Contextuality in Human Decision Making in the Presence of Direct Influences: A Comment on Basieva et al. (2019)
aa r X i v : . [ q - b i o . N C ] O c t Contextuality in Human Decision Making in the Presence of DirectInfluences: A Comment on Basieva et al. (2019).
James M Yearsley
Department of Psychology,City, University of London,EC1V 0HB, UK
Jonathan J Halliwell
Blackett Laboratory,Imperial College,London SW7 2AZ, UK
In a recent paper Basieva, Cervantes, Dzhafarov, and Khrennikov (2019) presented a series ofexperiments which they claimed show evidence for contextuality in human judgments. Thiswas based on a set of modified Bell-like inequalities designed to rule out e ff ects caused bysignalling. In this comment we show that it is, however, possible to construct a non-contextualmodel which explains the experimental data via direct influences , which we take to mean thata measurement outcome has a (model-specific) causal dependence on other measurements.We trace the apparent inconsistency to a definition of signalling which does not account for allpossible forms of direct influence. Further, we cast doubt on the idea that any experimental datain psychology could provide conclusive evidence for contextuality beyond that explainable bydirect influence. Introduction
As part of the broader quantum cognition programa number of researchers have considered whetherthere is evidence for violation of contextual inequal-ities in psychology (e.g., Aerts, Gabora, and Sozzo(2013), Asano, Hashimoto, Khrennikov, Ohya, and Tanaka(2014), Bruza, Kitto, Ramm, and Sitbon (2015),Bruza, Wang, and Busemeyer (2015)). Such inequali-ties are derived on the assumption that there exist hiddenjoint preference or probability states for the psychologicalobservables being measured. This is, loosely, equivalentto assuming that judgment processes giving rise to choicesbetween di ff erent options operate independently, which isan important constraint on the processes underlying humandecision making.One complicating factor is that it is hard to rule out thepossibility of direct influence between measurements, whichcan mimic the e ff ect of true contextuality. In other words,contextuality means the outcome of a judgment about ob-servable A has an apparent and unexpected dependence onwhat else is being measured, but that can also occur if theoutcome of the other measurements directly influence A.The absence of such direct influences must be justified orexplicitly tested in any particular application. In physics suchinfluences can sometimes be ruled out by reference to somephysical principle, but nothing equivalent in psychology canbe used to rule out direct influences a priori. The challengeof quantifying exactly when violations of contextual inequal-ities can be accounted for by direct influences, and when theycan only be explained by genuine contextuality, was taken upby Dzhafarov and Kujala (2015) who derived modified in- equalities which, they claim, allow the identification of truecontextuality not explainable by “signalling”, a specific formof direct influence which we will explain below.In a series of papers Dzhafarov and collaborators re-analyzed existing experimental claims of contextuality inpsychology and concluded they could all be explainedby signalling (Dzhafarov, Zhang, and Kujala (2016),Dzhafarov, Kujala, Cervantes, Zhang, and Jones (2016)).However in an elegant paper Cervantes and Dzhafarov(2018) presented an experiment which did satisfy theirmodified contextual inequality, and in a recent paperBasieva et al. (2019) followed this up with series of experi-ments, the majority of which produced data demonstratinggenuine contextuality, ie over and above that explicable bysignalling, according to Dzhafarov and Kujala’s modifiedinequalities.In this comment we explain why we nevertheless donot believe that the results presented by Basieva et al.(2019) provide conclusive evidence for contextuality in hu-man decision making. We only consider in detail theform of experiment conducted by Basieva et al. (2019) andCervantes and Dzhafarov (2018), but we use the insightgained to question whether contextuality could ever be ob-servable in human decision making. Outline of Basieva et al. (2019)
Consider one of the experiments in Basieva et al. (2019);“Alice wishes to order a two-course meal. For each courseshe can choose a high-calorie option (indicated by H) or alow-calorie option (indicated by L). Alice does not want bothcourses to be high-calorie nor does she want both of them tobe low-calorie.”
JAMES M YEARSLEY
Each participant was given two out of three courses(Starter, Main, Dessert) to choose from. Clearly participantsshould select options so that while the calorie content of, eg,the starter is undetermined, it is anti-correlated with the calo-rie content of the main (indeed this restriction was forcedon participants in the experiment.) We can easily see whyBasieva et al. (2019) expect to see evidence of contextuality- the underlying probability distribution for all three coursesneeds to have three binary anti-correlated variables, which isimpossible. In other words, the calorie content of at least twoof the dishes has to match, since there are three courses andonly two calorie options, but then one cannot have the threechoices anti-correlating.The specific inequality Basieva et al. (2019) test is an ex-ample of modified Bell-type inequalities derived by Dzha-farov and Kujala (2015). They use the contextuality by de-fault (CbD) approach, in which a set of random variables R n each taking values ± ff erent values in di ff er-ent measurement contexts. So R mn denotes the value of R n in measurement context labeled by m . In the above example n , m = , , R mn and R kn may be di ff erent.The modified Bell-type inequalities consist of standardBell-type inequalities but modified by terms of the form ∆ = | E [ R ] − E [ R ] | + | E [ R ] − E [ R ] | + | E [ R ] − E [ R ] | , (1)where E [ · ] denotes an expectation value. The quantity ∆ isheld to be a measure of signalling in this situation, which isthus seen to be defined in terms of an average of the degreeof direct influence, since it involves only terms of the form E [ R mn − R kn ]. Such modified inequalities permit the identifica-tion of contextuality beyond that explainable by signalling.In the specific model considered by Basieva et al. (2019) themodified Bell-type inequality has the simple form ∆ < , (2)here written in a notational form opposite to the traditionalform of the Bell inequalities so contextuality is deemed tobe present if this equation is satisfied. What Basieva et al.(2019) did was to show this inequality was satisfied by datacollected in a number of di ff erent experiments which werevariants of the one outlined above.However it is intuitively obvious how participants couldsolve the problem set by Basieva et al. (2019); given twocourses to select, eg starter and main, choose the caloriecontent of the first one randomly, then make the oppositechoice for the second course. This complies with the instruc-tions and is “non-contextual” in a colloquial sense, since itmakes no reference to measurement contexts. However itdoes not sit well with the idea of a pre-existing preference,since one of the judgments is made deterministically based on the other, with no reference to existing preferences. Wetherefore need to apply a more precise measure of contextu-ality. Also, as described, this strategy has the feature that itwill tend to produce equal preferences for each course, whichis not what was observed in Basieva et al. (2019). So we needto establish that this heuristic can generalise to cases wherethe preferences are not equal. A Possible Non-Contextual Account?
If the variables of interest are thought to be the same indi ff erent contexts (e.g. dish choice for main in the contextof starter, dish choice for main in the context of dessert)non-contextuality means that there exists a joint probabil-ity matching the set of marginal probabilities characteriz-ing the data. Contextuality is thus defined to be the ab-sence of such a distribution. This definition of contextual-ity is essentially the same as that frequently employed bothin physics (see for example Abramsky and Brandenburger(2011)) and in cognitive models in psychology (see for ex-ample, Oaksford and Chater (2007)). If the variables are al-lowed to take di ff erent values in di ff erent contexts, then theway this standard notion of non-contextuality is extended inthe CbD approach is to require that the variables vary as littleas possible across di ff erent contexts (Dzhafarov and Kujala,2015), about which more below.Let us begin with our idealized version of the experiment:assume participants solve the problem by choosing the calo-rie content of the first course randomly, then making the op-posite choice for the second course. The expectation value ofany of the variables therefore equals zero, regardless of thecontext in which it is measured. That means, ∆ = , (3)so Eq.(2) is satisfied and Basieva et al. (2019) would presum-ably claim genuine contextuality in this case, i.e. contextu-ality over and above that which could be explained by sig-nalling.However it is still possible to write down a probabilitydistribution on the variables R , R , R , R , R , R , which hasthese correlations and expectation values: p ( R , R , R , R , R , R ) =
164 (1 − R R )(1 − R R )(1 − R R ) . (4)Note E [ R , R ] = − E [ R ij ] = R on R etc, such that the valueof one random variable in a given context is set equal to mi-nus the value of the other one. This does not conform to thestandard notion of a non-contextual model in the CbD ap-proach since it involves probabilities on variables permitted OMMENT ON BASIEVA ET AL. (2019) ff erent values in di ff erent contexts, but without therequirement of minimal variation across di ff erent contexts.However, it is clearly still of interest since it gives a proba-bilistic explanation of the data in terms of the action of directinfluences.We note an interesting property of this probability distri-bution, which is that it clearly factorises as; p ( R , R , R , R , R , R ) = p ( R , R ) p ( R , R ) p ( R , R ) (5)One consequence is that correlation functions between thesame variable in di ff erent contexts are zero, eg E [ R R ] = R is ba-sically set by R , which is an independent random variable.So the e ff ect of the direct influence is to remove correlationsbetween the same variable in di ff erent contexts.This idealisation is interesting, because the fact the expec-tation values of all individual variables are all zero meansthe modified contextual inequality of Dzhafarov and Ku-jala’s (2015) reduces to the one in the absence of sig-nalling. In other words, although our account of this ex-periment involves direct influence between variables mea-sured in the same context, it does not involve the weakernotion of signalling. This suggests the origin of the discrep-ancy between the claims in Cervantes and Dzhafarov (2018)and Basieva et al. (2019) and our construction of a non-contextual model lies in the definition of signalling used byDzhafarov and Kujala (2015). We will explore this furtherbelow.Our idealisation of the experiments in Basieva et al.(2019) is informative, but the results they reported had non-zero expectation values for R , R and R . We can modifyour account to deal with this by taking the joint probabilityto have the same form as Eq.(5) above, where now, p ( R mi , R mj ) =
14 (1 + ( R mi − R mj ) E [ R mi ] − R mi R mj ) (6)where E [ R mm ] are the measured expectation values.This obviously has the correct values for the measured ex-pectation values and correlations. It also has the same in-terpretation, namely that there is a direct influence between,eg R and R , such that, on measuring the value of R , thevalue of R is set to minus this. The correlations betweenvariables measured in di ff erent contexts are no longer zero,however we have E [ R mi R ni ] = E [ R mi ] E [ R ni ], so they are stillindependent. There are no constraints on the E [ R mm ] in orderthat this construction be valid.We have therefore shown by explicitly constructing a jointprobability distribution that the experimental results reportedin Basieva et al. (2019) can be accounted for by a particu-lar type of non-contextual model which includes direct influ-ences. More precisely, a model is possible in which prefer-ences for the three dish choices are well defined at all times, but there is an explicitly modeled disturbance caused by elic-iting a preference which explains the apparently contextualdata. Signalling vs Direct Influence
The roots of our disagreement with Basieva et al. (2019)lie in the work of Dzhafarov and Kujala (2015) which maybe regarded as a generalization of the famous result of Fine(1982) who established the conditions under which certainsets of marginal probabilities possess a joint probability dis-tribution. A crucial assumption in Fine’s work is that over-lapping pairwise marginal probabilities are compatible witheach other, a condition referred to as marginal selectivity,which in the present application reduces to a set of simpleconditions of the form E [ R ji ] = E [ R ki ] , (7)in other words, the average values of all variables R ji are inde-pendent of context. Dzhafarov and Kujala (2015) essentiallydemonstrate how to extend Fine’s result to embrace the casein which marginal selectivity fails.This generalized Fine’s theorem leads to the set of Bell-type inequalities mentioned above which Dzhafarov and Ku-jala (2015) claim to be tests for contextuality in the presenceof signalling. As indicated, they define signalling as a failureof conditions such as Eq.(7), or more generally, by non-zerovalues of the quantity ∆ defined in Eq.(1). Since this defini-tion of signalling corresponds to the average degree of directinfluence, it leaves a residual degree of direct influence whichhas the power to explain apparent contextuality not explainedby signalling. The presence of direct influence is character-ized precisely by non-zero values of probabilities of the form p ( R ji , R ki ) (i.e, the probabilities that the same variable mea-sured in di ff erent contexts gives di ff erent results). This prob-ability can be non-zero even when Eq.(7) holds. Indeed thispossibility occurs in our model above where direct influenceis present trial to trial but averages to zero.The di ff erence between signalling and more general di-rect influence is not very apparent in the approach of Dzha-farov and Kujala (2015) since in their definition of non-contextuality, the underlying joint probability is required tochange as little as possible across di ff erent contexts. Theyimplement this by requiring that the probabilities of the form p ( R ji , R ki ) are minimized . The minimum then dependsonly on terms proportional to ∆ , Eq.(1), hence notions ofsignalling and more general direct influence coincide in thissituation.To put all this another way, in order to claim contextu-ality, it is necessary to show that there is no other possibleaccount of the correlations. In physics it is necessary to goto some lengths to be sure of this. The attitude one needsto adopt is of the “worst case scenario”, where the direct in-fluence is as hard to detect as possible. Only by ruling out JAMES M YEARSLEY this sort of stubborn direct influence can we be sure that anon-contextual account is impossible. In contrast, focussingon changes to the averages can be thought of as a “best casescenario”, where the direct influence is as easy to detect aspossible. Ruling out changes to the average distributions isnecessary, but not su ffi cient to rule out direct influences, be-cause one could imagine, for example, that the process ofmeasuring A changes the correlation between A and B, butnot the averages. This clearly implies a direct influence be-tween A and B, but one which is not detectable from theaverages alone.To give a simple example, imagine we have two coinswhich are tossed either together or independently and un-der either circumstance both have probability 1 / ff erence tothe weaker notion based just on signalling, which involvesreadily measurable quantities. In physics such measurementsare not hard to devise and higher order signalling conditionsthat detect direct influences beyond the average have beenproposed, e.g. by Clemente and Kofler (2015). This couldbe a lot harder in psychological experiments, which meansthat a definition of of contextuality phrased only in terms ofwhat can actually be measured is a reasonable one.In summary, we see that the claims of Dzhafarov & Ku-jala (2015) about the presence of contextuality beyond thatexplainable by signalling hinge on a notion of signalling asaverage direct influence, a notion weaker than that used inphysics (where “signalling” is more commonly associatedwith direct influences more generally). We have found that aparticular type of non-contextual model is possible if directinfluence beyond the average is taken into account. Discussion
The above results raise two interesting questions; first, isit ever possible to rule out direct influences in a psychologysetting? This remains an open question, but we suspect theanswer is negative. In physics one can always reproducethe results of quantum theory with a model which is non-contextual but non-local (Bohm, 1952). In physics such ac-counts can be ruled out on the basis of a physical principle,locality, but this is an additional assumption going beyondstatements about the statistics of measurements. There isnothing equivalent in psychology that would supply such aclear cut limit on the set of allowable models.Second, if we cannot rule out models involving direct in-fluence, does ruling out models involving the weaker notionof signalling tell us anything useful? In one sense the answer is clearly no - violations of contextual inequalities such asEq.(2) have been billed as tests of the necessity of a contex-tual (quantum) account of human decision making, and wehave seen that such violations do not in fact rule out all pos-sible non-contextual accounts, and therefore cannot defini-tively prove the necessity of a quantum model for such data.Does this mean contextual inequalities have nothing toteach us in psychology? Not necessarily. It has previ-ously been argued (Yearsley & Pothos, 2014) that data sat-isfying Dzhafarov and Kujala’s (2015) inequalities presentsus with a choice - either we can construct a model which isnon-contextual but which involves unobservable direct influ-ences, or we can construct a model which only involves ob-served quantities, but which combines them in a contextualway. The correct way to proceed is not fixed by any mathe-matical law, but depends on the goals of the researcher.Added note: after completion of this work we were madeaware that a number of other authors have made closely re-lated observations, including Atmanspacher and Filk (2019),Cavalcanti (2018) and Jones (2019). These connections willbe addressed in future publications.
Acknowledgments
We are grateful to Samson Abramsky, Jerome Busemeyer,Ehtibar Dzhafarov, Andrew Simmons and Rob Spekkens foruseful communications on this topic.References
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