The Axioms of Subjective Probability Author(s): Peter C. Fishburn Source: Statistical Science, Vol. 1, No. 3 (Aug., 1986), pp. 335-345 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2245466 . Accessed: 21/08/2013 08:28 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
[email protected].
.
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to Statistical Science.
http://www.jstor.org
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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.
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
335
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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]
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
=-not
(Bm> Am).
P. C. FISHBURN
340
monotonicity, This entails asymmetry, nonnegativity, and acyclicity,but not nontriviality,transitivity,or additivity.However,it does forbidadditivityreversals like E >- 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
- 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
XBm), Aj >-* Bj
Am) =o (B1,
forall j
* 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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
AXIOMS OF SUBJECTIVE PROBABILITY
341
of monotonicity, and a generalization nontriviality, strongadditivity.
PartitionAxiom3. For all A and B in sV,ifA >- B then S can be partitionedinto a finitenumberof
4. INFINITEALGEBRAS
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:
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.
events C1, *- *, Cmsuch that A >- B U Cj forall j.
[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).
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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) conditionsfora model givenecessaryand sufficient 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 ifA I B -o C I D thenthereis a C' in v such that C nlD C C ' and A IB ~o C' ID.
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
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.
This content downloaded from 137.222.114.241 on Wed, 21 Aug 2013 08:28:52 AM All use subject to JSTOR Terms and Conditions
