Quantitative Predictionsin Quantum Decision Theory

# Quantitative Predictions in Quantum Decision Theory

## Abstract

Quantum Decision Theory, advanced earlier by the authors, and illustrated for lotteries with gains, is generalized to the games containing lotteries with gains as well as losses. The mathematical structure of the approach is based on the theory of quantum measurements, which makes this approach relevant both for the description of decision making of humans and the creation of artificial quantum intelligence. General rules are formulated allowing for the explicit calculation of quantum probabilities representing the fraction of decision makers preferring the considered prospects. This provides a method to quantitatively predict decision-maker choices, including the cases of games with high uncertainty for which the classical expected utility theory fails. The approach is applied to experimental results obtained on a set of lottery gambles with gains and losses. Our predictions, involving no fitting parameters, are in very good agreement with experimental data. The use of quantum decision making in game theory is described. A principal scheme of creating quantum artificial intelligence is suggested.

Quantum decision theory, decision making, choice between lotteries, attraction index, quantitative predictions, game theory, artificial intelligence

## I Introduction

Classical decision making, based on expected utility theory [1], is known to fail in many cases when decisions are made under risk and uncertainty. Numerous variants of so-called non-expected utility theories have been suggested to replace expected utility theory by using other more complicated functionals. The long list of such non-expected utility models can be found in the review articles [2, 3, 4]. The non-expected utility theories are, by construction, descriptive. By introducing several fitting parameters, such theories can be calibrated to some given set of empirical data. However, it is often possible to have different theories fitting the same set of experiments equally well, so that it is difficult to distinguish which of the models is better [5]. Moreover, on the basis of such theories, it is impossible to account for the known paradoxes arising in classical decision making and to make convincing out-of-sample predictions of new sets of empirical data. The non-expected utility theories have been thoroughly analyzed in numerous publications confirming the descriptive nature of these theories and their inability to perform useful predictions (see, e.g., [6, 7, 8, 9, 10]). Thus, Birnbaum [6, 7] carefully studied the so-called rank dependent utility theory and cumulative prospect theory, concluding that, even with fitting parameters, these theories are not able to get rid of paradoxes and moreover create new paradoxes. Safra and Segal [8] state that none of the non-expected utility theories can explain all main paradoxes but, on the contrary, distorting the structure of expected utility theory, the non-expected utility theories result in several non-expected inconsistencies. Al-Najjar and Weinstein [9, 10] present a detailed analysis of non-expected utility theories, coming to the conclusion that any variation of expected utility theory ”ends up creating more paradoxes and inconsistences than it resolves”.

The same conclusions apply to the so-called stochastic decision theories [11, 12, 13] that are based on underlying deterministic theories, decorating them with the probability of making errors in the choice. Introducing such probabilities, caused by decision-maker errors, into the log-likelihood functional adds several more parameters in the calibration exercise that improve the description of the given set of data. But such a stochastic decoration does not change the structure of the underlying deterministic theory and does not make predictions possible.

Clearly, the possibility of making predictions can be strongly hindered by the presence of unknown or poorly formulated conditions accompanying decision making. For instance, there can exist an unknown stochastic environment [14] or a varying context [15]. It may also happen that the provided information is imprecise and only partially reliable [16] or preference relations are incomplete [17] requiring the use of fuzzy logic [18]. In such situations, any prediction is likely to be only partial and often merely qualitative.

But even when the posed problem is well defined, suggesting, e.g., a choice between explicitly presented lotteries, quantitative predictions as a rule are impossible. In particular, the non-expected utility theories mentioned above have been developed exactly for such seemingly simple choice between well defined lotteries. And, as is discussed above, in many cases, the given lotteries, although being explicitly formulated, contain uncertainty not allowing for predictions. It is important to also stress that, in some cases of well defined lotteries, predictions based on utility theory are qualitatively wrong, as has been demonstrated by Kahneman and Tversky [19].

In the present paper, we consider the situation when decision making consists in the choice between well defined lotteries. We develop an approach allowing for quantitative predictions in arbitrary cases, including those where utility theory fails, being unable to provide even qualitatively correct conclusions. It is important to emphasize that quantitative predictions in our approach can be realized without any fitting parameters. So, our approach is not a descriptive, but rather a normative, or prescriptive theory.

Our approach is based on Quantum Decision Theory (QDT), which we developed earlier [20, 21, 22, 23, 24, 25, 26]. There have been other attempts to apply quantum techniques to cognitive sciences, as is discussed in the books [27, 28, 29, 30] and review articles [31, 32, 33, 34]. However, these attempts were based on constructing some models for describing particular effects, with the use of several fitting parameters for each case. Our approach of QDT is essentially different from all those models in the following facets. First, QDT is formulated as a general theory applicable to any variant of decision making, but not as a special model for a particular case. Second, the mathematical structure of QDT is common for both decision theory as well as for quantum measurements, which has been achieved by generalizing the von Neumann [35] theory of quantum measurements to the treatment of inconclusive measurements and composite events represented by noncommutative operators [36, 37, 38, 39]. The third unique feature of QDT is the possibility to develop quantitative predictions without any fitting parameters, as has been shown for some simple choices in decision making [40].

The predictions concern the fractions of decision makers choosing the corresponding lotteries. In QDT, such fractions are predicted by their corresponding behavioral quantum probabilities, as follows from the frequentist interpretation of probabilities and the assumption that the population of decision makers are, to a first approximation, representative of a homogenous group of individuals making probabilistic choices. The scheme for calculating the quantum probabilities is based on our previous demonstration that it consists of two terms, called utility and attraction factors. The utility factor derives from the utility of each lottery, being defined on prescribed rational grounds. The attraction factor represents the irrational side of a choice. The value of the attraction factor for a single decision maker and for a given choice is random. However, for a society of decision makers, one can derive the quarter law, which estimates the non-informative prior for the absolute value of the average attraction factor as equal to . In simple cases, the signs of the attraction factors can be prescribed by the principle of ambiguity aversion. In more complicated situations, a criterion has been suggested [40] and applied to lotteries with gains.

Here, we extend Ref. [40] by considering lotteries with both gains and losses, and not just gains. We also improve on the quarter law based on the non-informative prior, by including available information on the level of ambiguity characterizing a given set of games, thus providing the potential for improved predictions. Moreover, we consider the cases for which our previously proposed criterion defining the signs of attraction factors does not allow for unique conclusions. We present a generalization of the criterion for the sign of the attraction factors that addresses these limitations and also applies to lotteries with losses.

The possibility of mathematically formalizing all steps of a decision process, allowing for quantitative predictions, is important, not merely for decision theory, but also for the problem of creating an artificial quantum intelligence that could function only if all operations are explicitly formalized in mathematical terms. We have previously mentioned [41] that QDT can provide such a basis for creating artificial quantum intelligence, since the QDT mathematical foundation is formulated in the same way as the theory of quantum measurements.

In the present paper, we overcome the limitations of our previous publication [40] by generalizing QDT along the following directions.

(i) A general method for defining utility factors is advanced, valid for lotteries with losses as well as for lotteries with gains, or mixed-type lotteries.

(ii) A criterion is formulated for the quantitative classification of attraction factors for all kinds of lotteries, whether with gains or with losses. In the case of games with two lotteries, this criterion uniquely prescribes the signs of attraction factors.

(iii) The quarter law is generalized by taking into account the ambiguity level for a given set of games. This defines the typical absolute value of the attraction factor more accurately than the quarter law following from non-informative prior.

(iv) A method for estimating attraction factors for games with multiple lotteries is described.

(v) The value of our theory is illustrated by comparing its prediction with empirical results obtained on a set of games containing lotteries with gains and with losses, for which expected utility theory fails. Our approach results in quantitative predictions, without fitting parameters, which are in very good agreement with empirical data.

(vi) It is shown how the QDT can be applied to game theory. An application is illustrated by the prisoner dilemma.

(vii) The general principles for creating artificial quantum intelligence are suggested. It is emphasized that artificial intelligence, mimicking the functioning of human consciousness, should be quantum.

## Ii Scheme of Quantum Decision Theory

In the present section, we briefly sketch the basic scheme of QDT in order to remind the reader about the definition of quantum probability used in decision theory. The technical details have been thoroughly expounded in the previous articles [20, 21, 22, 23, 24, 25, 26], which allows us to just recall here the basic notions.

As is mentioned in the Introduction, the mathematical scheme is equally applicable to quantum decision theory as well as to the theory of quantum measurements [36, 37, 38, 39]. An event can mean either the result of an estimation in the process of measurements, or a decision in decision making. In both the cases, there exist simple events that are operationally testable, that is, clearly observable, and inconclusive events that are either non-observable or even not well specified. The typical example in quantum measurements is the double-slit experiment, where the final registration of a particle by a detector is an operationally testable event, while the passage through one of the slits is not observable. In decision making, a straightforward example would be the choice between lotteries under uncertainty. The final choice of a lottery is an operationally testable event, while the deliberations on real or imaginary uncertainties in the formulation of the lotteries or in hesitations of the decision-maker can be treated as inconclusive events.

We consider a set of events labelled by an index . Each event is put into correspondence with a state of a Hilbert space , with the family of states forming an orthonormalized basis:

 An→|n⟩∈HA=span{|n⟩}. (1)

There also exists another set of events , labelled by an index , with each event being in correspondence with a state of a Hilbert space , the family of the states forming an orthonormalized basis:

 Bα→|α⟩∈HB=span{|α⟩}. (2)

A pair of events from different sets forms a composite event represented by a tensor-product state ,

 An⨂Bα→|n⟩⨂|α⟩∈H, (3)

in the Hilbert space

 H≡HA⨂HB=span{|n⟩⨂|α⟩}. (4)

An event is called operationally testable if and only if it induces a projector on the space . The event set is assumed to consist of operationally testable events.

A different situation occurs when we have an inconclusive event being a set

 B≡{Bα,bα:α=1,2,…} (5)

of events associated with amplitudes that are random complex numbers. An inconclusive event corresponds to a state in the space , such that

 B→|B⟩=∑αbα|α⟩∈HB. (6)

The states are not orthonormalized, because of which the operator is not a projector.

A composite event is termed a prospect. Of major interest are the prospects composed of an operationally testable event and an inconclusive event:

 πn=An⨂B. (7)

A prospect corresponds to a prospect state in the space ,

 πn→|πn⟩=|n⟩⨂|B⟩∈H, (8)

and induces a prospect operator

 ^P(πn)≡|πn⟩⟨πn|. (9)

The prospect states are not orthonormalized and the prospect operator is not a projector. The given set of prospects forms a lattice

 L={πn:n=1,2,…,NL}, (10)

whose ordering is characterized by prospect probabilities to be defined below. The assembly of prospect operators composes a positive operator-valued measure. By its role, this set is analogous to the algebra of local observables in quantum theory.

The strategic state of a decision maker in decision theory, or statistical operator of a system in physics, is a semipositive trace-one operator defined on the space . The prospect probability is the expectation value of the prospect operator:

 p(πn)=Tr^ρ^P(πn), (11)

with the trace over the space . To form a probability measure, the prospect probabilities are normalized,

 ∑np(πn)=1,0≤p(πn)≤1. (12)

Taking the trace in (11), it is possible to separate out positive-defined terms from sign-undefined terms, which respectively, are

 f(πn)=∑α|bα|2⟨nα|^ρ|nα⟩,
 q(πn)=∑α≠βb∗αbβ⟨nα|^ρ|nβ⟩. (13)

Then the prospect probability reads as

 p(πn)=f(πn)+q(πn). (14)

The appearance of a sign-undefined term is typical for quantum theory, describing the effects of interference and coherence.

Note that the decision-maker strategic state has to be characterized by a statistical operator and not just by a wave function since, in real life, any decision maker is not an isolated object but a member of a society [38, 40].

An important role in quantum theory is played by the quantum-classical correspondence principle [42, 43], according to which classical theory has to be a particular case of quantum theory. In the present consideration, this is to be understood as the reduction of quantum probability to classical probability under the decaying quantum term:

 p(πn)→f(πn),q(πn)→0. (15)

In quantum physics, this is also called decoherence, when quantum measurements are reduced to classical measurements. The positive-definite term , playing the role of classical probability, is to be normalized,

 ∑nf(πn)=1,0≤f(πn)≤1. (16)

From conditions (12) and (16) it follows

 ∑nq(πn)=0,−1≤q(πn)≤1, (17)

which is called the alternation law.

In decision theory, the classical part describes the utility of the prospect , which is defined on rational grounds. In that sense, a prospect is more useful than if and only if

 f(π1)>f(π2)(moreuseful). (18)

The quantum part characterizes the attractiveness of the prospect, which is based on irrational subconscious factors. Hence a prospect is more attractive than if and only if

 q(π1)>q(π2)(moreattractive). (19)

And the prospect probability (14) defines the summary preferability of the prospect, taking into account both its utility and attractiveness. So, a prospect is preferable to if and only if

 p(π1)>p(π2)(preferable). (20)

The structure of the quantum probability (14), consisting of two parts, one showing the utility of a prospect and the other characterizing its attractiveness, is representative of real-life decision making, where both these constituents are typically present. Quantum probability, taking into account the rationally defined utility as well as such an irrational behavioral feature as attractiveness, can be termed as behavioral probability.

It is worth stressing that QDT is an intrinsically probabilistic theory. This is different from stochastic decision theories, where the choice is assumed to be deterministic, while randomness arises due to errors in decision making. The probabilistic nature of QDT is not caused by errors in decision making, but it is due to the natural state of a decision maker, described by a kind of statistical operator. Upon the reduction of QDT to a classical decision theory, it reduces to a probabilistic variant of the latter, since decisions under uncertainty are necessarily probabilistic [44]. As mentioned above, the description of a decision maker strategic state by a statistical operator, and not by a wave function, emphasizes the fact that any decision maker is not an absolutely isolated object but rather a member of a society, who is subjected to social interactions [38, 40, 45]. When comparing theoretical predictions with empirical data, it follows from the logical structure of QDT that one has to compare the theoretically calculated probability (14) with the fraction of decision makers preferring the considered prospect.

## Iii General Definition of Utility Factors

In this section, we describe the general method for defining utility factors for a given set of lotteries containing both gains as well as losses.

Let a set of payoffs be given

 Xn={xi:i=1,2,…,Nn}, (21)

in which payoffs can represent either gains or losses, being, respectively positive or negative. The probability distribution over a payoff set is a lottery

 Ln={xi,pn(xi):i=1,2,…,Nn}, (22)

with the normalization condition

 ∑ipn(xi)=1,0≤pn(xi)≤1. (23)

The lotteries are enumerated by the index . Under a utility function , the expected utility of lottery is

 U(Ln)=∑iu(xi)pn(xi)(n=1,2,…,NL). (24)

Utility functions for gains and losses can be of different signs. Therefore, the expected utility can also be either positive or negative. When it is negative, one often uses the notation of the lottery cost

 C(Ln)≡−U(Ln)=|U(Ln)|(U(Ln)<0).

An expected utility is positive, when in its payoffs gains prevail. And it is negative, when losses overwhelm gains.

As has been explained in Ref. [40], the choice between the given lotteries in any game is always accompanied by uncertainty related to the decision-maker hesitations with respect to the formulation of the game rules, understanding of the problem, and his/her ability to decide what he/she considers the correct choice. All these hesitations form an inconclusive event denoted above as . Therefore a choice of a lottery is actually a composite event, or a prospect

 πn=Ln⨂B(n=1,2,…,NL). (25)

Here we denote the action of a lottery choice and a lottery by the same latter , which should not lead to confusion. The utility factor characterizes the utility of choosing a lottery . Since QDT postulates that the choice is probabilistic, it is possible to define the average quantity over the set of lotteries,

 U=NL∑n=1f(πn)U(Ln), (26)

playing the role of a normalization condition for random expected utilities [46].

The utility factor represents a classical probability distribution and can be found from the conditional minimization of Kullback-Leibler information [47, 48]. The use of the Kullback-Leibler information for defining such a probability distribution is justified by the Shore-Jonson theorem [49] stating that there exists only one distribution satisfying consistency conditions, and this distribution is uniquely defined by the minimum of the Kullback-Leibler information, under given constraints. The role of the constraints here are played by the normalization conditions (16) and (26). Then the information functional reads as

 I[f]=NL∑n=1f(πn)lnf(πn)f0(πn)+
 +γ[NL∑n=1f(πn)−1]+β[U−NL∑n=1f(πn)Un], (27)

where is a prior distribution, , and and are Lagrange multipliers.

As boundary conditions, it is natural to require that the utility factor of a lottery with asymptotically large expected utility would tend to unity,

 f(πn)→1(Un→∞), (28)

while the utility factor of a lottery with asymptotically large cost, would go to zero,

 f(πn)→0(Un→−∞). (29)

Also, the utility factors, as their name implies, have to increase together with the related expected utilities,

 δf(πn)δUn≥0. (30)

Minimizing the information functional (27) results in the utility factors

 f(πn)=f0(πn)eβUn∑nf0(πn)eβUn, (31)

with a non-negative parameter .

If one assumes that the prior distribution is uniform, such that , then one comes to the utility factors of the logit form. However, the uniform distribution does not satisfy the boundary conditions (28) to (29). Therefore a more accurate assumption, taking into account the boundary conditions, should be based on the Luce choice axiom [50, 51]. According to this axiom, if an -th object, from the given set of objects, is scaled by a quantity , then the probability of its choice is

 f0(πn)=λn∑nλn. (32)

In our case, the considered objects are lotteries and they are scaled by their expected utilities. So, for the non-negative utilities, we can set

 λn=Un(Un≥0), (33)

while for negative utilities,

 λn=1|Un|(Un<0). (34)

Expression (34) is chosen in order to comply with Luce’s axiom together with the ranking of preferences with respect to losses.

Generally, utilities can be measured in some units, say, in monetary units . Then we could use dimensionless scales defined as and for gains and losses, respectively. Obviously, expression (32) is invariant with respect to units in which is measured. Therefore, for simplicity of notation, we assume that utilities are dimensionless.

Thus, the utility factor (31), with prior (32), is

 f(πn)=λneβUn∑nλneβUn(β≥0). (35)

In particular, when gains prevail, so that all expected utilities are non-negative, then

 f(πn)=UneβUn∑nUneβUn(∀Un≥0). (36)

While, when losses prevail, and all expected utilities are negative, then

 f(πn)=|Un|−1e−β|Un|∑n|Un|−1e−β|Un|(∀Un<0). (37)

In the mixed case, where the utility signs can be both positive and negative, one has to employ the general form (35).

The parameter characterizes the belief of the decision maker with respect to whether the problem is correctly posed. Under strong belief, one gets

 f(πn)={1,  Un=maxnUn0,  Un≠maxnUn(β→∞), (38)

which recovers the classical utility theory with the deterministic choice of a lottery with the largest expected utility. In the opposite case of weak belief, when uncertainty is strong, one has

 f(πn)=λn∑nλn(β=0). (39)

To explicitly illustrate the forms of the utility factors, let us consider the often met situation of two lotteries under strong uncertainty, thus, considering the binary prospect lattice

 L={πn:n=1,2}(β=0), (40)

with zero belief parameter. Then, if in both the lotteries gains prevail, we have

 f(πn)=UnU1+U2(U1≥0,U2≥0). (41)

When losses are prevailing in the two lotteries, then

 f(πn)=1−|Un||U1|+|U2|(U1<0,U2<0). (42)

And if one expected utility is positive, say that of the first lottery, while the other utility is negative, then the utility factor for the first lottery is

 f(π1)=U1|U2|U1|U2|+1(U1>0,U2<0), (43)

respectively, .

In this way, the utility factors are explicitly defined for any combination of lotteries in the given game, with the payoff sets containing gains as well losses.

## Iv Classification of Lotteries by Attraction Indices

We now give a prescription for defining the attraction factors. By its definition, an attraction factor quantifies how each of the given lotteries is more or less attractive. The attractiveness of a lottery is composed of two factors, possible gain and its probability. It is clear that a lottery is more attractive, when it suggests a larger gain and/or this gain is more probable. In other words, a more attractive lottery is more predictable and promises a larger profit. On the contrary, a lottery suggesting a smaller gain or a larger loss and/or higher probability of the loss, is less attractive. A less certain lottery is less attractive, since it is less predictable, which is named as uncertainty aversion or ambiguity aversion. Below we give an explicit mathematical formulation of these ideas.

Let us introduce, for a lottery , the notation for the minimal gain

 gn≡mini{xi≥0:xi∈Ln} (44)

and for the minimal loss

 ln≡maxi{xi≤0:xi∈Ln}. (45)

These quantities characterize possible gains and losses in the given lotteries.

But payoffs are not the only features that attract the attention of decision makers. In experimental neuroscience, it has been discovered that, during the act of choosing, the main and foremost attention of decision makers is directed to the payoff probabilities [52]. We capture this empirical observation by considering different weights related to payoffs and to their probabilities in the characterization of the lottery attractiveness. Specifically, the weight of a payoff should be much smaller than the weight of its probability . We quantitatively formulate this by choosing weights proportional respectively to for the payoff versus for its probability. The later term is motivated by the decimal number system. This leads us to defining the lottery attractiveness

 an≡an(Ln)≡∑ixi10pn(xi). (46)

And the related relative quantity can be termed the attraction index

 αn=αn(Ln)≡an∑m|am|. (47)

The latter satisfies the normalization condition

 ∑n|αn|=1. (48)

The notion of the lottery attraction index makes it straightforward to classify all lotteries from the considered game onto more or less attractive. Thus a lottery is more attractive than , hence

 q(π1)>q(π2), (49)

when the attraction index of the first lottery is larger than that of the second,

 α1>α2. (50)

In the marginal case, when , the first lottery is more attractive if the probability of its minimal gain is smaller than that of the second lottery,

 α1=α2≥0,p(g1)

For short, this will be denoted as . And in the other marginal case, where , the first lottery is more attractive if the probability of its minimal loss is larger than that of the second,

 α1=α2<0,p(l1)>p(l2). (52)

This, for short, will be denoted as .

The criterion allows us to arrange all the given lotteries with respect to the level of their attractiveness.

For the particular case of a binary prospect lattice (40), the alternation property (17) reads as

 q(π1)+q(π2)=0. (53)

Therefore the attraction factors have different signs,

 q(π1)=−q(π2). (54)

The sign of each of the attraction factors is prescribed by the sign of the difference

 Δα≡α1−α2. (55)

If is positive, then the attraction factor of the first prospect is positive and that of the second is negative. On the contrary, if is negative, then the attraction factor of the first lottery is negative and that of the second is positive. In the marginal case, when , we shall use the notations accepted above and explained below (Eqs. (51) and (52)): If the first lottery is more attractive, we shall write , while when the second lottery is more attractive, this will be denoted as .

## V Typical Values of Attraction Factors

The criterion of the previous section allows us to classify all the lotteries of the considered game onto more or less attractive. But we also need to define the amplitudes of the attraction factors. According to QDT, these values are probabilistic variables, characterizing irrational subjective features of each decision maker. For different subjects, they may be different. They can also be different for the same subject at different times [13]. Different game setups also influence the values of the attraction factors [53]. However, for a probabilistic quantity, it is possible to define its average or typical value.

### V-a General considerations

We consider games, enumerated by , with lotteries in each, enumerated by . And let the choice be made by a society of decision makers, numbered by . In a -th game, decision makers make a choice between prospects . The typical value of the attraction factor is defined as the average

 ¯¯¯q≡1NGNG∑k=11NLNL∑n=1∣∣ ∣∣1NN∑j=1qj(πnk)∣∣ ∣∣. (56)

Denoting the mean value of the attraction factor for a prospect , as

 |q(πn)|≡1NGNG∑k=1∣∣ ∣∣1NN∑j=1qj(πnk)∣∣ ∣∣, (57)

we can write

 ¯¯¯q=1NLNL∑n=1|q(πn)|. (58)

For a large value of the product , the distribution of the attraction factors can be characterized by a probability distribution , which, in view of property (17), is normalized as

 ∫1−1φ(q)dq=1. (59)

The average absolute value of the attraction factor can be represented by the integral

 ¯¯¯q=∫10qφ(q)dq. (60)

This defines the typical value of the attraction factor that characterizes the level of deviation from rationality in decision making [54].

If there is no information on the properties and specifics of the given set of lotteries in the suggested games, then one should resort to a non-informative prior, assuming a uniform distribution satisfying normalization (59), which gives . Substituting the uniform distribution into the typical value of the attraction factor (60) yields , which was named the “quarter law” in the earlier paper [40].

However, it is possible to find a more precise typical value by taking into account the available information on the given lotteries. For example, it is straightforward to estimate the level of uncertainty of the lottery set.

### V-B Choice between two prospects

When choosing between two lotteries with rather differing utilities, the choice looks quite easy - the lottery with the largest utility is preferred. But when two lotteries have very close utilities, choosing becomes difficult. The closeness of the lotteries, corresponding to two prospects and , can be quantified by the relative difference

 δf(π1,π2)≡2|f(π1)−f(π2)|f(π1)+f(π2)×100%. (61)

When the choice is between just two prospects, whose utility factors are normalized according to condition (16), hence when , then the relative difference simplifies to

 δf=2|f(π1)−f(π2)|×100%(NL=2). (62)

There have been many discussions concerning choices between similar alternatives with close utilities or close probabilities, such that the choice becomes hard to make [55, 56, 57, 58]. We refer to such situations as “irresolute”. One of the major problems is how to quantify the similarity or closeness of the choices. Several variants of measuring the distance between the alternatives and have been suggested, including the linear distance , as well as different nonlinear distances , with . We propose that the value of that serves as an upper threshold, below which the lotteries are irresolute, should not depend on the exponent used in the definition of the distance. Therefore, in order for the exponent not to influence the boundary value, one has to require the invariance of the distance with respect to the exponent at the threshold, so that the critical threshold value should obey the equality: for any . The latter reads explicitly as

 [δfc(π1,π2)]m=δfc(π1,π2),

where is measured in percents. This equation is valid for arbitrary only for . Hence the critical boundary value equals . Thus the lotteries, for which the irresoluteness criterion

 δf(π1,π2)<1% (63)

is valid, are to be treated as close, or similar, and the choice between them, as irresolute.

The next question is how the irresoluteness in the choice influences the typical attraction factor. Suppose that the fraction of irresolute games equals . Then the following properties of the distribution over admissible attraction factors should hold.

In the presence of irresolute games for which the irresoluteness criterion holds true, the probability that the attraction factor is zero is asymptotically small,

 limq→0φ(q)=0(ν>0). (64)

In other words, this condition means that, on the manifold of all possible games, absolutely rational games form a set of zero measure.

If not all games are irresolute , the probability of the maximal absolute value of the attraction factor is asymptotically small,

 lim|q|→1φ(q)=0(ν<1). (65)

That is, on the manifold of all possible games, absolutely irrational games compose a set of zero measure.

Often employed as a prior distribution in standard inference tasks [59, 60, 61], the simplest distribution that obeys the two conditions (64) and (65) is the beta distribution that, under normalization (59), reads

 φ(q)=|q|ν(1−|q|)1−νΓ(1+ν)Γ(2−ν). (66)

Using this distribution, expression (60) gives the typical attraction factor value

 ¯¯¯q=1+ν6. (67)

Note that the average of given by (67) over the two boundary values and gives

 12(16+26)=14,

thus recovering the non-informative quarter law.

This expression (67) can be used for predicting the results of decision making. For example, in the case of a binary prospect lattice, the difference in the attraction indices (55) defines the signs of the attraction factors, making it possible to prescribe the attraction factors and to the considered prospects.

### V-C Choice between more than two prospects

When there are more than two prospects in the considered game, we propose the following procedure to estimate the attraction factors. Using the classification of the prospects by the attraction indices, as is described in the previous section, it is straightforward to arrange the prospects in descending order of attractiveness,

 q(πn)>q(πn+1)(n=1,2,…,NL−1). (68)

Let the maximal attraction factor be denoted as

 qmax≡q(π1)>0. (69)

Given the unknown values of the attraction factors, the non-informative prior assumes that they are uniformly distributed and at the same time they must obey the ordering constraint (68). Then, the joint cumulative distribution of the attraction factors is given by

 Pr[q(π1)<η1,...,q(πNL)<ηNL|η1≤η2≤...≤ηNL]=
 =∫η10dx1∫η2x1dx2....∫ηNLxNL−1dxNL , (70)

where the series of inequalities ensure the ordering. It is then straightforward to show that the average values of the are equidistant, i.e. the difference between any two neighboring factors, on average, is independent of , so that

 Δ≡⟨q(πn)⟩−⟨q(πn+1)⟩=const. (71)

Taking their average values as determining their typical values, we omit the symbol representing the average operator and use the previous equation to represent the -th attraction factor as

 q(πn)=qmax−(n−1)Δ. (72)

From the alternation property (17), it follows that

 qmax=NL−12Δ. (73)

The total number of lotteries can be either even or odd, leading to slightly different forms for the following expressions.

And the definition of the typical value (58) gives

 Δ=⎧⎪⎨⎪⎩4¯¯¯q/NL  (NLeven)4¯¯¯qNL/(N2L−1)  (NLodd). (74)

Then the maximal attraction factor (73) becomes

 qmax=⎧⎪⎨⎪⎩2¯¯¯q(NL−1)/NL  (NLeven)2¯¯¯qNL/(NL+1)  (NLodd). (75)

Therefore formula (72) yields the expressions for all attraction factors

 q(πn)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩2¯¯¯qNL+1−2nNL(NLeven)2¯¯¯qNL(NL+1−2n)N2L−1(NLodd). (76)

Let us denote the set of all attraction factors in the considered game as

 QNL≡{q(πn):n=1,2,…,NL}.

If there are only two lotteries, then we have

 Δ=2¯¯¯q,qmax=¯¯¯q(NL=2),

and the attraction-factor set is

 Q2={¯¯¯q,−¯¯¯q}.

In the case of three lotteries,

 Δ=32¯¯¯q,qmax=32¯¯¯q(NL=3),

and the attraction factor set reads as

In that way, all attraction factors can be defined.

## Vi Quantitative Predictions in Decision Making

In order to illustrate how the suggested theory makes it possible to give quantitative predictions, without any fitting parameters, let us consider the set of experiments performed by Kahneman and Tversky [19]. This collection of games, including both gains and losses, is a classical example showing the inability of standard utility theory to provide even qualitatively correct predictions as a result of the confusion caused by very close or coinciding expected utilities. Let us emphasize that the choice of these games has been done by Kahneman and Tversky [19] in order to stress that standard decision making cannot be applied for these games. This is why it is logical to consider the same games and to show that the use of QDT does allow us not only to qualitatively explain the correct choice, but also that QDT provides quantitative predictions for such difficult cases.

In the set of games described below, each game consists of two lotteries , with . The number of decision makers is about .

Recall that, as is explained in Sec. III, the choice between lotteries corresponds to the choice between prospects (25) including the action of selecting a lottery under a set of inconclusive events representing hesitations and irrational feelings. Therefore the choice, under uncertainty, between lotteries is equivalent to the choice between prospects . The choice under uncertainty for the case of a binary lattice can be characterized by the utility factors (41) to (43). We take the linear utility function, whose convenience is in the independence of the utility factors from the monetary units used in the lottery payoffs. The attraction factors are calculated by following the recipes described in Sec. IV and Sec. V.

We compare the prospect probabilities , theoretically predicted by QDT, with the empirically observed fractions [19]

 pexp(πn)≡N(πn)N

of the decision makers choosing the prospect , with respect to the total number of decision makers taking part in the experiments.

### Vi-a Lotteries with gains

Game 1. The lotteries are

 L1={2.5,0.33|2.4,0.66|0,0.01},L2={2.4,1}.

For this game, we shall show explicitly the related calculations, while omitting the intermediate arithmetics in the following cases.

The utilities of these lotteries are

 U(L1)=2.5×0.33+2.4×0.66+0×0.01=2.409,
 U(L2)=2.4×1=2.4.

Their sum is

 U(L1)+U(L2)=2.409+2.4=4.809.

The utility factors are close to each other,

 f(π1)=2.4094.809=0.501,f(π2)=2.44.809=0.499.

For the lottery attractiveness (46), we find

 a1=2.5×100.33+2.4×100.66+00.1=16.32,
 a2=2.4×101=24,

which gives

 a1+a2=16.32+24=40.32.

The attraction indices (47) become

 α1=16.3240.32=0.405,α2=2.440.32=0.595.

Then the attraction difference (55) is

 Δα=0.405−0.595=−0.19.

The negative attraction difference tells us that the first lottery is less attractive, , which suggests that the second lottery is preferable, . The experimental results confirm this, displaying the fractions of decision makers choosing the respective lotteries as

 pexp(π1)=0.18,pexp(π2)=0.82.

Thus, although the first lottery is more useful, having a larger utility factor, it is less attractive, which makes it less preferable.

Game 2. The lotteries are

 L1={2.5,0.33|0,0.67},L2={2.4,0.34|0,0.66}.

The following procedure is the same as in the first game. Calculating the utility factors

 f(π1)=0.503,f(π2)=0.497,

we again see that the lottery utilities are close to each other, so it is difficult to make the choice. For the lottery attractiveness, we have

 a1=16.57,a2=5.25,

giving the attraction indices

 α1=0.759,α2=0.241,

and the attraction difference

 Δα=0.518.

Now the latter is positive, showing that the first lottery is more attractive, , which suggests that the first lottery is preferable, . The experimental data for the related fractions are

 pexp(π1)=0.83,pexp(π2)=0.17,

in agreement with the expectation that the first lottery is preferable.

Game 3. The lotteries are

 L1={4,0.8|0,0.2},L2={3,1}.

We calculate in the prescribed way the utility factors

 f(π1)=0.516,f(π2)=0.484,

lottery attractiveness,

 a1=25.24,a2=30,

and the attraction indices

 α1=0.457,α2=0.543.

The negative attraction difference

 Δα=−0.086

implies that the first lottery is less attractive, , which tells us that the second lottery should be preferable, . Again this is in agreement with the experimental results

 pexp(π1)=0.2,pexp(π2)=0.8.

The first lottery is less preferable, despite it is more useful, having a larger utility factor.

Game 4. The lotteries are

 L1={4,0.2|0,0.8},L2={3,0.25|0,0.75}.

Calculating the utility factors

 f(π1)=0.516,f(π2)=0.484,

lottery attractiveness

 a1=6.34,a2=5.33,

and the attraction indices

 α1=0.543,α2=0.457,

we find the positive attraction difference

 Δα=0.086.

Hence the first lottery is more attractive , which suggests that the first lottery is preferable, . The experimental data

 pexp(π1)=0.65,pexp(π2)=0.35

confirm this expectation.

Game 5. The lotteries are

 L1={6,0.45|0,0.55},L2={3,0.9|0,0.1}.

The utility factors

 f(π1)=0.5,f(π2)=0.5

turn out to be equal, which makes it impossible to decide in the frame of classical decision theory based on expected utilities. Then we calculate the lottery attractiveness

 a1=16.91,a2=23.83,

and the related attraction indices

 α1=0.415,α2=0.585.

The negative attraction difference

 Δα=−0.17

means that the first lottery is less attractive, , thence the second lottery is expected to be preferable, . This is confirmed by the empirical data

 pexp(π1)=0.14,pexp(π2)=0.86.

Game 6. The lotteries are

 L1={6,0.001|0,0.999},L2={3,0.002|0,0.998}.

Again their utility factors are equal to each other,

 f(π1)=0.5,f(π2)=0.5.

The lottery attractiveness values

 a1=6.01,a2=3.01

yield the attraction indices

 α1=0.666,α2=0.334,

whose positive attraction difference

 Δα=0.332

implies that the first lottery is more attractive, , which suggests that the first lottery should be preferable, . The experimental results are

 pexp(π1)=0.73,pexp(π2)=0.27,

in agreement with the expectation.

Game 7. The lotteries are

 L1={6,0.25|0,0.75},L2={4,0.25|2,0.25|0,0.5}.

Their equal utility factors,

 f(π1)=0.5,f(π2)=0.5,

do not allow us to make a choice based on their utility. We calculate the lottery attractiveness

 a1=10.67,a2=10.67

and the attraction indices

 α1=0.5,α2=0.5.

Here the attraction difference is zero, , with the attraction indices being positive. Therefore, we resort to criterion (51), for which the minimal gains are . We find that

 p1(gmin1)=0.75>p2(gmin2)=0.5.

According to definitions (51) and (52), the marginal case, when and , is denoted as . This proposes that the first lottery is less attractive, according to the negative sign

 Δα=−0.

Thus we find that , which suggests that the second lottery is preferable, . The experimental results give

 pexp(π1)=0.18,pexp(π2)=0.82.

### Vi-B Lotteries with losses

In the previous seven games, the lotteries with gains were considered. We now turn to lotteries with losses.

Game 8. The lotteries are

 L1={−4,0.8|0,0.2},L1={−3,1}.

Following the same general procedure, we find the utility factors

 f(π1)=0.484,f(π2)=0.516,

lottery attractiveness

 a1=−25.24,a2=−30,

and the attraction indices

 α1=−0.457,α2=−0.543.

The positive attraction difference

 Δα=0.086

means that the first lottery is more attractive, , because of which, we expect that the first lottery is preferable, . The experiments give

 pexp(π1)=0.92,pexp(π2)=0.08,

confirming that the first lottery is preferable, although its utility factor is smaller.

Game 9. The lotteries are

 L1={−4,0.2|0,0.8},L2={−3,0.25|0,0.75}.

With the utility factors

 f(π1)=0.484,f(π2)=0.516,

lottery attractiveness

 a1=−6.34,a2=−5.33,

and the attraction indices

 α1=−0.543,α2=−0.457,

the attraction difference is negative,

 Δα=−0.086.

Thence the first lottery is less attractive, , and we expect that the second lottery is preferable, . The empirical data are

 pexp(π1)=0.42,pexp(π2)=0.58.

Game 10. The lotteries are

 L1={−3,0.9|0,0.1},L2={−6,0.45|0,0.55}.

The utility factors are equal,

 f(π1)=0.5,f(π2)=0.5,

hence both lotteries are equally useful. But the lottery attractiveness is different,

 a1=−23.83,a2=−16.91,

yielding the attraction indices

 α1=−0.585,α2=−0.415.

The negative attraction difference

 Δα=−0.17

signifies that the first lottery is less attractive, , which hints that the second lottery is preferable, . The experimental results are

 pexp(π1)=0.08,pexp(π2)=0.92.

Game 11. The lotteries are

 L1={−3,0.002|0,0.998},
 L2={−6,0.001|0,0.999}.

The utility factors are again equal to each other,

 f(π1)=0.5,f(π2)=0.5,

which makes it impossible to employ the classical utility theory. But the lottery attractiveness

 a1=−3.01,a2=−59.86

and the attraction indices

 α1=−0.048,α2=−0.952

show that the attraction difference is positive,

 Δα=0.904.

Therefore the first lottery is more attractive, , which suggests that the first lottery is preferable, . The experimental data are

 pexp(π1)=0.7,pexp(π2)=0.3.

Game 12. The lotteries are

 L