We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Ríos Insua (Theory Decis 33:83–100, 1992) and Alon and Lehrer (J Econ Theory 151:476–492, 2014) is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott (J Math Psychol 1:233–247, 1964), relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature

Harrison-Trainor, M

Holliday, WH

Icard, TF

Harrison-Trainor, Matthew

Holliday, Wesley H

Icard, Thomas F

eScholarship - University of California

UC BerkeleyUC Berkeley Previously Published WorksTitleA note on cancellation axioms for comparative probabilityPermalinkhttps://escholarship.org/uc/item/2sf7x271JournalTheory and Decision, 80(1)ISSN0040-5833AuthorsHarrison-Trainor, MatthewHolliday, Wesley HIcard, Thomas FPublication Date2016DOI10.1007/s11238-015-9491-2 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaA Note on Cancellation Axiomsfor Comparative ProbabilityMatthew Harrison-Trainor ·Wesley H. Holliday ·Thomas F. Icard, IIIAbstract We prove that the generalized cancellation axiom for incompletecomparative probability relations introduced by Ŕıos Insua (1992) and Alonand Lehrer (2014) is stronger than the standard cancellation axiom for com-plete comparative probability relations introduced by Scott (1964), relative totheir other axioms for comparative probability in both the finite and infinitecases. This result has been suggested but not proved in the previous literature.Keywords cancellation axioms · comparative probability · qualitativeprobability · incomplete relationsLet % be a binary relation on an algebra ⌃ of events over a nonempty statespace S. The intended interpretation of E % F is that event E is at least aslikely as event F . Say that a pair of sequences hE1, . . . , Eki and hF1, . . . , Fkiof events is balanced i↵ for all s 2 S, the cardinality of {i | s 2 Ei} is equal tothe cardinality of {i | s 2 Fi}. Consider the following axioms on %:Reflexivity – for all E 2 ⌃, E % E.Completeness – for all E,F 2 ⌃, E % F or F % E.Positivity – for all E 2 ⌃, E % ;.Non-triviality – it is not the case that ; % S.Preprint of November 2014. Forthcoming in Theory and Decision.Matthew Harrison-TrainorGroup in Logic and the Methodology of Science, University of California, BerkeleyE-mail: matthew.h-t@berkeley.eduWesley H. HollidayDepartment of Philosophy and Group in Logic and the Methodology of Science, Universityof California, BerkeleyE-mail: wesholliday@berkeley.eduThomas F. Icard, IIIDepartment of Philosophy and Symbolic Systems Program, Stanford UniversityE-mail: icard@stanford.edu2 Harrison-Trainor, Holliday, and IcardFinite Cancellation (FC) – for all balanced pairs of sequenceshE1, . . . , En, Xi and hF1, . . . , Fn, Y i of events from ⌃, if Ei % Fi for all i,then Y % X.Generalized Finite Cancellation (GFC) – for all balanced pairs ofsequenceshE1, . . . , En, X, . . . , X| {z }r timesi and hF1, . . . , Fn, Y, . . . , Y| {z }r timesiof events from ⌃, if Ei% Fifor all i, then Y % X.FC was introduced by Scott (1964) as a reformulation of axioms from Kraftet al. (1959). For a finite state space, Scott showed that FC, Completeness,Positivity, and Non-triviality are necessary and su cient for the existenceof an additive probability measure µ on ⌃ such that for all E,F 2 ⌃:E % F i↵ µ(E)   µ(F ).GFC was introduced by Ŕıos Insua (1992) and again by Alon and Lehrer(2014).1 For a finite state space, both papers showed that GFC, Reflexivity,Positivity, and Non-triviality are necessary and su cient for the existenceof a nonempty set P of additive probability measures on ⌃ such that for allE,F 2 ⌃:E % F i↵ for all µ 2 P, µ(E)   µ(F ).Clearly GFC implies FC, and assuming Completeness (X % Y or Y %X), FC implies GFC. In the papers by Ŕıos Insua (p. 89) and Alon andLehrer (p. 481), it is suggested but not proved that GFC is stronger thanFC for incomplete relations, i.e., relative to Reflexivity, Positivity, andNon-triviality.2 The following establishes the correctness of their claim.Proposition 1 Let S = {a, b, c, d} and define % such that for all E,F ✓ S,E % F i↵ one of the following holds:(i) E ◆ F ;(ii) {a, c} ✓ E and F ✓ {b, d};(iii) {a, d} ✓ E and F ✓ {b, c}.Then % satisfies Reflexivity, Positivity, Non-triviality, and FC, but notGFC.Proof Reflexivity, Positivity, and Non-triviality are obvious. To see thatGFC fails, note that h{a, c}, {a, d}, {b}, {b}i and h{b, d}, {b, c}, {a}, {a}i arebalanced, so with {a, c} % {b, d} from (ii) and {a, d} % {b, c} from (iii), GFCrequires {a} % {b}, which is not permitted by (i)-(iii).1 We adopt Alon and Lehrer’s name for the GFC axiom and Ŕıos Insua’s equivalentformulation of the axiom.2 In correspondence with the authors of both papers, we verified that proving the claimwas an open problem.A Note on Cancellation Axioms for Comparative Probability 3To see that FC holds, assume that hE1, . . . , En, Xi and hF1, . . . , Fn, Y i arebalanced and Ei% Fifor all i. By (i)-(iii), if a 2 Fi, then a 2 Ei. Thus, bythe balancing assumption, there is at most one j such that a 2 Ejand a 62 Fj(in which case a 2 Y ). Suppose there is no such j. Then by (i)-(iii) and theassumed relationships, Ei◆ Fifor all i, which with the balancing assumptionimplies Y ◆ X and hence Y % X by (i). Suppose there is one such j, sayj = 1. Then by (i)-(iii) and the assumed relationships, Ei◆ Fifor all i > 1. IfE1 ◆ F1, then by the argument above, we have Y % X. Otherwise, E1 6◆ F1,and so the reason for E1 % F1 is (ii) or (iii). If it is (ii), then since Ei ◆ Fifor all i > 1, the balancing assumption implies {a, c} ✓ E1   F1 ✓ Y andX ✓ F1   E1 ✓ {b, d}. Thus, by (ii), Y % X. The case for (iii) is similar. ⇤Alon and Lehrer (2014) also considered the case where the state space Smay be infinite. The representation theorem in this case requires one addi-tional axiom, analogous to Savage’s (1954, §3.3) axiom P6, but for incompleterelations. For A,B 2 ⌃, let A    B i↵ there is a finite partition {G1, . . . , Gr}of S such that A Gi% B [Gjfor all i and j. The additional axiom is:Non-atomicity – if A 6% B then there is a finite partition of B,{B1, . . . , Bm}, such that for all i, Bi    ? and A 6% B  Bi.Alon and Lehrer (2014) showed thatGFC,Reflexivity,Positivity,Non-triviality, and Non-atomicity are necessary and su cient for the existenceof a nonempty, compact,3 and uniformly strongly continuous4 set P of finitelyadditive probability measures on ⌃ such that for all E,F 2 ⌃:E % F i↵ for all µ 2 P, µ(E)   µ(F ).In the case where ⌃ is a  -algebra, Alon and Lehrer (2014) showed that wemay replace ‘finitely additive’ with ‘countably additive’ in the previous resultif we add the axiom:Monotone Continuity – for any sequence E1 ◆ E2 ◆ . . . withTnEn=; and any F    ;, there is some n0 such that for all n > n0, F % En.In this setting, we again show that GFC is stronger than FC, now relativeto Reflexivity, Positivity, Non-triviality, Non-atomicity, and Mono-tone Continuity.To prepare for Proposition 2, let S = {a, b, c, d}⇥ [0, 1]. Given E ✓ S andx 2 {a, b, c, d}, let Exbe the fiber {y 2 [0, 1] | hx, yi 2 E} over x. Let ⌃ bethe  -algebra consisting of sets E ✓ S where Exis Lebesgue measurable foreach x 2 {a, b, c, d}. Let µx(E) = µ(Ex), where µ is the Lebesgue measure on3 By compact, Alon and Lehrer mean that P is weak⇤ compact, i.e., compact in the spaceof pointwise convergence.4 A set P of measures is uniformly strongly continuous i↵ both of the following hold:1. for all µ, µ0 2 P and B 2 ⌃, µ(B) > 0 i↵ µ0(B) > 0;2. for all ✏ > 0, there is a finite partition {G1, . . . , Gr} of S such that for all j, µ(Gj) < ✏for all µ 2 P.4 Harrison-Trainor, Holliday, and Icardthe interval [0, 1]. A weight function w on {a, b, c, d} is an assignment, to eachx 2 {a, b, c, d}, of a value wx2 [1, 2]. Define % such that for all E,F 2 ⌃,E % F i↵ one of the following holds:(i) for all weight functions w,Xxwxµx(E)  Xxwxµx(F ),where the sum is taken over x 2 {a, b, c, d};(ii) for the weight function w(ii) which gives a and c weight 2 and b and dweight 1, Xxw(ii)xµx(E) Xxw(ii)xµx(F )   2;(iii) for the weight function w(iii) which gives a and d weight 2 and b and cweight 1, Xxw(iii)xµx(E) Xxw(iii)xµx(F )   2.Note that in (i), it su ces to take wx= 1 when µx(E)   µx(F ) and wx= 2when µx(F ) > µx(E). One can view % as just defined as a modification of therelation from Proposition 1 by assigning measures in [0, 1] to each of a, b, c, andd. For example, from (i), (ii), and (iii) above we can derive three particularcases (i0), (ii0), and (iii0) below under which E % F . These correspond to (i),(ii), and (iii) from Proposition 1:(i0) for all x 2 {a, b, c, d}, µx(E)   µx(F );(ii0) µa(E) = µc(E) = 1 and µa(F ) = µc(F ) = 0;(iii0) µa(E) = µd(E) = 1 and µa(F ) = µd(F ) = 0.The relation %0 given by (i0), (ii0), and (iii0) does not satisfy Non-atomicity.The relation % adds to %0 the ability to “exchange” measure from some x 2{a, b, c, d} to some other y, but at an exchange rate of two to one. This exchangeis required to getNon-atomicity. So, for example, by (i) we have {a}⇥[0, 1] %{b}⇥ [0, 12 ] because we can exchange one measure of a for half a measure of b.This relation % separates FC and GFC relative to the other axioms.Proposition 2 The relation % on ⌃ satisfies Reflexivity, Positivity, Non-triviality,Non-atomicity,Monotone Continuity, and FC, but notGFC.Proof Positivity and Non-triviality are obvious. Reflexivity follows from(i). To see thatMonotone Continuity holds, consider a sequence E1 ◆ E2 ◆. . . withTnEn= ; and any F    ;. Recall that F    ; means that thereis a finite partition {G1, . . . , Gr} of S such that F   Gi % Gj for all i and j.Since the partition is finite, we can pick Gksuch thatPxµx(Gk) > 0. Nowgiven F   Gk% Gk, this holds for reason (i), (ii), or (iii). In each case, itfollows thatPxµx(F ) > 0. Then since limn!1Pxµx(En) = 0, there is ann0 such thatPxwxµx(F ) >Pxwxµx(En0) for any set of weights in [1, 2], soF % En0 by (i), which clearly implies that for all n > n0, F % En.A Note on Cancellation Axioms for Comparative Probability 5To see that GFC fails, note that the example from Proposition 1 stillworks, replacing {a} by {a}⇥ [0, 1], {b} by {b}⇥ [0, 1], and so on.That % satisfies Non-atomicity and FC is the content of the next twolemmas which complete the proof of the proposition.Lemma 1 The relation % on ⌃ satisfies Non-atomicity.Proof Suppose that E 6% F . By (i), there is a weight function w such thatXxwxµx(E) <Xxwxµx(F ).Let ✏1 > 0 be such thatXxwxµx(F ) Xxwxµx(E) > ✏1.By (ii), we haveXxw(ii)xµx(E) Xxw(ii)xµx(F ) < 2.(Recall that w(ii)xand w(iii)xare the weight functions defined in (ii) and (iii)respectively.) Let ✏2 > 0 be such thatXxw(ii)xµx(E) Xxw(ii)xµx(F ) < 2  ✏2.Similarly, by (iii),Xxw(iii)xµx(E) Xxw(iii)xµx(F ) < 2.Let ✏3 > 0 be such thatXxw(iii)xµx(E) Xxw(iii)xµx(F ) < 2  ✏3.Now since E 6% F , for some x, µx(F ) > 0, so we can partition F intofinitely many sets Fisuch that0 < 2Xxµx(Fi) < ✏jfor each j = 1, 2, 3. Fix Fi; we must show that Fi   ? and that E 6% F  Fi.We start by showing that Fi   ?. Partition S into finitely many sets{G1, . . . , Gr} such that5Xxµx(Gj) <Xxµx(Fi).6 Harrison-Trainor, Holliday, and Icardfor each j = 1, . . . , r. Thus for any weight function v, and j1, j2,Xxvxµx(Fi Gj1)  Xxµx(Fi)  2Xxµx(Gj1)>35Xxµx(Fi)> 2Xxµx(Gj2) Xxvxµx(Gj2)and so Fi Gj1 % Gj2 by (i). Thus Fi    ?.Now we show that E 6% F   Fi. Let w be the weight from above for whichwe chose ✏1. ThenXxwxµx(F   Fi) Xxwxµx(E) Xxwxµx(F ) Xxwxµx(E) Xxwxµx(Fi)> ✏1  Xxwxµx(Fi)  ✏1   2Xxµx(Fi)> 0and hence (i) does not imply E % F   Fi. Also,Xxw(ii)xµx(E) Xxw(ii)xµx(F   Fi)Xxw(ii)xµx(E) Xxw(ii)xµx(F ) + 2Xxµx(Fi)< 2  ✏2 + ✏2= 2and hence (ii) does not imply E % F   Fi. The case of (iii) is similar. ThusE 6% F   Fi. ⇤Lemma 2 The relation % on ⌃ satisfies FC.Proof To see that FC holds, assume that hE1, . . . , En, Xi and hF1, . . . , Fn, Y iare balanced and Ei% Fifor all i. By the balancing assumption, for allx 2 {a, b, c, d}, we haveµx(X) +nXi=1µx(Ei) = µx(Y ) +nXi=1µx(Fi).A Note on Cancellation Axioms for Comparative Probability 7First, suppose that for all i, Ei% Fiby (i). Let w be a weight function.Then from the balancing assumption,Xxwxµx(Y ) Xxwxµx(X) =nXi=1Xxwx(µx(Ei)  µx(Fi)).Each summandPxwx(µx(Ei)  µx(Fi)) is non-negative, soXxwxµx(Y )  Xxwxµx(X)and hence Y % X.Now suppose that for some j, say j = 1, (i) is not satisfied. Then E1 % F1by either (ii) or (iii). Suppose that it is by (ii) (the case of (iii) is similar).Then Xxw(ii)xµx(E1) Xxw(ii)xµx(F1)   2.Consider the sumXxw(ii)xµx(Y ) Xxw(ii)xµx(X) =nXi=1Xxw(ii)x(µx(Ei)  µx(Fi)).For i = 1,Pxw(ii)x(µx(Ei)  µx(Fi))   2. For each other i, we will show thatPxw(ii)x(µx(Ei)   µx(Fi))   0. We have three cases. If Ei% Fiby (i), thenPxw(ii)x(µx(Ei)   µx(Fi))   0. If Ei% Fiby (ii), thenPxw(ii)x(µx(Ei)  µx(Fi))   2. Finally, suppose that Ei% Fiby (iii), so we have thatXxw(iii)xµx(Ei) Xxw(iii)xµx(Fi)   2.ThenXxw(ii)xµx(Ei) Xxw(ii)xµx(Fi) =Xxw(iii)xµx(Ei) Xxw(iii)xµx(Fi)+ (µc(Ei)  µc(Fi))  (µd(Ei)  µd(Fi)).Since the measures µcand µdtake values in [0, 1], we getXxw(ii)xµx(Ei) Xxw(ii)xµx(Fi)   0.This completes the third case. So we have shown that for this particular weightfunction w(ii), we haveXxw(ii)xµx(Y ) Xxw(ii)xµx(X)   2.Hence Y % X by (ii). This completes the proof of the lemma. ⇤8 Harrison-Trainor, Holliday, and IcardThus, in both the finite and infinite cases, GFC is stronger than FC rela-tive to the other axioms for comparative probability without Completeness.As Fine (1973) remarks about Completeness, “The requirement that allevents be comparable is not insignificant and has been denied by many carefulstudents of probability including Keynes and Koopman” (p. 17). In light ofPropositions 1 and 2, those sympathetic to the denial of Completeness havereason to expand the study of cancellation axioms for comparative probabilitybeyond the standard focus on FC to include a study of GFC as well.Acknowledgements The first author was partially supported by the Berkeley Fellowshipand an NSERC PGS-D award. The authors wish to thank an anonymous referee for helpfulcomments.ReferencesAlon S, Lehrer E (2014) Subjective multi-prior probability: A representation of a partiallikelihood relation. Journal of Economic Theory 151:476–492Fine TL (1973) Theories of Probability. Academic Press, New YorkKraft C, Pratt J, Seidenberg A (1959) Intuitive probability on finite sets. The Annals ofMathematical Statistics 30(2):408–419Ŕıos Insua D (1992) On the foundations of decision making under partial information.Theory and Decision 33(1):83–100Savage LJ (1954) The Foundations of Statistics. John Wiley and SonsScott D (1964) Measurement structures and linear inequalities. Journal of MathematicalPsychology 1(2):233–247

A note on cancellation axioms for comparative probability

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