Download The Axioms of Subjective Probability

StatitsticalScience
1986, Vol. 1, No. 3, 3:35-358
The AxiomsofSubjectiveProbabili
Peter C. Fishburn
Abstract.This surveyrecountscontributionsto the axiomatic foundations
of subjective probabilityfromthe pioneering era of Ramsey, de Finetti,
Savage, and Koopman to the mid-1980's. It is designed to be accessible to
readers who have little prior acquaintance with axiomatics. At the same
time, it provides a fairlycomplete picture of the present state of the
foundationsof subjectiveprobability.
measurement-theoretic
Key words and phrases: Axioms for comparative probability,numerical
representations,additivity,countable additivity.
The probabilityof any event is the ratio between the
value at which an expectationdependingon the happening of the event ought to be computed,and the
value of the thing expected upon its happening.
(Bayes, 1763, page 376)
We are driventhereforeto the second suppositionthat
the degreeof a beliefis a causal propertyof it, which
we can express vaguelyas the extentto whichwe are
preparedto act on it. (Ramsey, 1931, page 170)
... the degree of probabilityattributedby an individual to a given event is revealed by the conditions
under which he would be disposed to bet on that
event. (de Finetti, 1937; fromKyburg and Smokler,
1964,page 101)
The intuitive thesis in probability holds that ...
probabilityderivesdirectlyfromthe intuition,and is
prior to objective experience ... (Koopman, 1940a,
page 269)
Personalisticviewshold that probabilitymeasuresthe
confidencethat a particularindividualhas in the truth
of a particularproposition,forexample, the proposition that it will rain tomorrow.(Savage, 1954,page 3)
1. INTRODUCTION
The theory of subjective probabilityattempts to
make precise the connectionbetweencoherentdispositionstowarduncertaintyand quantitativeprobabilityas axiomatized by Kolmogorov(1933) and others.
It accommodatesthe classical interpretationsofprobabilityin Bayes (1763) and Laplace (1812), the intuiPeter C. Fishburn is a memberof the Mathematical
Studies Department in the Mathematical Sciences
Research Center,AT&T Bell Laboratories,600 Mountain Avenue,MurrayHill, New Jersey07974.
335
tive views of Koopman (1940a, 1940b) and Good
(1950), and the decision-orientedapproach of Ramsey
(1931), de Finetti (1931, 1937), and Savage (1954).
The aim of this essay is to recount the axiomatic
developmentof subjectiveor personal (Savage) probabilityfrom1926,whenFrank P. RamseywroteTruth
and Probability,to the present. My hope is that this
will not only provide a useful currentperspectiveon
subjectiveprobabilityper se but that it will also promote appreciation of a vital part of the Bayesian
approach to statistical decision theorypioneered by
Good (1950) and Savage (1954) and furtherdeveloped
by Schlaifer (1959), Raiffa and Schlaifer (1961),
DeGroot (1970), and Hartigan (1983) among others.
A briefbut veryinformativeintroductionto problems
and perspectivesof the Bayesian approach is givenby
Sudderth(1985).
The axioms of subjective probabilityreferto assumedpropertiesof a binaryrelationis moreprobable
than,or its nonstrictcompanion is at least as probable
as, on a set of propositionsor events. This relation,
oftenreferredto as a qualitative or comparativeprobabilityrelation,can be taken either as an undefined
primitive(intuitive views) or as a relation derived
from a preference relation (decision-oriented approach). In the lattercase, to say that you regardrain
as more probable tomorrowthan shine, or that you
believe the pound sterlingis more likelyto fall than
rise against the dollar next year means roughlythat
you would ratherbet on the first-namedevent fora
valuable prize that you will receive if your chosen
event obtains.
Comparativeprobabilityaxioms are usuallythought
of as criteriaof consistencyand coherencefora person's attitudes toward uncertainty.The prevailing
view has been that these criteria do not purportto
describe actual behavioral attitudes so much as they
characterizethe partial beliefsof a rational,idealized
individual.Lately, however,some traditionalaxioms
P. C. FISHBURN
336
have been weakened to yield theoriesthat accommodate observeddeparturesfromthose axioms, and the
lines between the ideal and the actual have become
blurred.I shall say more about this later.
Except when noted otherwise,I adopt the strict
comparative probability relation >- as basic. Read
A >- B as "eventA is (regardedby the individual as)
more probable than event B." When >- is basic, its
symmetriccomplement - and nonstrictcompanion
> are definedby
A -B
if neither A >-B
nor B >A,
A >B
if either A >-B
or A - B.
We usually view - as a comparativeequiprobability
relation in either a precise sense or an approximate
"no significantdifference"sense.
On the otherhand, when > ("is at least as probable
as") is taken as basic, then - and >- are definedfrom
it by
A-B
A>-B
if A-B
if A -B
and B>A,
andnot
(B>-A).
This has the advantage of identifyingA and B as
noncomparable(Keynes, 1921) when neitherA > B
nor B > A. In the >- basic case, noncomparabilityis
embeddedin - and thereis no obvious way to distinguish between it and comparative equiprobability
when such a distinctionis desired.
It is easily checked that, when >- is basic, the definitions in the precedingparagraphs agree with each
other if and only if >- is asymmetric,i.e., whenever
A >- B then not (B >- A). Alternatively,if > is basic,
the definitionsagree if and only if > is complete,i.e.,
A > B or B > A for everypair of events. We shall
generally assume that >- is asymmetricwhen it is
taken as basic, but will considercases where> is not
completewhen it is taken as basic (see Section 3).
The set on which >- or > is definedis assumed to
be a Boolean algebra v of subsets A, B, * of a
universalset S. We referto each A in v as an event.
The emptyevent is 0, the universalevent is S, and
0 C A C S for everyA in V. We recall that v is a
S\A C W,and A,
Boolean algebraif S C W, A GEv
B E
i ; A U B E V1.The complementS\A of A in
S will also be writtenas AC.
A probabilitymeasureon v is a real valued function
P on v such that P(S) = 1 and, forall A and B in X,
U B) = P(A) + P(B).
P(A) -O,and A n B = 0 =P(A
P
is finitelyadditive:
that
This last propertyimplies
ifA1, ***,An are mutuallydisjoint events in W,then
P(UjAj) = E JP(Aj). Countable additivityand v-algebras, which are taken for granted in some standard
workson probability(Loeve, 1960; Feller, 1966), will
be discussed later.
We say that P on v
partiallyagreeswith>-
if A >- B i P(A) > P(B),
almostagreeswith>
if A > B i P(A) - P(B),
agreeswith>- (or >)
if A >- B * P(A) > P(B),
for all A and B in S. Partial agreement requires
asymmetryfor >-, but almost agreement does not
require completeness for >, so > is the appropriate
basis forconsiderationof almost agreement.
The most demandingaxiomatizationsof subjective
probabilityspecifyconditions for >- on v that are
necessary and sufficient,or perhaps only sufficient,
forthe existenceof a probabilitymeasure that agrees
with >-. Less demanding theories seek only partial
agreement or almost agreement. We also mention
cases intermediatebetweenpartial and fullagreement
and note others in which additivityof P must be
replaced by a weaker concept to obtain a suitable
numericalrepresentation.
What axioms for>- or > on v are so obvious and
uncontroversialas to occasion no serious criticism?
A fewthat mightqualifyforthis distinctionare
asymmetry: If A
>-
nontriviality: S
0;
>-
B
then not (B
>-
A);
nonnegativity: A > 0;
monotonicity: If A D B
then A > B;
inclusionmonotonicity:
or (A>-B,B
If (ADB,B>-C)
then A >- C.
C)
One mightalso nominate
transitivity: If A >- B and B
then A >-C;
>-
C
nC
additivity: If A n C=0=B
then A>-B.A AUC>-BUC;
complementarity:If A >- B
then not (Ac> Bc),
but caution is advised here since these assumptions
have not gone unchallenged. Examples appear in
Section 3.
In any event,axioms likethese formthe foundations
oftheoriesofsubjectiveprobabilityand ofour remarks
about those theories.We begin with agreementwhen
the event set is finite,then consider weaker finite
representations,finitelyand countably additive representationswithinfinitealgebras,and conclude with
commentson conditionalprobability.
Excellent sources for articles of historical interest
for subjective probabilityare Kyburg and Smokler
(1964) and Savage, Hacking, and Shimony (1967).
Technical surveysare included in Luce and Suppes
AXIOMS OF SUBJECTIVE PROBABILITY
(1965), Krantz et al. (1971), Fine (1973), and Roberts
(1979). Other interpretationsof probabilityare discussed in Savage (1954), Good (1959), Fishburn
(1964), Fine (1973), de Finetti (1978), and Walley and
Fine (1979).
337
1 c i c n, the numberof Aj that contain i equals the
numberof Bj that contain i. In otherwords,the sums
ofthe indicatorfunctionsoverthe twoeventsequences
are identical.Then, forany real numbersPi, *, Pn,
(Al,, - Am) =o (Bi,
2. FINITE AGREEMENT
This section and the next assume that v is finite.
With no loss in generality,let v be the familyof all
subsets of S = $1,2, * **, n}. Each i in S is a state and
S is the set of states or the set of states of the world
(Savage, 1954).
We take >- as basic in the present section. For
convenience,let
Pi = PO i 1),
i = 1, *--, n,
so that P agrees with>- if,forall A and B in sl,
(1)
A
B
, Pi>
iEA
$2, 3$ >- $1, 4$,
{1, 5$ >- $3,4$,
$1, 3, 4$ >- $2, 5$.
If (1) holds then
P4 > PI + P3,
+
Pl +
P3 > P
+ P4,
5 > P3 +P4,
Pl + P3 + P4 > P2 + P5-
But these are inconsistentsince addition and cancellation leaves us with0 > 0.
What Kraft, Pratt, and Seidenberg discovered is
that a much strongeradditivitycondition is needed
for (1). To motivateit, let (A1, ,d Am)=o (B1, *-Bi) mean that the A, and Bj are in v and, foreach
.
m
iEAj
Pt=E
j=
1iEBj
Pi.
Consequently, if (1) holds and (A1, *.., Am) =0
(B1, * , Bm),then we cannot have A1> B1 foreveryj
along withAj >- Bj forat least one j. In the preceding
example,
(A1, A2, A3, A4) = ($4$, $2,3$, $1, 5$, {1, 3, 4$),
(B1, B2, B3, B4), = ($1, 3$, $1,4$, $3,4$, $2,5$).
We shall say that >- on v is stronglyadditiveif,for
all m -2 and all Aj and Bj,
E pi.
141>- {1, 3$,
P2
j=1
[(A1, *
iEB
This requires>- to be asymmetricand transitiveand,
since A B
P(A) = P(B), it also requires - to be
transitive.We referto >- as a weak orderwhen it has
these properties.
What else besides weak order is needed for agreement? Some time ago de Finetti (1931) noted that
(1) also entails nontriviality(S >- 0), nonnegativity
(A > 0 foreach A), and additivity(An c = 0 =
Bfn C=> [A >- B4=>A U C>-B U C]). The question
of whetherthese axioms are sufficientforagreement
remainedopen until it was settled in the negativeby
Kraft,Pratt,and Se'denberg (1959). They constructed
an example withn = 5 that satisfiesde Finetti's basic
axioms and includes the comparisons
,Bm)
m
'E
,
A.) =o (B1,
(2)
foreach
Bm) Aj - Bj
-,
i
< m]
=
not (Am> B).
As just noted,this is necessaryfor(1). Its strengthis
suggestedby the fact that it implies weak order as
well as additivity.For example,
(A,B)=o(B,A),
so A>-B =not
(B>-A);
(A,B, C) =o (B, C,A),
so
(A >B,B
>C)=-A
>C.
But strongadditivityimplies much more. Namely
(Fishburn,1970, Chapter 4), >- on v is stronglyadditive if and ornlyif there are real numbersPI, * ,Pn
that satisfy(1). In the special case of subjectiveprobabilitywithpi ' 0 and E pi = 1, it is enoughto assume
that >- is nontrivial and nonnegative as well as
stronglyadditive. Proofs that these or equivalent axioms are necessary and sufficientfor agreementappear in Kraft, Pratt, and Seidenberg (1959), Scott
(1964), and Krantz et al. (1971).
Those proofs,and ones for theorems in the next
sectionthatuse conditionssimilarto strongadditivity,
are based on solution theory for systems of linear
inequalities (Kuhn and Tucker, 1956). Relevant theorems are referredto Farkas's lemma, the theorem
of the alternative, Fan's theorem, and separatinghyperplane lemmas. The basic algebraic results of
linear solution theory are also used by Heath and
Sudderth(1972) and Buehler (1976) among othersto
expand on de Finetti's ideas on coherentsystemsof
bets.
A crudebut instructivecomparisonbetweenbetting
and strongadditivitycan be made as follows.Suppose
P. C. FISHBURN
338
strongadditivityfails with (A1, * , Am) =o (B1,
Bi), Aj > Bj forall j < m, and Am> Bin. Because of
Am> B, an individualwiththese comparisonswould
presumablybe willingto pay some positiveamount to
engagein the followinggame. For each Aj that obtains
he receives $1, and foreach Bj that obtains he pays
out $1. (Event A obtains if the true state is in A.)
However,thiswouldbe foolish,forregardlessofwhich
state obtains, (A1, *--, Am) =o (B1, mBin) ensures
that his net take, exclusive of stake, will be precisely
zero. I shall say more about decision models shortly.
Assumingagreementas in (1) with nonnegativepi
that sum to 1, it is trivialto note that thepi are unique
up to transformationsthat preserve nonnegativity,
the sum of 1, and the weak orderof partial sums induced by >-. Of more interestis the question of when
the pi in (1) are unique. A partial answer, sufficient
for uniqueness (Luce, 1967; Krantz et al., 1971) is
that wheneverevents A, B, C, and D are such that
A n B = 0, A >- B, and B > D, then thereare events
C', D', and E such that
E-A
UB,
C' n D' = 0,
(3)
D'CE
C'
C,
and D'
- D.
This is a difficultcondition to unravel and it is
not necessary even in simple cases. An example not
coveredby the conditionis, forn = 5,
$1,21 - 141,
{1141 - 129319
EEli
3/22,4/22,5/22,8/22).
In general,givenPi > 0 forall i, the pi are unique if
and only if there are n - 1 event pairs (Aj, Bj) with
A, n B1 = 0 and A, - B1 such that the corresponding
n - 1 linear equations
E
iEA,
Pi - E pi= ,
iEB,
j1,
*-n
(a) ui (a) >
S
i(a) ui(a) ,
i=1 aEM
where l(a) is the probability that lottery I yields
outcomea.
Subjective probabilities emerge from this utility
representation(Fishburn, 1967) if we adopt an independence axiom which says that preferencesover
lotteries within each nonnull state are essentially
to
is indifferent
l
identical. [State i is null if (11,
,n)
(l', ...,*nln) wheneverlj = 1I forallj? i.JWe can then
replace u- by piu under suitable rescalingof the ui to
obtain
(1l
,
-1,
are linearlyindependent.
Other avenues to uniqueness are available in the
finitestates settingif W is embedded in or extended
to a richerstructureand somethingmore than (1) is
required of P that will ensure its uniqueness. For
example, Suppes and Zanotti (1976) and Luce and
In)
. n.,
n
i=1 aEM
Under (1) these yieldP1 + P2 = P4, Pi + P4 = P2 + P3,
P2 + p4 = p5, and p2 + A5 = P1 + p3 + p4. The unique
solutionwithnonnegativepi that sum to 1 is
= (2/22,
4,) is preferredto (1,
(11,*
n
$2,41 - 15},
$2,5} - {1, 3, 4}.
(P1, P2, p3, p4, p5)
Narens (1978) extend V to infinitestructureswith
correspondingextensions of >- such that the representation required of the probabilitymeasure for
the extendedsystemimpliesthat it is unique. If P for
(1) is the restrictionto V of this measure,then P is
unique,but onlybecause ofthe conditionsimposedon
its parent. In particular,the linear independenceconditionof the precedingparagraphneed not hold for on s. Extendabilityis treatedin detail in Kaplan and
Fine (1977).
A moretraditionalrouteto uniqueness arises when
v is embeddedin a decision structureand extraneous
scaling probabilities (Anscombe and Aumann, 1963;
Fishburn, 1967, 1970; Myerson, 1979) or canonical
lotteries(Pratt, Raiffa,and Schlaifer,1964) are used
to derivea subjectiveexpectedutilitymodel in which
the state probabilities are unique. One might, for
example, use probabilitiesassociated with events for
a random device to construct lotteries on a set of
monetaryoutcomes.We then consideran individual's
preferencerelationon the set 1(11,...* ,): each li is a
lottery),where(11,..., In) is a "lotteryact" that selects
lotteryli if state i obtains or is the "true state." The
selected lotteryis then played to determinethe final
outcome.If the individual'spreferencerelationon the
set of lotteryacts satisfiesthe expectedutilityaxioms
of von Neumann and Morgenstern(1944), then for
each i in S thereis a real valued utilityfunctionui on
the outcome set M such that, for all pairs of lottery
acts,
ln)
to (1{,
is preferred
n
n
4*
E
i=1
,)
.
Pi
aEM
4(a)u(a) > E pi
i=1
E
aEM
lf(a)u(a),
with the pi unique and pi = 0 if and only if state i is
null.
Additional details of this and other lottery-based
theoriesare givenin Fishburn(1981, Section 8). Other
decision-orientedtheories that derive P on finiteor
AXIOMS OF SUBJECTIVE PROBABILITY
infiniteW as part of the utilityrepresentation
for
preferences
are describedtherealso.
3. OTHER FINITE MODELS
Finitemodelsweakerthan (1) arisewhenwe relax
of
strongadditivity,
including
perhapsits implication
weaker
weakorder.Plausiblereasonsforconsidering
modelswillbe illustrated
by examplesbeforewe get
I omitasymmetry,
nonintospecificrepresentations.
fromtheexamples.
triviality,
and monotonicity
3.1 Noncomparability(Incompleteness of -)
A = Mexico City's populationwill exceed
20,000,000
by 1994;
B = The firstcard drawnfromthisold and
probablyincomplete
bridgedeckwillbe
a heart.
As Keynes (1921) mighthave argued,A and B are
sufficiently
disparateto discouragea reasonableperson fromjudgingeitherA -B orB > A.
3.2 Nontransitive
For a slightly
bentcoin,
A = The next 101 flipswill giveat least 40
heads;
B = The next100 flipswill giveat least 40
heads;
C = The next1000flipswillgiveat least460
heads.
A - C, C - B, and A >- B do notseem
The judgments
unreasonable.
3.3 Nontransitive>- (May, 1954; Tversky,1969;
Fishburn,1983a)
Cyclicpatternscould arise fromcomparisonsbetweenmultidimensional
events.Sue will meetMike
X. Smithat a partynextSaturday.She knowsnothing
abouthimexceptthathe hasjustpublisheda hotnew
book on winningstrategiesin real estate.Her judgforMr. Smithinclude
mentson separateattributes
Height: 6'-0"
>-
6'-1" >. 6'-2",
Age:40 >- 50 >- 60,
Hair color:brunette>- red>- blonde.
forMr. Smithare
Threecomposites
A = 6'-0" 60-year-old
redhead;
B = 6'-1" 40-year-old
blonde;
C = 6'-2" 50-year-old
brunette.
Sue considersone of thesemorebelievablethan anotherifthefirstis moreprobableon twoofthethree
attributes.
HenceA >- B, B >- C, and C >- A.
3.4 Nonadditivity
(Ellsberg, 1961)
An urncontains90 coloredballs, 30 of whichare
ecru. The other60 are red and navy in unknown
339
proportion.
One ball willbe chosenat randomfrom
theurn.
E = chosenball is ecru;
R = chosenball is red;
N = chosen ball is navy.
An individualcan earn a valuableprizeby guessing
the color of the chosenball. Two comparisonsare
made: E versusR, and E U N versusR U N. Ellsberg's
experimentsindicate that E >- R and R U N >- E U N
are likely.In each case the"moreprobable"eventhas
to E,
greaterspecificity:
exactly30 balls correspond
and exactly60 correspondto R U N. The noted
judgments
clearlyviolateadditivity.
Theyalso violate
complementarity.
sectionwillbe
Two relativesof=o in thepreceding
conditions
and sufficient
for
usedto expressnecessary
severalweakermodels.First,
(Al9..* Am)--zo(Big ..
* Bm)
meansthat,foreach 1 c i c n,thenumberofAjthat
containi equals or exceedsthe numberof Bj that
containi. Second,
(Al,
..
.. - Am) >O (B1, *
*
9 Bm)
meansthatforeveryi thenumberofAjthatcontaini
exceedsthenumberofBj thatcontaini. In bothcases,
theAj and Bj are understood
to be in S.
We firstconsideralmostagreement
with' as basic.
In thiscase (Kraft,Pratt,and Seidenberg,
1959)there
is a probability
measureP on v that almostagrees
with' ifand onlyif,forall m -1 and all Aj and Bj,
[(A1
...*
Am) >o (B1
...*
Bim),Bj , Aj
foreachj < m]
(4)
=not
(Bm , Am).
This is a very weak condition.Its only implication form = 1 is not (0 z S). For m = 2, since
(A, AC) >o (0, 0), it implies that we cannot have
both 0 > A and 0 > AC. Indeed,the weaknessof
almostagreementis seen by the factthat it does
not forbidcyclic>-. For example, if S = $1,2, 31 with
I1
>-
$21>- $31>- $1, almost agreementcould hold
withPi = P2 = p3 =
1/3.
Partial agreementwith >- as basic forbidscyclic >-
and seems somewhatmoredemandingthan almost
In thiscase (Adams,1965;Fishburn,
agreement.
1969)
measureP on v thatpartially
thereis a probability
agreeswith>- if and onlyif,forall m -1 and all Aj
and Bj,
[(Al ..
*
(5)
XAm)--'o(Bi
..
*
Bm)gBj >- Aj
forallj < m]
=-not
(Bm> Am).
P. C. FISHBURN
340
monotonicity,
This entails asymmetry,
nonnegativity,
and acyclicity,but not nontriviality,transitivity,or
additivity.However,it does forbidadditivityreversals
likeE >- R and R U N >- E U N in our fourthexample.
Strongerversions of partial agreementhave been
developedto explicitlyinclude discriminatorythresholds in "if and only if" representations.They were
motivatedby Luce's (1956) semiordermodel and Fishburn's (1970, 1985) intervalordermodel in preference
orderif it is asymmetric
theory.We call >- an interval
and satisfies(A >- B and C >- D) => (A >- D or C >- B),
forall A, B, C, ana D. If, in addition, (A >- B >- C)
(A >- D or D >- C), forall A, B, C, and D, then >- is a
semiorder.
Examples like those givenearliershow that
these conditions can be violated by reasonable judgments in some circumstances.For instance,if A D B
and C D D with A and B of a substantiallydifferent nature than C and D (c.f. A versus B in our
firstexample), then A >- B, C >- D, A
C - B, and
A - D - B mightobtain.
The semiorderpartial agreementrepresentationis
a constantthresholdmodel with
A>- B
P(A) > P(B) + 65
0c
<6
1,
and the intervalorderpartial agreementrepresentation is a variable thresholdmodel with
A >- B4P(A)
> P(B) + a(B),
a>.0.
In both cases, P is a probabilitymeasure on s. Necessary ana sufficientaxioms (Fishburn, 1969; Domotorand Stelzer,1971) forthe semiorderrepresentation
are asymmetry,
inclusion monotonicity,
nontriviality,
and
* B2m),
[(Al, *., A2m) =o (B1, ...
Bj >- A. fori = 1, * , m, and
Aj>-B.for j= m+ 1, ..., 2m-1]
=
not (A2m> B2m).
Sufficientaxioms (Fishburn, 1969) for the interval
orderrepresentationare asymmetry,(A >- B and C >D) =* (A >- D or C >- B), inclusion monotonicity(not
strictlynecessary),and
[(A1,
...X
Am) =o (B1,
forall j
XBm), Aj >-* Bj
<
m]
= not
(Am >* Bm)
whereA >* B means that, forall C, A >- C if B >- C,
and forsome C, A >- C and not (B >- C). The displayed
conditions essentially reflectthe differencebetween
the constantand variable thresholdnotions.
The intervalorderrepresentationbears comparison
to models of upper and lower probabilitiesthat arise
in Koopman (1940a), Smith (1961), Good (1962),
Dempster (1967, 1968), Suppes (1974), Williams
(1976), Walley and Fine (1979), and Kumar (1982)
among others. See also Shafer (1976) on the concept
of belieffunctions.One model of this typehas
A >- B t-*P*(A) > P*(B),
where P* and P* are monotonicbut not necessarily
additive measures on v with P*(S) = P*(S) = 1,
P*(0) = P*(0) = 0, and P*(A) - P*(A) foreach A. If
we let v(A) = P*(A) - P*(A), then
A > tB - P*(A) > P*(B) + a(B),
which is similar to the intervalorder model. Axioms
forthis case consist of the intervalorder conditions
plus suitablenonnegativity
and monotonicityassumptions. It requires a special additivitycondition only
if there is to be a probabilitymeasure P for which
Davidson and Suppes (1956) and Schmeidler(1984)
axiomatizeutilitymodels in whichP is monotonicbut
not necessarilyadditive. Davidson and Suppes adopt
Ramsey's (1931) measurementapproach. They first
use an event with subjective probability1/2to scale
outcomeutilitiesand then use these to scale probabilitiesofotherevents.Their outcomeutilitiesare evenly
spaced and P(A) is definedas
P(A)
=
u(d *) - u(d)
u(d*) -u(d) + u(c)
-
u(c*)
when u(c) # u(c*), u(d) # u(d*), and the act that
yields c if A and d if Ac is indifferent
to the act that
yieldsc * ifA and d * ifAC.This gives P(A) + P(AC) =
1, and theirP is unique if n - 5.
Schmeidler (1984) weakens the lotteryact axioms
forthe model at the end of the precedingsection to
obtain a modifiedexpected utilityrepresentationin
whichP is monotonicand unique. Because P need not
be additive,his definitionof expectation(integration)
is special. Schmeidler's model is designed to accommodate Ellsberg-typeviolationsof additivity.
Anothernonladditive
model,axiomatizedin both the
intuitiveand decisionmodes (Fishburn,1983a, 1983b),
allows intransitivities.Unlike preceding representations, it uses a two-argumentfunctionp on v X _v
with
A >- B
p(A, B) > O
along withp(Q, 0) = 1, p(B, A) = -p(A, B), A D B
p(A,B)>-0, and
A nB=0
= p(A UB, C) + p(0,C)
= p(A,C) + p(B, C).
The last propertyis a nonseparable generalizationof
additivity.Necessary and sufficientconditions are
AXIOMS OF SUBJECTIVE PROBABILITY
of
monotonicity,
and a generalization
nontriviality,
strongadditivity.
4. INFINITEALGEBRAS
This and thenexttwosectionsassumethatv is an
inflnite
BooleanalgebraofsubsetsofS. The present
and special axioms
sectionintroducesterminology
developedforthiscase. It also notesa partialagreementaxiomatization
thatdoes not fitinto the next
two sections,whichdiscuss agreementand almost
when>- is a weakorder.The nextsection
agreement
focuseson finiteadditivity;
Section6 looksat countablyadditivemeasures.We take >- as basic throughout.
The Boolean algebrav is a a-algebraifA is in v
wheneverA is the union of a countablenumberof
measureP is countably
eventsin X. A probability
additiveif
P
A)i
00
=
P(A-)
wheneverthe A- are mutuallydisjointeventswhose
unionis also in -V.Notethatv does nothaveto be a
a-algebraforP to be countablyadditive.
FollowingSavage (1954), > is fine if, whenever
A
0, S can be partitionedinto events B1, *--, Bm
withA > Bj forall j. Events A and B are almost
>
equivalent
if,forall C and D in -,
[C >- 0, D >- 0, A n C =0 = B n D]
=* A U C
B and B U D ~tA;
A and B are almost
and >- is tightifA B whenever
equivalent.Savage '(1954, page 37) subsequently
of fineby replacingA >
his definition
strengthened
in the
Bj byA >- B, forallj. We referto >- as superfine
lattercase.
FollowingVillegas (1964), eventA is arnatom if
A >- 0 and A >- B >- 0 forno eventB C A. The algebra
ifithas no atom.It shouldbe notedthat
v is atomless
thesedefinitions
dependon thebehaviorof>-.
Threeoftheearliestaxiomsforinfinite
algebrasare,
dueto de Finetti(1931,1937)and Savage (1954).The
firstof the followingis fromde Finetti,and also
Bernstein(1917) and Koopman (1940a). The other
of
twoare Savage's.All threesuggestthe possibility
of
events
with
into
a
number
v
verylarge
partitioning
verysmallprobabilities.
Partition
Axiom1. For each n > 29S can be partin
tionedinto equallylikely(-) events.
Partition
Axiom2. For each n > 2, S can be partitionedinton eventssuchthattheunionofno r events
is moreprobable than the union of any r + 1.
341
PartitionAxiom3. For all A and B in sV,ifA >- B
then S can be partitionedinto a finitenumberof
events C1, *- *, Cmsuch that A >- B U Cj forall j.
PartitionsforAxiom1 are uniform
partitions,
and
thoseforAxiom2 arealmostuniform
partitions.
Given
de Finetti'sbasic axioms (Section 2), Savage notes
that Axiom3 impliesthat > is both superfineand
tightand thatit impliesAxioms1 and 2.
PartitionAxiom3 is used by Fishburn(1975) to
axiomatizepartialagreement
witha finitely
additive
measure.Sufficient
axiomsforpartialagreement
are
inclusioninonotonicity,
Axiom 5 with =0 in place
of-O, PartitionAxiom3, and the following
part of
additivity:
[A n C=0
=B n C,A U C>-B U C]J=A >-B.
is defendedby the contention
Partialadditivity
that
ifA U C >- B U C when(A U B) n C = 0, thenthe
betweenthe twoeventswillbe evenmore
difference
apparentwhenthe commonpartC is removed.The
fourth
sectionquestionsits
examplein thepreceding
generalacceptability.
Because of the possibilityof atoms,none of the
foragreement.
axiomsis necessary
We therepartition
forenotethreemoreconditions
thatare necessaryfor
theexistenceofa probability
measurethatagreeswith
>-. These are referredto as Archimedeanaxioms be-
cause theylead to realrepresentations
withno infinitesimal nonstandardnumbersor infinitelylarge
numbers.
A fewpreliminary
definitions
are needed.
A subsetW of v is orderdensein _v if,whenever
A >- B and neitherA nor B is in ', then A ~t C t B
forsome C in W. A countable (finiteor denumerable)
sequenceofeventsA1,A2, ... is a standardsequence
relativeto eventA if foreach Ai thereare disjoint
eventsBi and C- such thatA1 = B1, Ai+1= Bi U C1
(wheni + 1 is present),B1 A, Ci A, and Bi Ai.
The upshotof this is that if P agreeswith>- then
P(Ai) = iP(A) foreach i.
1. v includesa countableorder
ArchimedeanAxiom
densesubset.
Axiom2. ForeacheventA > 0, every
Archimedean
standardsequencerelativeto A is finite.
Axiom3. Forall m > 2 and all events
Archimedean
A1, ** Am,B1, . * Bin: if A1>- B1 then there is a
positiveintegerN suchthat,whenever(k0, nA1,A2,
* Am)=o (kS, nBi, B2, .. . Bm) with k >0 , n > 1,
andA.1t Bj forall j > 2, thenk/n> 1/N.
The firstArchimedeanaxiom,whichdates from
is necCantor's(1895) workon transfinite
numbers,
in
with
weak
order
and
sufficient
essary
conjunction
on
fortheexistenceofa realvaluedfunction
f v such
that, for all A and B in XV,A
>-
B o f(A)
>
f(B)
(Birkhoff,
1967;Fishburn,1970;Krantzet al., 1971).
342
P. C. FISHBURN
Axiom2, fromLuce (1967),is one of
Archimedean
severalstandardsequenceconditionsused in Krantz
et al. (1971). To be applicable,it requiresstructural
of such seconditionsthat allow the construction
quences.Neitherof the firsttwo axiomsrelatesto
whichmustenterthrough
of probability,
additivity
anotheraxiom.
is builtintoArchimedean
Axiom
Strongadditivity
3 via k = 0 and n = 1. In its statement, due to
Chateauneufand Jaffray
(1984), kA denotesA repeated k times.Its necessityforagreementfollows
and itshypotheses
fromthefactthatagreement
imply
k = n[P(Al)
>
-
P(B1)] + , [P(Aj)
j?2
-
P(Bj)]
n[P(A) -P(Bj)],
so k/n- P(A1) - P(Bj). Since P(A1) - P(B1) > 0 by
A1 >- B1, k/ncannot be arbitrarily
small without
violatingagreement.
5. FINITE ADDITIVITY
The basic axioms of de Finetti,i.e., weak order,
and additivity,
have benontriviality,
nonnegativity,
come so standardthat theyare used moreor less
foragreedirectlyin all infinitev axiomatizations
foralmostagreement,and in mostaxiomatizations
ment.Savage (1954) defines> (or >-) as a qualitative
whenit satisfiesde Finetti'saxioms,and
probability
thisnamehas persistedin moreor less thisform.
Since all axiomatizations
in this sectionand the
nextuse de Finetti'sbasic axioms,theywillbe taken
forgranteduntilthe finalsectionof the paper.The
presentsectionis concernedsolelywiththeexistence
offinitely
additiveprobability
measuresthatagree,or
almostagree,with>- whensv is infinite.
We consider
almostagreement
first.
Savage (1954) provesthat thereexists a P that
almostagreeswith>- whenPartitionAxiom2 holds
(almostuniform)
and v is thecr-algebra
ofall subsets
of S. Moreover,P is unique,and if P(A) > 0 and
0 < X < 1, then P(B) = XP(A) forsome B C A. The
sameconclusionsfollowif>- is superfine
and,in this
case, P(A) > 0 ifA >- 0 and P(A) = P(B) if and only
ifA and B are almostequivalent.See also Wakker
(1981,Lemmas3-5).
Related resultsfor almost agreementhave been
obtainedbyotherswhenv is merelyassumedto be a
Booleanalgebra.Niiniluoto(1972)andWakker(1981)
notethatthereis a uniquealmostagreeingP when>is fine.Narens(1974) and Wakker(1981) showthat
almost agreement(not necessarilyunique) follows
from(4).
The earliestaxiomatizations
foragreement(Bernstein,1917;de Finetti,1931,1937;Koopman,1940a)
used PartitionAxiom1 (uniform
partitions).Savage
(1954),againwithv as thefamilyofall subsetsofS,
obtainsa uniqueagreeingP withPartitionAxiom3.
SincetheP(B) = XP(A)property
holdshere,v must
be atomless.A relatedresultis obtainedin French
(1982) by adjoiningan auxiliaryexperiment
(extraneous scalingprobabilities)to S. This approach,of
enrichinga possiblyfiniteS witheventsgenerated
froma randomdevice,was used earlierby DeGroot
(1970), and indeeddid not escape noticeby Savage
(1954,pages33 and 38).
Otheraxiomatizations
foragreement
do notassume
that.w is a a-algebra.Luce (1967),whoseaxiomsapply
also to finiteS, showsthata uniqueagreeing
P follows
from(3) and Archimedean
Axiom2. Roberts(1973)
obtainsunique agreementfromPartitionAxiom2
and Archimedean
Axiom1, aftera similarresultin
Fine(1971).Wakker(1981)notesthat,when>- is fine,
uniqueagreement
holdsifand onlyifeither>- is tight
or sv has an atom.
Necessaryand sufficient
conditionsforagreement
are discussedby Domotor(1969) and Chateauneuf
andJaffray
(1984).Domotor'scomplexconditions
will
not be recalledhere.Chateauneufand Jaffray
show
thatifv is countable,e.g.,thefamily
offinitesubsets
of 1, 2,
...-
and their complements,then some P
agreeswith> if and onlyif Archimedean
Axiom3
holds.Because theirconditionsare necessaryas well
as sufficient,
theydo notimplyuniqueness.Indeed,a
pricepayedforuniquenessis nonnecessity
sinceconditionsthat guaranteeunique agreementcannotbe
whollynecessary.
Furtherreferenceson mathematicaldetails of
finitelyadditivemeasuresincludeBochner (1939),
Sobczykand Hammer (1944), Yosida and Hewitt
(1952), Hewittand Savage (1955), Dubins (1974),
Purves and Sudderth(1976), Bhashara Rao and
BhaskaraRao (1983),and Schervish,
and
Seidenfeld,
Kadane (1984). Contributions
of special interestin
decisiontheoryincludeDubins and Savage (1965),
de Finetti (1975), Heath and Sudderth (1978),
Lane and Sudderth (1978), and Seidenfeldand
Schervish(1983).
6. COUNTABLEADDITIVITY
We continueto assume de Finetti'sfour basic
axioms.
the keyassumptionon >
Villegas(1964) identifies
neededforcountableadditivity.
Modifiedslightlyto
accommodateBoolean algebras that are not also
his axiomis
a-algebras,
Monotonecontinuity:
Forall A, B, A1,A2, ... in X,
if A1 C A2 ..., A = UA and B t Ai forall i,
thenB > A.
Thus, if the nondecreasing
Ai convergeto a limit
eventA, then the judgmentthat B is at least as
343
AXIOMS OF SUBJECTIVE PROBABILITY
probableas A-forall i cannotbe reversedin thelimit
by a jump to A > B. Monotonecontinuity
is quite
foragreement
witha countappealingandis necessary
ably additiveprobability
measure.In fact,Villegas
(1964) provesthat if v is a a-algebraand if P is a
finitely
additiveprobability
measurethatagreeswith
>-, thenP is countablyadditiveif and only if > is
monotonelycontinuous.Chateauneufand Jaffray
(1984) remarkthatthisremainstrueevenwhenv is
nota a-algebra.
Consequently,
foreveryaxiomatization
in theprecedingsectionthat impliesthe existenceof P that
agreeswith>-, such a P is countablyadditiveif and
onlyif> is monotonely
continuous.
Otheraxiomatizations
have been developedspecificallyforcountableadditivity
whenv is a a-algebra.
All assume monotonecontinuity.Villegas (1964)
provesthat if v is atomlessthenthereis a unique
countablyadditiveP that agreeswith>. DeGroot
(1970) and French(1982) obtainthesameconclusion
withthe explicituse of an auxiliaryexperiment
that
itselfis atomlessbutallowsatomsin theoriginalstate
set.Similarly,
Chuaquiand Malitz(1983)adoptstrong
toobtainuniqueagreement
whenever
v has
additivity
a nontrivial
atomlesssubalgebra.Theyalso consider
thepurelyatomiccase,whereP is a discretemeasure,
andprovean almostagreement
theorem
understrong
additivity.
Finally,Chateauneufand Jaffray
(1984)
show that ArchimedeanAxiom3 is necessaryand
sufficientfor agreement.Their complete set of
axiomsforcountably
additiveagreement
whenv is a
a-algebrais equivalentto de Finetti'saxioms,monotone continuity,
and ArchimedeanAxiom 3. Since
theseare necessaryas wellas sufficient,
theirP is not
necessarilyunique.
7. CONDITIONALPROBABILITY
A typicalversionofBayes' formulaforconditional
state
whenS = 0 x Z and theunderlying
probability
orparameter
space0 and samplespaceZ are finiteis
P1(Iz)l z) =
P10
P2 (Z I 0) PO(O)
X'P2 (Z IO')P(OW)~
HereP1(i * z) andP2(* I 0) are conditional
probability
mass functions,
and Po is a (marginal)probability
massfunction
on 0. In theusual subjectivistic
interand P1 is the
pretation,
Po is the priordistribution
posterioron 0 givenobservationz. Each of Po, P1,
and P2 maybe basedon a jointdistribution
overS as
describedin thenextparagraph.
Ifthereis a uniqueprobability
measureP on v that
agreesor almostagreeswith> or t, then subjectiveconditional
can be definedunambigprobabilities
uouslyin theusualwayas
(6)
P(A IB) = P(A fl B)/P(B) when P(B) >0O,
thusputtingBayes' theoremand othermathematical
machinery
of conditionalprobability
at our disposal.
The theoretical
foundations
ofBayesiandecisiontheorycan be understood
in thiswaywhenall uncertainties, includingthose associated with experiment's
outcomes, are embedded in X. At least in theory,
if not in commonpractice,priorsand otherspecial
probabilities
can be extracted
fromtheglobalP measure by the usual operations.For the settingof the
preceding
paragraph,
P0(0) = P({0} x Z ), P1(OIz) =
x
z)})/P(O
P({(0,
{z}), and P2(z I 0) = P({(0, Z)})/P({0}
x Z) whenthedenominators
do notvanish.
A potentialproblemwith(6) ariseswhenP is not
unique,as forexamplein Section2, sinceconditional
probabilities
mightnot be uniquelyordered.For example,one agreeingP maygiveP(A IB) > P(C ID)
by (6), whileanothergivesP(C ID) > P(A IB).
Partlyfor this reason,and partlybecause some
authors,includingKoopman(1940a, 1940b),believe
that conditionaljudgmentsare basic to subjective
probability,
therehas been interestin axiomatizing
This is usuallydone
conditionalprobability
directly.
by means of a comparativeconditionalprobability
relationto on v X -o, wheres0 is thesetofnonnull
eventsin v and A IB 0oC ID is interpreted
as "A
givenB is at least as probableas C givenD." When
;0ois completeand 0 is definedby
A IB -o C I D
if AIB>-oCID
and CID
-oAIB,
a null eventis an A in v forwhichA IS - 0 1 S.
forcomparative
conditional
A typicalrepresentation
probability
is,forall A IB and C ID in v X -o,
(7)
A IRB o CID
P(A n B)/P(B) :: P(C n D)/P(D),
measureon -V.
whereP is a probability
Domotor(1969) includesnecessaryand sufficient
on ;0ofor(7) whenv is finite.His axioms
conditions
extendthe approachof Kraft,Pratt,and Seidenberg
(1959)forfiniteagreement.
SuppesandZanotti(1982)
and
sufficient
conditionsfora model
givenecessary
like(7) forgeneralv byusingextensionsofindicator
in a mannersimilarto Suppesand Zanotti
functions
for(7) and
axiomatizations
Other
sufficient
(1976).
in
relatedrepresentations
appear Koopman (1940a,
1940b),Aczel (1961, 1966),Luce (1968),and Krantz
et al. (1971). The last of theseprovidesa nice descriptionof previouswork (pages 221-228). Their
main axiomatizationis based on Luce (1968) and
consistsof sevenconditionsnecessaryfor(7) plus a
nonnecessary
solvabilityconditionwhichsays that
B
C
ifA I
-o I D thenthereis a C' in v such that
C nlD C C ' and A IB ~o C' ID.
344
P. C. FISHBURN
Conditionalprobabilitiesalso arise directlyfromthe
preference-basedaxiomatizations of Ramsey (1931),
Pfanzagl (1967, 1968), Luce and Krantz (1971), Fishburn (1973), and Balch and Fishburn (1974). Reviews
of theirtheoriesare includedin Fishburn (1981).
IDUBINS, L. E. and SAVAGE, L. J. (1965). How to Gamble if You
Must. McGraw-Hill,New York. Dover edition,1976.
ELLSBERG, D. (1961). Risk, ambiguity,and the Savage axioms.
Quart.J. Econ. 75 643-669.
FELLER, W. (1966). An Introductionto ProbabilityTheoryand Its
Applications2. Wiley,New York.
FINE, T. (1971). A note on the existenceof quantitativeprobability.
Ann. Math. Statist.42 1182-1186.
REFERENCES
ACZEL, J. (1961). Uber die Begrundungder Additions-und Multi-
plikationsformeln von bedingten Wahrscheinlichkeiten.
Magyar Tud. Akad. Mat. Kutat6 Int. Kozl. 6 110-122.
ACZEL, J. (1966). Lectures on Functional Equations and Their
Applications.Academic,New York.
ADAMS, E. W. (1965). Elements ofa theoryof inexactmeasurement.
Philos. Sci. 32 205-228.
ANSCOMBE, F. J. and AUMANN, R. J. (1963). A definitionof subjective probability.Ann. Math. Statist. 34 199-205.
BALCH, M. and FISHBURN, P. C. (1974). Subjective expectedutility
for conditional primitives.In Essays on Economic Behavior
under Uncertainty57-69. North-Holland,Amsterdam.
BAYES, T. (1763). An essay towards solving a problem in the
doctrine of chances. Philos. Trans. Roy. Soc. 53 370-418.
Reprintedin FacsimilesofTwo Papers ofBayes (W. E. Deming,
ed.). Departmentof Agriculture,Washington,D. C., 1940.
BERNSTEIN, S. N. (1917). On the axiomatic foundationof probabilitytheory(in Russian). Soob. Khark. ob-va 15 209-274.
BHASHARA RAO, K. P. S. and BHASKARA RAO, M. (1983). Theory
ofCharges.Academic, New York.
BIRKHOFF, G. (1967). Lattice Theory. 3rd ed. Amer. Math. Soc.,
Providence,R. I.
BOCHNER, S. (1939). Additiveset functionson groups.Ann. Math.
40 769-799.
BUEHLER, R. J. (1976). Coherent preferences.Ann. Statist. 4
1051-1064.
CANTOR, G. (1895). Beitrage zur Begrundungder transfiniteMengenlehre.Math. Ann. 46 481-512; 49 (1897) 207-246. (English
translationin Contributionsto the Founding of the Theoryof
TransfiniteNumbers.Dover, New York, n.d.)
CHATEAUNEUF, A. and JAFFRAY,J.-Y. (1984). Archimedeanqualitativeprobabilities.J. Math. Psychology28 191-204.
CHUAQUI, R. and MALITZ, J. (1983). Preorderingscompatiblewith
probabilitymeasures. Trans. Amer. Math. Soc. 279 811-824.
DAVIDSON, D. and SUPPES, P. (1956). A finitisticaxiomatizationof
subjectiveprobabilityand utility.Econometrica24 264-275.
DE FINETTI, B. (1931). Sul significatosoggettivodella probabilita.
Fund. Math. 17 298-329.
DE FINETTI, B. (1937). La prevision ses lois logiques, ses sources
subjectives. Ann. Inst. H. Poincare 7 1-68. (Translated by
H. E. Kyburgin Kyburgand Smokler (1964).)
DE FINETTI, B. (1975). TheoryofProbability.Wiley,New York.
DE FINETTI, B. (1978). Probability:interpretations(II). InternationalEncyclopediaofStatistics744-754. Free Press, New York.
DEGROOT, M. H. (1970). Optimal Statistical Decisions. McGrawHill, New York.
DEMPSTER, A. P. (1967). Upper and lowerprobabilitiesinduced by
a multivaluedmapping.Ann. Math. Statist. 38 325-339.
DEMPSTER, A. P. (1968). A generalizationof Bayesian inference.
J. Roy. Statist.Soc. Ser. B 30 205-247.
DOMOTOR, Z. (1969). Probabilistic relational structuresand their
applications.Technical Report 144, InstituteforMathematical
Studies in the Social Sciences, StanfordUniv.
DOMOTOR, Z. and STELZER, J. (1971). Representationof finitely
additive semiordered qualitative probability structures. J.
Math. Psychology8 145-158.
DUBINS, L. E. (1974). On Lebesgue-like extensions of finitely
additivemeasures.Ann. Probab. 2 456-463.
FINE, T. (1973). TheoriesofProbability.Academic,New York.
FISHBURN, P. C. (1964). Decision and Value Theory.Wiley, New
York.
FISHBURN, P. C. (1967). Preference-baseddefinitionsof subjective
probability.Ann. Math. Statist. 38 1605-1617.
FISHBURN, P. C. (1969). Weak qualitativeprobabilityon finitesets.
Ann. Math. Statist. 40 2118-2126.
FISHBURN, P. C. (1970). UtilityTheoryforDecision Making. Wiley,
New York. Kriegeredition,1979.
FISHBURN, P. C. (1973). A mixture-set
axiomatizationofconditional
subjectiveexpectedutility.Econometrica41 1-25.
FISHBURN, P. C. (1975). Weak comparativeprobabilityon infinite
sets. Ann. Probab. 3 889-893.
FISHBURN, P. C. (1981). Subjective expected utility:a review of
normativetheories.Theoryand Decision 13 139-199.
FISHBURN, P. C. (1983a). A generalizationof comparativeprobabil-
ityon finitesets. J. Math. Psychology27 298-310.
FISHBURN, P. C. (1983b). Ellsberg revisited:a new look at compar-
ative probability.Ann. Statist. 11 1047-1059.
FISHBURN, P. C. (1985). IntervalOrdersand IntervalGraphs.Wiley,
New York.
FRENCH, S. (1982). On the axiomatisation of subjectiveprobabili-
ties. Theoryand Decision 14 19-33.
GooD, I. J. (1950). Probabilityand the WeighingofEvidence.Griffin,
London.
GOOD, I. J. (1959). Kinds of probability.Science 129 443-447.
GOOD, I. J. (1962). Subjective probabilityas the measure of a nonmeasurable set. Logic, Methodologyand Philosophyof Science
319-329. StanfordUniv. Press, Stanford,Calif.
HARTIGAN, J. A. (1983). Bayes Theory.Springer,New York.
HEATH, D. C. and SUDDERTH, W. D. (1972). On a theoremof de
Finetti,oddsmaking,and game theory.Ann. Math. Statist. 43
2072-2077.
HEATH, D. C. and SUDDERTH, W. D. (1978). On finitelyadditive
priors,coherence,and extended admissibility.Ann. Statist. 6
333-345.
HEWITT, E. and SAVAGE, L. J. (1955). Symmetricmeasures on
Cartesian products.Trans. Amer.Math. Soc. 80 470-501.
KAPLAN, M. and FINE, T. (1977). Joint orders in comparative
probability.Ann. Probab. 5 161-179.
KEYNES, J. M. (1921). A Treatise on Probability.Macmillan, New
York. Torchbookedition,1962.
der WahrscheinlichkeitsKOLMOGOROV, A. N. (1933). Grundbegriffe
rechnung.Springer,Berlin. Chelsea edition (English), 1950.
KOOPMAN, B. 0. (1940a). The axioms and algebra of intuitive
probability.Ann. Math. 41 269-292.
KOOPMAN, B. 0. (1940b). The bases of probability.Bull. Amer.
Math. Soc. 46 763-774. Reprinted in Kyburg and Smokler
(1964).
KRAFT, C. H., PRATT, J. W. and SEIDENBERG, A. (1959). Intutitive
probabilityon finitesets. Ann. Math. Statist. 30 408-419.
KRANTZ, D. H., LUCE, R. D., SUPPES, P. and TVERSKY A. (1971).
FoundationsofMeasurement1. Academic, New York.
KUHN, H. W. and TUCKER, A. W., eds. (1956). Linear Inequalities
and Related Systems,Ann. Math. Study 38. Princeton Univ.
Press, Princeton,N. J.
KUMAR, A. (1982). Lower probabilitieson infinitespaces and instabilityofstationarysequences. Ph.D. dissertation,CornellUniv.
KYBURG, H. E., JR. and SMOKLER,H. E., eds. (1964). Studies in
Probability.Wiley,New York.
SXubjective
AXIOMS OF SUBJECTIVE PROBABILITY
LANE, D. A. and SUDDERTH,
W. D. (1978). Diffuse models
for sampling and predictiveinference.Ann. Statist. 6 13181336.
LAPLACE, P. S. (1812). TheorieAnalytiquedes Probabilites.Paris.
Reprintedin OeuvresCompletes7 1947.
LoEVE, M. (1960). Probability Theory. 2nd ed. Van Nostrand,
Princeton,N. J.
LUCE, R. D. (1956). Semiordersand a theoryof utilitydiscrimination. Econometrica24 178-191.
LUCE, R. D. (1967). Sufficientconditions for the existence of a
finitelyadditive probabilitymeasure. Ann. Math. Statist. 38
780-786.
LUCE, R. D. (1968). On the numericalrepresentationof qualitative
conditionalprobability.Ann. Math. Statist. 39 481-491.
LUCE, R. D. and KRANTZ, D. H. (1971). Conditionalexpectedutility.
Econometrica39 253-271.
LUCE, R. D. and NARENS, L. (1978). Qualitative independence in
probabilitytheory.Theoryand Decision 9 225-239.
LUCE, R. D. and SUPPES, P. (1965). Preference,utility,and subjective probability. Handbook of Mathematical Psychology 3,
249-410. Wiley,New York.
MAY, K. 0. (1954). Intransitivity,utility,and the aggregationof
preferencepatterns.Econometrica22 1-13.
MYERSON, R. B. (1979). An axiomatic derivation of subjective
probability,utilityand evaluation functions.Theoryand Decision 11 339-352.
NARENS, L. (1974). Minimal conditions for additive conjoint
measurementand qualitative probability.J. Math. Psychology
11 404-430.
NIINILUOTO, I. (1972). A note on fineand tightqualitative probabilities.Ann. Math. Statist. 43 1581-1591.
PFANZAGL, J. (1967). Subjective probability derived from the
Morgenstern-von Neumann utility concept. In Essays in
Mathematical Economics 237-251. Princeton Univ. Press,
Princeton,N. J.
PFANZAGL, J. (1968). TheoryofMeasurement.Wiley,New York.
PRATT, J. W., RAIFFA, H. and SCHLAIFER, R. (1964). The foundations of decision under uncertainty:an elementaryexposition.
J. Amer.Statist.Assoc. 59 353-375.
PURVES, R. A. and SUDDERTH, W. D. (1976). Some finitelyadditive
probability.Ann. Probab. 4 259-276.
RAIFFA, H. and SCHLAIFER, H. (1961). Applied StatisticalDecision
Theory.Harvard Univ., Boston.
RAMSEY, F. P. (1931). Truth and probability.In The Foundations
of Mathematicsand Other Logical Essays. Humanities Press,
New York. Reprintedin Kyburgand Smokler (1964).
ROBERTS, F. S. (1973). A note on Fine's axioms for qualitative
probability.Ann. Probab. 1 484-487.
ROBERTS, F. S. (1979). Measurement Theory. Addison-Wesley,
Reading,Mass.
345
SAVAGE, L. J. (1954). The Foundations of Statistics. Wiley, New
York. Dover edition,1972.
SAVAGE, L. J., HACKING, I. and SHIMONY, A. (1967). A panel
discussionof personal probability.Philos. Sci. 34 305-332.
SCHERVISH, M. J., SEIDENFELD, T. and KADANE, J. B. (1984). The
extentofnon-conglomerability
offinitelyadditiveprobabilities.
Z. Wahrsch.Verw.Gebiete66 205-226.
SCHLAIFER, R. (1959). Probabilityand StatisticsforBusiness Decisions. McGraw-Hill,New York.
SCHMEIDLER, D. (1984). Subjective probabilityand expectedutility
vithoutadditivity.Preprint84, InstituteforMathematics and
Its Applications,Univ. Minnesota.
SCOTT, D. (1964). Measurementstructuresand linear inequalities.
J. Math. Psychology1 233-247.
SEIDENFELD, T. and SCHERVISH, M. J. (1983). A conflictbetween
finite additivityand avoiding Dutch book. Philos. Sci. 50
398-412.
SHAFER, G. (1976). A MathematicalTheoryof Evidence. Princeton
Univ. Press, Princeton,N. J.
SMITH, C. A. B. (1961). Consistency in statistical inferenceand
decision.J. Roy. Statist. Soc. Ser. B 23 1-37.
SOBCZYK, A. and HAMMER, P. C. (1944). A decompositionof additive set functions.Duke Math. J. 1 1 839-846.
SUDDERTH, W. D. (1985). Review of Hartigan (1983). Bull. Amer.
Math. Soc. 12 294-297.
SUPPES, P. (1974). The measurementof belief.J. Roy. Statist.Soc.
Ser. B 36 160-175.
SUPPES, P. and ZANOTTI, M. (1976). Necessary and sufficient
conditionsforexistence of a unique measure strictlyagreeing
with a qualitative probability ordering.J. Philos. Logic 5
431-438.
SUPPES, P. and ZANOTTI, M. (1982). Necessary and sufficient
qualitative axioms for conditional probability.Z. Wahrsch.
Verw.Gebiete60 163-169.
TVERSKY, A. (1969). Intransitivity
of preferences.Psychol.Rev. 76
31-48.
VILLEGAS, C. (1964). On qualitative probabilitya-algebras. Ann.
Math. Statist. 35 1787-1796.
VON NEUMANN, J. and MORGENSTERN, 0. (1944). Theory of
Games and Economic Behavior. Princeton Univ. Press,
Princeton,N. J.
WAKKER, P. (1981). Agreeingprobabilitymeasuresforcomparative
probabilitystructures.Ann. Statist.9 658-662.
WALLEY, P. and FINE, T. L. (1979). Varieties of modal (classificatory)and comparativeprobability.Synthese41 321-374.
WILLIAMS, P. M. (1976). Indeterminate probabilities. Formal
Methods in the Methodologyof Empirical Sciences 229-246.
Ossolineumand Reidel, Dordrecht,Netherlands.
YOSIDA, K. and HEWITT, E. (1952). Finitely additive measures.
Trans. Amer.Math. Soc. 72 46-66.