦ
^{s} = [ｦ _{1}*, ..., ｦ
_{s-1}*, ｦ _{s}, ｦ
_{s+1}*, .., ｦ _{n}*], thus ｦ ^{s} is identical to ｦ except
for the sth position. Doing so for all s ﾎ S, then we obtain a
collection of n functions, {ｦ ^{s}}_{sﾎ S}. Now, observe that:

・/font>

_{sﾎ S}ｦ^{s}= ｦ + (n-1)ｦ *

where ｦ = [ｦ _{1},
ｦ _{2}, .., ｦ _{n}].
To see this heuristically, let n = 3. Then:

ｦ |
ｦ |
ｦ |
|||

・/font> |
ｦ |
+ |
ｦ |
+ |
ｦ |

ｦ |
ｦ |
ｦ |

or rearranging:

ｦ |
2ｦ |
||||

・/font> |
ｦ |
+ |
2ｦ |
= |
ｦ + 2ｦ * |

ｦ |
2ｦ |

Thus, in general, for any n, we see ・/font> _{sﾎ S} ｦ ^{s} = ｦ + (n-1)ｦ *. Now, dividing through by
n:

(1/n)・/font>

_{sﾎ S}ｦ^{s}= (1/n)ｦ + ((n-1)/n)ｦ *

Now, by affinity of U: F ｮ R:

(1/n)・/font>

_{sﾎ S}U(ｦ^{s}) = (1/n)U(ｦ ) + ((n-1)/n)U(ｦ *)

or:

(1/n)U(ｦ ) = (1/n)・/font>

_{sﾎ S}U(ｦ^{s}) - ((n-1)/n)U(ｦ *)

Now, let us turn to the following. For any p ﾎ
D (X), let us define state-dependent U_{s}(p) as:

U

_{s}(p) = U(ｦ_{1}*, .., ｦ_{s-1}*, p, ｦ_{s+1}*, .., ｦ_{n}*) - ((n-1)/n)U(ｦ *)

Letting p = ｦ _{s} ﾎ
D (X) then obviously:

U

_{s}(ｦ_{s}) = U(ｦ^{s}) - ((n-1)/n)U(ｦ *) - ((n-1)/n))U(ｦ *)

by the definition of ｦ ^{s}. Thus,
summing up over s ﾎ S and dividing through by n:

(1/n)・/font>

_{sﾎ S}U_{s}(ｦ_{s}) = (1/n)・/font>_{sﾎ S}U(ｦ^{s}) - ((n-1)/n)U(ｦ *)

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

(1/n)U(ｦ ) = (1/n)・/font>

_{sﾎ S}U_{s}(ｦ_{s})

or simply:

U(ｦ ) = ・/font>

_{sﾎ S}U_{s}(ｦ_{s})

thus we have a representation of the utility of the action ｦ U(ｦ ) expressed as the sum of
state-dependent utility function over lotteries, U_{s}(ｦ
_{s}), as we had intimated before. Thus, we know that as U represents preferences,
then:

ｦ ｳ

_{h}g ﾛ U(ｦ ) ｳ U(g) ﾛ ・/font>_{sﾎ S}U_{s}(ｦ_{s}) ｳ ・/font>_{sﾎ S}U_{s}(ｦ_{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(ｦ _{1}*, .., ｦ _{s-1}*,
p, ｦ _{s+1}*, .., ｦ _{n}*)
- ((n-1)/n)U(ｦ *), then for any p, q ﾎ
D (X), then:

U

_{s}(a p + (1-a )q) = U(ｦ_{1}* .., ｦ_{s-1}*, a p + (1-a )q, ｦ_{s+1}*, .., ｦ_{n}*) - ((n-1)/n)U(ｦ *)

= U(a ｦ

_{1}* + (1-a )ｦ_{1}* .., a p + (1-a )q, .., a ｦ_{n}* + (1-a )ｦ_{n}*) - ((n-1)/n)U(a ｦ * + (1-a )ｦ *)

or as U is affine, then we obtain:

U

_{s}(a p + (1-a )q) = a [U(ｦ_{1}*, .., p, .., ｦ_{n}*) - ((n-1)/n)U(ｦ *)] + (1-a )[U(ｦ_{1}*, .., q, .., ｦ_{n}*) - ((n-1)/n)U(ｦ *)]

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 von
Neumann-Morgenstern theorem, since D (X) is a set of simple
lotteries and d _{x} ﾎ D (X), then there is a function u_{s}: X ｮ
R such that:

U

_{s}(ｦ_{s}) = ・/font>_{xﾎ X}ｦ_{s}(x)u_{s}(x)

As this is true for any s ﾎ S, then:

U(ｦ ) = ・/font>

_{sﾎ S}U_{s}(ｦ_{s}) = ・/font>_{sﾎ S・/font> xﾎ X}ｦ_{s}(x)u_{s}(x)

thus we conclude that for any ｦ , g ﾎ F, then:

ｦ ｳ

_{h}g ﾛ U(ｦ ) ｳ U(g) ﾛ ・/font>_{sﾎ S}U_{s}(ｦ_{s}) ｳ ・/font>_{sﾎ S}U_{s}(g_{s})ﾛ ・/font>

_{sﾎ S・/font> xﾎ X}ｦ_{s}(x)u_{s}(x) ｳ ・/font>_{sﾎ S・/font> xﾎ 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 ｳ 0 is *state-independent*.ｧ

Now, so far we have obtained an additive representation of U(ｦ ) with *state-dependent* elementary utility functions on*
outcomes*, u_{s}: X ｮ R. Our aim, however, is to
derive an additive representation with a *state-independent* elementary utilities on
outcome, u:X ｮ 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 ﾎ S is anull stateif (ｦ_{1}, .., ｦ_{s-1}, p, ｦ_{s+1}, .., ｦ_{n}) ~_{h}(ｦ_{1}, .., ｦ_{s-1}, q, ｦ_{s+1}, .., ｦ_{n}) for all p, q ﾎ 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 ｦ , g ﾎ F such that ｦ >_{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, (ｦ
_{1}, ｦ _{2 ..,ｦ n})
~_{h} (ｦ _{1｢ }, ｦ _{2}, .., ｦ _{n}) ~_{h}
(ｦ _{1｢ }, ｦ _{2｢ }, .., ｦ _{n}) ~_{h} .... ~_{h} (ｦ _{1｢ }, ｦ
_{2｢ }, .., ｦ _{n｢ }). But, (ｦ _{1｢ }, ｦ _{2｢
}, .., ｦ _{n｢ }) can
be any g ﾎ F. Thus, ｦ ~_{h}
g for all g ﾎ F, or there is no g ﾎ
F such that ｦ >_{h} g. Thus, (A.5) is contradicted.

Let us now turn to a rather important axiom:

(A.6)

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

(ｦ

_{1}, ..., ｦ_{s-1}, p, ｦ_{s+1}, .., ｦ_{n}) >_{h}(ｦ_{1}, ..., ｦ_{s-1}, q, ｦ_{s+1}, .., ｦ_{n})then, for

everynon-null state t ﾎ S:(ｦ

_{1}, ..., ｦ_{t-1}, p, ｦ_{t+1}, .., ｦ_{n}) >_{h}(ｦ_{1}, ..., ｦ_{t-1}, q, ｦ_{t+1}, .., ｦ_{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 ﾎ S, then p ｳ _{h} q at *any*
non-null state t ﾎ 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 = [s_{1}, .., s_{n}] and let D (X) be a set of simple probability distributions on X. Let ｳ_{h}be a preference relation on the set F = {ｦ | ｦ :S ｮ D (X)}. Then ｳ_{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 ｮ R such that for every ｦ , g ﾎ H:ｦ ｳ

_{h}g if and only if ・/font>_{sﾎ Sp }(s)・/font>_{xﾎ X}ｦ_{s}(x)u(x).ｳ ・/font>_{sﾎ Sp }(s)・/font>_{xﾎ X}g_{s}(x)u(x).Moreover, (p , u) is unique (p ｢ , v) is another probability measure on S and if v: X ｮ R represents ｳ

_{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 ｮ R}_{sﾎ S} such that for every ｦ , g ﾎ F:

ｦ ｳ

_{h}g if and only if ・/font>_{sﾎ S}・/font>_{ xﾎ X}ｦ_{s}(x)u_{s}(x).ｳ ・/font>_{sﾎ S}・/font>_{ xﾎ X }g_{s}(x)u_{s}(x).

Now, let s ﾎ S be a non-null state (which we
know exists by non-degeneracy axiom (A.5)). Consider now two actions, ｦ
^{s} = (ｦ _{1}, .., ｦ
_{s-1}, p, ｦ _{s+1}, .., ｦ
_{n}) and g^{s} = (ｦ _{1}, .., ｦ _{s-1}, q, ｦ _{s+1},
.., ｦ _{n}) where p, q ﾎ D (X). Then by the above representation, notice that ｦ ^{s} ｳ _{h} g^{s}
if and only if ・/font> _{s・/font> x} ｦ _{s}^{s}(x)u_{s}(x).ｳ
・/font> _{s・/font> x }g_{s}^{s}(x)u_{s}(x).which
reduces to ｦ ^{s} ｳ _{h}
g^{s} ﾛ ・/font> _{x}
p(x)u_{s}(x).ｳ ・/font> _{x }q(x)u_{s}(x).
But we know by state-independence axiom (A.6) that if ｦ ^{s}
ｳ _{h} g^{s} for non-null s ﾎ S, then ｦ ^{t} ｳ _{h} g^{t} for *all* non-null t ﾎ S. Thus, it is *also* true that if t is non-null, then ｦ ^{t} ｳ _{h} g^{t}
ﾛ ・/font> _{x} p(x)u_{t}(x).ｳ ・/font> _{x }q(x)u_{t}(x).
But recall that the von Neumann-Morgenstern representation argued that if U(p) ｳ U(q), then there is a real-valued function u: X ｮ R such that ・/font> _{x} p(x)u(x).ｳ ・/font> _{x }q(x)u(x) and if any v:
X ｮ 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_{s}u + a_{s} and u_{t} = b_{t}u + a_{t}.
This will be true for any non-null s, t ﾎ S. If, however, s is
null, then b = 0. Thus, substituting into our earlier expression:

ｦ ｳ

_{h}g ﾛ ・/font>_{sﾎ S}・/font>_{ xﾎ X}ｦ_{s}(x)(b_{s}u + a_{s})(x).ｳ ・/font>_{sﾎ S}・/font>_{ xﾎ X }g_{s}(x)(b_{s}u + a_{s})(x).

or:

ｦ ｳ

_{h}g ﾛ ・/font>_{sﾎ S }b_{s}・/font>_{ xﾎ X}ｦ_{s}(x)u(x).ｳ ・/font>_{sﾎ S}b_{s}・/font>_{xﾎ X }g_{s}(x)u(x).

so, defining B = ・/font> _{sﾎ
S}b_{s} > 0 (by non-degeneracy (A.5), there is at least one such S), then
dividing through by B:

ｦ ｳ

_{h}g ﾛ ・/font>_{sﾎ S }(b_{s}/B) ・/font>_{xﾎ X}ｦ_{s}(x)u(x).ｳ ・/font>_{sﾎ S }(b_{s}/B) ・/font>_{xﾎ X }g_{s}(x)u(x).

so, finally, defining p (s) = b_{s}/B
and it will be noted that ・/font> _{sﾎ
S} b_{s}/B = ・/font> _{sﾎ
S} p (s) = 1, then:

ｦ ｳ

_{h}g ﾛ ・/font>_{sﾎ S p }(s) ・/font>_{xﾎ X}ｦ_{s}(x)u(x).ｳ ・/font>_{sﾎ S p }(s) ・/font>_{xﾎ X }g_{s}(x)u(x).

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

We have now obtained the state-independent utility function u: X ｮ R and expressed preferences over actions via this expected utility
decomposition. To see the expected utility composition more clearly, recall that U_{s}(ｦ _{s}) = ｦ _{s}(x)u_{s}(x)
= ｦ _{s}(x)u(x) = U(ｦ _{s})
by our last result. Thus this becomes:

ｦ ｳ

_{h}g ﾛ ・/font>_{sﾎ S p }(s)u(ｦ_{s}).ｳ ・/font>_{sﾎ S p }(s)u(g_{s}).

Thus, we have obtained an expected utility representation of preferences
over actions, ｦ : X ｮ D (X). Thus, a particular action ｦ is
preferred to another g *if* the expected utility of action ｦ
is greater than the expected utility of action g. Note the terms we use. The term ・/font> _{sﾎ S} p
(s)u(ｦ (s)) is the expected utility of action ｦ because it sums up the utility of the consequences of this action
(u(ｦ (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 ｦ 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.

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