**1. Introduction**

Why is the famous P&G (Procter and Gamble) 2010 "Thank you, Mom" advertisement [1] showing devoted mothers supporting young athletes, among the most successful ads of all time? The pervasiveness of informationally irrelevant messaging in advertising is stunning. In this paper, we present and provide experimental evidence for an application of quantum-like decision-making theory that explains why distraction—i.e., addressing informationally irrelevant issues—can be a powerful manipulation technics.

The idea that people are being influenced and manipulated by a systematic exploitation of non rational psychological factors rather that by providing information that is rationally processed, was first forcefully put forward in the seminal book of Vance Packard (1957) "The Hidden Persuaders" [2]. His thesis is that persuador relies on psychiatric and psychological technics to address their message to our "wild and unruly subconscious". Later Cialdini developed a "science of persuasion" based on a general behavioral principles (e.g., bias for reciprocity) that can be exploited to influence people's choice (see e.g., [3,4]). Closer to our approach which focuses on information processing, is an early work by Festinger and Maccoby [5]. They published the first experiment showing that distraction can induce attitude change: a message has larger persuasion power among respondents subjected to distraction. Their idea is that distraction in the course of information processing makes attempts to provide counter arguments less successful. This in turn makes

**Citation:** Lambert-Mogiliansky, A.; Calmettes, A. Phishing for (Quantum-Like) Phools—Theory and Experimental Evidence. *Symmetry* **2021**, *13*, 162. https://doi.org/ 10.3390/sym13020162

Academic Editor: G. Jordan Maclay Received: 14 December 2020 Accepted: 11 January 2021 Published: 21 January 2021

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people more vulnerable to the persuador's message. Later, we saw the development of a broad literature in psychology showing that distraction may decrease attention, impair learning and remembering opening up for manipulation [4,6–11]. Failures in information processing are also a the heart of Nobel-prize winner Kahneman's best selling book "Thinking Fast and Slow" [12]. In the last section, we discuss how they relate to our approach to distraction.

More recently, Akerlof and Shiller (2015) provided loads of evidence showing that people are systematically "phished" in economic transactions. The authors suggest that this is due to the significance of the story people tell themselves when making decision i.e., the "narratives" or as they also write "the focus of the mind". They conclude "just change people's focus and you can change the decisions they make" (p. 173 [13]). Emphasizing the significance of the "narratives" is closely related to a rich literature in psychology on framing effects (see among others [14–21]).

Quantum cognition offers an approach to the concept of narratives in terms of perspectives on reality [22]. Formally, a perspective is a coordinate system of a state space and there exists a number of equally valid alternative coordinate systems. Different perspectives can be simultaneously true but not compatible with each other. In this paper, we rely on a formalisation of the concept of narratives in line with the general theory of quantum decision-making. As shown in [23], that theory delivers a power of distraction in the terms of Akerlof and Shiller. The power of distraction arises from non-Bayesian information processing reflecting the mathematical structure of the quantum model. Experimental evidence (see e.g., [24,25]) shows that people oftentimes systematically depart from Bayes' rule when confronted with new information. Cognitive sciences propose a number of alternatives to Bayesianism (see e.g., [26–29]). The attractiveness of the quantum approach is partly due to the fact that quantum mechanics has properties that reminds of the paradoxical phenomena exhibited in human cognition. In addition, quantum cognition has been successful in explaining a wide variety of behavioral phenomena such as disjunction effect, cognitive dissonance or preference reversal (see among others [30–35]). Importantly, there exists by now a fully developed decision theory in the context of non-classical (quantum) uncertainty. Different formulations of that theory exist, including that by Aerts et al. [36]. In this paper, we rely on the formulation developed by Danilov et al. [37,38]. Clearly, the mind is likely to be even more complex than a quantum system, but our view is that the quantum cognitive approach already delivers interesting new insights in particular with respect to persuasion.

In quantum cognition, the object of interest is the decision-maker's mental representation of the world. It is modelled as a quantum-like system represented by its state—a cognitive state which is the equivalent of beliefs in the classical context. In quantum cognition, the decision relevant uncertainty is consequently of non-classical (quantum) nature. As argued in [22] this modelling approach allows capturing widespread cognitive limitations in information processing. The key quantum property that we use is the "Bohr complementarity" of characteristics (properties) of the mental object (representation of the world). The decision-maker cannot consider all properties simultaneously i.e., they cannot have a definite value in his mind. Instead, the decision-maker processes information sequentially moving from one perspective to the other and order matters.

As in the classical context our rational decision-maker uses new information to update her beliefs. A rational quantum-like decision-maker is a decision-maker who has preferences over mental objects representing items (or actions). These mental objects are modelled as quantum-like systems. Her preferences satisfy a number of axioms that secure that they can be represented by an expected utility function. In [38] we learned that a dynamically consistent quantum-like decision-maker updates her beliefs according to Lüder's postulate which in Quantum Mechanics governs state transition following the measurement of a system. In two recent papers, important theoretical results were established. First, it is shown in [39] that in the absence of constraints (on the number of operations that trigger updating), full persuasion applies: Sender can always persuade Receiver to believe anything that he

wants. Next, in [23] the same authors investigate a short sequence of operations but in the frame of a simpler task that they call "targeting". The object of "targeting" is the transition of a belief state into another specified target state. The main result of relevance to our issue is that distraction i.e., a test or question that generates irrelevant but "Bohr complementary" information has significant persuasion power. In contrast, a Bayesian decision-maker does not update her beliefs when the information is not relevant to her concern and thus cannot be persuaded in this manner to change her decision.

In the present paper we first formulate a model of quantum-like decision-making in the context of a choice between two uncertain alternatives. The model is used to derive the impact of relevant respectively distractive information on choice behavior. The results are contrasted with those of Bayesian persuasion. A contribution of the paper is to provide a first (illustrative) experimental test of the model's predictions. We opted for a more basic treatment of the data because the quantum persuasion model is so rich that a rigorous estimation of the relevant parameters is beyond the scope of the present paper. The experimental situation that we consider is the following. People are invited to choose between two projects aimed at saving endangered species (elephants and tigers). The selected project will receive a donation of 50 euros (one randomly selected respondent will determine the choice). We consider two perspectives of relevance for the choice: the urgency of the cause and the honesty of the organization that manages the donations. As a first step and in a separate experiment we establish that the two perspectives are incompatible by exhibiting a significant order effect which is the signature of incompatible measurements (see [40]). In the main experiment 1253 respondents are divided into three groups: a control and two treatment groups. They all go through a presentation of the projects and some questions about their preferences. The difference between the groups is that the first treatment group is invited to answer a question about their beliefs of direct relevance to their choice while the second must answer a question that distracts them from what is relevant to their choice. We find that, at the population level, the results are in accordance with the predictions of the quantum model: the distractive question has a significant impact on the respondents choices as compared with both the control group and the other treatment group. The pattern of reactions is disconnected from the thematic content of the distractive information screen which is to be expected when the two perspectives are incompatible. In contrast the question on decision relevant beliefs had no significant impact compared to the control group. The data reveal some significant variation between subgroups with respect to their responsiveness to distraction. In particular, we find that people who care about the urgency of the cause are more responsive to distraction. We argue that this is consistent with the quantum model under the reasonable assumption that those people are more passionate about the issue. The quantum-like working of the mind is expected to be more pronounced for passionate people. This is because standard rational thinking which denies the contextuality of mental representations tends to constrain that spontaneous drive. We conclude with a discussion on rationality in information processing and relate our approach to other prominent behavioral theories.

This paper contributes to the economic literature on persuasion initiated by Kamenica and Gentskow's seminal article "Bayesian Persuasion [41]. More precisely, it contributes to its recent development which introduces various kinds of imperfections in information processing one example is Bloedel and Segal's "Persuasion with rational inattention" [42], which show how Sender optimally exploits Receiver's inattention. Another example is Lipnowsky and Mathevet [43]. Their focus is on how Sender responds to Receiver's problem with temptation and self-control by adapting the signal structure. More closely related to our work is Galperti "Persuasion—the art of changing worldviews" [44]. The author is interested on how a better informed Sender can modify Receiver's incorrect worldview with "surprises" that trigger a change in the support of Receiver's beliefs. Our approach is different because we do not assume that there is a single correct worldview. As shown in [45], a large number of non-bayesian rules systematically distort updated beliefs. However, Kamenica and Gentzkow's concavification argument for optimal persuasion extends

to such rules which boil down to introducing some form of bias. Kamenica and Gentzkow's result entails that Senders payoff is concave in Receivers's belief so that Sender's problem can be formulated as the choice of Receiver's posteriors. Our contribution departs more fundamentally from Kamenica and Gentzkow because quantum cognition relies on nonclassical (quantum) uncertainty (contextuality) so in particular that result does not apply. This approach reveals a powerful role for distraction in persuasion and we provide some experimental evidence for it.

Our results also contribute to the literature in psychology by offering a novel explanation for the well documented distraction-persuasion nexus. Our study provides support to the thesis that people's propensity to be persuaded is due to the contextuality (intrinsic indeterminacy) of their representation of the world rather than to limited cognitive capacity or to some bias. In so doing our paper contributes to the growing literature in quantum cognition (see recent contributions in [46–49] for other examples on how the (quantum) contextuality approach offers a new paradigm for explaining a variety of behavioral phenomena.

The paper is organized as follows. We first briefly remind of the classical Bayesian persuasion approach. Next we provide a quantum-like model of choice between two uncertain alternatives. We formulate the predictions related to the impact of information on choice behavior. In the second part of the paper, we first describe the experimental set-up used to test the predictions. We thereafter report and inteprete the results from the analysis of the data. We conclude with a discussion of our results in view of some of the existing literature.

#### **2. Quantum Persuasion**

#### *2.1. Bayesian Persuasion*

Let us first briefly describe Bayesian persuasion an approach developed by Kamenica and Gentzkow [41] in a classical uncertainty setting. The subject matter of the theory of Bayesian persuasion is the use of an "information structure", we shall refer to it a "measurement" (in practical terms, it corresponds to an investigation, a test or a question), that generates new information in order to modify a person's state of beliefs with the intent of making her act in a specific way.

More precisely the setting involves two players Sender and Receiver. Receiver chooses an action among a set of alternatives with uncertain consequences. An action yields consequences for both players. Sender may try to influence Receiver so she chooses an action that is most valuable to him. A crucial element of the Bayesian persuasion approach is that Sender does not choose the information Receiver obtains. If he did that would raise issues of strategic concealment and revelation. Instead Sender chooses an "information structure" (IS) or a measurement that is a test, an investigation or a question. Sender is committed to truthfully reveal the outcome of the IS (e.g., he does not control the entity that performs the study). One example is in lobbying. A pharmaceutical company commissions to a scientific laboratory a specific study of a drug impact, the result of which is delivered to the regulator. Another example, closer to our application here, is a question to Receiver: do you believe (Yes or No) that politicians' climate inaction will lead to global catastrophe under this century? The outcome of any IS is information. In our examples above, it is information about the impact of a drug or about the opinion (beliefs) of Receiver on the responsibility of politicians. This information generally affects Receiver's beliefs which in turn may affect her evaluation of the uncertain choice alternatives and therefore the choices she makes. Sender chooses an IS to move Receiver's (expected) choice closest to his own preferred choice. In the classical context Receiver updates her beliefs using Bayes rule and therefore the power of Sender is constrained by Bayesian plausibility: the fact the expected posteriors must equal the priors.

#### *2.2. The Quantum Persuasion Approach*

The quantum persuasion approach has been developed in the same vein as Bayesian persuasion: we are interested in how Sender can use an IS to influence Receiver's choices. A central motivation is that persuasion seems much more influential than what comes out of the Bayesian approach. So instead of assuming that agents are classical Bayesian, it has been proposed that their beliefs are quantum-like. A first line of justification is that people do not make decision based on reality but based on a representation of that reality, a mental object. In quantum cognition, the decision-maker's mental representation is modeled as a quantum-like system and characterized by a cognitive state. The decision relevant uncertainty is therefore of a non-classical (quantum) nature. A second line of motivation is that, as argued in e.g., [30,38], this modeling approach allows capturing widespread cognitive limitations. In particular, the fact that people face difficulties in combining different types of information into a stable picture. Instead, the picture (mental object) that emerges depends on the order in which information is processed. The key quantum property that we appealed to is 'Bohr complementarity' of attributes i.e., that some attributes (or properties) of a mental object may be incompatible in the decision-maker's mind: they cannot have definite value simultaneously. A central implication is that measurements (new information) modifies the cognitive state in a non-Bayesian well-defined manner: the mental representation evolves in response to new information in accordance Lüder's rule which, as shown in [38], secures the dynamic consistency of preferences. As shown in [23,39] a rational quantum-like decision-maker can be manipulated well beyond the limits imposed by Bayesian plausibility. In particular, Sender can exploit the incompatibility properties of certain attributes in Receiver's mind by providing distracting information to modify her representation and consequent choice.
