Quantum dynamical mode (QDM): A possible extension of belief function

Quantum dynamical mode (QDM): A possible extension of belief function

Fuyuan Xiao School of Computer and Information Science, Southwest University, China,
No.2 Tiansheng Road, BeiBei District, Chongqing, 400715,P.R.China
Abstract

Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. Besides, it has been proven that the quantum theory has powerful capabilities of solving the decision making problems, especially for modelling human decision and cognition. However, due to the inconsistency of the expression, the classical Dempster–Shafer evidence theory modelled by real numbers cannot be integrated directly with the quantum theory modelled by complex numbers. So, how can we establish a bridge of communications between the classical Dempster–Shafer evidence theory and the quantum theory? To answer this question, a generalized Dempster–Shafer evidence theory is proposed in this paper. The main contribution in this study is that, unlike the existing evidence theory, a mass function in the generalized Dempster–Shafer evidence theory is modelled by a complex number, called as a complex mass function. In addition, compared with the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between two evidences is released in the generalized Dempster’s combination rule so that it is more general and applicable than the classical Dempster’s combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. This generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Numerical examples are illustrated to show the efficiency of the generalized Dempster–Shafer evidence theory. Finally, an application of an evidential quantum dynamical model is implemented by integrating the generalized Dempster–Shafer evidence theory with the quantum dynamical model. From the experimental results, it validates the feasibility and effectiveness of the proposed method.

keywords:
Dempster–Shafer evidence theory, Generalized Dempster–Shafer evidence theory, Belief function, Quantum theory, Complex number
journal: Artificial Intelligence In Medicine

1 Introduction

How to measure the uncertainty has been an attracting issue in information fusion area. The amount of theories had been proposed and extended for measuring the uncertainty, including the rough sets theory (1), fuzzy sets theory (2; 3; 4; 5), evidence theory (6; 7; 8; 9), Z numbers (10; 11), D numbers (12; 13; 14; 15; 16; 17), evidential reasoning (18; 19; 20; 21), and so on (22; 23).

As an uncertainty reasoning tool, Dempster–Shafer evidence theory was firstly presented by Dempster (6) in 1967 year. Soon afterwards, it had been developed by Shafer (24) in 1976 year. Thanks to the flexibility and effectiveness in modeling uncertainties without prior information, Dempster–Shafer evidence theory has been widely used in various fields of applications, like decision making (25; 26; 27; 28; 29; 30), pattern recognition (31; 32; 33; 34), risk analysis (35; 36; 37), supplier selection (38), fault diagnosis (39; 40; jiang2017failure; 41), and so on (42; 43; 44; 45; 46; 47; 48). Although Dempster–Shafer evidence theory is a very useful uncertainty reasoning tool, the fusing of highly conflicting evidences may result in counter-intuitive results (49). To address this issue, two main kinds of methodologies have been studied (50; 51; 52). One methodology focus on modifying Dempster’s combination rule (53; 54; 55), while the other one focus on pre-processing the bodies of evidences (56; 57).

Currently, the quantum theory has became an interesting and hot topic in solving the decision making problems. As justified in literatures (58; 59; 60), the quantum theory can better describe the way humans make judgments towards uncertainty and decisions under conflict environment. It has been known that the quantum theory is represented by complex probability (61). So the question remains, can we leverage the complex probability to express the Dempster–Shafer evidence theory in the same way? As a pioneer, Deng (62) first proposed a meta mass function expressed by complex numbers in Dempster–Shafer evidence theory. Inspired by his research work, a generalized Dempster–Shafer evidence theory is proposed in this study. The proposed method is both orthogonal and complementary to Deng (62)’s method. Specifically, a mass function in the generalized Dempster–Shafer evidence theory is modelled by a complex number, called as a complex mass function. Furthermore, compared with the classical Dempster’s combination rule, the condition in terms of the conflict coefficient between two evidences is released in the generalized Dempster’s combination rule. Hence, the proposed method is more general and applicable than the classical Dempster’s combination rule. In particular, when the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between two evidences is less than 1. In this context, the generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Consequently, several numerical examples are provided to illustrate the efficiency of the generalized Dempster–Shafer evidence theory. Besides, an application of an evidential quantum dynamical model is implemented by integrating the generalized Dempster–Shafer evidence theory with the quantum dynamical model. The experimental results validate the feasibility and effectiveness of the proposed method.

The remaining content of this paper is organised below. Section 2 introduces the preliminaries of this paper briefly. In Section 3, a generalized Dempster–Shafer evidence theory is proposed. Section 4 gives numerical examples to illustrate the effectiveness of the proposal. In Section 5, an application of an evidential quantum dynamical model is implemented. Finally, Section 6 gives the conclusion.

2 Preliminaries

2.1 Complex number (61; 62)

A complex number is a number of the form,

(1)

where and are real numbers and is the imaginary unit, satisfying .

Give two complex numbers and , the addition is defined as follows:

(2)

The subtraction is defined as follows:

(3)

The multiplication is defined as follows:

(4)

The division is defined as follows:

(5)

An important parameter is the absolute value (or modulus or magnitude) of a complex number is

(6)

where if is a real number (i.e., ), then .

The square of the absolute value is

(7)

where is the complex conjugate of , i.e., .

2.2 Dempster–Shafer evidence theory (6; 24)

Dempster–Shafer evidence theory is extensively applied to handle uncertain information that belongs to the category of artificial intelligence. Because Dempster–Shafer evidence theory is flexible and effective in modeling the uncertainty regardless of prior information, it requires weaker conditions compared with the Bayesian theory of probability. When the probability is confirmed, Dempster–Shafer evidence theory degenerates to the probability theory and is considered as a generalization of Bayesian inference. In addition, Dempster–Shafer evidence theory has the advantage that it can directly express the “uncertainty” via allocating the probability into the set’s subsets, which consists of multi-objects, instead of a single object. Furthermore, it is capable of combining the bodies of evidence to derive new evidence. The basic concepts and definitions are described as below.

Definition 1

(Frame of discernment)

Let be a nonempty set of events that are mutually-exclusive and collectively-exhaustive, defined by:

(8)

in which the set denotes a frame of discernment.

The power set of is represented as , where:

(9)

and is an empty set.

When is an element of the power set of , i.e., , is called a hypothesis or proposition.

Definition 2

(Mass function)

In the frame of discernment , a mass function is represented as a mapping from to [0, 1] that is defined as:

(10)

which meets the conditions below:

(11)

The mass function in the Dempster–Shafer evidence theory can also be called a basic belief assignment (BBA). When is greater than zero, as the element of is named as a focal element of the mass function, where the mass function indicates how strongly the evidence supports the proposition or hypothesis .

Definition 3

(Belief function)

Let be a proposition where ; the belief function of the proposition is defined by:

(12)

The plausibility function of the proposition is defined by:

(13)

where is the complement of , such that .

Apparently, the plausibility function is equal to or greater than the belief function , where the belief function is the lower limit function of the proposition , and the plausibility function is the upper limit function of the proposition .

Definition 4

(Dempster’s rule of combination)

Let two basic belief assignments (BBAs) be and on the frame of discernment where the BBAs and are independent; Dempster’s rule of combination, defined by , which is called the orthogonal sum, is represented as below:

(14)

with

(15)

where and are also the elements of and is a constant that presents the conflict coefficient between the BBAs and .

Notice that Dempster’s combination rule is only practicable for the BBAs and under the condition that .

Definition 5

(Pignistic probability transformation)

Let be a basic belief assignment on the frame of discernment and be a proposition where , the pignistic probability transformation function is defined by

(16)

where represents the cardinality of .

3 Generalized Dempster–Shafer evidence theory

Let be a nonempty set of events that are mutually-exclusive and collectively-exhaustive, defined by:

(17)

in which the set denotes a frame of discernment.

The power set of is represented as , where:

(18)

and is an empty set.

Definition 6

(Complex mass function)

In the frame of discernment , a complex mass function is modelled as a complex number:

(19)

with

(20)

and is represented as a mapping from to , denoted as:

(21)

which meets the conditions below:

(22)

The complex mass function modelled as a complex number in the generalized Dempster–Shafer evidence theory can also be called a complex basic belief assignment (CBBA). When is greater than zero, as the element of is named as a focal element of the generalized mass function, where the mass function indicates how strongly the evidence supports the proposition or hypothesis . Note that is the magnitude of which can be calculated based on Eq. (6).

Definition 7

(Complex belief function)

Let be a proposition where ; a complex belief function of the proposition is defined by:

(23)

The complex plausibility function of the proposition is defined by:

(24)

where is the complement of , such that .

Apparently, the plausibility function is equal to or greater than the belief function , where the belief function is the lower limit function of the proposition , and the plausibility function is the upper limit function of the proposition .

Definition 8

(Generalized Dempster’s rule of combination)

Let two complex basic belief assignments (CBBAs) be and on the frame of discernment where the CBBAs and are independent; the generalized Dempster’s rule of combination, defined by , which is called the orthogonal sum, is represented as below:

(25)

with

(26)

where and are also the elements of and is a constant that presents the conflict coefficient between the CBBAs and .

Remark 1

Generalized Dempster’s combination rule is only practicable for the CBBAs and under the condition that the conflict coefficient .

Remark 2

Compared with the classical Dempster’s combination rule, the condition in terms of the conflict coefficient is released in the generalized Dempster’s combination rule so that it is more general and applicable than the classical Dempster’s combination rule.

Remark 3

When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient .

An example is given to prove that the condition is ignored in the generalized Dempster’s combination rule and depicts the variation of the magnitude of conflict coefficient between two CBBAs where can be calculated based on Eq. (6).

Example 0

Supposing that there are two CBBAs and in the frame of discernment , and the two CBBAs are given as follows:

, ;
, .

According to Definition 6, the parameters and are set within [-1, 1] satisfying the conditions that and at the same time.

Figure 1: An example of the variation of between two CBBAs from the front and the side angles.

Fig. 1 show the results of the magnitude of conflict coefficient between the two CBBAs and from different angles.

In particular, as shown in Fig. 1, in the case that and , we can obtain the and . The conflict coefficient is calculated as ; then the magnitude of conflict coefficient between the two CBBAs and is 0.7071.

When and , the and can be obtained. The conflict coefficient is calculated as ; then the magnitude of conflict coefficient between the two CBBAs and is 0.7071 which shows the same result as the case that and .

In the case that and , we can obtain the and . The conflict coefficient is calculated as ; then the magnitude of conflict coefficient between the two CBBAs and is 0.

When and , the and can be calculated. The conflict coefficient is calculated as ; then the magnitude of conflict coefficient between the two CBBAs and is 1.3660.

In the case that and , the and can be calculated. The conflict coefficient is calculated as ; then the magnitude of conflict coefficient between the two CBBAs and is 0.3660.

Definition 9

(Complex pignistic probability transformation)

Let be a complex basic belief assignment on the frame of discernment and be a proposition where , the complex pignistic probability transformation function is defined by

(27)

where represents the cardinality of .

4 Numerical examples

In this section, several numerical examples are illustrated to show the effectiveness of the generalized Dempster–Shafer evidence theory.

Example 0

Supposing that there are two CBBAs and in the frame of discernment , and the two CBBAs are given as follows:

, , ;
, , .

Then, the fusing results are calculated by utilising Eq. (25) as follows:

= 0.0979 + 0.0186i,
= 0.9031 - 0.1820i,
= -0.0010 + 0.1634i.

It is verified that + + = 1 in this example.

Example 0

Supposing that there are two CBBAs and in the frame of discernment , and the two CBBAs are given as follows:

, , .
, , ;

The fusing results by utilising Eq. (25) are calculated as follows:

= 0.0979 + 0.0186i,
= 0.9031 - 0.1820i,
= -0.0010 + 0.1634i.

It is obvious that + + = 1 in this example.

Through Example 2 and Example 3, it proves that the generalized Dempster–Shafer evidence theory satisfies the commutative law.

Example 0

Supposing that there are two CBBAs and in the frame of discernment where they are degenerated to real numbers, and the two CBBAs are given as follows:

, ;
, .

On the one hand, by utilising Eq. (25) of the generalized Dempster’s rule of combination, the fusing results are generated as follows:

,
;

On the other hand, based on Eq. (14) of the classical Dempster’s rule of combination, the fusing results are calculated as follows:

,
;

It is easy to see that the fusing results from the generalized Dempster’s rule of combination is exactly the same as the fusing results from the classical Dempster’s rule of combination. In this example, the conflict coefficient is 0.2600.

This example proves that when the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster’s combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1.

Example 0

Supposing that there are two highly conflicting CBBAs and in the frame of discernment where they are degenerated to real numbers, and the two CBBAs are given as follows:

, ;
, .

By utilising Eq. (25) of the generalized Dempster’s rule of combination, the fusing results are generated as follows:

= 1.

Based on Eq. (14) of the classical Dempster’s rule of combination, the fusing results are calculated as follows:

= 1.

In this example, highly conflicts with , because has a great belief value 0.99 on the object , while has a great belief value 0.99 on the object . However, as shown in the results, we can notice that when fusing the highly conflicting evidences, counter-intuitive results occur no matter we use the the generalized Dempster’s rule of combination or the classical Dempster’s rule of combination.

Example 0

Supposing that there are two highly conflicting CBBAs and in the frame of discernment , and the two CBBAs are given as follows:

, ;
, .

The fusing results are calculated by utilising Eq. (25) of the generalized Dempster’s rule of combination as follows:

= 1.0000 + 0.0000i.

Through Example 5 and Example 6, it is implied that counter-intuitive results occur when fusing the highly conflicting evidences modelled by either real numbers or complex numbers via the generalized Dempster’s rule of combination.

5 Application

In this section, the proposed method is incorporated in quantum dynamical model, where the experimental data sets in (63; 64) are used for the comparison with the related methods.

5.1 Problem statement

A new paradigm was presented by Townsend et al. (65) in 2000 year to investigate the interactions between categorisation and decision-making. Initially, this new paradigm was utilised to test a Markov model. Afterward, it was extended for comparisons of Markov and quantum dynamical models by Busemeyer et al. (63) in 2009. In a categorisation (C) - decision (D) task, two different distributions of faces were utilised and shown to participants on each trial. In particular, for a “narrow” face distribution, it had a narrow width and thick lips on average as shown in Fig. 2(a); for a “wide” face distribution, it had a wide width and thin lips on average as shown in Fig. 2(b). The participants were requested to categorise the faces as a “good” guy or “bad” guy group, and/or they were requested to decide whether to take an “attack” or “withdraw” action. The participants were notified that “narrow” faces had a 0.60 probability or chance to come from the “bad” guy population, while “wide” faces had a 0.60 probability or chance to come from the “good” guy population. Thereinto, two test conditions, namely, a C-then-D condition and a D-alone condition were implemented to each participant across a series of trials. Under the C-then-D condition, participants were requested to categorise the faces first, then made an action decision. Differ with the C-then-D condition, participants only were requested to make an action decision without categorisation under the D-alone condition. The experiment included 26 participants in total, of which for the C-then-D condition, each participant given 51 observations producing = 1326 observations, and for the D-alone condition, each person provided 17 observations producing = 442 total observations.

(a) Narrow category
(b) Wide category
Figure 2: Example faces used in a categorisation-decision experiment.
  Type face
  Wide 0.84 0.35 0.16 0.52 0.37 0.39 0.5733
  Narrow 0.17 0.41 0.83 0.63 0.59 0.69 2.54
Table 1: Experimental results of a categorisation-decision task.

The experimental results were shown in Table 1. The column labeled denotes the probability of categorising the face as a “good” guy; the column labeled shows the probability of attacking when the face was categorised as a “good” guy. The column labeled represents the probability of categorising the face as a “bad” guy; the column labeled shows the probability of attacking when the face was categorised as a “bad” guy. Then, the column labeled represents the total probability of attacking as

(28)

On the other hand, the column labeled denotes the probability of attacking when this decision was made alone.

In accordance with the law of total probability, the probability of attacking was supposed to be equal under two conditions. Nevertheless, some deviation between and were generated for both faces as shown in Table 1. Especially, for the narrow faces, the most pronounced deviation arose that caused a large positive interference effect. Through a paired -test to measure the significance of the difference between and , the results indicated that the mean interference effect was statistically significant for the narrow faces, but not for the wide faces. In this study, therefore, the interference effect is investigated and analysed in terms of attacking actions towards the narrow faces.

5.2 Implementation

In this section, the proposed method is integrated into quantum dynamical model to model the human decision making process in an evidential framework.

5.2.1 Representation of beliefs and actions

In an evidential quantum dynamical model, the categorisation (C) - decision (D) experiment involves a set of six exhaustive outcomes , , , , , , where, for instance, symbolises the event in which the participant believes the face as a “good” (G) guy, but the participant intends to take an “attack” (A) action, while symbolises the event in which the participant is skeptical or hesitating of the face as a “good” or “bad” (B) guy that is in a an uncertain (U) condition, but the participant intends to act by withdrawing (W). The evidential quantum dynamical model assumes that these six events correspond to six basis belief-action states of the decision maker , , , , , . All the possible transitions between the six basis states in an evidential quantum dynamical model are depicted in Fig. 3.

Figure 3: Transition diagram in an evidential quantum dynamical model.

At the beginning of a categorisation-decision task, the participant has some possibilities to be in every basis state in Fig. 3. Hence, the state of a participant is a superposition of the six orthonormal basis states, denoted by

(29)

An amplitude distribution corresponding to the initial state is denoted by the following column matrix,

(30)

where represents the probability of observing basis state initially in which and . The squared length of must be equal to one, such that , where is the conjugate of . Here, the probability of initial state is assumed to be distributed averagely.

5.2.2 Inferences based on prior information

In the course of decision making process, the initial state with regard to the participant’s beliefs at time is turned into a new state at time . For the evidential quantum dynamical model, the categorisation of faces are decided by participants under the C-then-D condition. When the face is classified as a “good” guy, the amplitude distribution across the basis states becomes

(31)

in which represents the initial probability of categorising the face as a “good” guy. This matrix has a squared length that is equal to one. It is a conditional amplitude distribution across actions under the situation where the face is classified as a “good” guy.

When the face is categorised as a “bad” guy, the amplitude distribution across the basis states states turns into

(32)

in which is the initial probability of categorising the face as a “bad” guy. This matrix has a squared length that is equal to one. It is a conditional amplitude distribution across actions under the situation where the face is classified as a “bad” guy.

When the face cannot be categorised as a “good” or “bad” guy due to the skepticism or hesitation of participant, the amplitude distribution across the basis states becomes

(33)

in which denotes the initial probability that the participant cannot categorise the face as a “good” or “bad” guy because of lacking sufficient information. This matrix has a squared length that is equal to one. It is a conditional amplitude distribution across actions under the case where the face cannot be classified and it is in an uncertain situation.

Under the D alone condition, because the participant is not requested to categorise the faces before taking an action, there is no new information involved in terms of categorisation. Therefore, the amplitude distribution across the basis states remains the same as the initial one

(34)

where it represents the initial state under a condition without categorisation as a superposition which is a weighted sum of the amplitude distributions for the two conditions.

5.2.3 Strategies based on payoffs

In order to choose an appropriate action, a decision maker needs to assess the payoffs, so that it turns the previous state at time into a new state at time . The state evolution during this time period corresponds to the thought process resulting in a decision. For the evidential quantum dynamical model, the evolution of the state obeys a Schrdinger equation during the decision making process which is driven by a Hamiltonian matrix :

(35)

where is a Hermitian matrix: that will be discussed below.

It has the following matrix exponential solution for ,

(36)

where represents the amplitude distribution across states after evolution by evaluating the payoffs, and a unitary matrix is defined by

(37)

which determines the transition probabilities.

Here, the Hamiltonian matrix is defined by

(38)

where

(39)

When the face is categorised as a “good” guy by the participant, the Hamiltonian matrix is supposed to be utilised, while when the face is categorised as a “bad” guy by the participant, the Hamiltonian matrix should be used. If the participant cannot categorise the face as a “good” or “bad” guy which is in an uncertain state, the Hamiltonian matrix will be applied. To be specific, the parameter is a function of the difference between the payoffs for attacking with respect to withdrawing when categorising the face as a “good” guy; the parameter is a function of the difference between the payoffs for attacking with respect to withdrawing when categorising the face as a “bad” guy; the parameter is a function of the difference between the payoffs for attacking with respect to withdrawing when the participant cannot categorise the face. The Hamiltonian matrix transforms the state probabilities to favor either attacking or withdrawing according to the payoff in terms of each belief state.

Afterwards, the state of the participant at time can be obtained. In the C-then-D condition, when the face is classified as a “good” guy, the state at time changes into the state at time by

(40)

When the face is classified as a “bad” guy, the state at time turns into the state at time by

(41)

When the participant cannot categorise the face as a “good” or “bad” guy, the state at time becomes the state at time by

(42)

On the other hand, in the D alone condition, the state at time turns into the state at time by

(43)

where it expresses the state at time under unknown categorisation condition as a superposition which is a weighted sum of the amplitude distributions for the two cases.

5.2.4 Predictions of the evidential quantum dynamical model

In the evidential quantum dynamical model, the interference effect can be predicted based on the state evolution of the participant. In order to predict a state of attacking with regard to a certain categorisation of face, a measure matrix is defined by

(44)

where when the face is categorised as a “good” guy by the participant, the measure matrix is supposed to be utilised; when the face is categorised as a “bad” guy by the participant, the Hamiltonian matrix should be used; if the participant cannot categorise the face as a “good” or “bad” guy, the Hamiltonian matrix will be applied.

In the C-then-D condition, for measuring the belief of attacking with respect to the situation where the face is categorised as a “good” guy, the measure matrices , and are set as follows

(45)

Then, the prediction of the attacking belief for the three cases, i.e., , and can be obtained by the following equations, respectively

(46)
(47)
(48)

Then, on the basis of Eq. (27), the belief of uncertain case that the face cannot be classified by the participant will be assigned to another two certain cases equally, denoted by

(49)
(50)

where represents the conditional amplitude of attacking when the face is categorised as a “good” guy with the involvement of uncertain information, which has a squared length equal to ; represents the conditional amplitude of attacking when the face is categorised as a “bad” guy involving the uncertain information, which has a squared length equal to .

After that, the prediction of total probability for attacking under the C-then-D condition can be calculated by

(51)

In the D alone condition, in order to measure the belief of attacking without categorisation, the measure matrices and are set as follows

(52)

Thus, the prediction of total probability for attacking without categorisation can be computed by

(53)

We can notice that the prediction of total probability for attacking under the C-then-D condition is different comparing with the D alone condition. The difference of probability between these two conditions indicates the interference effect caused by the interactions between categorisation and decision-making. Specifically, the belief of uncertain state is modelled and transferred into another two certain states under the C-then-D condition. Whereas, the function is not generated under the D alone condition, since the action of attacking is taken without categorisation. As a results, the interference effect resulted from the interactions between categorisation and decision-making can be predicted under these two different conditions.

5.3 Experimental results

5.3.1 Parameter setting

In the experiments, based on literatures (66; 59; 63; 64), on account of realising the predictions for the evidential quantum dynamical model, the time process parameter is set as to allow the selection probability to achieve maximum across time. or is set as the same with the relevant observed experimental results. Meanwhile, three free parameters, , and are estimated under the C-then-D condition, while two free parameters, and are estimated under the D alone condition. These free parameters are fitted by minimising the sum of squared errors (SSE) between the predicted and observed mean probability judgments for each of the two conditions.

5.3.2 Comparisons of different models

In order to validate the feasibility and effectiveness of the proposed evidential quantum dynamical model, it is compared with the Markov belief-action (MBA) model (65), the quantum belief-action entanglement (QBAE) model (63), and the evidential Markov (EM) model (66). The comparisons of prediction results for the categorisation-decision task under two above-mentioned conditions (i.e., the C-then-D condition and the D alone condition) are shown in Table 2, in which denotes the observed experimental results from the literatures (63; 64).

The columns labeled and represent the predicted probabilities of attacking when the faces are classified as a “good” guy and a “bad” guy under the C-then-D condition, respectively. The column labeled denotes the predicted total probability of attacking under the C-then-D condition, while the column labeled represents the predicted total probability of attacking under the D alone condition.

  Literature Method