Sep 8, 2017 - we shall use the Hilbert space model of QM to represent the beliefs of an ... After this intermezzo, they return to the emergency issue and our. MP does not feel as convinced as before and chooses to vote down the law.
Targeting in Quantum Persuasion Problems
arXiv:1709.02595v1 [physics.soc-ph] 8 Sep 2017
V. I. Danilov∗and A. Lambert-Mogiliansky† 28.08.17
Abstract In this paper we investigate the potential for persuasion linked to the quantum indeterminacy of beliefs. We focus on a situation where Sender only has few opportunities to influence Receiver. We do not address the full persuasion problem but restrict attention to a simpler one that we call targeting i.e., inducing a specific belief state. The analysis is develop within the frame of a n−dimensional Hilbert space model. We find that when the prior is known, Sender can induce a targeted belief with a probability of at least 1/n when using two measurements. This figure climbs to at least 1/2 when both the target and the belief are pure states. A main insight from the anlysis is that a strategy of distraction is used as a first step to confuse Receiver. We thus find that distraction rather than the provision of relevant arguments is an effective means to achieve persuasion. This is true under the hypothesis that Receiver’s belief exhibits quantum indeterminacy which is a formal expression of well-known cognitive limitations. We provide an example from political decision-making.
1
Introduction
In this paper we build further on a theoretical result (forthcoming in [5]). The result establishes the power of unconstrained belief manipulation or persuasion in the context of nonclassical (quantum) uncertainty. This result was a first step in the development of Quantum ∗ †
CEMI RAS, Moscow Paris School of Economics.
1
persuasion theory in the spirit of ”Bayesian Persuasion” by Kamenica and Gentskow [9], hereafter KG. Bayesian persuasion should not be confused with the theory of persuasion in communication games where an informed party (Sender) with certifiable information chooses what pieces of information to reveal to a uninformed decision-maker (Receiver). The subject matter of Bayesian persuasion is the choice by Sender of an information structure (or a measurement device). In particular in KG, Sender is not better informed so they are not concerned with strategic revelation (or concealment) of information. The only choice variable of Sender is the measurement itself which is performed publicly so the outcome is automatically fully revealed. And the question is how much can be achieved in terms of modifying a rational Receiver’s belief by a proper choice of measurement device. The ultimate goal is to influence Receiver’s decision to act which depends on her beliefs about the world. KG investigate the issue in the classical uncertainty framework. Receiver’s beliefs are expressed as a probability distribution over the set of states of the world and updating follows Bayes rule. However as amply documented the functioning of the mind is more complex and often people do not follow Bayes rule. Cognitive sciences propose various alternatives to Bayesianism. One avenue of research within cognitive sciences appeals to the formalism of quantum mechanics. A reason is that QM has properties that reminds of the paradoxical phenomena exhibited in human cognition. But the motivation for turning to QM is much richer. First, the two fields share fundamental common features namely that the object of investigation and the process of investigation cannot always be separated. This similarity was already put forward by the fathers of quantum mechanics. Indeed its mathematical formalism was developed to respond to that epistemological challenge. In addition, quantum cognition has shown successful in explaining a wide variety of behavioral phenomena (see for a survey Bruza and Busemeyer [3]). Finally, there exists by now a fully developed decision theory relying on the principle of quantum cognition (see [4, 6]). Therefore in the following we shall use the Hilbert space model of QM to represent the beliefs of an individual and capture the impact of new information on those beliefs. 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, we distinguish between the ”world” and the decision-maker’s 2
mental representation of it which is the basis for her decision. This representation of the world is modelled as a quantum-like system and characterized by a cognitive state. The decision relevant uncertainty is of non-classical (quantum) nature. As argued in Dubois et al. ([7]) this modeling approach allows capturing widespread cognitive difficulties that people exhibit when constructing a mental representation of a ‘complex’ alternative. The key quantum property that we use is that some characteristics (or properties) of a complex mental object may be ”Bohr complementary” that is 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 kind of situation we have in mind can be expressed in the following story. A member of parliament (MP) is considering voting for a law to introduce a state of emergency in order to fight against terrorism. The threat of terror act can be either severe or moderate. If the threat is severe enough, our MP will support the law but not otherwise. Initially, she believes that the threat is severe with a high probability, so she would support the law. But a civil liberty activist wants her to vote down the law. Instead of trying to bring forward arguments about the actual threat, he brings up another topic: the EU mode of decisionmaking and its intrusive regulatory policies. In particular, he proposes to find out whether the EU is regulating the weight of cucumbers - our MP represents farm owners. EU spent many months reflecting on cucumbers and ended up with a strict regulation which is broadly perceived as nonsensical. After this intermezzo, they return to the emergency issue and our MP does not feel as convinced as before and chooses to vote down the law. In section 3 we show how this story can be formalized and explained in terms of quantum persuasion. Our result in [5] shows that with an unbounded sequence of measurements, the belief state of a quantum-like Receiver can be brought into any desirable state starting from any initial unknown state. Interestingly, the idea of turning to a sequence of measurements never arises in the classical context because any sequence of measurements can be merged into a single one (possibly very hard to implement in practice though). In the non-classical context however, it is generally not possible to merge measurements unless they are compatible in which case we are back to the classical uncertainty setting. The non-classical framework allows to account for incompatible measurements which can therefore not be merged. This incompatibility is an expression of the fact that measurements modify the system. How 3
new information modifies the cognitive state is established in [6]. Updating follows the von Neumann-Luders rule which generalizes Bayes rule to quantum context. Our result about full manipulability suggests that the power of quantum persuasion is much larger than in the classical context where it is constrained by Bayesian plausibility.1 This first finding calls for further elaboration in order to extract results of more practical significance. Here, we are interested in the polar case when only a limited sequence of measurement is feasible; more precisely we shall focus on the case of one or two measurements. In the present paper we focus on persuasion, i.e. on a manipulation of belief and we refer to [6] for a complete description of decision-making under non-classical uncertainty. Moreover, because of the highly non linear structure of the problem, we are not able to derive results about optimal measurement(s). Instead, we deal with a simpler task that we call ”targeting”. The object of ”targeting” is the transition of a belief state into a specified target state that induces the desired action. This also imply that we focus on a binary decision situation for Receiver. Our main result (Theorem 2) shows that if the initial belief state (prior) is known, any target can be reached with a probability of at least 1/n (where n is the dimension of the corresponding Hilbert space). This conclusion is also true for unknown prior if the target is a pure state. When the prior and the target are pure states, the probability for successful manipulation jumps to at least 1/2. This is because one can then reduce the problem to a two dimensional one. The path of measurements is of interest on its own. The second measurement fully determines Receiver’s decision: we ask Receiver to determine herself with respect to the decision relevant uncertainty. For instance: do you think this used smartphone is worth more than 400 euro (target) or less when the selling price is 400 euro. Clearly, if she thinks it is worth less than 400 euros, it is equivalent to a decision not to buy. However in a nonclassical context, measuring the beliefs changes the state so it is not equivalent to making the decision on the basis on the non elicited (unmeasured) belief.2 What we find most interesting is the characterization of the first measurement which aims at creating a state of uniform 1 2
Bayesian plausibility means that the expected posteriors is equal to the priors. The impact of eliciting beliefs on decision-making has been exhibited in experimental set-ups see, e.g.
[8].
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uncertainty. We show in Theorem 1 that Sender can always transform known Receiver’s prior into uniform uncertainty which explains appearance of probability 1/n. One interpretation is that the power of quantum persuasion is related to the possibility of creating confusion in Receiver’s mind. The strategy involves exploiting the incompatibility of perspectives (non commutativity of measurements). Basically, this is a strategy of ”calculated diversion” to instigate confusion. Once Receiver is confused, it is much easier to persuade her to do what Sender wants. The seminal paper by Kamenica and Gentskow [9] analyzes the general problem of persuading a rational agent by controlling her information environment. It was followed by a series of other papers by the same authors addressing competition in persuasion, costly of persuasion, and most recently Bayesian persuasion with multiple senders. It also gave rise to a number of applications, see for instance ”Persuading Voters” by Alonso Camara [2]. Our contribution extends that literature with a theoretical development to non-classical uncertainty. Moreover Akerlof and Schiller (2015), argue that manipulation is determinant to the functioning of markets. They suggest that it goes well beyond Bayesian updating: ”change the focus of people mind and you will change the way the decisions they make” (p.173). In this paper we show that quantum persuasion is more powerful than Bayesian persuasion even for short sequences of measurements and that it delivers a mechanism where changing the focus of the mind affects decision-making. The paper proceeds as follows. In section 2 we expose the persuasion problem in the classical setting. Thereafter we formulate the corresponding problem in the quantum context. In section 3 we analyze the targeting task and derive our main results. Thereafter we provide an illustration in terms of the example presented in the Introduction. We conclude with some final remarks.
2
The model
We shall formulate the persuasion problem in terms of a communication game as we want to facilitate the comparison with Kamenica and Genskow (KG) [9]. There are two players called respectively Receiver and Sender. We are interested in persuasion aimed at influencing Receiver’s choice over uncertain alternatives which we model as quantum lotteries following 5
[6].
2.1
The classical setting
Classical, or Bayesian, persuasion has been well described by KG. There is a set Ω of states of Nature. Receiver’s belief is a probability distribution or measure β on Ω, β (ω) ≥ 0, P β ω (ω) = 1 (below we denote β as B). For the sake of simplicity we shall assume that the set Ω is finite. So that we have the simplex ∆ (Ω) of probability distributions on Ω. For a given belief β Receiver chooses an action that maximizes her expected utility a∗ (β). Each action a brings Sender utility u(a). (Here we assume that Sender’s utility only depends on the action chosen by Receiver and not on the state of the world). Sender tries to influence receiver’s choice by means of additional information. In order to obtain information, Sender performs some measurement on the state of the world (selects an information structure). An information structure (or a measurement device) is a map ϕ : Ω → ∆ (S) , where S is a set of signals (outcomes) of our measurement device. In a state ω ∈ Ω this device gives (randomized) signal ϕ (ω) ∈ ∆ (S) . If we write this more carefully such a device is given by a family (fs , s ∈ S) of function fs : Ω → R+ ; fs (ω) gives the probability of obtaining signal P s in a state ω ∈ Ω. Of course all the functions fs must be nonnegative and their sum ( s fs ) must yield the unity function 1Ω on Ω. If Receiver holds belief β = (β (ω) , ω ∈ Ω) ∈ ∆(Ω) then she can easily compute the probabilities of the signals. Namely, the probability of receiving signal s given prior β, is P P equal to ps = ϕ (β) (s) = ω β(ω)ϕ(ω)(s) = ω β (ω) fs (ω) . More important is that upon receiving a signal s, Receiver, as a rational DM, updates her beliefs by Bayes’ rule, i.e. forms the posterior β s ∈ ∆ (Ω) , given as β s (ω) = fs (ω) β (ω) /ps . Therefore, she will choose action a∗ (β s ) and Sender will receive utility u (a∗ (β s )). On average P if Sender uses such an information structure he receives ‘expected’ utility ps u (a∗ (β s )). And so we can ask which is the best measurement device for Sender? Kamenica and Genskow characterize the optimal measurement.
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2.2
The quantum setting
The description of a quantum system starts with the fixation of a Hilbert space (in our case a finite dimensional) over the field of complex or real numbers. The choice of field does not affect our results so for the sake of simplicity we choose to limit ourselves to real numbers (the complex case obtains with minor changes). In this case H is simply the Euclidian space equipped with a symmetric scalar product (.,.). We shall be interested not so much in the Hilbert space H as in operators (that is linear maps from H to H). For such an operator A we denote by A∗ its conjugate which is defined by the following condition: (v, Aw) = (A∗ v, w) for all u, w in H. Self-conjugate operators (for which we have A = A∗ ) are called Hermitian 3 will play most important role in our analysis. Hermitian operators play the role of random variables on Ω. More precisely, we consider them as a (quantum) lotteries, see [6]. Utility of a lottery A for a DM depends on a belief this DM about state of considered quantum system. We elaborate on this below, after introducing two additional notions. 1) A Hermitian operator A is non-negative if (x, Ax) ≥ 0 for any x ∈ H. 2) The notion of trace is a central instrument in what follows. The trace T r of a matrix can be defined as the sum of its diagonal elements. We collect in the following proposition the properties of the trace that we shall make use of. Proposition 1. 1) Tr(A) is linear over A; 2) Tr(AB) = Tr(BA); 3) If an operator is represented by a square matrix A = (aij ), then Tr(A) = a11 +...+ann , that is equal to the sum of diagonal terms of the matrix. 4) Tr(E)=n= dim H. 5) If (in Dirac’s notations) A = |a >< b|, where a and b are elements of the Hilbert space H, then Tr(A) = Tr(< b|a >) = (b, a). 6) If A is a non-negative operator then Tr(A) ≥ 0, and equal to 0 only for A = 0. 3
More correctly we should speak here about symmetric operators; we use the term Hermitian holding in
mind the extension to the complex case.
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Definition. A state is a non-negative (Hermitian) operator S with Tr(S) = 1. The notion of state replaces the concept of probability distribution in the classical context. The nonnegativity of the operator S is analogous to the nonnegativity of a probability measure, and the trace 1 to the sum of probabilities which equals 1. As in the classical case, if the state of beliefs of a DM is (operator) B she evaluates the (expected) utility of ‘lottery’ A as Tr(BA). Examples of states. 1. S0 = E/n. This is a state of ‘uniform uncertainty’ or ‘completely mixed’ state. In a sense, this state is a central point of the set S of states. 2. Pure states. Let e be an element of H with length 1 (that is (e, e) = 1). And let P re be the orthogonal projector on e, that is P re (x) = (x, e)e for any x ∈ H. In Dirac’s notation, P re = |e >< e|, so that Tr(P re ) = (e, e) = 1. Therefore P re is a state; such states are called pure. This operator is a one-dimensional (or the rank 1) projector. One can show that any one-dimensional projector is a pure state. 3. Let S be a state, and U a unitary operator (that is U −1 = U ∗ ), then the operator USU −1 is a state as well. Indeed, it is non-negative, and Tr(USU −1 ) = Tr(U −1 US) = Tr(S) = 1. Note that the state S0 is invariant under any unitary conjugation. 4. The set of states S is a convex space. If S1 , ..., Sr are states, and p1 , ..., pr are nonnegative real numbers such that p1 +...+pr = 1, then the convex combination p1 S1 +...+pr Sr is a state as well. The extreme points of this convex space are exactly the pure states. Any states can be represented as a convex combination of pure states; therefore non-pure states are also called mixed states. The utility of a quantum lotteries A is an affine function (B 7→ Tr(BA)) on the convex space S. We understand (cognitive) states as Receiver’s beliefs on the state of some relevant quantum system.4 We consider the actions of Receiver as quantum lotteries. In state of belief B she chooses a lottery a∗ (B) that is optimal for her. As in the classical case, Sender tries to change receiver’s beliefs with the help of some measurement in order to persuade her to choose an action that is better for him (than the one Receiver would choose based on her 4
In our context, the relevant system is the represented world, see [7].
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prior). Below we define what we mean by measurements and measurement devices. Essentially, this is the main and only difference between the quantum and the classical problem of persuasion. Measurements. We shall not consider the most general measurements, we restrict ourselves to POVMs (positive operator valued measurement). Definition. A (quantum) Measurement Device (MD) is a (finite) collection (or family) P (Qi , i ∈ I) of non-negative operators Qi such that i Qi = E. The elements of the set I are considered as the outcomes (of the MD); the operators Qi are considered as the events (or fuzzy-events) that accompany the outcomes i. When we apply a MD (Qi , i ∈ I) on a quantum system that is in a state S, we obtain P P an outcome i with probability pi = T r(Qi S). (Indeed, pi ≥ 0 and i pi = T r( i Qi S) = T r(S) = 1.) We assume that in this case (if we obtain outcome i, consequently, pi > 0) the √ √ √ √ √ system transits into the state Si = Qi S Qi /T r( Qi S Qi ). Here and further Q denotes the square root of non-negative operator Q, that is a unique non-negative operator such that √ √ √ √ √ √ Q Q = Q. Note that Tr( Qi S Qi ) = Tr( Qi Qi S) = Tr(Qi S) = pi . Physicists take this formula as a postulate generalizing von Neumann-L¨ uders rule (see [11]). In [6] we show that this physical postulate is compatible with the updating of beliefs of a rational DM (of course, under some natural hypothesis about preferences of the DM on quantum lotteries). Since this is a central assumption in the remaining of this paper, we state it once more: if we operate measurement M = (Qi , i ∈ I) on a system that is in a state S then √ √ a) outcome i occurs with probability pi = Tr(Qi S) = Tr( Qi S Qi ), and b) the system transits into state
Si =
p p p p Qi S Qi /Tr( Qi S Qi ).
(1)
A MD (Qi , i ∈ I) is direct if all Qi are one-dimensional projectors. This means that there exists an ortho-normal basis (e1 , ..., en ) of the Hilbert space H (so that (ei , ej ) = δ ij ), such that Qi = P rei . If the operator B has (in this basis) coefficients bij , then (if an outcome i is observed) the system transits into pure state Pi = P rei with probability pi = (ei , Bei ) = bii .
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Let us return to the persuasion problem, that is to a problem of changing Receivers’ belief. Let the initial state of beliefs of Receiver be represented by prior B. And assume that Sender uses of a measurement device given by a POVM (Qi , i ∈ I), and performs the corresponding measurement. If the obtained result is i ∈ I, and Receiver is informed about √ √ it, her prior-belief B is updated into posterior Bi = ( Qi B Qi )/pi , where pi = Tr(Qi B) = √ √ Tr( Qi B Qi ). She chooses a new optimal action a∗ (Bi ). As a consequence, using this MD, P Sender can expect to get utility i pi u(a∗ (Bi )). Note that Sender has one more possibility – to perform the measurement and inform Receiver about the fact that the measurement was performed but to refrain from reporting its outcome. We talk about blind measurement to refer to this procedure. If the measurement P is carried out blindly, the posterior is equal to the expected posterior B ex = i pi Bi = P √ √ Qi B Qi . With posterior B ex Receiver chooses action a∗ (B ex ) and Sender’s expected i utility is equal to u(a∗ (B ex )). Since the utility u as well as the optimal action a∗ are nonP linear, that latter utility can differ from i pi u(a∗ (Bi )) which is the expected utility for the case the outcome is revealed. Since Sender aims at maximizing his expected utility, the question of selecting the optimal MD arises. It turns out that this problem is rather difficult due to the high degree of nonlinearity. Whether or not there exists an optimal measurement is far from obvious due to the lack of compactness of the set of possible POVMs. If we restrict ourselves to direct measurements, the existence of an optimal measurement should not be an issue. However, we cannot formulate the criterion for optimality. We above considered a single measurement. But Sender could choose to perform a sequence of measurements. In the classical context this has no value because any sequence can be merged into a single measurement (cf KG). In the quantum context, opting for a sequence increases Sender’s persuasion power but the problem of finding an optimal sequence becomes significantly more complex. We here want to mention an important result obtained in [5]. Namely, if Sender is not constrained in the number of measurements, then starting from any prior he can with near certainty have Receiver’s beliefs transit into any possible targeted belief state. Of course this result has mostly a theoretical value because it shows that in contrast with the classical case there exists in principle no obstacle to persuasion5 . 5
In classical case such an obstacle is given by the so called Bayesian plausibility, see KG or footnote 1.
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In practice however the value of this result is strongly limited because Receiver is not likely to accept a large number of attempts to be persuaded. Moreover performing measurements is likely to be time and resource consuming. Therefore is it interesting to investigate what Sender can achieve when strongly limiting the number of feasible measurements. Below we consider a problem closely related to persuasion which we call the targeting.
3
Targeting
As mentioned above we do not address the question of optimal persuasion. In fact, we shall confine ourselves to a simpler task. We start with some prior B and some ‘target’ T , and evaluate the probability for a transition from B into T with one measurement or a smaller series of sequential measurements. The solution to this task can give a better understanding on how to solve the full persuasion problem. The connection between the two problems is most direct when considering a situation when Receiver chooses between two actions ‘Good’ and ‘Bad’ (from the point of view of Sender). We can denote by Sg the set of ‘good’ states and by Sb the set of ‘bad’ ones. We can also assume that the prior B lies in the domain Sb , because in the opposite case no measurement is needed. Then Sender chooses some point T in the ‘good’ domain Sg and looks for measurements which gives him this target T with the largest possible probability p. This number and associated expected utility is a lower bound for what can be achieved in the full persuasion problem. We leave open the important question as to how to select the point T in the ‘good’ domain Sg . Intuitively, T should be as close as possible to the prior B. We shall say that we can transform a prior B into the target T , if there exists a MD (Qi , i ∈ I) such that some its posterior Bi is equal to T . The probability pi = T r(Qi B) is called the probability for transition. More generally, we transform B into T with help of a MD (Qi , i ∈ I), if there exists a p P p subset J ⊂ I such that the operator j∈J Qj B Qj is proportional to T , that is pT = p p P p P p P Qj B Qj . Obviously, in this case p = Tr( j∈J Qj B Qj ) = j∈J j Tr(Qj B) = P Tr(( j Qj )B) and it is the probability for the transition. Thus, we shall be dealing with the following problem. Given an initial state (a prior) B and a final state (target) T , we search for a sequence of MDs which transform B into T with 11
the largest possible probability. Earlier we recalled that with a large enough sequence of measurements, starting from any priors, the probability for transfer can be made arbitrary close to 1. Here we shall show that this probability can be made larger than 1/n with two measurements only. For this aim we consider below two cases: when only one measurement can be made and when two measurements are allowed.
3.1
Transition with one measurement
Since we are interested in transforming B into T , we need not consider all the events Qi of a MD, but only those which can give the desired target T . So it is useful to introduce the notion of partial measurement. Definition. A partial measurement is a family (Qj , j ∈ J) of non-negative operators Qj P such that j Qj ≤ E. Of course, any partial measurement can be completed to a ‘complete’ MD, for example P by adding the complementary event E − j Qj . In a partial measurement, the absence of outcome is understood as a NO outcome, i.e., none of the outcomes under consideration (Qj , j ∈ J) was obtained. Definition. A partial measurement (Qj , j ∈ J) transforms B into T if the operator p p p P p Qj B Qj is proportional to the target T , pT = Qj B Qj . The number j∈J j∈J p P p P p = Tr( j∈J Qj B Qj ) = j Tr(Qj B) is called the probability of the transition. Thus, P
the measurement transform B into the state T =(
Xp j∈J
Of course, if p =
P
j
Qj B
p
Qj )/(
X
Tr(Qj B)).
j
Tr(Qj B) = 0 one can say that this measurement does not give the
target. It is intuitively clear that if B and T are close to each other then the probability of a transition of B into T can be made close to 1. Conversely, in [6] we proved that if p is close to 1 then B and T are close. On the other hand, this probability p can vanish. For example, if B and T are pure and orthogonal states, one can expect that the probability is equal to 0. Indeed, the straightforward (or naive) partial measurement T gives the posterior T BT 12
which is proportional to T , but the probability of the transition B → T is Tr(T B) = 0. Sender can use less trivial POVM to achieve his target but the following general statement is true. Proposition 2. Suppose that prior B and target T are orthogonal operators (that is BT = 0). Then no single (partial) measurement can transform B in T . Proof. Suppose that a collection (Qj , j ∈ J) is a partial measurement, and that pT = p p Qj B Qj for some p > 0. Multiplying this equation by B, we have the equality j∈J p p p P 0 = BpT = j∈J B Qj B Qj . For brevity, we write Rj = Qj . Applying the trace, P P we obtain the equality 0 = Tr( j∈J BRj BRj ) = j∈J Tr(BRj BRj ). Each term of this p p p p sum has the form Tr(BRj BRj ) and can be rewritten as Tr(B Rj Rj B Rj Rj ) = p p p p p p Tr( Rj B Rj Rj B Rj ). If we denote Rj B Rj as Aj , we can write Tr(Aj Aj ). Since
P
Aj is non-negative, as well as A2j , Proposition 1 gives us that Tr(A2j ) ≥ 0. And we obtain that 0 is the sum of non-negative numbers Tr(Aj Aj ). Hence, all these numbers are equal p p to 0. From this follows (see Proposition 1) that A2j = 0 and Aj = Rj B Rj = 0 for p any j ∈ J. Multiplying this equality by Rj (on the left and on the right) we obtain that p p P Rj BRj = 0. Consequently, p = j Tr( Qj B Qj ) = 0. On the contrary, if B and T are non-orthogonal, then one can transform B to T with help of one measurement. We do not prove this statement in full generality here. Instead of, we consider three particular cases. I. Suppose that T is a pure state that is the projection on unit vector t, so that T t = t. In this case we can use the ”naive” partial measurement Q = T . Such a measurement transforms prior B into the state T BT /p, where p = Tr(T B). If T and B are non-orthogonal then the probability p is equal to Tr(T B) and is strictly positive. Moreover, we assert that pT = T BT . Indeed, both operators are Hermitian with image proportional to the vector t. Therefore, we need only check the equality of numbers (t, pT t) and (t, T BT t). The first number is (t, pt) = p = T r(BT ) = (t, Bt). The second one is (T t, BT t) = (t, Bt), from which it follows that pT = T BT . II. Suppose that B = S0 . In order to transit into a target T , we again use the naive
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partial measurement Q = T . In this case the posterior B ′ is equal to √
√ √ √ T S0 T /Tr(T S0 ) = T (E/n) T /(Tr(T )/n) = T.
The target T is reached with probability p = Tr(T S0 ) = 1/n. However, we can do better. Let (a1 , ..., an ) be the spectrum of operator T. This means that for some ortho-normal basis e1 , ..., en we have the spectral representation T = a1 P re1 + ... + P ren . Clearly, all ai ≥ 0 and a1 + ... + an = 1. Let a = max(ai ) be the maximal
eigenvalue of T (the spectral radius of T ). Denote Tb = T /a; it is clear that 0 ≤ Tb ≤ E.
Therefore, we can select Tb as Q for a partial measurement. For such a partial measurement p p the posterior is TbS0 Tb/Tr(TbS0 ) = T . The probability of the transition B → T is
Tr(TbS0 ) = Tr(Tb/n) = 1/an. Since a ≥ 1/n, this probability can be significantly larger than
1/n.
Let us note that in point II above operators B and T are compatible (commuting). Therefore this situation is essentially classical. The situation is completely different when the target T is equal to S0 . III. Suppose now that the target T = S0 is the completely mixed state, and the prior B is an arbitrary state. We show that (in contrast with the classical case) there exists a (complete and blind) measurement which transforms B into T = S0 with probability 1. The following particular case of Horn’s wonderful theorem (see, for example, the book Marshal and Olkin [10, Theorem 9.B.2]) provides the foundation of this assertion. Namely, the following is true
Lemma. Let p1 , ..., pn be non-negative numbers, the sum of which is equal to 1. Then there exists a symmetric n × n matrix A = (aij ) with spectrum p1 , ..., pn and diagonal terms 1/n (that is aii = 1/n for all i). Corollary. Let B be a state. Then there exists an orthonormal basis e1 , ..., en of the space H such that (ei , Bei ) = 1/n for any i = 1, ..., n. Consider the (complete) measurement M = (P1 , ..., Pn ), where Pi are projectors on the P P basis vectors ei from Corollary. The (expected) posterior is i (ei , Bei )Pi = 1/n i Pi = E/n. The probability of this transition is equal to Tr(E/n) = 1. Thus, we proved the following 14
Theorem 1. Any state B can be transformed by one ‘blind‘ direct measurement into the completely mixed state S0 with probability 1. This result is quite surprising because it shows that even in the case with a single measurement, the quantum setting yields clearly distinct results compared with the classical. Indeed in the classical context the support of a posterior must be in the support of the prior distribution. What is crucial here is that we use a measurement Q which is incompatible with prior B. We shall see next that this result which by itself is of little practical value since T seldom is equal to S0 is very useful for targeting with two measurements. We also note that in contrast with the classical case the naive measurement T (which can be understood as a direct question on beliefs) because it changes the state has a power of persuasion as we illustrate in the example below.
3.2
The case of two measurements: instigating confusion
A. Let us consider the following sequence of two measurements.
First, we transform the
prior B into completely mixed state S0 applying measurement M defined above in point III. Thereafter, we apply partial measurement T to transform S0 into the target T . Thus, we proved the following Theorem 2. Let prior B and target T be arbitrary states known to Sender. Then there exists a sequence of two measurements transforming B into T with probability at least 1/n. B. Note that in order to construct the measurement M, we need to know the prior B. An interesting question is: what is achievable when the prior B is unknown to Sender. Below we provide one particular result in this direction. Namely, suppose that our target T is a pure state, that is the projector on a normalized vector t. Due to the Corollary of the Lemma above (applied now to operator T ), there exists an orthonormal basis e1 , ...en such that (ei , T ei ) = 1/n for any i. Since T ei = (ei , t)t, we have 1/n = (ei , (ei , t)t) = (ei , t)2 . In other words, in basis e1 , ...en vector t has the form √ √ (1/ n, ..., 1/ n). Let Pi be the projector on vector ei . As the first measurement M we take the direct measurement in the basis (e1 , ..., en ). Applying this measurement, the prior B transits into
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posteriors Bi = Pi BPi /pi with probabilities pi = Tr(Pi B) = (ei , Bei ) because
√
Pi = Pi .
Note that Pi BPi = pi Pi so that Bi = Pi . What concerns the second measurement, we take it as the (naive) partial measurement T . √ √ This measurement transits any posterior state Bi = Pi into the state T Bi T /Tr(T Bi ) = T with probability Tr(T Bi ) = Tr(T Pi ) = (t, ei )2 = 1/n. In the average, we transit in the state P T with probability i pi · 1/n = 1/n. Thus we proved Proposition 3. If the target T is a pure state then there exists a sequence of two measurements (depending on T ) which transits any known or unknown to Sender prior B into T with probability at least 1/n. C. Let us consider more in detail the case with known prior. As before the dimension of the Hilbert space H is equal to n. Moreover, we suppose that prior B and target T are pure states, given by the normalized vectors b and t. Without loss of generality, we can assume that these vectors have coordinates t = (1, 0, 0..., 0) and b = (cos(ϕ), sin(ϕ), 0, ..., 0). As the first measurement M we take the direct measurement with orthonormal basis (e1 , e2 , ...en ), where e1 = (cos(ϕ/2), sin(ϕ/2), 0..., 0) and e2 = (sin(ϕ/2), − cos(ϕ/2), 0, .., 0) and e3 = (0, 0, 1, 0, ..., 0) , e4 = (0, 0, 0, 1, 0, ..., 0) and so on. After this measurement, the prior b transits into the state e1 with the probability p1 = (e1 , b)2 = cos2 (ϕ/2) and into the state e2 with the (complementary) probability sin2 (ϕ/2). Under an impact of the second (naive) measurement (T, E − T ) the state e1 transits into t with probability (e1 , t)2 = cos2 (ϕ/2) whereas the state e2 transits into t with probability (e2 , t)2 = sin2 (ϕ/2). As the result, with this sequence of two measurements, the prior b transits into the target t with probability cos4 (ϕ/2) + sin4 (ϕ/2). Since cos2 (ϕ/2) = 1/2 + cos(ϕ)/2 and sin2 (ϕ/2) = 1/2 − cos(ϕ)/2, we can rewrite
cos4 (ϕ/2) + sin4 (ϕ/2) as 1/2 + cos2 (ϕ)/2. Obviously, this number is no less than 1/2. Summing up we have the following proposition Proposition 4. Assume the prior and the target are known pure states. Then we can transform the prior b into the target t with the probability (1 + cos2 (ϕ))/2 = (1 + (b, t)2 )/2, where ϕ is the angle between the vectors b and t. In accordance with intuition, we learn from Proposition 4 that the closer the target 16
to the prior, the higher the probability of persuasion. This can be seen in the expression (1 + cos2 (ϕ))/2 which has its minimum when b and t are orthogonal, it is then equal to 1/2. Comparing with Theorem 2, we find that when the belief and the target are pure states, the relevant state space becomes a plan spanned by those vectors so the problem reduces to a 2 dimensional one. It becomes then possible to use more efficient measurement devices. Interestingly, in dimension 2, Theorem 2 implies a probability of persuasion of at least 1/2 as well for arbitrary belief and target. So the crucial feature of the purety assumption of proposition 4 is that it allows reducing the dimensionality of the problem.
3.3
Illustration: Persuading the MP to vote No
We now return to the story we told in the Introduction about a MP’s decision to support or not a law about introducing a state of emergency to combat terrorism. We below show how the theory developed above can be used to analyze the activist successful persuasion of the MP. Let H be a two-dimensional Hilbert space with an orthonormal basis e1 , e2 . The MP (Receiver) has to choose between two actions: Yes and No. We consider these actions as (quantum) lotteries (since their utility for MP depends on her belief aboutstate of world 1 0 . The ). We assume that the lottery No is given by the Hermitian operator N = 0 −2 utility of the lottery in the state e1 is equal to 1, and is equal to -2 in the state e . The lottery 2
−1 0 a b . If our MP has a belief B = , Y es is given by Hermitian operator Y = 0 1 b 1−a then utility of No is T r(NB) whereas utility of Yes is T r(Y B). If a ≤ 3/5 our MP votes Yes; if a > 3/5 then MP votes for No (to the pleasure of Sender). 1/5 2/5 . Under this belief We assume that our MP’s initial belief (the prior) is B = 2/5 4/5 MP votes Yes. The activist (Sender) receives utility 1 when succeeding in rallying an MP to the NO vote whatever the true level of threat and 0 otherwise. With the current belief of the MP, the activist gets the utility 0. Can he persuade Receiver to vote NO by selecting an appropriate measurement? We next show that he indeed can induce her to reject the law with probability
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1/2 as according to Theorem 2. But to start with let us suppose that the activist naively asks MP whether she believes the state is moderate (that is e1 ) or severe (that is e2 ). In the other words, he makes the direct measurement (e1 , e2 ). Already this simple and naive question changes the belief of our MP: with probability p1 = T r(BP1 ) = 4/5 her posterior becomes P1 , and with probability 1/5 her posterior becomes P2 . So even this simple question increases the (expected) utility of the activist up to 1/5. We shall next see that with two measurements Sender can do much better. Let us consider another perspective on the law which we call quality of public decisionmaking (cf. EU decision in the Introduction). A state of emergency gives extended new powers to public officials. The quality property is ”tested” bythe followingdirect measure 1/2 1/2 1/2 −1/2 and Q2 = . ment (Q1 , Q2 ) with two possible outcomes Q1 = 1/2 1/2 −1/2 1/2 Such a measurement represents a basis of the state space (of the mental representation of the issue) different from the basis (e1 , e2 ) , a 45◦ rotation of (e1 , e2 ). This means that (P1 , P2 ) and (Q1 , Q2 ) are two non-commuting measurements. Or equivalently (P1 , P2 ) and (Q1 , Q2 ) correspond to properties of the system that are incompatible in the mind of Receiver. She can think in terms of either one of the two perspectives but she cannot synthesize (combine in stable way) pieces of information from the two perspectives. Assume now that Sender brings the discussion to the quality perspective and performs measurement (Q1 , Q2 ) that is Receiver learns whether or not the EU regulates cucumber. With some probability p the belief B transits into Q1 (and with the complementary probability into Q2 ). Thereafter, the activist again asks her whether she believes the state is moderate or severe i.e., performs measurement (P1 , P2 ). If posterior B ′ is Q1 then with probability T r (Q1 P1 ) = T r (Q2 P1 ) = 1/2 the new cognitive state becomes the target P1 . Similarly if the posterior is B ′′ = Q2 the with probability T r (Q2 P1 ) = T r (Q2 P1 ) = 1/2 it becomes the target B ′ = P1 . In the state of belief P1 our MP votes NO. So with a probability 1/2 the target is reached and Sender gets an expected utility of 1/2 (instead of 0 without a persuasion or 1/5 in the case of a direct question). Above we consider the case when the target was taken as P1 . But we can take as
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√ 3/5 6/5 . This target (as a belief state of our MP) a target another state T = √ 6/5 2/5 also induces action NO. As asserted in Proposition 4, Sender can transits into this target √ √ √ √ √ √ state with probability 1/2 + (b, t)2 /2. Since b = (1/ 5, 2/ 5), and t = ( 3/ 5, 2/ 5), √ √ (b, t) = 3/5 + 8/5, this probability is equal to 0.912! In our example we assume that the danger perspective (threat level) which appeals to geopolitical arguments and emotional ones (fear) and the perspective on the quality of decision-making in public administrations which appeals to arguments about bureaucratic senselessness and the intrusiveness of the state, are two incompatible perspectives in the MP’s mind. She can think in either perspective but find it difficult to deal with them simultaneously although they are both perspectives on the issue at stake. The example illustrates how Sender can exploit the quantum indeterminacy of the cognitive state (expressed in the incompatibility of the two perspectives) to persuade the rational quantum-like decision-maker. By performing a measurement on an incompatible perspective, the cognitive state is modified such that beliefs with respect to the severeness of the threat are updated so Receiver prefers to vote NO with probability larger than 1/2.
4
Concluding remarks
The theory of Bayesian persuasion has established that it is often possible to influence a rational decision-maker by selecting a suitable information structure and making the corresponding measurement. However, in the classical uncertainty environment the persuasion power is strongly limited by Bayesian plausibility. The reality of persuasion or manipulation seems however far more extended. Therefore, we have here extended it to the non-classical (quantum) uncertainty environment. This corresponds to investigating persuasion with a quantum cognitive approach - an avenue of research that has experienced rapid growth under the latest 20 years. In a recent paper [5] forth-coming in Theoretical Computer Sciences we establish that provided Sender can make as many measurement as he wishes, full persuasion is always feasible. Sender can bring Receiver to believe whatever he wants. This theoretical result suggests that quantum persuasion is powerful indeed compared with Bayesian persuasion. But no one expects Receiver to let herself be the object of attempts to persuasion 19
under an indefinite time. This paper investigates what can be achieved with short sequences of measurements. While Bayesian plausibility imposes a constraint, it also allows to simplify the persuasion problem so as to allow quite easily to characterize an optimal persuasion policy. In the nonclassical context, no such simplification is available and the issue of optimality cannot be directly addressed. A simpler task that we call targeting is investigated and its results can be viewed as a lower bound for what can be achieved in a full persuasion problem. Our results show that even with a short sequence of two measurements, targeting is quite powerful. In particular when the prior is known (and is a pure state), whatever the initial belief, any target belief can be reached at least half of the time. Most interesting is to understand how this is made possible. Interestingly the key mechanism amounts to first creating what we would like to call confusion (formally uniform uncertainty). This is possible precisely because any quantum belief state (even pure) can always be ”broken” by using a non-commuting measurement. In quantum cognition this is related to the hypothesis according to which people have difficulties to process different kinds of information into a single and stable representation of the world. Our analysis suggests that this very feature makes a person easily influencable. And that a person can be influenced more effectively by a well-designed distraction rather than by arguments of informational value to the decision. Distraction occurs when the mind is turned toward a perspective that Receiver finds hard to combine with her current perspective. It is a common experience that sellers use this tactic intuitively sensing that it is quite efficient. We have here provided an argument based on purely informational considerations that justifies a distracting policy in influence seeking activities.
References [1] Akerlof G., and R. Schiller (2015) Phishing for Phools - the economics of manipulation and deception, Princeton University Press. [2] Alonso, Ricardo, and Odilon Cˆamara. ”Persuading voters.” The American Economic Review 106.11 (2016): 3590-3605.
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[3] Bruza P., and J. Busemeyer (2012) Quantum Cognition and Decision-making, Cambridge University Press. [4] Danilov V. I., and A. Lambert-Mogiliansky (2010). Expected Utility under Non-classical Uncertainty. Theory and Decision, 68, 25-47. [5] Danilov V. I., and A. Lambert-Mogiliansky (2017) ”Preparing a (quantum) belief system” PSE WP 2017-20 and ArXiv http://arxiv.org/abs/1708.08250, forth-coming in Theoretical Computer Sciences [6] Danilov V. I., A. Lambert-Mogiliansky, and V. Vergopoulos (2016) Dynamic consistency of expected utility under non-classical (quantum) uncertainty. PSE Working Papers 2016-12, and ArXiv https://arxiv.org/abs/1708.08244, Forth-coming in Theory and Decision [7] Dubois F., and A. Lambert-Mogiliansky (2016). Our (represented) world and quantumlike object, in Contextuality in Quantum Physics and Psychology, ed. Dzafarof et al, World Scientific, Advanced Series in Mathematical Psychology Vol. 6, 367-387. [8] Erev, Ido, Gary Bornstein and Thomas S. Wallsten. 1993. The negative effect of probability assessments on decision quality. Organizational Behavior and Human Decision Processes 55:78-94. [9] Kamenica E., and M. Gentzkow (2011). Bayesian Persuasion. American Economic Review, 101(6): 2590-2615. [10] Marshall A.W., and I. Olkin (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. [11] Nielsen M.A., and I.L. Chuang (2010) Quantum Computation and Quantum Information. Cambridge Univ. Press, Cambridge.
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