The Conjunction Fallacy in Quantum Decision Theory

Abstract

The conjunction fallacy is a renowned violation of classical probability laws, which is persistently observed among decision makers. Within Quantum decision theory (QDT), such deviations are the manifestation of interference between decision modes of a given choice prospect. We propose a novel QDT interpretation of the conjunction fallacy, which cures some inconsistencies of a previous treatment, and incorporates the latest developments of QDT, in particular the representation of a decision-maker’s state of mind with a statistical operator. Rather than focusing on the interference between choice options, our new interpretation identifies the origin of uncertainty and interference between decision modes to an entangled state of mind, whose structure determines the representation of prospects. Thus, on par with prospects, the state of mind can be a source of uncertainty and lead to interference effects, resulting in characteristic behavioral patterns. We present the first in-depth QDT-based analysis of an empirical study (the touchstone experimental investigations of Shafir et al. [6]), which enables a data-driven exploration of its underlying theoretical construct. We link typicality judgements to probability amplitudes of the decision modes in the state of mind, and quantify the level of uncertainty and the relative contributions of prospect’s interfering modes to its probability judgement. This enables inferences about the key QDT interference “attraction” q-factor with respect to different types of prospects—compatible versus incompatible. We propose a novel empirically motivated “QDT indeterminacy (or uncertainty) principle,” as a fundamental limit of the precision with which certain sets of prospects can be simultaneously known (or assessed) by a decision maker, or elicited by an experimental procedure. For any type of prospects, we observe a general tendency for the q-factor to converge to the same negative range $q\in (-0.3,-0.1)$ in the presence of high uncertainty, which motivates the hypothesis of an universal “aversion” q. The “aversion” q is independent of the (un-)attractiveness of a prospect under more certain conditions, which is the main difference with the previously considered “QDT quarter law”. The universal “aversion” q substantiates the previously proposed “QDT uncertainty aversion principle” and clarifies its domain of application. The universal “aversion” q provides a theoretical basis for modelling different risk attitudes, such as aversions to uncertainty, to risk or to losses.

Appendices

Appendices

Appendix 1: Experimental Results from [6]

See Appendix Table 8.

Table 8 Each of the 14 instances I (description of a subject) was presented to decision makers on separate occasions with one of 4 categories: either incompatible (i) conjunction and its constituent, or compatible (c) conjunction and its constituent

Appendix 2: Previous Interpretation of the Conjunction Fallacy Within Quantum Decision Theory

1.1 Summary of the Previous Interpretation

The previous interpretation of the conjunction fallacy within Quantum decision theory (QDT), based on the experimental results of [6], was proposed in [13]. The current section condenses the arguments of that interpretation, and provides citations from the original.¹ Parts of sentences in italic correspond to pieces of text extracted from [13]. Within these quotes, we indicate within parentheses when we had to adapt the text to make it understandable with our present conventions.

Consider the following two intentions. One intention, with just one representation, is “to decide whether the (su)bject (which is described in an instance I) has the feature B.” The second intention is “to decide about the secondary feature” which has two representations, when one decides whether “the (su)bject has the special characteristic” ($A_1$) or “the (su)bject does not have this characteristic” ($A_2$). Thus, according to the notation (B = “bankteller”, $A_1$ = “feminist”, $A_2$ = “non-feminist”), prospects are formulated as follows:

• for conjunction AB (e.g. AB(i) “feminist bankteller”):

  $$\begin{aligned} \pi _1 = BA_1\text {, which is represented by the state } | \pi _1 \rangle = a_{11}| BA_1 \rangle , \end{aligned}$$

  (57)

• for its constituent B (e.g. B(i) “bankteller”):

  $$\begin{aligned}&\pi _2 = BA = B(A_1+A_2)\text {, which is represented by the state }\nonumber \\&| \pi _2 \rangle = a_{21}| BA_1 \rangle +a_{22}| BA_2 \rangle . \end{aligned}$$

  (58)

The following general scheme is applied to calculate the probability of the prospect with a constituent category B:

$$\begin{aligned} p(\pi _2)= & {} p(BA) = p(BA_1) + p(BA_2) + q(BA) \nonumber \\= & {} p(B|A_1)p(A_1) + p(B|A_2)p(A_2) + q(BA) \end{aligned}$$

(59)

This is a typical situation where a decision is taken under uncertainty. The uncertainty-aversion principle requires that the interference term q(BA) should be negative ($q(BA) < 0$).

For the set of compatible (c) pairs of characteristics, it turned out that the average probabilities were p(BA) = 0.537 and $p(BA_1)$ = 0.567, with statistical errors of 20%. Hence, within this accuracy, p(BA) and $p(BA_1)$ coincide and no conjunction fallacy arises for compatible characteristics. From the view point of QDT, this is easily interpreted as due to the lack of uncertainty: since the features B and $A_1$ are similar to each other, one almost certainly yielding the other, there is no uncertainty in deciding, hence, no interference, and, consequently, no conjunction fallacy.

For incompatible (i) categories, the simplest and most natural mathematical embodiment of the property of “incompatibility” is to take the probabilities of possessing B, under the condition of either having or not having $A_1$, as equal, that is, $p(B|A_j) = 0.5$. For these incompatible (i) categories, equation (59) reduces to

$$\begin{aligned} p(BA) = \frac{1}{2} + q(BA). \end{aligned}$$

(60)

For incompatible (i) categories, the average values of the reported probabilities are $p(BA) = 0.220$ and $p(BA_1) = 0.346$ [6].

Given the observed values of p(BA) for each of the 14 constituents of incompatible categories (i) [6] and Eq. (60), the observed interference terms are found fluctuating around a mean of $-0.28$, with a standard deviation of $\pm 0.06$:

$$\begin{aligned} q(BA) = -0.28 \pm 0.06. \end{aligned}$$

(61)

The conjunction error is found to be:

$$\begin{aligned} \varepsilon (BA_1) \equiv p(BA_1) - p(BA) = 0.126. \end{aligned}$$

(62)

From Eq. (59), the average value of $p(BA_2)$ is equal to 0.154. In addition, the proposed assumption that $p(B|A_j)=0.5$ leads to $p(A_1) = p(BA_1)/0.5 = 0.692$, and similarly $p(A_2) = p(BA_2)/0.5 = 0.308$.

[13] conclude: QDT interprets the conjunction effect as due to the uncertainty underlying the decision, which leads to the appearance of the intention interferences. The interference of intentions is caused by the hesitation whether, under the given primary feature (B), the (su)bject possesses the secondary feature $(A_1)$ or does not have it $(A_2)$. The term q(BA) is negative, reflecting the effect of deciding under uncertainty (according to the uncertainty-aversion principle). Quantitatively, we observe that the amplitude |q(BA)| is in agreement with the QDT interference-quarter law.

1.2 Weaknesses of the Previous Interpretation

As summarised in the previous Sect. 7, the interpretation of the conjunction fallacy in [13] rests on two assumptions:

• 1. the formulation of a prospect for a constituent category B such that it includes uncertainty about a secondary feature $(A_1+A_2)$, i.e. Eq. (58);

• 2. the independence of incompatible features, which underlies Eq. (60).

The current section analyses these two assumptions and demonstrates the existence of some inconsistencies.

1.2.1 Formulation of the Prospect for a Constituent Category (B): Intention Concerning a Primary Feature and Undetermined Sign of q (Eqs. 57–59)

For a secondary characteristic (e.g. “feminist”), the uncertainty about its presence (i.e. an undecided attribution of this feature to the subject from an instance I) is represented as a composite action A, which is a sum of two action modes $A_1$ (“the subject has a secondary feature”) and $A_2$ (“the subject does not have a secondary feature”). However, for a primary characteristic (e.g. “bankteller”), a simple action with one action mode B (“to decide whether the subject has a primary feature”) is suggested. This formulation of intention B as an active decision concerning a primary feature is necessary to justify the negative sign of the attraction factor q for a corresponding prospect with a constituent category (B), if a level of uncertainty about a secondary feature ($A_1+A_2$) is introduced in this prospect, such that:

$$\begin{aligned} \pi _2 = B(A_1+A_2)~\text {, i.e.}~~| \pi _2 \rangle = a_{21}| B \otimes A_1 \rangle + a_{22}| B \otimes A_2 \rangle , \end{aligned}$$

(63)

where the symbol $\otimes$ represents the tensor product operator (see [13] for details).

Thus, under an assumption of passivity in the presence of uncertainty, it is proposed that making a decision concerning a primary feature B (e.g. “bankteller”) is not attractive, i.e. $q(\pi _2)\le 0$.

The following two arguments reveal an inconsistency in the above formulation of a constituent intention B.

First, it is natural to assume that, for a primary characteristic, an intention complementary to B should exist, which is denoted for simplicity notB and stands for “not to decide whether the subject has a primary feature”). Action notB reflects an undecided attribution of a primary feature and, similar to action A for a secondary feature, can be presented as a sum of two action modes: $B_1$ (“the subject has a primary feature”) and $B_2$ (“the subject does not have a primary feature”). Continuing this analogy, in formulating the prospects for both categories, i.e. for a conjunction and its constituent, action mode $B_1$ should be used to represent a consideration (attribution) of a primary feature (e.g. “bankteller”). However, the introduction in [13] of action B as described above remains unclear. In fact, an interpretation of action B as an active decision (“to decide whether the object has a primary feature”) seems unrealistic, as in the experiment participants were exposed to the predefined categories and had to judge the corresponding probabilities, i.e. participants were not asked to decide whether to make the judgement or not.

Second, if the intention B represents a single action mode of possessing a primary feature that is equivalent to $B_1$ (“the subject has a primary feature”), then a complementary action mode $B_2$ (“the subject does not have a primary characteristic) exists and also requires an active decision of a decision maker about the possession (or absence) of a primary feature in the subject. Thus, the sign of the attraction factor q for a constituent category cannot be determined, even when assuming the presence of uncertainty concerning a secondary feature.

These two inconsistencies can thus be summarised as follows:

• an intention concerning a primary feature B (e.g. “bankteller”) should be formulated similarly to an intention about a secondary feature A (e.g. “feminist”), in the form of “the subject has a feature”;

• the sign of the attraction factor q for a constituent category cannot be determined, even when assuming the presence of uncertainty with respect to a secondary feature in this prospect.

1.2.2 Formulation of the Prospect for a Constituent Category (B): Uncertainty About a Secondary Feature (Eq. 58)

In order to fully understand the assumptions underlying the QDT interpretation of [13], it is important to note that the experiments that have investigated the conjunction fallacy have been performed under several distinct treatments, which introduce subtle but important differences for their theoretical interpretation. The following three main classes of experiment treatment have been used.

• 1. Indirect tests, when subjects were exposed to a description of a subject (an instance I) and only one of the categories (either conjunction AB, or its constituent B) at a time. In this setup, the judgements about the probability of an instance I with respect to each category—AB or B—were made separately and were not juxtaposed. For example, in [8], an indirect between-subjects comparison was conducted, when the probability of the conjunction was evaluated by one group and the probability of its constituent was evaluated by another group. In [6], judgments of probabilities for each category—AB or B—were performed separately, but by the same decision makers.

• 2. Direct-subtle tests, when, following an instance I, participants are exposed to both a conjunction and its constituent category, but the inclusion relation is not made apparent. For example, in [8], the two categories of interest are shown simultaneously, but are camouflaged among five additional filler items.

• 3. Direct-transparent tests, when an instance I, a conjunction and its constituent are presented together to highlight the connection between the categories.

In the current QDT treatment [13], the conjunction fallacy, i.e. $p(AB) > p(B)$, is explained by the negative attraction factor of a prospect with constituent category B. For a negative q to appear, a judgement about the prospect with a constituent B is assumed to be influenced by the existence of a level of uncertainty about the presence of a secondary feature A. In other words, a judgment about a primary feature B is saddled with an added degree of uncertainty about the attribution of a secondary feature A, even when absent in the judged category B (Eq. 58).

However, in the indirect test design as presented in [6] and analyzed in [13], the judgement about the probability of an instance I with respect to a constituent category B was made separately, without exposition to a conjunction category AB. In this indirect setup, it is not obvious that a secondary feature, which is present only in a conjunction, has any influence on a judgment about a constituent category. Thus, there is no evidence that a secondary feature should be included in the formulation of the QDT prospect for a constituent category B (e.g. “bankteller”), which instead can be simply represented by

$$\begin{aligned} \pi _2 = B~\text {, i.e.}~~| \pi _2 \rangle = a_{21}| B \rangle , \end{aligned}$$

(64)

instead of Eq. 58.

Importantly, for indirect tests, with this formulation of a prospect for a constituent B, there is no uncertainty about the presence of a secondary feature A in a constituent category. Thus, the QDT uncertainty aversion principle ought not to be invoked to explain the observed conjunction fallacy, and another mechanism is required.

For the direct test designs, which allow for a direct comparison of a conjunction and a constituent categories, the formulation of prospects proposed in [13] is more plausible. It could be expected though that a more profound manifestation of uncertainty for a constituent category, which is associated with the presence of a secondary feature, would increase the absolute value of a negative attraction factor of this prospect ($| \pi _2 \rangle$), amplifying the conjunction fallacy. However, the opposite results are observed in experiments [8]. Probably, the most convincing evidence was obtained in a direct-transparent test, where the probability judgment about an instance I (the description of Linda) was made with respect to the following two categories:

• 1. Linda is a bank teller whether or not she is active in the feminist movement: B(i) (versus $B(A_1+A_2)$ in [13]);

• 2. Linda is a bank teller and is active in the feminist movement: AB(i) (versus $BA_1$ [13]).

In this example, the degree of uncertainty about the presence of a secondary “feminist” feature ($A_1+A_2$) is made explicit in a constituent category B(i), which provides a good match for the formulation of a prospect with a constituent in [13]. However, contrary to what one could expect from the formulation of [13], the portion of decision makers who committed the conjunction fallacy dropped from above 80% (observed in both indirect and direct tests) to 57% (for the direct-transparent test) [8]. This finding signals that the recognition by decision makers of a level of uncertainty about a secondary feature in a constituent category does not make this prospect less attractive, but rather emphasizes the inclusive relation between the two categories (conjunction and its constituent) and facilitates the correct application of the conjunction rule.

Furthermore, [8] outlined that the representativeness heuristic may be at the heart of the persistent conjunction fallacy. Even when provided with a valid and clear explanation of the inclusion of a conjunction category into a constituent, the majority of subjects choose to stick to an “emotional” resemblance argument.

This suggests an alternative QDT mechanism for the explanation of the conjunction fallacy: rather than a negative attraction of a constituent category due to the uncertainty of a secondary feature, the key ingredient is a higher attraction to a conjunction prospect, if a secondary feature is compatible with the description of an instance I.

1.2.3 Independence of (Incompatible-Type) Prospects (Eq. 60)

Equation (60), which is a key result in [13], is based on two underlying assumptions:

• the “incompatibility” of constituents in a conjunction category is treated as leading to their “independence”, i.e. the probability of possessing a primary characteristic B is assumed to be independent from having a secondary characteristic $A_j$, yielding $p(B|A_1) = p(B|A_2) = p(B)$;

• with no prior information, it is assumed that $p(B) = 0.5$.

A first general criticism can be raised about the assumption that the existence of uncertainty about an independent category would lead to a negative attraction factor when deciding about another independent category. If this was the case, any decision would then be associated with $q<0$, as it is impossible to create a completely certain environment in our complex uncertain world, as there are always many variables that remain uncertain around us.

Secondly, in [13], the “compatibility” of categories is associated with an absence of uncertainty, i.e. the possession of one of the compatible characteristics yields the other one, which implies positive correlation close to 1. In this context, “incompatibility” is more likely to imply negative correlation close to $-1$, rather than independence with correlation close to 0.

Thirdly, an assumption that a high degree of “compatibility” (or “incompatibility”) is associated with very low uncertainty, and thus leads to $q\rightarrow 0$, has to be tested. For example, it is plausible that the subjective estimation of a high “compatibility” (“incompatibility”) of categories may increase the ‘subjective’ confidence of a decision maker, which could be reflected in a high positive (negative) value of the attraction factor, and make a choice deviate from an ‘objective’ judgement. Empirical evidence should be gathered to support this hypothesis.

We are thus led to suggest two alternative propositions replacing the two assumptions of [13] discussed above:

• Independent intentions do not interfere. Thus, the existence of uncertainty about one intention does not influence the probability that another intention will be realized, if these intentions are independent.

• Equation (60) requires revision: if (incompatible) intentions are treated as independent, they should not interfere and $q=0$; if (incompatible) intentions interfere and $q\ne 0$, then $p(B|A_1) = p(B|A_2) = p(B)=0.5$ can not be assumed.

1.2.4 Partial Use of Data

[13] did not make use of experimental data on compatible conjunctions and their constituents, and on typicality judgements. Most importantly, the description of a subject—instance I—is a key element of the experiment, which consists in framing participants prior the choice (judgement). However, this was ignored in many theoretical interpretations, including [13].

1.3 Synthesis

The previous interpretation of the conjunction fallacy within QDT [13] aimed at explaining it for the most clearcut cases of one type of incompatible prospects with a constituent category B(i). Agreement between partial empirical data and the general QDT relation p=f+q was obtained. However, the needed underlaying assumptions have been shown to be unsubstantiated.

In particular, the representation of prospects, which is needed to justify the application of the uncertainty-aversion principle and the corresponding negative sign of the attraction factor q, leads to serious inconsistencies. The interference effects argued to occur for a single constituent category B, as formulated in [13], have a shaky foundation. As discussed above in details, within an indirect test setup [6], the inclusion of uncertainty about a secondary feature A from a conjunction category AB should not be relevant to a separate judgement regarding B. Another essential assumption about the independence of incompatible categories, upon which the proposed interpretation rests, is arbitrary. Importantly, the influence of framing, i.e. the pre-exposure of decision makers to an instance I (the description of a subject), is disregarded. Last but not least, the available empirical data is used only partially (just for one out of four judged categories).

Since the attempt of [13], the theoretical construct of QDT has been significantly enriched. We use this opportunity to ‘cure’ the above mentioned weaknesses, to propose a genuine rationalisation and quantitative explanation of the conjunction fallacy within QDT, which allows us to further explore the limits of QDT.

Appendix 3: Verification of the Necessary Conditions for the Emergence of the QDT Attraction Factor

The necessary conditions for the attraction factor to be non-zero are: (a) entanglement in a strategic decision-maker state, and (b) entanglement of a prospect, i.e. a decision is to be made under uncertainty. Concerning the former condition (a), a strategic decision-maker state can be separable (not entangled) only if the measurements of observables are not temporally correlated. Section 3.3.1 demonstrates the evolution of a strategic decision-maker state through a sequence of channels, which produces an entangled state $\hat{\rho }_{ABIM}$ (resp., $\hat{\rho }_{BIM}$) just prior to a decision. For the second condition (b), to determine whether prospect $\pi _{(AB)_1I}$ from (20) is entangled, we investigate the separability of the corresponding operator $\hat{P}_{(AB)_1I}$ (22), as proposed in [16, 18]. For this, we introduce two Hilbert-Schmidt spaces below. The first one is defined by

$$\begin{aligned} \widetilde{\mathcal {A}\mathcal {B}} \equiv \{\mathcal {A}\mathcal {B}, \mathcal {H}_{AB}, \sigma _{AB}\} \end{aligned}$$

(65)

where $\mathcal {A}\mathcal {B}= \{\hat{P}_{(AB)_i}\}$ is an operator algebra (or an algebra of local observables), acting on the Hilbert space $\mathcal {H}_{AB}$, while $\sigma _{AB}$ is the scalar product $\sigma _{AB}:\mathcal {A}\mathcal {B}\times \mathcal {A}\mathcal {B}\rightarrow \mathbb {C}$ that is defined as

$$\begin{aligned} \sigma _{AB}:(\hat{P}_{(AB)_1},\hat{P}_{(AB)_2}) = \mathrm{Tr}_{AB}\hat{P}_{(AB)_1}^\dagger \hat{P}_{(AB)_2} \end{aligned}$$

(66)

and generates the Hilbert-Schmidt norm $|| \hat{P}_{(AB)} ||\equiv \sqrt{(\hat{P}_{(AB)_i},\hat{P}_{(AB)_i})}$, $i\in {1,2}$.

Similarly, for the second Hilbert-Schmidt space:

$$\begin{aligned} \widetilde{\mathcal {I}} \equiv \{\mathcal {I}, \mathcal {H}_{I}, \sigma _{I}\} \end{aligned}$$

(67)

where $\mathcal {I}= \{\hat{P}_{I_i}\}$ is an operator algebra (or an algebra of local observables) on the Hilbert space $\mathcal {H}_{I}$, the scalar product is a map $\sigma _{I}:\mathcal {I}\times \mathcal {I}\rightarrow \mathbb {C}$ that is defined as

$$\begin{aligned} \sigma _{I}:(\hat{P}_{I_1},\hat{P}_{I_2}) = \mathrm{Tr}_{I}\hat{P}_{I_1}^\dagger \hat{P}_{I_2} \end{aligned}$$

(68)

and generates the Hilbert-Schmidt norm $|| \hat{P}_{I} ||\equiv \sqrt{(\hat{P}_{I_i},\hat{P}_{I_i})}$, $i\in {1,2}$.

Now, a composite Hilbert-Schmidt space can be introduced as a tensor-product space

$$\begin{aligned} \widetilde{\mathcal {A}\mathcal {B}} \otimes \widetilde{\mathcal {I}} = \{\mathcal {A}\mathcal {B}, \mathcal {H}_{AB}, \sigma _{AB}\} \otimes \{\mathcal {I}, \mathcal {H}_{I}, \sigma _{I}\}. \end{aligned}$$

(69)

An operator acting on this composite Hilbert-Schmidt space $\widetilde{\mathcal {A}\mathcal {B}} \otimes \widetilde{\mathcal {I}}$ is called separable (or disentangled) if and only if it can be represented as a sum

$$\begin{aligned} \sum \limits _{i}\hat{P}_{(AB)_i} \otimes \hat{P}_{I_i}~~~~~(\hat{P}_{(AB)_i}\in \widetilde{\mathcal {A}\mathcal {B}},~~\hat{P}_{I_i}\in \widetilde{\mathcal {I}}) \end{aligned}$$

(70)

Importantly, the operator $\hat{P}_{(AB)_1I}$ cannot be represented in the separable form (70), because the last term $| I_k \rangle \langle I_l |$ in (22) does not pertain to $\widetilde{\mathcal {I}}$. Thus, we conclude that the corresponding composite prospect $\pi _{(AB)_1I}$ in (20) is entangled.

Following the same procedure, the operator $\hat{P}_{B_1I}$ in (23) is non-separable, i.e. it cannot be represented as a sum

$$\begin{aligned} \sum \limits _{i}\hat{P}_{B_i} \otimes \hat{P}_{I_i}~~~~~(\hat{P}_{B_i}\in \widetilde{\mathcal {B}},~~\hat{P}_{I_i}\in \widetilde{\mathcal {I}}) \end{aligned}$$

(71)

and the related composite prospect $\pi _{B_1I}$ in (20) is entangled as well.

Thus, the necessary conditions for the emergence of the QDT attraction factor in (28), i.e. $q(\pi _{(AB)_1I})\ne 0$ and $q(\pi _{B_1I})\ne 0$, are satisfied.

Appendix 4: Squared Probability Amplitudes $|\gamma _i|^2,~i\in \{1,2\}$, of Prospects’ Decision Modes

See Appendix Tables 9, 10 and 11.

Table 9 Adjusted coefficients $|\gamma _i|^2, i\in \{1,2\}$, and their sums, for the Minimum interference case

Table 10 Adjusted coefficients $|\gamma _i|^2, i\in \{1,2\}$, and their sums for the maximum interference case

Table 11 Results of the generalized (extreme Studentized deviate) ESD test${}^\text {a}$ to detect $i=1...k$ outliers in samples of $\gamma$ coefficients, approximated by the normal distribution