Decision Making for Inconsistent Expert Judgments Using Negative Probabilities

Decision Making for Inconsisten t Exp ert Judgmen ts Using Negativ e Probabilities J. A cacio de Barros Liberal Studies Program, San F ra ncisco State Un iv ersit y , 1600 Hollow a y A ve., San F rancisco, CA 94132 Abstract. In this paper w e provide a simple random-va riable ex am- ple of inconsistent informatio n, and analyze it using three different ap- proac hes: Ba yesian, quantum-like, and negativ e probabilities. W e t hen sho w that, at least for t h is particular example, b oth the Bay esian and the quantum-lik e approaches ha ve less normative p o w er than the nega- tive probabilitie s one. 1 In tro duction In r ecen t years the quantum-mec hanical for malism (mainly fro m non-relativistic quantum m echanics) has be en used to mo del economic and decision- ma king pro- cesses (see [1,2] a nd references therein). The success of such models may origina te from several related issues . First, the quantum formalism leads to a prop ositional structure that do es not confor m to cla s sical log ic [3]. Sec o nd, the probabilities of quan tum obser v a bles do not sa tisfy Kolmogorov’s axioms [4]. Third, quantum mechanics describ es exp eriment al outcomes that are highly cont extual [5,6,7,8,9]. Suc h issues are connected beca use the log ic of quant um mechanics, re presen ted b y a quantum la ttice structure [3], lea ds to upp er pro babilit y distr ibutions and th us to non-Kolmogor o vian mea sures [10,11,12], while contextualit y leads the nonexistence of a joint proba bilit y distribution [13,14]. Both from a foundationa l and from a pr actical p oint of view, it is importa n t to a sk which asp ects of quantum mechanics are actually needed for so cial-science mo dels. F or instance, the Hilbert space formalism leads to non-sta ndard logic and pro babilities, but the conv erse is not true: o ne cannot der iv e the Hilbert space for malism s olely fr o m weak er axioms o f probabilities or fro m quantu m la t- tices. F urthermore, the quantum for malism yields non- tr ivial res ults such a s the impo s sibilit y of sup erluminal signaling with entangled states [1 5]. These types of results ar e not necessary for a theory of so cial phenomena [16], and w e s ho uld ask what are the minimalistic ma thematical structures suggested by quantum mechanics that repro duce the relev an t features o f quantum-lik e behavior. In a previous article, we used reasona ble neurophysiological assumptions to created a neural-oscillato r mo del of behavioral Stimulus-Respo ns e theory [17]. W e then sho wed how to use such mo del to r eproduce quantum-lik e behavior [18]. Finally , in a subsequent article, w e remar k ed that the same neural-os c illa tor mo del could b e used to repres e nt a set of obse r v a bles that co uld not corr espond 2. INCONSISTENT INF ORMA TION to quantum mechanical o bserv ables [19], in a sense that we later on fo r malize in Section 3. These results suggest that one of the main quant um featur e s r elev ant to s ocial modeling is contextu ality , represe n ted by a non-Kolmogo ro vian proba- bilit y measure, and that impos ing a quantum formalism ma y be to o r estrictiv e. This non- K olmogo ro vian characteristic would come when t wo contexts providing incompatible infor mation ab out observ a ble quant ities w ere present. Here we fo cus on the incompatibility o f c o n texts as the sour ce o f a violation of standard probability theory . W e then ask the following question: what formalisms are norma tiv e with resp ect to such incompatibility? This question comes from the fact that, in its orig in, pr obabilit y was devised as a normative theory , a nd not descriptive. F or instance, Richard Jeffrey [2 0] explains that “the term ’proba ble’ (Latin pr ob able ) meant app r ovable , and was applied in that sense, univocally , to opinion a nd to action. A probable actio n o r opinion was one such as sensible peo ple would undertake or hold, in the cir cumstances.” Thus, it sho uld come as no sur prise that humans actually violate the rules of pro babilit y , as shown in many psychology exp erimen ts. How ever, if a per son is to b e considered “rational,” according to Bo ole, he/she should follow the rules of probability theory . Since inco nsisten t information, as ab o ve ment ioned, violates the theory of probability , how do we provide a normative theory o f rational decisio n-making? There are many a pproaches, suc h as Bay esian mo dels o r the Demps ter -Shaffer theory , but here we focus on tw o non-sta ndard ones: quantum-lik e and nega tiv e probability mo dels. W e star t first by pr esen ting a s imple case where exp ert judgmen ts lea d to inco nsistencies. Then, we approach this problem fir st with a standard Bayesian pro babilistic method, follow ed b y a quantum mo del. Finally , we use negative probability distributions as a third alternative. W e then compare the different o utcomes of ea c h appro ac h, and show tha t the use o f negative probabilities seems to provide the most normative p ower among the three. W e end this paper with some commen ts. 2 Inconsisten t Information As mentioned, the use o f the quant um formalism in the so cial sciences originates from the observ ation that Kolmogorov’s axioms are vio la ted in many situations [1,2]. Such violations in decisio n-making seem to indica te a departure from a rational view, and in particular to thoug h-proc e s ses that may in volv e irra tional or contradictory reasoning, as is the case in non-monotonic reasoning . Thus, when dealing with quantum-lik e so cial phenomena, we are frequently dealing with some t yp e of inconsistent infor mation, usually arrived at as the end result of some non-classica l (or incor rect, to some) reaso ning. In this section w e examine the case where incons is tency is present from the b eginning. Though in everyda y life inconsistent information a bounds, standard class ical logic has difficulties dealing with it. F or instance, it is a well know fa c t that if we have a contradiction, i.e. A & ( ¬ A ) , then the log ic b ecomes trivial, in the sense that an y formula in such logic is a theo r em. T o deal with s uch difficult y , logicians hav e pr opos e d mo dified logical systems (e.g. para consisten t log ic s [21 ]). 2 2. INCONSISTENT INF ORMA TION Here, we will discuss how to deal with inconsistencies not fro m a logical p oin t of view, but instead from a probabilistic one. Inconsistencies of exp ert judgments are o ften represented in the proba bilit y literature by measures corr esponding to the ex p erts’ sub jective beliefs [22]. It is frequently argued that this sub jective nature is necess ary , as each exp ert makes statement s abo ut outcomes that are, in principle, av ailable to all exp erts, and disagreements co me not fro m sampling a certain pr obabilit y space, but fr o m per sonal b eliefs. F or example, let us assume that t wo exp erts, Alice and Bo b, are examining whether to recommend the pur chase of sto cks in company X , and each gives different recommendations . Suc h differences do not emerge fro m an ob jective data (i.e. the actua l future prices of X ), but fro m each exper t’s in terpretatio ns of current market conditions and of company X . In so me cases the inconsis tencies are evident, as when, say , Alice Alice recommends buy , and Bob recommends sell; in this case the decision maker would hav e to reco ncile the discrepancies. The ab ov e example provides a simple case. A more subtle one is when the exp erts hav e incons isten t b eliefs that seem to b e co nsisten t. F or ex a mple, each exp ert, with a limited acc e s s to information, may for m, based on different con- texts, lo cally consistent b eliefs without directly cont radicting o ther exp erts. But when we take the totality of the information provided by a ll of them a nd try to arr iv e at p ossible inferences, w e re a c h contradictions. Here w e want to create a simple ra ndom-v ariable model that incor pora tes exp e r t judgments that ar e lo cally co nsisten t but g lobally inconsistent. This mo del, inspired b y quantum ent anglement, w ill b e used to show the main features of negative pro babilities as applied to decision making. Let us start with three ± 1 -v alued random v aria bles, X , Y , a nd Z , with zero exp ectation. If suc h ra ndom v ariables have co r relations that are to o str o ng then there is no joint pro babilit y distribution [13]. T o see this, imagine the extreme case where the co r relations betw een the random v a r iables are E ( XY ) = E ( YZ ) = E ( XZ ) = − 1 . Imagine that in a given trial w e draw X = 1 . F r o m E ( XY ) = − 1 it follows that Y = − 1 , and from E ( YZ ) = − 1 that Z = 1 . But this is in contradiction with E ( XZ ) = − 1 , which r e quires Z = − 1 . Of course, the problem is not that there is a mathematical inco ns is tency , but that it is not po ssible to find a proba bilistic sample space fo r which the v ar iables X , Y , and Z ha ve such strong corr elations. Another way to think about this is that the the X mea sured together with Y is not the same one as the X measured with Z : v alues of X depend on its context. The a bov e example pos its a deterministic relationship b et ween all rando m v aria bles , but the inconsistencies pe r sist ev en when w eaker cor r elations exist. In fact, Supp es and Zanotti [13] pr oved that a joint probability dis tr ibution for X , Y , and Z exists if and only if − 1 ≤ E ( XY ) + E ( YZ ) + E ( XZ ) ≤ 1 + 2 min { E ( XY ) , E ( YZ ) , E ( XZ ) } . (1) The ab ove case violates inequality (1). 3 3. QUANTUM APPR OACH Let’s us now consider the example we w ant to analyze in deta il . Imagine X , Y , a nd Z as corresp onding to future outcomes in a company’s sto c ks. F or instance, X = 1 corresp onds to an increas e of the s to ck v a lue o f company X in the following day , while X = − 1 a decrea se, a nd so on. Three exp erts, Alice ( A ), Bob ( B ), a nd Carlo s ( C ), hav e the follo wing b eliefs ab out those sto cks. Alice is an exper t o n co mpanies X and Y , but knows little or nothing ab out Z , so she only tells us wha t we don’t know: her expected co rrelation E A ( XY ) . Bob (Carlos), on the other hand, is only an exper t in companies X and Z ( Y and Z ), and he too only tells us ab out their co rrelations. Let us ta k e the cas e where E A ( XY ) = − 1 , (2) E B ( XZ ) = − 1 2 , ( 3) E C ( YZ ) = 0 , (4) where the subscripts refer to each exp erts. F or suc h case, the sum of the corr e la - tions is − 1 1 2 , a nd according to (1 ) no joint proba bilit y distribution exists. Since there is no joint, how can a rational decision-maker decide wha t to do w hen faced with the question of how to b et in the market? In particula r, ho w can she get information ab out the join t probability , a nd in particular the unknown triple moment E ( XYZ ) ? In the next sections w e will show how we can try to answer these questions using three p ossible a pproaches: qua n tum, Bay esian, and signed probabilities. 3 Quan tum Approach W e star t with a comment ab out the quantum-lik e na ture of cor relations (2)-(4). The r andom v ar iables X , Y , and Z with cor relations (2)-(4) cannot b e r epre- sented b y a quantum state in a Hilb ert space fo r the observ ables co rresp onding to X , Y , and Z . This claim ca n be expressed in the form of a simple propo sition. Prop osition 1. L et ˆ X , ˆ Y , and ˆ Z b e thr e e observables in a Hilb ert sp ac e H with eigenvalues ± 1 , let them p airwise c ommute, and let the ± 1 - value d r andom variable X , Y , and Z r epr esent the outc omes of p ossible exp eriments p erforme d on a qu antum system | ψ i ∈ H . Then, t her e exists a joint pr ob ability distribution c onsistent with al l the p ossible outc omes of X , Y , and Z . Pr o of. Because ˆ X , ˆ Y , and ˆ Z ar e observ ables and they pairwise commute, it follows that their co m binations, ˆ X ˆ Y , ˆ Y ˆ Z , ˆ X ˆ Z , and ˆ X ˆ Y ˆ Z ar e also o bserv ables, and they co mm ute with each other. F or instance,  ˆ X ˆ Y ˆ Z  † = ˆ Z † ˆ Y † ˆ X † = ˆ X ˆ Y ˆ Z . F urthermore, [ ˆ X ˆ Y ˆ Z , ˆ X ] = [ ˆ X ˆ Y ˆ Z , ˆ Y ] = · · · = [ ˆ X ˆ Y ˆ Z , ˆ X ˆ Z ] = 0 . 4 4. BA YESIAN APPR OA CH Therefore, quantum mechanics implies that all three observ ables ˆ X , ˆ Y , and ˆ Z can b e sim ultaneously measured. Since this is true, for the same state | ψ i w e can create a full data table with all three v alues of X , Y , and Z (i.e., no mis s ing v alues), which implies the existence o f a joint. So, how would a quantum-lik e mo del of correlations (2)–(4) be like? The ab ov e result depe nds on the use o f the same quantum state | ψ i throughout the many runs of the exp eriment, and to cir cum ven t it we would need to u s e different states for the system. In other words, if we wan t to use a qua n tum for malism to describ e the corre la tions (2)-(4), a | ψ i would have to b e selected for ea ch run s uc h that a different state would be used when w e measure ˆ X ˆ Y , e.g. | ψ i xy , than when we measure ˆ X ˆ Z , e.g. | ψ i xz . Then, the quant um description c o uld be accomplished b y the state | ψ i = c A | A i ⊗ | ψ i xy + c B | B i ⊗ | ψ i xz + c C | C i ⊗ | ψ i y z . This state would mo del the corre lations the following wa y . When Alice mak es her choice, she uses a pr o jector in to her “state of knowledge” ˆ P A = | A ih A | , and gets the co rrelation E A ( XY ) , and similarly for Bob a nd Carlos . In the ab o ve example, all correla tio ns a nd exp ectations are given, and the only unknown is the triple moment E ( XYZ ) . F urthermo r e, since we do not hav e a joint probability distribution, we cannot compute the range of v alues fo r such moment bas e d o n the exp ert’s beliefs. But the question still remains as to what would be o ur best b et g iv en what we k no w, i.e., what is our best guess for E ( XYZ ) . The quan tum mec hanical appro a c h do es not addres s this question, as it is not clear how to get it fro m the fo r malism giv en that any sup erp osition of the states preferred by Alice, Bob, and Car los ar e acceptable (i.e., we can choo se any v alues of c A , c B , and c C ). 4 Ba y esian Approac h Here we fo cus a gain on the unknown triple moment . As we ment ioned befor e, there a re many differen t ways to appro ac h this problem, s uc h a s par aconsis- ten t logics, conse ns us r eac hing, or information revision to restore consistency . Common to all those appr oach es is the complexity of how to r esolve the incon- sistencies, often with the aid of ad ho c assumptions [22]. Here we show how a Bay esian approa c h would deal with the iss ue [2 3,24]. In the Bayesian a pproach, a decision maker, Dea nna ( D ), needs to access what is the joint pro babilit y distribution from a set o f inconsistent ex p ecta- tions. T o set the notation, let us first lo ok at the case when there is only one exp ert. Let P A ( x ) = P A ( X = x | δ A ) b e the proba bilit y assigned to even t x b y Alice conditioned on Alice’s knowledge δ A , a nd let P D ( x ) = P D ( X = x | δ D ) be Deanna’s prior distribution, a ls o conditioned on her knowledge δ D . F urthermore, let P A = P A ( x ) be a contin uous r a ndom v a riable, P A ∈ [0 , 1 ] , such that its out- come is P A ( x ) . The idea behind P A is that consulting an exp ert is similar to conducting an exp eriment wher e we sample the exp erts opinion b y observing a 5 4. BA YESIAN APPR OA CH distribution function, and ther efore we can talk ab out the proba bilit y tha t an exp ert will give an a nsw er for a s pecific sample p oint. Then, for this c ase, Bay es’s theorem can b e written as P ′ D ( x | P A = P A ( x )) = P D ( P A = P A ( x )) P D ( x ) P D ( P A = P A ( x )) , where P ′ D ( x | P A = P A ( x )) is Deanna ’s pos ter ior distribution r evised to take in to account the exp e r t’s opinion. As is the c ase with Bayes’s theor em, the difficulty lies on determining the likelihoo d function P D ( P A ) , a s well as the prio r. This likelihoo d function is, in a certain sense, Deanna’s mo del of Alice, as it is what Deanna believes are the likelihoo ds of each of Alice’s beliefs. In other words, she should ha ve a mo del of the exp erts. Such mo del of experts is akin to giving e a c h exp ert a certain measure of credibilit y , since an exp ert whose mo del do esn’t fit Deanna’s w ould be ass igned low er probability than an exper t whose mo del fits. The extension for our case of three ex perts and three random v a riables is cum b ersome but straightforw ard. F or Alice, Bo b, and Carlos, Deanna needs to hav e a mo del for each one of them, bas ed on her pr io r kno wledge a bout X , Y , and Z , as well as Alice, Bob, and Car los. F o llo wing Morris [23], w e construct a set E consisting of our three ex p erts join t priors: E = { P A ( x, y ) , P B ( y , z ) , P C ( x, z ) } . Deanna’s is no w faced with the problem o f determining the p osterior P ′ D ( x | E ) , using Ba yes’s theorem, given her new knowledge of the exp ert’s pr io rs. In a Bayesian appr oach , the decision maker should s tart with a pr ior b elief on the stocks of X , Y , and Z , based o n her knowledge. There is no recip e for choosing a prior, but let us start with the simple case where Dea nna’s lack o f knowledge ab out X , Y , a nd Z means she starts with the initial be lief that all combinations of v alues for X , Y , and Z ar e equipr o bable. Let us use the following notation for the probabilities of each atom: p xy z = P ( X = +1 , Y = +1 , Z = + 1) , p xy z = P ( X = +1 , Y = +1 , Z = − 1) , p x y z = P ( X = − 1 , Y = +1 , Z = − 1) , and so on. Then Deanna’s prior probabilities for the atoms are p D xy z = p D xy z = · · · = p D xy z = 1 16 , where the sup erscript D refers to Deanna. When rea soning abo ut the likeliho od function, Deanna asks w hat w ould b e the probable distribution o f re s ponses o f Alice if somehow she (Deanna) c o uld see the future (say , by consulting an Or acle) and find out that E ( X Y ) = − 1 . F or s uch case, it would b e r easonable for Alice to think it mo re probable to hav e , say , xy than xy , s ince she was consulted as an exp e r t. So, in terms of the correla tio n ǫ A , Deanna co uld as sign the following lik eliho od function: P D ( ǫ A | xy ) = P D ( ǫ A | xy ) = 1 4 (1 − ǫ A ) 2 , (5) P D ( ǫ A | xy ) = P D ( ǫ A | xy ) = 1 − 1 4 (1 − ǫ A ) 2 , (6) 6 4. BA YESIAN APPR OA CH where the min us sign r epresen ts the negative, i.e. p A xy · = p xy · = 1 4 (1 + ǫ A ) and p xy · = p xy · = 1 4 (1 − ǫ A ) . Notice that the c hoice of lik eliho o d function is a rbi- trary . Deanna’s p osterior, once s he knows that Alice though t the co rrelation to b e zero (cf. (2)), constitutes, a s we men tioned ab ove, an exp eriment. T o illustrate the computation, we find its v alue b elow, from Alice’s exp ectation E A ( XY ) = − 1 . F rom Bay es’s theorem p D | A xy z = k  1 − 1 4 (1 − ǫ A ) 2  1 8 = 1 4  1 − 1 4 (1 − ǫ A ) 2  = 3 16 , where the no rmalization constant k is given b y k − 1 =  1 − 1 4 (1 − ǫ A ) 2  1 8 +  1 4 (1 − ǫ A ) 2  1 8 +  1 4 (1 − ǫ A ) 2  1 8 +  1 − 1 4 (1 − ǫ A ) 2  1 8 +  1 4 (1 − ǫ A ) 2  1 8 +  1 4 (1 − ǫ A ) 2  1 8 +  1 − 1 4 (1 − ǫ A ) 2  1 8 +  1 − 1 4 (1 − ǫ A ) 2  1 8 , and we use the notation p D | A to explicitly indicate that this is Deanna’s p osterior probability informed b y Alice’s expectatio n. Similarly , we hav e p D | A xy z = p D | A xyz = p D | A x yz = p D | A x yz = 1 16 , and p D | A xy z = p D | A xy z = p D | A xyz = p D | A xyz = 3 16 . If we a pply Bay es’s theor em t wice more, to take in to account Bob’s a nd Carlo s ’s opinions given by correlations (3) and (4), using likelihoo d functions similar to the o ne ab o ve, w e compute the follo wing p osterior join t pro babilit y distribution, p D | A B C xy z = p D | A B C x yz = p D | A B C xyz = p D | A B C xy z = 0 , p D | A B C xyz = p D | A B C x y z = 7 68 , and p D | A B C xy z = p D | A B C xyz = 27 68 . Finally , fro m the joint , we can co mpute all the moments, including the triple moment , and obtain E ( XYZ ) = 0 . It is interesting to no tice that the triple moment from the po sterior is the same as the o ne from the prior. This is no co incidence. Because the revisio ns from Bay es’s theorem o nly mo dify the v alues of the corr elations, nothing is c hanged 7 5. NEGA TIVE PR OBABILITIES with r e spect to the triple moment. In fact, if we compute Deanna’s pos ter ior distribution for any v alues o f the cor relations ǫ A , ǫ B , and ǫ C , we obtain the same triple moment, a s it comes solely from Deanna’s prior distribution. Thus, the Bay esian appr oach , tho ug h pr o viding a pro per distribution for the atoms, do es not in any w ay provide further insights on the triple moment. 5 Negativ e Probabilities W e now w ant to see how we can use neg ativ e probabilities to approach the in- consistencies from Alice, Bob, and Carlo s. The fir st pers on to seriously consider using nega tiv e probabilities was Dira c in his Bakerian Lectures o n the physical in terpretatio n of relativistic qua n tum mec hanics [25]. Ever since, many physi- cists, most no tably F eynman [26], tried to use them, with limited success, to describ e ph ysical pro cesses (see [27] or [28] and reference s therein). The main problem with negative probabilities is its lack of a clear in terpretation, which limits its us e as a purely computational to ol. It is the goal of this sectio n to show that, at least in the co n text of a simple example, negative probabilities can provide useful normative information. Before we discuss the example, let us int ro duce negative proba bilities in a more for mal wa y 1 . Let us prop ose the following mo difications to Kolmogorov’s axioms. Definition 1. L et Ω b e a finite set , F an algebr a over Ω , p and p ′ r e al-value d functions, p : F → R , p ′ : F → R , and M − = P ω i ∈ Ω | p ( { ω i } ) | . Then ( Ω , F , p ) is a ne gative pr ob ability sp ac e if and only if: A. ∀ p ′ M − ≤ X ω i ∈ Ω | p ′ ( { ω i } ) | ! B. X ω i ∈ Ω p ( { ω i } ) = 1 C. p ( { ω i , ω j } ) = p ( { ω i } ) + p ( { ω j } ) , i 6 = j. R emark 1. If it is po ssible to define a prop er join t pro babilit y distribution, then M − = 0 , and A-C are equiv alent to Kolmogorov’s axioms. Going bac k to o ur example, we hav e the following equations for the atoms. p xy z + p x y z + p x y z + p xy z + p x yz + p x y z + p xy z + p xyz = 1 , (7) p xy z + p x y z + p x y z + p xy z − p x yz − p x y z − p xy z − p xyz = 0 , (8) p xy z + p x y z − p x y z + p xy z − p x yz + p x y z − p xy z − p xyz = 0 , (9) p xy z + p x y z + p x y z − p xy z − p x yz − p x y z + p xy z − p xyz = 0 , (10) p xy z − p xy z − p xy z + p xy z − p xyz − p xyz + p xyz + p xyz = 0 , (11) 1 W e limit our discussio n to finite spaces. 8 5. NEGA TIVE PROBABILITIES p xy z − p xy z + p xyz − p xy z − p xyz + p xyz − p xyz + p xyz = − 1 2 , (12) p xy z + p x y z − p x y z − p xy z + p x yz − p x y z − p xy z + p xyz = − 1 , (13) where (7) co mes from the fa c t that all proba bilities must sum to one, (8)-(10) from the zero exp ectations for X , Y , and Z , a nd (11)-(13) from the pairwise correla tio ns. Of co ur se, this problem is underdetermined, as we have seven equa- tions and eight unknowns (we don’t know the unobserved triple moment). A general solution to (7)-(10) is p xy z = − p x y z = − 1 8 − δ, (14) p xyz = p xy z = 3 16 , (15) p xy z = p xyz = 5 16 , (16) p x yz = − p xyz = − δ, (17) where δ is a real n umber. F rom (14)–(17) it follows that, for any δ , some probabilities are negative. First, we notice that we ca n use the jo int pr o ba- bilit y distribution to compute the e xpectation of the triple momen t, which is E ( XYZ ) = − 1 4 − 4 δ. Since − 1 ≤ E ( XYZ ) ≤ − 1 , it follows that − 1 1 4 ≤ δ ≤ 3 4 . Of course, δ is not determined b y the low er moments, as we should exp ect, b ut axiom A requires M − to b e minimized. So, to minimize M − , we fo cus only on the terms that co n tribute to it: the neg ativ e ones. T o do so, let us split the problem into several different sectio ns. Let us s ta rt with δ ≥ 0 , which gives M − δ ≥ 0 = − 1 8 − 2 δ, having a minimum o f − 1 8 when δ = 0 . F or − 1 / 8 ≤ δ < 0 , M − − 1 8 ≤ δ< 0 = δ − 1 8 + δ = − 1 8 , which is a constant v alue. Finally , fo r δ < − 1 / 8 , the mass for the negative terms is given by M − δ< − 1 8 = 1 8 − 2 δ. Therefore, negative mass is minimized when δ is in the following range − 1 8 ≤ δ ≤ 0 . Now, g o ing back to the triple co rrelation, we see that by imposing a minimization of the negative mass we res trict its v alues to the following ra nge: − 1 4 ≤ E ( XYZ ) ≤ 1 2 . But eq uations (7)-(13) and the fact that the random v ariables a re ± 1 -v alued allow any corr e lation b et w een − 1 and 1 , a nd we s ee that the minimization of the negative mass o ffers further constraints to a dec ision maker. Before we pro ceed, we need to addres s the meaning of nega tiv e probabil- ities, as well as the minimization of M − . W e saw from Remark 1 that when M − is zero we obtain a sta nda rd proba bilit y mea sure. Thus, the v alue of M − is a mea sure of how far p is fro m a prop er join t pr o babilit y distribution, and minimizing it is equiv alent to as king p to be a s close as pos sible to a prop er 9 6. CONCLUSIONS joint , while at the same time keeping the marginals. This p oint in itself sho uld be sufficient to sugg est s ome normative use to negative probabilities: a negative probability (with M − minimized) gives us the most ra tional b et we can mak e given inconsis tent information. But the q ues tion r e ma ins a s to the meaning of negative probabilities. T o give them mea ning, let us r e define the probabilities from p to p ∗ such that p ∗ ( { ω i } ) = 0 when p ( { ω i } ) ≤ 0 . It follows from this redefinition that P ω i ∈ Ω p ∗ ( { ω i } ) ≥ 1 . T his newly defined proba bilit y would no t v iolate Kol- mogorov’s nonnegativity a xiom, but instead would viola te B a b ov e. The p ∗ ’s corresp onds to de Finetti’s upp er probability measures , and axio m A ab ov e guarantees that such upp er is as close to a pro per distribution as possible. Th us, according to a sub jective interpretation, the negative probability atoms co rre- sp o nd to imp ossible even ts, and the po sitiv e o nes to a n upp er pr o babilit y mea - sure cons isten t with the ma rginals. Once again, the triple moment corresp onds to o ur b est bet. 6 Conclusions The q uan tum mechanical for malism has b een successful in the so cial sciences. How ever, one of the questio ns we rais ed elsewhere w as whether s ome minimal- ist versions o f the quantum formalism whic h do not include a full version of Hilber t spa c e s and observ ables co uld b e relev ant [19]. In this pap er we a dapted the example mo deled with neura l oscillator s in [1 9] to a different c a se where each random v a riable could b e interpreted a s outcomes of a market, and where the inconsistencies betw een the corr elations could be interpreted as inconsistencies betw een exp erts’ b eliefs. Such inco ns is tencies result in the impo ssibilit y to define a standar d pr obabilit y measure that allows a decision-maker to select an exp ec- tation for the triple moment . The computation of the triple moment from the inconsistent information was do ne in this paper using three differen t approa c hes: Bay esian, quantum-lik e, a nd nega tiv e probabilities. With the Bay esian a pproach, w e show e d tha t no t o nly do es it re ly on a mo del o f the exp erts (the lik eliho od function), but also that no new infor mation is gained from it, a s the triple moment from the prior is not changed by the application of B a yes’s r ules. Therefor e, the Bay esian appro ac h had nothing to say abo ut the triple momen t. Similar to the Bayesian, the quantum appr oac h also had nothing to say ab out the triple moment, as the arbitrarines s of c hoices for qua n tum sup erpo sitions (without any additional constra in ts) results in all v alues of triple moments b eing po ssible. In fa c t, the quantum appro ac h ab ov e c o uld b e similarly implement ed using a c o n textual theory . F or instance, Dzhafarov [29] prop oses the use of a n extended probability space where different random v ariables (say , X z and X y ) are used, and where we then as k how similar they ar e to ea c h o ther (for instance, what is the v alue of P ( X z 6 = X y ) ). How ever, as with the quantum case, the meaning given to P ( X = 1) in our example do es not fit w ith this model, a s it corresp onds to the e x pectation o f an increase in the sto ck v a lue of co mpan y X 10 6. CONCLUSIONS in the future, a nd the X that Alice is talking a bout is exactly the same o ne for Bob and Ca rlos, as it corresp onds to the increa se in the ob jective v alue (in the future) of a sto ck in the same company . F urthermore, as exp ected due to its similar features, this approach has the same pr oblem as the quantum one in terms of dealing with the triple mo men t, but it has the adv a n tage o f making it clearer what the problem is: the tr iple moment do es not exist beca use w e have nine random v a riables ins tea d of three, as we have three differen t con texts. The negative probability approach, on the other hand, led to a nontrivial constraint to the p o ssible v alues of the triple momen t. When used as a computa- tional tool, a jo in t pro babilit y distribution, and with it the triple moment, could be o btained. T og ether with the minimization of the nega tiv e ma ss M − , this join t leads to a non trivial r ange of pos sible v alues for the triple moment. Given the in terpretatio n o f negative probabilities with resp ect to upp ers, it follows that this range is o ur best guess as to wher e the v alues o f the triple moment should lie, given our incons isten t information. Th us, negative probabilities provide the decision maker with some nor mativ e information that is unav ailable in either the Bay es ian or the quantum-lik e approaches. A cknow le dgments. Many of the details ab out negative probabilities were devel- op ed in co lla bora tion with Patrick Suppes , Gar y Oas, and Claudio Carv alhaes on the context of a seminar held at Stanford Univ ersity in Spring 2 011. I am indebted to them as well as the seminar pa rticipan ts for fruitful discuss ions. I also like to thank T a nia Magdinier, Niklas Damiris, Newton da Costa, and the anonymous referees for co mmen ts and sugges tio ns. References 1. Khrenn ik o v, A.: Ubiquitous Quantum Structure. Springer V erlag, Heidelb erg (2010) 2. 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