*2.3. A 2* × *2 Quantum Decision Model*

We next present a simplified model that we formulate in the terms of our experiment, that is a choice between two uncertain options with two attributes each. For a general and detailed exposition of the formal framework of quantum-like decision making see [23].

#### 2.3.1. The Representation of a Choice Alternative

We have two animal protection projects Tiger Forever (TF) and Elephant Crisis Fund (ECF). The initial information is incomplete so their (utility) value is uncertain. Our decision-maker (DM) is endowed with a cognitive state which encapsulates the probability distribution for every possible state of the world. The notion of cognitive state is similar to the notion of belief. However, in contrast to beliefs, a cognitive state is not an (imperfect) image of the objective world. It is a mental construct, a representation of the objective world that evolves in a way that reflects the cognitive constraints that we focus on: namely that our DM cannot consider all perspectives (attributes) simultaneously. There exist perspectives that are not compatible in her mind. They are incompatible or Bohr complementary. The notion of Bohr complementarity is a central feature of Quantum Mechanics. It relates to properties of a physical system that cannot have definite values simulatneously. In quantum cognition, it relates to properties of the mental object (the represented choice item). When the object has a determinate value i.e., the individual is subjectively certain about e.g., the issue of urgency of a cause, her beliefs about an incompatible characteristics e.g., honesty of the NGO is necessarily mixed. As a consequence, the picture (representation) that arises depends on the order in which different pieces of information are processed.

The DM has an initial representation of the projects. To each project we associate a vector that captures the initial representation i.e., the cognitive state with respect to that project (hereafter we refer simply to *project-state* or simply *state*). We use Dirac's notation which allows easily connecting with the geometrical illustration, the initial states are denoted |*T* and |*E* with *T* for Tiger Forever and *E* for Elephants Crisis Fund. The two states are modelled as independent systems meaning that we assume that a measurement operation

on one system has no impact on the other i.e., we have no entanglement. Each (represented) project is characterized by two properties (or characteristics) which are assumed incompatible with each other (in the DM mind). We call them Urgency (of the cause) and Honesty (of the NGO managing the project). The Urgency property (or perspective) is represented by a two dimensional space spanned by pure (subjective certainty) states <sup>|</sup>*U*, and *U*<sup>⊥</sup> corresponding to the property of the project being Urgent respectively not-Urgent. The Honesty perspective is represented by an alternative basis of the same state space (|*H*, *H*⊥ ) corresponding to Honest respectively not-Honest NGO. The fact that the two bases span the same space is the geometrical expression of the (subjective) incompatibility of the two properties.

#### 2.3.2. Preferences

Individual preferences are captured by the utility value attributed by the DM to the projects in the possible pure states e.g., <sup>|</sup>*U* or *U*⊥ . The utility of an uncertain state is calculated as a linear combination of those values. In addition, individual preferences are characterized by a "preferred perspective" corresponding to the (most) decision relevant characteristics of the item for the individual (e.g., the Urgency of the cause). When two or more perspectives are incompatible in her mind, the individual uses her preferred perspective to evaluate the expected utility of a project. This means that whereas our DM is capable of looking at alternative perspectives on the same item, when it comes to evaluation, she evaluates utility from one and the same perspective throughout the game. This secures that in any belief state, the utility value is uniquely defined while other incompatible perspectives affect choice through their impact on the belief state (see below). In the following, we assume that the preferred perspective is the same for the two projects.

This is illustrated in Figure 1. The two projects are represented each by two distinct states |*T* and |*E*.

The figure reads as follows. For a DM endowed with preferences that define Urgency (U) as her preferred perspective (a U-individual), the expected utility value of contributing 50 euros to the Elephant Crisis Fund in project-state |*E* is denoted *u*(*ECF*; |*E*, *U*). It depends on two things: 1. her beliefs (about the cause's urgency) encapsulated in state |*E* and 2. her valuation of contributing to an urgent respectively non-urgent ECF project. We denote these values *xU* <sup>∈</sup> <sup>R</sup> respectively *xU*<sup>⊥</sup> <sup>∈</sup> <sup>R</sup>. It is useful to express an U- individual's preferences as *EU* = *xU* 0 0 *xU*<sup>⊥</sup> . As shown in [38] the utility value of project ECF is given as follows

$$\ln\left(ECF; \left|E\right>\mathcal{U}\right) = Tr(E\_{\mathcal{U}}, E) = \mathbf{x}\_{\mathcal{U}} \cdot \left| \left< E \left| \mathcal{U} \right> \right|^2 + \mathbf{x}\_{\mathcal{U}^\perp} \cdot \left| \left< E \left| \mathcal{U}^\perp \right> \right|^2. \tag{1}$$

Our DM is risk neutral: her expected utility of the project is as usual the utility associated with the possible states multiplied by the (subjective) probability for those states. So, for instance, with *xU* = 1 and *xU*<sup>⊥</sup> = 0, the expected utility value of contributing to ECF when the individual has U-preferences is equal to her subjective probability that the elephant cause is urgent. That probability is calculated according to Born's rule which is the formula for calculating probability in a quantum setting. It corresponds to the square of the correlation coefficient *E*|*U*; also called amplitude of probability. Graphically, the probability amplitudes are read off in the diagram as the orthogonal projection (yellow and blue thin doted lines) of vector |*E* on the basis vectors ( |*U*, *U*⊥ , for a U-individual). Similarly, we have for an individual with H-preferences:

*<sup>u</sup>*(*ECF*; <sup>|</sup>*E*, *<sup>H</sup>*) = *Tr*(*EH*, *<sup>E</sup>*) = *xH* · |*E*|*H*|<sup>2</sup> <sup>+</sup> *xH*<sup>⊥</sup> · |*E*|*H*⊥|<sup>2</sup> (2)

where *EH* the utility matrix is defined in the (preferred) (*H*, *H*⊥) basis: *EH* = *xH* 0 0 *xH*<sup>⊥</sup> . The corresponding expected utility values of the TF project for individual having U- and H-preferences respectively are:

$$\begin{aligned} \mu(TF; |T\rangle, \mathcal{U}) &= \quad \text{Tr}(T\_{\mathcal{U}\prime}T) = \mathcal{Y}\_{\mathcal{U}} \cdot |\langle T|\mathcal{U}\rangle|^{2} + \mathcal{Y}\_{\mathcal{U}^{\perp}} \cdot |\langle T|\mathcal{U}^{\perp}\rangle|^{2} \\ \mu(TF; |T\rangle, \mathcal{H}) &= \quad \text{Tr}(T\_{H\prime}T) = \mathcal{Y}\_{\mathcal{T}} \cdot |\langle T|H\rangle|^{2} + \mathcal{Y}\_{T^{\perp}} \cdot |\langle T|H^{\perp}\rangle|^{2} \end{aligned}$$

where *TU* = *yU* 0 0 *yU*<sup>⊥</sup> (as defined in the (*U*, *U*⊥) basis) and *TH* = *yH* 0 0 *yH*<sup>⊥</sup> (as defined in the (*H*, *H*⊥) basis) are the operator representing the utility value of choosing TF for a U-individual respectively a H-individual.

#### 2.3.3. Choice

The individual makes her choice by comparing the expected utility of each project and selecting the one that yields the highest expected utility. For the sake of illustration take *xU* = *yU* = 1 and *xU*<sup>⊥</sup> = *yU*<sup>⊥</sup> = 0, reading directly from the figure we see that

$$\mu(TF; |T\rangle, \mathcal{U}) = |\langle T|\mathcal{U}\rangle|^2 > \mu(ECF; |E\rangle, \mathcal{U}) = |\langle E|\mathcal{U}\rangle|^2$$

and similarly setting *xH* = *yH* = 1 and *xH*<sup>⊥</sup> = *yH*<sup>⊥</sup> = 0

$$\mu(ECF; |E\rangle, H) = |\langle E|H\rangle|^2 > \mu(TF; |T\rangle, H) = |\langle T|H\rangle|^2$$

Which means that in our example a U-individual prefers to contribute to the TF project while a H-individual prefers to contribute to the ECF project.

#### 2.3.4. Persuasion

Persuasion is about modifying the cognitive state, i.e., the (mental) project-states. This is achieved by means of an informational structure (IS) which we define as an operation that triggers the resolution of (subjective) uncertainty with respect to some aspect. This corresponds to complete (projective) measurements of the state. In a two-dimentional case, such measurements yield maximal (but not complete) knowledge. It is important to keep in mind that we are dealing with mental objects (represented projects). So, in our context, an IS can be a question that the individual puts to herself (alternatively an IS is an

investigation of the outside world that determines whether the threat of extinction is real or not). The outcome is generally some level of conviction (subjective certainty).

In quantum persuasion, an IS is decomposed into two parts: a measurement device (MD) and an information channel (IC) (see [23] for details). An IC translates outcomes into signals. In the present context, we confine ourselves to trivial IC, where the signals are the outcomes of the MD. A MD is defined by a set of possible outcomes *I*, a collection of probabilities *pi* to reach these outcomes where *pi* depends on the (cognitive) project-state, *pi* = *Tr*(*PiEPi*) where |*E* is the initial belief state (our project-state),the prior and *Pi* is the projector corresponding to outcome *i*. Upon obtaining outcome *i*, the belief-state transits (is updated) into *Ei* = *PiEPi pi* according to Lüders' rule (a behavioral justification for this rule is provided in [38]). In line with the general theory, we focus on direct (or projective) measurements, that is, MDs whose outcomes transit the prior into a pure cognitive state, i.e., a state of full conviction (subjective certainty with respect to some aspect).

As in the standard persuasion problem, Sender chooses the MD. In our experiment Sender chooses the question put to Receiver. For instance "how urgent do you think it is to protect elephants from the threat of extinction". Such an MD is similar to a procedure that actualizes (rather than "elicit") the individual's beliefs about the severity of the threat. The distinction between eliciting and actualizing is that in the first case, it is assumed that the beliefs pre-existed the questioning, it is simply revealed. In contrast, actualizing means that the revealed beliefs were a potential among others which were made actual by the operation of questioning—they did not pre-exist. The statistical distribution of answers expresses the (mixed) beliefs. A crucial point that we emphasize here is that simply eliciting beliefs does not provide any *informational* justification for modifying those beliefs (project-state). In the classical context, belief elicitation has no impact. Yet, as we next shall see, as Sender asks such a question Receiver's cognitive state changes which is the signature of its intrinsic indeterminacy.

#### 2.3.5. The Impact of Measurements

In the development, below we focus exclusively on introspective measurements, i.e., questions put to the decision-maker about the (represented) state of the world. This is in accordance with the experiment that follows. We distinguish between two types of measurements. Those that are compatible with each other, they correspond to commuting operations on the project-state. And those that are incompatible which correspond to noncommuting operations. In a similar way we speak of measurements that are compatible (incompatible) with the preferred perspective.

#### Compatible Measurements

The performance of a measurement of the (represented) projects in the individual's preference perspective corresponds to actualizing decision-relevant beliefs. The U-question: do you think that the cause is urgent YES/NO? is for a U-individual compatible with her preferred perspective. The question is formulated as a binary choice YES/NO, the initial mixed project state generates the probabilities for the responses.

In the classical (Bayesian) context this type of questioning is inconsequential (see below). This contrast with the quantum context where it modifies the project-state (beliefs). A compatible YES/NO question induces the 'collapse' of the (mixed) state onto one of the pure states. Consider a U-individual when questioned about her belief regarding the urgency of the elephant cause her prior |*E* collapses onto |*E* = |*UE* with probability |*E*|*UE*|<sup>2</sup> and onto <sup>|</sup>*<sup>E</sup>* <sup>=</sup> *U*⊥ *E* with probability *E U*⊥ *E* <sup>2</sup> and similarly if questioned about the urgency of the Tiger cause |*T* → |*T* = |*UT* or |*T* <sup>=</sup> *U*⊥ *T* where the subscript informs about the project and are neglected when no confusion arises i.e., it is clear which project we talk about. The measurement of the two project-states (corresponding to ECF respectively TF) generates 4 possible combinations of project-states e.g., (|*E*, |*T*) transits onto (|*UE*, <sup>|</sup>*UT*) with a probabilities given by |*E*|*U*|<sup>2</sup> |*<sup>T</sup>* <sup>|</sup>*U*|<sup>2</sup> .

We can now examine the impact of a measurement on the DM's choice in the graphical example above. Recall that in the absence of measurement our U-individual is selecting TF with probability 1 and ECF with probability 0. When the choice is preceded by the Urgency question, with probability *T U*⊥ 2 |*E*|*U*|<sup>2</sup> <sup>&</sup>gt; 0 the resulting states are |*UE*, *U*⊥ *T* . In this event, she selects ECF because *<sup>u</sup>*(*TF*; <sup>|</sup>*<sup>T</sup>* , *U*)*T*=*U*<sup>⊥</sup> *T* = *Tr*(*TUU*<sup>⊥</sup> *<sup>T</sup>* ) = 0 < *u*(*ECF*; |*E* , *U*)*E*=*UE* = *Tr*(*EUUT*) = 1. So, we find that her choice behavior is affected by the question. Similarly, our H-individual will, after answering the compatible question, select TF with positive probability. Thus we have shown that even a "naive" question about the individual's decision relevant beliefs can induce a change in the expected revealed preferences i.e., we already have some "persuasion".

The impact of the mere actualization of beliefs underlines a distinction between the quantum and the classical framework. In the quantum world measurements generally change the state of the measured system (here the beliefs or project-states). This is an expression of the fundamental distinction with the classical world where it is assumed that reality preexists any measurement that merely reveals it. In the quantum world reality is contextual which means that measurements contribute in determining the state i.e., they do not reveal a preexisting state, they contribute in shaping that state. This is called contextuality (see [50] for a rich collection of contributions on contextuality). Another distinction with the classical case is the difference in the impact of compatible versus incompatible measurements as we show next.

### Incompatible Measurement: Distraction

We now turn to distraction which we define as the actualization of beliefs with respect to features not directly relevant to decision-making i.e., belonging to a perspective that is incompatible with the preferred perspective. Below we depict the case when addressing a U-preference individual. Distraction corresponds to putting a H-question e.g., do you believe WWF (managing ECF) is honest YES/NO? As in the compatible case the question triggers the collapse of the project-state |*E* in the basis corresponding to the question, here *H*, *H*⊥ . The state |*E* transits into |*E* equal to either <sup>|</sup>*HE* or *H*⊥ *E* and it does so with probability |*E*|*H*|<sup>2</sup> and *E H*⊥ 2 . And similarly for the H-question regarding TF (the NGO managing the Tiger project). We illustrate this in Figure 2 with the green lines for E the cognitive state representing ECF.

In contrast with the compatible measurement case, after having answered the incompatible question the resulting cognitive state does not allow the DM to evaluate the expected utility associated with the choice alternatives. She needs to project it back into her preferred perspective. The expected utility of ECF for a U-individual in initial project-state |*E* subjected to the H-question, is obtained by considering a sequence of two non-commuting operations. First distraction, the state is projected onto the *H*, *H*⊥ basis. Then the resulting state (*H* or *H*⊥) is projected back onto the preferred basis (*U*, *U*⊥) in order to evaluate the project:

$$\begin{split} \mu(\boldsymbol{E}\boldsymbol{C}\boldsymbol{F};\lvert\boldsymbol{E}\rvert,\boldsymbol{U}) &= \label{squeetive} \left[ |\langle\boldsymbol{E}\rvert\boldsymbol{H}\rangle|^{2} |\langle\boldsymbol{H}\rvert\boldsymbol{U}\boldsymbol{I}\rangle|^{2} + |\langle\boldsymbol{E}\rvert\boldsymbol{H}^{\perp}\rangle|^{2} \Big| \langle\boldsymbol{H}^{\perp}\rvert\boldsymbol{U}\boldsymbol{I}\rangle|^{2} \right] \mathbf{x}\_{\boldsymbol{U}} + \\ & \quad \Big[ |\langle\boldsymbol{E}\rvert\boldsymbol{H}\rangle|^{2} |\langle\boldsymbol{H}\rvert\boldsymbol{U}^{\perp}\rangle|^{2} + |\langle\boldsymbol{E}\rvert\boldsymbol{H}^{\perp}\rangle|^{2} \Big| \langle\boldsymbol{H}^{\perp}\rvert\boldsymbol{U}^{\perp}\rangle|^{2} \Big] \mathbf{x}\_{\boldsymbol{U}^{\perp}}. \end{split}$$

Example

Consider the following numerical example for a H-individual where we simplify the matter by assuming |*T* = |*E* = |*D*, that is the two projects are represented by the same project-state meaning that they are subjectively perceived as equally urgent and honest. Let this state in the H-perspective be

$$D = \begin{pmatrix} 4/5 & 2/5 \\ 2/5 & 1/5 \end{pmatrix}$$

Consider the following utility values *xH* = 8, *xH*<sup>⊥</sup> = 7, *yH* = 10, *yH*<sup>⊥</sup> = 4. Using the formula in (2) we obtain:

$$\mu(ECF; |D\rangle, H) = 4/5 \cdot 8 + 1/5 \cdot 7 = 39/5^{\circ}$$

And similarly

$$\mu(TF; |D\rangle, H) = 4/5 \cdot 10 + 1/5 \cdot 4 = 44/5$$

Which means that this H-individual chooses to donate to TF.

**Figure 2.** Distraction.

Let us now consider a distraction toward the Urgency perspective that we model for simplicity as a 45◦ rotation of the H-basis (which corresponds to the case when the pure states are statistically uncorrelated across perspectives) *U* = 1/2 1/2 1/2 1/2 and *U*⊥ = 1/2 <sup>−</sup>1/2 <sup>−</sup>1/2 1/2 .

The distractive procedure is as follows: first the H-individual in project-state |*D* is asked whether she thinks the Elephant respectively Tiger cause is Urgent or not Urgent which takes the state |*D* onto *UE*(*T*) or *U*⊥ *E*(*T*) . Then, our H-individual evaluates her expected utility value in the *H*, *H*⊥ perspective. With a 45◦ rotation the computation simplifies greatly because whether distraction takes the states to *U*or *U*⊥, the probability for *H* respectively *H*⊥ is the same:

$$\begin{aligned} \left(\mu(\text{ECF};|D\rangle,H)\right)^{\perp} &= \left[|\langle D|\,|\mathcal{U}\rangle|^{2} + |\langle D|\,|\mathcal{U}^{\perp}\rangle|^{2}\right] 1/2\mathbf{x}\_{H} + \left[|\langle D|\,|\mathcal{U}\rangle|^{2} + |\langle D|\,|\mathcal{U}^{\perp}\rangle|^{2}\right] 1/2\mathbf{x}\_{H^{\perp}} \\ &= 1/2\mathbf{x}\_{H} + 1/2\mathbf{x}\_{H^{\perp}} = \mathbf{4} + 3.5 = 7.5 \end{aligned}$$

and similarly

$$\begin{array}{rcl} \mu(TF; |D\rangle, T) &=& \left[|\langle D| | \boldsymbol{l} \boldsymbol{l} \rangle|^{2} + |\langle T | \boldsymbol{l} \boldsymbol{l}^{\perp} \rangle|^{2}\right] \mathbf{1}/2y\_{H} + \left[|\langle T | \boldsymbol{l} \boldsymbol{l} \rangle|^{2} + |\langle T | \boldsymbol{l} \boldsymbol{l}^{\perp} \rangle|^{2}\right] \mathbf{1}/2y\_{H^{\perp}} \\ &=& 1/2y\_{H} + 1/2y\_{H^{\perp}} = \dots = 2 = 7. \end{array}$$

So after the distraction our individual chooses to donate to the ECF project with probability 1 instead of TF in the absence of distraction. So we note that the impact of distraction can be a total reversal of the choice. This is in contrast with the impact of decision-relevant belief actualization (compatible measurement) which only triggers some partial reversal. For a general theoretical argument on the persuasion power of a distractive IS as compared with a compatible IS see [23].

Before moving to the experiment let us briefly remind ourselves of the classical subjective uncertainty approach in our example.

#### 2.3.6. The Classical Uncertainty Approach

The classical uncertainty framework is nested in the quantum setting. It corresponds to the case when all properties of an item (perspectives) are compatible and therefore Lüder's rule for updating is equivalent to Bayesian updating. The individual can simultaneously considers Urgency and Honesty and combine them to obtain her expected utility value. Assuming a separable and additive utility function, we write

$$\begin{cases} u(T) &= \left. a \left( p\_{ll}^{0}(T) \mathbf{x}\_{lI} + \left( 1 - p\_{ll}^{0}(T) \right) \mathbf{x}\_{lI^{\perp}} \right) + (1 - a) \left( p\_{T}^{0}(T) \mathbf{x}\_{H} + \left( 1 - p\_{T}^{0}(T) \right) \mathbf{x}\_{H^{\perp}} \right) \right|\_{t} \\ u(E) &= \left. a \left( p\_{ll}^{0}(E) \mathbf{y}\_{lI} + \left( 1 - p\_{lI}^{0}(E) \right) \mathbf{y}\_{lI^{\perp}} \right) + (1 - a) \left( p\_{T}^{0}(E) \mathbf{y}\_{T} + \left( 1 - p\_{T}^{0}(E) \right) \mathbf{y}\_{T^{\perp}} \right) \end{cases}$$

where *p*<sup>0</sup> *<sup>U</sup>*(*T*) is the subjective probability in state |*T* that the TF project is urgent (and *c* <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>0</sup> *<sup>U</sup>*(*T*)*c* that it is not urgent) and similarly for the other probabilities. The superscript refers to time *t* = 0 (initial beliefs). The *α* is the relative preference weight given to urgency ((1 − *α*) the relative weight of honesty). An individual for whom honesty is determinant is an individual with *α* < 1/2 and similarly for U-individuals (*α* ≥ 1/2).

Recall that the measurements that we consider are exclusively introspective i.e., no information appealing to the outside world is called upon. In other words, the questions correspond to eliciting Receiver's beliefs.

When asked "do you believe the tiger cause is urgent YES/NO. With probability *p*<sup>0</sup> *<sup>U</sup>*(*T*) Receiver answers YES and with probability <sup>1</sup> <sup>−</sup> *<sup>p</sup>*<sup>0</sup> *<sup>U</sup>*(*T*) she answers NO. But her beliefs do not change, they remain mixed. Since beliefs are unchanged so is the expected utility from the two projects. As a consequence Receiver's choice is not affected by Sender's question. To put it differently, an introspective measurement has no persuasion power whatsoever in the classical context. The classical prediction contrasts starkly with the quantum model where an introspective measurement with respect to both compatible and incompatible perspectives has impact on decision-making. Those predictions appears more consistent with numerous experimental works that exhibit a significant impact of belief elicitation on decision-making see most recently [51]). In addition, as illustrated above, incompatible introspective measurements have the strongest potential to affect decision-making. TIt is precisely this prediction that we aim at testing with the next following experiment.

#### **3. Experimental Design**

Our main experiment features the choice to donate to either one of two projects concerned with the protection of endangered species. It uses the property of Bohr complementarity of mental perspectives. More precisely it relies on the hypothesis that two perspectives on the projects are incompatible in the mind of Receiver. The two perspectives that we consider are "the urgency of the cause" and "the honesty of the organization that manages the funds' (the terms "honesty" and "trustworthiness", or "trust", are used interchangeably). As a first step we provided experimental support for the incompatibility hypothesis. We know that when two properties are incompatible measuring them in different orders yields different outcomes. Therefore, we started with an experiment to check whether order matters for the response profile obtained. Note that even in Physics, there is no theoretical argument for establishing whether two properties are compatible or not. This must be done empirically.

#### *3.1. Testing for the Incompatibility of Perspectives*

At the time we conducted our study, the world was confronted with a severe refugee crisis in Myanmar. The situation actualized quite sharply the two perspectives we wanted to test. On the one hand, the urgency of the humanitarian crisis and, on the other hand, the uncertainty about the reliability/honesty of the NGOs on the ground.

We recruited 295 respondents through Amazon's Mechanical Turk, for which data quality has been confirmed by different studies (e.g., [52,53]). The respondents completed the short survey below on the website Typeform. They were paid \$0.1 and spent on average 0:17 minutes to complete the survey.

The participants were first presented a screen with a short description of the situation of refugees in Myanmar including a mention of the main humanitarian NGO present in the field:

*"About a million refugees (a majority of women and children) escaped persecution in Myanmar. Most of them fled to Bangladesh. The Bengali Red Crescent is the primary humanitarian organization that is providing help to the Rohingyas. They are in immediate need of drinkable water, food, shelter and first medical aid."*

They were then asked to evaluate the urgency of the cause and the honesty to the NGO on a scale from 1 ("Not urgent" or "Do not trust") to 5 ("Extremely urgent" or "Fully trust"). The order of presentation of the two questions was randomized so that half of participants responded to the urgency question before trust (U-T), and the other half conversely (T-U).

We prove the existence of order effects by showing that the responses are drawn from two different distributions. We do this using both a difference in means test (i.e., two-sample *t*-test with *t* = −2.54 and *p*-value = 0.011) and a nonparametric test of the two sample distributions (i.e., two-sample Kolmogorov-Smirnov test with *D* = 0.11 and *p*-value = 0.047) in *R*.

The results are consistent with the hypothesis that the two perspectives are incompatible in the mind of people. As well-known there exist other theories for order effects. We observe that the value of the responses to the Trust question tend to be lower when that question comes first (i.e., T-U), whereas the responses to the Urgency question tends to have a higher value when that question comes first (i.e., U-T). Therefore, we can reject both the hypothesis of a recency biais and that of a primacy bias. This strengthens our quantum interpretation. We next proceed to the main experiment using those two perspectives.

### *3.2. Main Experiment*

1253 participants completed the survey on the website Typeform, they were recruited through Amazon's Mechanical Turk. They were paid either \$1 to \$0.75 depending on the condition.

The participants were divided into three groups. Two treatment groups and a control group as explained below. All three groups were presented a screen with an introductory message, informing them that the questionnaire is part of a research project on quantum cognition and that they will contribute in deciding which one of two NGOs projects will receive a -50 donation. The decision will be made by randomly selecting a respondent and implementing his or her decision. Presumably, this created an incentive to respond truthfully. The respondents were next asked to click on a button that randomly assigned them to a specific condition. In all conditions, participants were shown a short text about the situation of elephants respectively tigers and of ongoing actions of two NGOs working for their protection the Elephant Crisis Fund (ECF) and Tiger Forever (TF). The order of presentation of the text was reversed for half of the subjects. This aimed at isolating order effects not relevant to our main point. The screen displayed the following two texts:

*"Elephant crisis fund: A virulent wave of poaching is on-going with an elephant killed for its tusks every 15 min. The current population is estimated to around 700,000 elephants in the wild. Driving the killing is international ivory trade that thrives on poverty, corruption, and greed. But there is hope. The Elephant Crisis Fund closely linked to World Wildlife Fund (WWF) exists to encourage collaboration, and deliver rapid impact on the ground to stop the killing, the trafficking, and the demand for ivory."*

"*Tiger Forever: Tigers are illegally killed for their pelts and body parts used in traditional Asian medicines. They are also seen as threats to human communities. They suffer from large scale habitat loss due to human population growth and expansion. Tiger Forever was founded 2006 with the goal of reversing global tiger decline. It is active in 17 sites with Non-Governmental Organizations (NGOs) and government partners. The sites host about 2260 tigers or 70% of the total world's tiger population"*

It is worth mentioning that the descriptions were formulated so as to slightly suggest that the elephants' NGO (EFC) could be perceived as more trustworthy (because of its link with of WWF, a well-known NGO). In contrast, the text about tigers suggested a higher level of urgency (the absolute number of remaining tigers is significantly lower than the number of remaining elephants). Thereafter, all respondents were confronted with a choice:

*"When considering donating money in support of a project to protect endangered species, different aspects may be relevant to your choice. Let us know what counts most to you:*

*-The urgency of the cause: among the many important issues in today's world, does the cause you consider belong to those that deserve urgent action? or*

*-The honesty of the organization to which you donate: do you trust the organization managing the project to be reliable; i.e., do you trust the money will be used as advertised rather than diverted."*

The objective was to elicit an element of their preferences namely their preferred perspective, see Section 2. The rest of the questionnaire depended on which one of the three groups the participants belonged to.

In the control condition (baseline), they were next asked whether or not they wanted to read the first descriptions again or if they wanted to make their final decision i.e., to make their choice between supporting the Elephant Crisis Fund or Tiger Forever both represented by an image of an adult elephant respectively adult tiger (presented in random order on the same screen).

In the first treatment condition, the respondents were redirected to a screen with general information compatible with the aspect they indicated as determinant to their choice when making a donation. Importantly, the information did not directly or indirectly favor or disfavor any of the two projects. The information was aimed at triggering a measurement as they were invited to determine themselves with respect to which of the species was most urgent to save respectively which NGO was most trustworthy. We below return to the role and expected impact of the general information screens. Those who cared most honesty saw a screen with the following text:

"*Did you know that most Elephant and Tiger projects are run by Non-Governmental Organizations (NGOs)? But NGOs are not always honest! NGOs operating in countries with endemic corruption face particular risks. NGOs are created by enthusiastic benevolent citizens who often lack proper competence to manage both internal and external risks. Numerous scandals have shown how even long standing NGOs had been captured by less scrupulous people to serve their own interest. So a reasonable concern is whether Tiger Forever respectively Elephant Crisis Fund deserves our trust."*

Those who cared most for urgency saw:

*"Did you know that global wildlife populations have declined 58% since 1970, primarily due to habitat destruction, over-hunting and pollution. It is urgent to reverse the decline! "For the first time since the demise of the dinosaurs 65 million years ago, we face a global mass extinction of wildlife. We ignore the decline of other species at our peril–for they are the barometer that reveals our impact on the world that sustains us." —Mike Barrett, director of science and policy at WWF's UK branch. A reasonable concern is how urgent protecting tigers or elephants actually is."*

Thereafter the respondents were offered the opportunity to read again the descriptions before making their image choice between ECF and TF.

In the main treatment condition (distraction), participants were redirected to a screen with general information on the aspect they did not select as determinant to their choice; this is what we call a distraction. So, those who selected honesty (resp. urgency) saw the screen on global wildlife decline (resp. NGO's scandals). Thereafter, the respondents were offered the opportunity to read again the initial project description before making their image choice.

Finally, information about their age, gender, education and habits of donation to NGOs was collected before the thank-you message ending of the experiment.

Before presenting the results, we wish to address a feature of the experimental design absent from the theoretical model.

#### The General Information Screens

First, we note that the theoretical model does not account for anything like a general information screen. The connection with the model is with the questions that follow the general information. Indeed, general information plays no role for persuasion since it conveys no new data on the relative urgency or honesty of the specific projects. Therefore it should not affect the choice between the two projects. So, what is the role of those screens?

Our justification for the general information screens is to be found in the quantum approach to cognition. Quantum cognition recognizes that people consider project from different perspectives some of which may be incompatible but not mutually exclusive. They are Bohr complementary which implies that the questions related to those perspectives do not commute i.e., order matters. This in turn is an expression of the fact that the cognitive state is modified by responding to a question. Our intuition is that there can be some inertia. Consider a person who declared that Honesty is her priority which we interpret as her being in the Honesty perspective. If you abruptly ask whether the elephant cause is urgent, she might not make the effort to switch perspective in order to respond faithfully. In contrast if you softly accompany her into the switch with an engaging short text, she will find herself capable of responding truthfully without particular effort.

Hence the point with those screens is to accompany the change in perspective. Clearly, that is only justified in the distraction treatment, but for the sake of symmetry, we have a similar screen in the treatment where no change of perspective is required.

Next, one may wonder why we do not, in the experiment, simply ask people for their beliefs e.g., do you trust that NGO? But instead we suggest a questioning: "So a reasonable concern is whether Tiger Forever respectively Elephant Crisis Fund deserves our trust". The reason is that we wanted to avoid that the response would influence the respondent beyond the impact under investigation. Additional impact can be expected because of perceived dissonance. Assume the individual cares for the urgency of the cause and since there are only 2700 tigers left, so she is most likely to choose TF. If she is explicitly asked whether she believes that the TF NGO is honest and decides that she does not trust them much, then it becomes psychologically difficult to select TF. Since we do not put an explicit question but use a general text to induce the measurement, people are expected to be less likely to perceive dissonance and choose more spontaneously. These precautions are among the difficult decisions we have to make in quantum cognition when trying to exhibit quantum effect in behavior. Individuals are, in contrast with particles, thinking systems endowed with, among other things, a drive toward consistency which can interfere with the intrinsic indeterminacy of preferences (see e.g., [26,54]) and Discussion in Section 5).

#### *3.3. Theoretical Predictions*

Before getting into the results and their interpretation, let us remind of the main theoretical predictions:


treatment compared with the control group. It should be emphasized that since we lack information about the correlation coefficients between the two perspectives and the utility values, we do not have quantitative predictions. Generally, the less correlated two perspectives (in the example of Section 2 they were fully uncorrelated) the larger the expected impact in terms of switching the choice for given utility values.

#### **4. Results**

#### *4.1. Descriptive Statistics*

Data were processed, cleaned and analyzed with statistical software R. The number of acceptable observations was 1253 (114 participants were removed from the data due to a technical error which created a risk that some participants might have responded twice). 58.5% of all respondents were male, the average age was 35.6 years and the average education level was undergraduate. Overall, 71.1 % of the participants declared that the Honesty of the NGO rather than the Urgency of the cause is what counts most to their choice. Across the three conditions, 54.4% chose to support with their donation the Elephants Crisis Fund (ECF) and 45.6 the Tiger Forever project (TF). Looking into the different treatment groups, we find that 59% of the respondents in the control condition chose ECF and 54% in the compatible information treatment group. In contrast, in the distraction treatment group, only 47% chose ECF. Conditional on revealed preferences, 50% of the respondents who valued Urgency most chose to support TF, whereas 56% of those who valued Honesty most chose to support ECF. Overall, 87.8% made their final decision without reading the description of the projects a second time. They spent, on average, 1:33 minutes to complete the experiment.

We divided up the respondents into a number of subgroups based on their preferences and the detailed treatment they received. Figure 3 represents the number of participants who chose TF respectively ECF conditional on their preference (Honesty vs. Urgency) and the order of the presentation they have been exposed to (ECF-TF vs. TF-ECF). Note that for all conditions but "Honesty-ET", a majority of participants chose ECF in the control condition and TF in the incompatible condition. This is particularly striking for "Honesty-TE" and "Urgency-ET". At first glance, we find a clear reversal in three of the subgroups.

**Figure 3.** Descriptive Histograms.
