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
5.1. The algebraic SEU hypothesis The first aim of this section is to provide sufficient conditions like those of Anscombe
S S
and Aumann for the preference ordering on DY ; F to have a subjective expected
S S
utility SEU representation which, for each l [ DY ; F, takes the form
S S
S S
U l 5
O
l y
O
p v y 1
s s
S S
s [S y [Y
for a unique cardinal equivalence class of von Neumann–Morgenstern utility functions NMUFs v:Y
→ F, and for suitable unique subjective probabilities p [ F
satisfying
s 1
o p 5 1. The difference from Anscombe and Aumann is that the subjective
s [S s
probabilities p and utilities v y may belong to the general ordered field F, as may the
s S
S S
S
objective probabilities l y determining the lottery l [ DY ; F. In addition, the
subjective probabilities p must be positive rather than merely non-negative. This
s S
particular characterization of the preference ordering will be called the algebraic SEU hypothesis.
A second aim is to ensure that the utility function v is real-valued, even when the field
146 P
.J. Hammond Mathematical Social Sciences 38 1999 139 –156
F is non-Archimedean. To do so, I will introduce a new state-independent continuity axiom SIC. Then only the probabilities, and not the utilities, are allowed to be
non-Archimedean.
5.2. Reversal of order In order to derive subjective probabilities, Anscombe and Aumann added three more
axioms to their basic hypothesis that ‘‘roulette lotteries’’ would be chosen to maximize objective expected utility.
S S
S
For each y [ Y and s [ S, define Y y: 5 hy [ Y uy 5 yj as the set of CCFs yielding
s s
S S
the particular consequence y in state s. Given any lottery l [ DY ; F, any state s [ S
and any consequence y [ Y, let
S S
l y: 5
O
l y
s
S S
y [
Y y
s
denote the marginal probability that y occurs in s. Then the probabilities l y y [ Y
s
specify the marginal distribution in state s [ S. The first of the three additional axioms is:
S S
S
• RO Reversal of Order
. Whenever l , m [ DY ; F have marginal distributions
S S
S
satisfying l 5 m for all s [ S, then l | m .
s s
This condition owes its name to the fact that there is indifference between: i having
S S
the roulette lottery l
determine the random CCF y before the horse lottery that
resolves which state s [ S and which ultimate consequence y occur; and ii resolving
s
the horse lottery first, before its outcome s [ S determines which marginal roulette lottery
l generates the ultimate consequence y.
s
5.3. The sure thing principle The second of Anscombe and Aumann’s additional axioms concerns any event E , S,
E
together with the product space Y : 5
P
Y of contingent CCFs taking the form
s [E s
E E
y 5 ky l
[ Y , and the existence of an associated contingent preference ordering
s s [E E
E E
. Here it is natural to assume that is defined on DY ; F, the space of F-valued
E
probability distributions, instead of only on DY , the space of real-valued probability
distributions. So the second extra axiom becomes: •
STP Sure Thing Principle . Given any event E , S, there exists a contingent
E E
preference ordering on DY ; F satisfying
E E
E E
S \E S
E S \E
l m ⇔
l , n m , n
E E
E S \E
S \E E
S \E
for all l , m [ DY ; F and n
[ DY
; F, where l , n
denotes the
E S \E
combination of the conditional lottery l if E occurs with n
if S\E occurs, and
E S \E
similarly for m , n
.
P .J. Hammond Mathematical Social Sciences 38 1999 139 –156
147
However, following an idea due originally to Raiffa 1961 and then used by BBD, it is easy to show:
S
Lemma 3. Suppose that the three axioms O , I, and RO are all satisfied on DY ;
F. Then so is STP.
Proof. See Hammond 1999b. h
Because of this result, STP will not be imposed as an axiom, but it will often be used in the ensuing proofs.
5.4. State independence For each s [ S, condition STP ensures the existence of a contingent preference
hsj
ordering on
DY ; F 5 DY; F. The last axiom is:
s
• SI State Independence
. Given any state s [ S, the contingent preference ordering
hsj
over DY; F is independent of s.
S S
Let denote this state-independent preference ordering. When
on DY ; F
satisfies conditions O, I and AC, so too must on
DY; F, because of STP. 5.5. Subjective probabilities
The five axioms O, I, AC, RO, and SI are assumed throughout the following, as is condition STP.
E E
E
Lemma 4. a Suppose that E , S is any event and that l ,m [ DY ; F satisfy
E E
E E
E E
l m for all s [ E. Then l m . b If in addition l | m , then l | m for
s s
s s
every state s [ E.
Proof. By induction on the number of states in E. h
From now on, exclude the trivial case of universal indifference by assuming that there ]
] exist two roulette lotteries
l, l [ DY; F with l s l. ]
]
E
Given any event E , S and any lottery l [ DY; F, let l1 denote the lottery in
E
DY ; F whose marginal distribution in each state s [ E is l 5 l, independent of s.
s
Lemma 5. The ordering on
DY; F is represented by a utility function U :DY;
S S
F →
F satisfying U l 5 U l1 for all l [ DY; F.
S S
Proof. Because on DY ; F and on DY; F both satisfy conditions O, I
and AC, Lemma 2 of Section 4.2 implies that they can be represented by normalized
S S
utility functions U : DY ; F
→ F and U :DY; F
→ F satisfying AMP and also
148 P
.J. Hammond Mathematical Social Sciences 38 1999 139 –156
] ]
S S
S S
U l1 5 0, U l1 5 1,
and U
l 5 0, U l 5 1 2
] ]
S S
S
Next, Lemma 4a implies that l m
⇒ l1 m1 . On the other hand, Lemma 4b
S S
S S
implies that m s l
⇒ m1 s l1 . Because
and are complete orderings, the
S S
S S
S S
reverse implication l1 m1
⇒ l m follows. Hence, l1 m1
⇔ l m. So
S S
U l1 and U l must be cardinally equivalent functions of l on the domain DY; F.
Because of the two normalizations in 2, the result follows immediately. h
S S
Lemma 6. For each l [ DY ; F one has
S S
U l 5
O
q U l
3
s s
s [S
]
S S \
hsj
where q : 5 U l1
, l [ F
for all s [ S, implying that o
q 5 1.
s 1
s [S s
]
Proof. The proof by Fishburn 1970, Theorem 13.1, p. 176 of the corresponding result
S S
for the space DY of real-valued probabilities applies to DY ; F, with no change
except that AMP replaces the usual mixture preservation property. See also Hammond 1999b.
h
Lemma 7. There exists a unique cardinal equivalence class of NMUFs v:Y
→ F and,
unless there is universal indifference , unique subjective probabilities p s [ S such that
s S
S
the ordering on DY ; F is represented by the subjective expected utility function
S S
U l
;
O
p
O
l yv y 4
s s
s [S y [Y
Proof. By Lemma 6, with p replacing q in 3, one has
s s
S S
U l 5
O
p U l 5
O
p
O
l yv y 5
s s
s s
s [S s [S
y [Y
where v y: 5 U 1 for all y [ Y, while the second equality in 5 follows because l
y s
is the finite mixture o
l y1 and U satisfies AMP. As in Lemma 2, the NMUF
y [Y s
y
v could be replaced by any cardinally equivalent v: Y
→ F. But this is equivalent to
S S
S
replacing U by a cardinally equivalent V :
DY ; F →
F. Any such transformation leaves the ratio
]
S S \
hsj S
S
U l1
, l 2 U l1
] ]
]]]]]]] p 5
]
s S
S S
S
U l1 2 U l1
] of expected utility differences unaffected. This ensures that the subjective probabilities
are unique. h
5.6. Real-valued utility Lemmas 5 and 7 imply that the state-independent preference ordering
on DY; F
is represented by the expected value of each F-valued state-independent NMUF v:
Y →
F in a unique cardinal equivalence class. Where F is non-Archimedean, v could be
P .J. Hammond Mathematical Social Sciences 38 1999 139 –156
149
also. Yet in game theory, the motivation for non-Archimedean probabilities is that players’ deviations from their presumed best responses can be given infinitesimal
subjective probability. Where there is no uncertainty of this kind, but only risk in the form of specified objective probabilities, there is no good reason to depart from classical
expected utility theory, which requires preferences over real-valued probability dis- tributions to be continuous, and expected utility to be real-valued – as in BBD, for
example. To ensure that v is real-valued, on
DY; R is assumed to satisfy an extra standard continuity axiom for real-valued probability distributions.
3
• SIC State Independent Continuity
. For all
l,m,n [ DY; R satisfying l s m s
n, there exist a9, a0 [ 0, 1 such that a9l 1 1 2 a9n s
m and
m s a0l 1 1 2 a0n
Theorem 8. Suppose that the six axioms O , I, AC , RO , SI and SIC are all
S
satisfied on DY ; F. Then, unless there is universal indifference, there exist unique
subjective probabilities p [ DS; F and a unique cardinal equivalence class of
s 1
real-valued NMUFs v: Y
→ R such that
S S
S
l m ⇔
O
p
O
l yv y
O
p
O
m yv y
s s
s s
s [S y [Y
s [S y [Y
Proof. By Lemma 7, there exist unique subjective probabilities p and a unique cardinal
s
equivalence class of NMUFs v:Y →
F. Because there is not universal indifference, v cannot be constant. So it can be normalized within the same cardinal equivalence class
] ]
]
to make vy 5 1 and vy 5 0 for some pair y, y [ Y satisfying 1 s 1 .
y y
S
] ]
S ]
Next, let denote the restriction of the ordering
to the space DY; R of
R S
real-valued probability distributions. Note that satisfies axioms O, I, SIC,
R S
RO and SI. By a standard result in utility theory, also satisfies AC, and can be
R
represented by the expected value of a real-valued NMUF v which can be normalized
R S
] to satisfy v y 5 1 and v y 5 0. The expected values of both v and v represent
R R
R R
] on
DY; R, so these two NMUFs are cardinally equivalent. Because of the common normalization, they must be identical. That is, v y 5 v y [
R for all y [ Y, implying
R
that v: Y →
R. h
3
This is sometimes called the Archimedean axiom – see, for instance, Karni and Schmeidler 1991, p. 1769. For obvious reasons I avoid this name here. Note that imposing axiom AC – or the counterpart of condition
S S
C as stated in Section 7.2 – on restricted to DY ; R would imply that all subjective probabilities are
S
real-valued. This explains why , and not , is assumed to be continuous w.r.t. changes in real-valued probabilities.
150 P
.J. Hammond Mathematical Social Sciences 38 1999 139 –156
`
6. The ordered field R e as a complete metric space
