P .J. Hammond Mathematical Social Sciences 38 1999 139 –156
143
sums n 5 1 1 1 1 ? ? ? 1 1 of n copies of the element 1 [ F. Obviously, it follows that F
includes all rational numbers. The ordered field
F is Archimedean if, given any x [ F , there exists such a positive
1
integer n for which nx . 1. For any x [ F, let uxu denote x if x 0 and 2 x if x , 0. Say
that any x [ F\h0j is infinitesimal iff uxu , 1 n for every large positive integer n.
Evidently, a field is non-Archimedean iff it has infinitesimal elements, and is Archime- dean iff it is isomorphic to a subset of the real line
R. See Royden 1968, for example. In the rest of the paper,
F will denote any ordered field that may or may not be Archimedean. Also, in the ensuing analysis, it will be convenient to use the notation 0,
1 to indicate the interval
hx [ Fu0 , x , 1j. This notation helps to distinguish this
F
possibly non-Archimedean interval from the usual real interval 0, 1, which will often be denoted by 0, 1 .
R
4. Algebraic expected utility
4.1. Consequences and states of the world Let Y be a fixed domain of possible consequences, and S a fixed finite set of possible
states of the world. No probability distribution over S is specified. An act, according to Savage 1972, is a mapping a: S
→ Y specifying what consequence results in each
possible state. Inspired by the Arrow 1953, 1964 and Debreu 1959 device of ‘‘contingent’’ securities or commodities in general equilibrium theory, I refer instead to
contingent consequence functions, or CCFs for short. Also, each CCF will be considered
S
as a list y 5 ky l
of contingent consequences in the Cartesian product space
s s [S S
Y : 5
P
Y , where each Y is a copy of the consequence domain Y.
s [S s
s
Anscombe and Aumann 1963 allowed subjective probabilities for the outcomes of ‘‘horse lotteries’’ or CCFs to be inferred from expected utility representations of
preferences over compounds of horse and ‘‘roulette lotteries’’. Formally, Savage’s
S
1972 framework is extended to allow preferences over DY ; R, the space of all
S S
S
finitely supported simple roulette lotteries defined on Y . Each l [ DY ; R specifies
S S
S S
a real number l y as the objective probability that the CCF is y [ Y .
S S
Instead of DY ; R, this paper considers the space DY ; F of F-valued simple
S S
S S
S S
lotteries on Y . Each l [ DY ; F satisfies l y 5 0 iff y is outside the finite
S S
S S
S S
support F , Y of l . On F each probability l y [ F , and o
l y 5 1. That
S
1 y [F
S S
S S
S
is, l y . 0 for all y [ F, even though l y could be infinitesimal.
4.2. Objective expected utility Anscombe and Aumann directly assumed expected utility maximization for roulette
lotteries. BBD, like Fishburn 1970 and many others, laid out axioms implying expected utility maximization rather than assuming it directly. Apart from being more
fundamental, here such an approach is essential because non-Archimedean expected
144 P
.J. Hammond Mathematical Social Sciences 38 1999 139 –156
utility maximization should be deduced before subjective probabilities are inferred. Accordingly, I assume:
• O Ordering
. There exists a reflexive, complete and transitive preference ordering
S S
on DY ; F.
S S
S S
• I Strong Independence
. If l , m , n [ DY ; F and a [ 0, 1 , then
F S
S S
S S
S S
S
l m ⇔
al 1 1 2 an am 1 1 2 an
2 S
S S
S
• AC Algebraic Continuity
. Given any three lotteries
l , m , n [ DY ; F
S S
S S
S S
S
satisfying l s m and m s n , there exists a [ 0, 1
such that al 1 1 2
F S
S S
an | m . Axioms O and I obviously extend standard conditions for real-valued probability
S S
distributions in DY ; R to F-valued probability distributions in DY ; F. Even the
S
version of axiom AC for DY ; R has appeared in the literature. However, the latter is
usually derived from a more fundamental continuity assumption like the counterpart of condition C in Section 7.2 – see, for example, Hammond 1999a, b. For this reason,
condition AC is discussed further in the special non-Archimedean framework of Sections 6 and 7.
S
Lemma 1. Suppose that axioms O and I are satisfied on DY ; F. Then, for any
S S
S S
S S
pair of lotteries l , m [ DY ; F with l s m and any a9, a0 [ 0, 1 with a9 . a0,
F
one has
S S
S S
S S
S S
S
l s a9l 1 1 2 a9m s a0l 1 1 2 a0m s m
Proof. If
a9, a0 [ 0, 1 with a9 . a0, then there exists d [ 0, 1 such that a0 5 da9.
F F
Now one has
S S
S S
S S
S
d [a9l 1 1 2 a9m ] 1 1 2 d m 5 da9l 1 1 2 da9m 5 a0l 1 1 2 a0m
S S
S S
S S
Repeated application of axiom I gives l s a9l 1 1 2 a9m s m , and also
S S
S S
S S
S S
S
a9l 1 1 2 a9m s d [a9l 1 1 2 a9m ] 1 1 2 d m s dm 1 1 2 d m
S
5 m
Together these statements clearly imply the result. h
S S
S S
Say that the the utility function U : DY ; F
→ F represents on DY ; F if for
S S
S
every pair of lotteries l , m [ DY ; F one has
S S
S S
S S
S
l m ⇔
U l U m
2 S
S
In the following, s and |
respectively denote the strict preference and indifference relations corre-
S
sponding to .
P .J. Hammond Mathematical Social Sciences 38 1999 139 –156
145
S
Say too that U satisfies the algebraic mixture preservation condition AMP provided
that, whenever a [ 0, 1 , then
F S
S S
S S
S S
U al 1 1 2 am 5 aU l 1 1 2 a U m
S S
S S
Note how property AMP implies that for every l [ DY ; F one can write U l in
the algebraic expected utility form
S S
S S
S
U l 5
O
l y v y
S S
y [Y S
S
where v y : 5 U 1 is the utility of the degenerate lottery 1 which attaches
S S
y y
S S
probability 1 to the particular CCF y [ Y .
S S
S
Finally, say that the two functions U , V : DY ; F
→ F are cardinally equivalent
S S
S S
provided there exist constants r [ F and d [ F such that V l 5 d 1 rU l for all
1 S
S S
S
l [ DY ; F. Obviously, if U and V are cardinally equivalent, they must also be ]
S S
S S
S S
identical if they satisfy the common normalizations U l 5V l 5 0 and U l 5
] ]
] ]
S S
S S
S
V l 5 1 for some l , l [ DY ; F.
] The following Lemma is a simple adaptation of a result which is familiar for
real-valued probabilities. Accordingly, the proof will not be provided here. For details see, for example, Fishburn 1970 or Hammond 1999a.
S
Lemma 2. Suppose that axioms O , I, and AC are all satisfied on DY ; F. Then
S S
there exists a unique cardinal equivalence class of utility functions U : DY ; F
→ F
S
which represent and satisfy AMP.
5. Algebraic subjective probability
