An evidential Markov decision making model

arXiv:1705.06578v1 [] 10 May 2017

Zichang Hea , Wen Jianga,∗
a

School of Electronics and Information, Northwestern Polytechnical University, Xi’an,
Shaanxi, 710072, China

Abstract
The sure thing principle and the law of total probability are basic laws in
classic probability theory. A disjunction fallacy leads to the violation of
these two classical laws. In this paper, an Evidential Markov (EM) decision making model based on Dempster-Shafer (D-S) evidence theory and
Markov modelling is proposed to address this issue and model the real human
decision-making process. In an evidential framework, the states are extended
by introducing an uncertain state which represents the hesitance of a decision
maker. The classical Markov model can not produce the disjunction effect,
which assumes that a decision has to be certain at one time. However, the
state is allowed to be uncertain in the EM model before the final decision is
made. An extra uncertainty degree parameter is defined by a belief entropy,
named Deng entropy, to assignment the basic probability assignment of the
uncertain state, which is the key to predict the disjunction effect. A classical
categorization decision-making experiment is used to illustrate the effectiveness and validity of EM model. The disjunction effect can be well predicted
Corresponding author at Wen Jiang: School of Electronics and Information, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China. Tel: (86-29)88431267.
E-mail address: jiangwen@nwpu.edu.cn, jiangwenpaper@hotmail.com
∗

Preprint submitted to Elsevier

May 19, 2017

and the free parameters are less compared with the existing models.
Keywords: Evidential Markov model; Markov decision making model;
Dempster-Shafer evidence theory; the sure thing principle; the law of total
probability; disjunction effect

1. Introduction

The sure thing principle introduced by Jim Savage[1] is fundamental in economics and probability theory. It means that if one prefers action A over
B under state of the world X, while action A is also preferred under the
complementary state ¬X, then it can be concluded that one will still prefer
action A over B under the state is unknown. The law of total probability
is a fundamental rule of Bayesian probability relating marginal probabilities
to conditional probabilities. It expresses the total probability of an outcome
which can be realized via several distinct events. However, many experiments and studies have shown that the sure thing principle and the law of
total probability can be violated due to the disjunction effect[2–4]. The disjunction fallacy is an empirical finding in which the proportion taking the
target gamble under the unknown condition falls below both of the proportions taking the target gamble under each of the known conditions. The
same person takes the target gamble under both known conditions, but then
rejects the target gamble under the unknown condition[5].
To explain the disjunction fallacy, many studies have been proposed. The
original explanation was a psychological idea based on the failure of consequential reasoning under unknown condition[5]. A Markov decision making
model was proposed by Townsend et al. (2000)[3]. A Markov process can be
used to model a random system that changes states according to a transition
2

rule that only depends on the current state. Markov models has been used in
many applications[6–9], especially in the prediction[10, 11], such as rainfall
prediction[12, 13], economy prediction[14, 15] and so on. However, it failed to
predict the disjunction effect, which will be introduced later. More recently,
the theory of quantum probability has been introduced in the cognition and
decision making process. Quantum probability is an effective approach to
psychology and behavioristics [16–19]. It has been widely applied to the fields
of cognition and decision making fields to explain the phenomena in classical
theory, like order effect[20–22], disjunction effect[4], the interference effect of
categorization[23], prisoner’s dilemma[24], conceptual combinations[25, 26],
quantum game theory[27–29] and so on. To explain the disjunction fallacy,
many quantum models have been proposed, such as a quantum dynamical
(QD) model[30, 31], quantum-like models[32–34] quantum prospect decision
theory[35–38] and quantum-like Bayesian networks[39] etc. In a quantum
framework, the decision in human brain is deemed as a superposition of several decisions before the final one is made. Although the quantum model
works for explaining the disjunction fallacy, the major problem is the additional introduction of quantum parameters.
Decision making and optimization under uncertain environment is normal
in reality and is heavily studied[40–42]. It is still an open issue for uncertain information modeling and processing[43–45]. Some methods have
been proposed to handle the uncertainty like probability theory[46], fuzzy
set theory[47] , Dempster-Shafer evidence theory[48, 49], rough sets[50], and
D-numbers[51]. Dempster-Shafer (D-S) evidence theory[48, 49] is a powerful tool to handle the uncertainty. It has been widely used in many fields,
like risk analysis[52–54], controller design[55, 56], pattern recognition[57–59],
fault diagnosis[60–62], multiple attribute decision making[63, 64] and so on.
Also D-S theory is widely combined with Markov models[65–68].

3

In this paper, an evidential Markov (EM) decision making model based on
D-S evidence theory and Makov modelling is proposed to model the decision
making process. In the paradigm of studying the disjunction fallacy, generally
a decision making process is consisted of beliefs and actions. The EM model
assumes that people’s decision can be uncertain before the final one is made
while a it is assumed that people is decision has to be certain at one time.
In an evidential framework, the actions states are extended by introducing
an uncertain state to represent the hesitance of a decision maker. An extra
uncertainty degree is determined by a belief entropy, named Deng entropy,
which is the key to predict the disjunction effect in our model. The EM
model is used in an categorization decision-making experiment. The model
results is discussed and compared, which show the effectiveness and validity
of our model.
The rest of the paper is organized as follows. In Section 2, the preliminaries of
basic theories employed are briefly introduced. The classical Markov decision
making model is introduced in Section 3. Then a categorization decisionmaking experiment is illustrated in Section 4. Our EM model is proposed
and applied to explain the experiment result in Section 5. The model result
is discussed and compared in Section 6. Finally, Section 7 comes to the
conclusion.

2. Preliminaries

2.1. Dempster-Shafer evidence theory
In D-S theory, a finite set of N mutually exclusive and exhaustive elements,
called the frame of discernment (FOD), which is symbolized by Θ= {H1 , H2, . . . , HN }.
4

Let us denote P (Θ) as the power set composed of 2N elements A of Θ:
P (Θ) = {∅, {H1 } , {H2 } , . . . , {HN } , {H1 ∪ H2 } , {H1 ∪ H3 } , . . . , Θ}
A basic probability assignment (BPA) is a mapping from P (Θ) to [0, 1],
defined as m : P (Θ) → [0, 1] satisfying[48, 49]:
P

m (A) = 1,
(1)

A∈P (Θ)

m (∅) = 0.
The mass function m represents supporting degree to focal element A. The
elements of P (Θ) that have a non-zero mass are called focal elements. A
body of evidence (BOE) is the set of all the focal elements, defined as
n
o
(ℜ, m) = [A, m (A)] ; A ∈ P (Θ) , m > 0 ,

A mass function corresponds to a belief (Bel) function and a plausibility (P l)

function respectively. Given m : P (Θ) → [0, 1], Bel (A) function represents
the whole belief degree to A, defined as
Bel (A) =

P

m (B) ; ∀A ⊆ P (Θ)

P

m (B) ; ∀A ⊆ P (Θ)

B⊆A

(2)

P l function represents the belief degree of not denying A, defined as
P l (A) =

B∩A6=∅

(3)

It should be noticed that when all the focal sets of m are singletons, m is
said to be Bayesian; Bel and P l then degenerate into the same probability
measure.
In the following, an game of picking ball will be used to show the D-S theory’s
ability of handling uncertainty[42]. There are two boxes filled with some balls
as shown in Figure 1. Left box is contended with red balls and right box
5

is contended with blue balls. The number of balls in each box is unknown.
Now, a ball is picked randomly from two boxes. The probability of picking
from left box P 1 is known as 0.4 while picking from right box P 2 is known
as 0.6. It is easy to obtain that the probability of picking a red ball is
0.4 while picking a blue ball is 0.6 based on probability theory. Now, the

Figure 1: A game of picking ball which can be handled by probability theory

situation changes as shown in Figure 2. The left box is contended with right
balls while the right box is contended with red and blue balls. The exact
number of the balls in each box and the ratio of red balls with blue balls
are completely unknown. The probabilities of selecting from two boxes keep
the same, P 1 = 0.4 and P 2 = 0.6. The question is what the probability
that a red ball is picked is. Due to the lack of information, the question can
not be addressed in probability theory. However, D-S evidence theory can
effectively handle it. We can obtain two mass functions that m (R) = 0.4
and m (R, B) = 0.6. Then the uncertainty is well handled in the frame of
D-S theory.

6

Figure 2: A game of picking ball where probability theory is unable but D-S evidence
theory is able to handle

2.2. Pignistic probability transformation
The term ”pignistic” proposed by Smets is originated from the word pignus,
meaning bet in Latin. Pignistic probability transformation (PPT) has a
wide application in decision making. Principle of insufficient reason is used
to assign the probability of a multiple-element set to singleton sets. In other
word, a belief interval is distributed into the crisp one determined as[69]:
X
m (B)
Bet (A) =
(4)
A⊆B |B|

where m is a mass function, B is the focal element of m and |B| is the
cardinality of B which denotes the number of elements in set B.

2.3. Deng entropy
Deng entropy is proposed to measure uncertainty, which is defined as follows[42]:
X
m (A)
,
(5)
m (A) log2 |A|
Ed = −
2
−
1
A∈X
7

where m is a mass function defined on FOD X, A is the focal element of m
and |A| is the cardinality of A. Especially, for a BOE (ℜ, m), if all the focal
elements are singletons, Deng entropy degenerates into Shannon entropy.
Ed = −

X

m (A) log2

A∈X

X
m (A)
=
−
m (A) log2 m (A)
21 − 1
A∈X

(6)

3. Classical Markov decision-making model

Townsend et al. (2000)[3] originally proposed a Markov model for studying
the disjunction effect. For simplicity, we use a two-dimensional Markov model
to illustrate the decision making process.

3.1. State representation
The model assumes that the perceptual system can be in one of two states at
each moment in time: a plus state denoted |+i or a minus state denoted |−i
representing the other orientation. A sample path by the Markov process
starts in some initial state, either |S (0)i = |+i or |S (0)i = |−i, and jumps
back and forth across time from a clear plus state to a clear minus state
(Figure 3). In general, however, we do not know which it is for any given

Figure 3: Two-state transition digram

sample path, and so the uncertainty is represented by assigning an initial
8

probability distribution across these two possibilities, denoted φ+ (0) for plus
and φ− (0) for minus, and φ+ (0) +φ− (0) =0.

3.2. State transition
The initial state is defined by a probability distribution represented by a 2×1
matrix
φ (0) =

"

φ+ (0)
φ− (0)

#

.

If the system is known to start from the plus state, then φ+ (0) = 1; if the
state is known to start from the minus state, then φ− (0) = 1; and if the
state is unknown, then φ+ (0) + φ− (0) = 1. After some period of time t1 , the
perceptual system can transit to a new state. Hence, a probability transition
matrix is determined as
T (t1 , 0) =

"

T+,+ (t1 , 0) T+,− (t1 , 0)
T−,+ (t1 , 0) T−,− (t1 , 0)

#

,

where, for example, T+,− (t1 , 0) represents the probability of transferring from
the minus state at time zero to the plus state at time t1 . It is called a
stochastic matrix because all the elements are nonnegative and sum to unity
with a column. The matrix product of the transition matrix times the initial
probability distribution at time zero produces the updated distribution across
the two states at time t1 :
φ (t1 ) = T (t1 , 0) · φ (0) .
Given the state at time t1 , we can update again using the transition matrix
T (t2 , t1 ) that describes the probabilities of transitions for the period of time
from t1 to t2 :
φ (t2 ) = T (t2 , t1 ) · φ (t1 ) = T (t2 , t1 ) · T (t1 , 0) · φ (0) .
9

It is reasonable to assume that the transition matrix remains stationary for
many applications. Hence, the transition matrix can be simply write as
T (t), in which t is the duration time period. In this case, the probability
distribution at t2 equals
φ (t2 ) = T (t2 ) · φ (0) = T (t2 − t1 ) · T (t1 ) · φ (0) .

(7)

It means that the transition matrix obeys the semi-group property of dynamic
systems,
T (t + u) = T (t) · T (u) = T (u) · T (t) .
The time evolution of the transition matrix should obey the following differential equation, called the Kolmogorov forward equation.
dT (t)
= K · T (t)
dt

(8)

Also the time evolution of the probability distribution over states should
obey:
dφ (t)
= K · φ (t) .
(9)
dt
The 2 × 2 matrix K is called the intensity matrix, which is used to construct
the transition matrix as a function of processing time duration. The matrix
must satisfy the following constraints to guarantee that the solution is a
transition matrix: the off-diagonal elements must be nonnegative but the
sum to zero with a column. The solution of Kolmogorov equation is a matrix
exponential function:
T (t) = etK ,
φ (t) = etK · φ (0) .

(10)

3.3. Response probabilities
A response refers to a measurement that the perceiver maker or reports. To
derive response probabilities at various time points from the Markov model,
10

measure matrixes are defined as
"
#
"
#
1 0
0 0
M+ =
and M− =
,
0 0
0 1
where M+ is applied to measure the plus state and M− is applied to measure
the minus state. Let R (t) = + denote the response that the system is in the
plus state at time t. In the following, it will be convenient to use the 1 × 2
matrix L = [1, 1] perform summation across states. Using these matrixes,
the probability if observing a ”plus” at time t equals
p (R (t) = +) = L · M+ · T (t) · φ (0) .

(11)

3.4. The law of total probability
Assume that the initial state is known to ”plus” or ”minus”, the probability
of R (t) = + equals
p (R (t) = +|+) = [1, 1] ·

"

1 0

p (R (t) = +|−) = [1, 1] ·

"

1 0

0 0
0 0

# "
·

# "
·

T+,+ (t) T+,− (t)
T−,+ (t) T−,− (t)
T+,+ (t) T+,− (t)
T−,+ (t) T−,− (t)

#"

1

#"

0

0
1

#

= T+,+ (t) ,

#

= T+,− (t)

respectively. Assume that the initial state is unknown, then the probability
of R (t) = + equals
p (R (t) = +|U) = [1, 1] ·

"

1 0
0 0

# "
·

T+,+ (t) T+,− (t)
T−,+ (t) T−,− (t)

#"

φ+ (0)
φ− (0)

#

= T+,+ (t) φ+ (0) + T+,− (t) φ− (0) .
The last line expresses that the probability of being the plus state for the
unknown condition equals to the probability of reaching it by two different
11

paths (see Figure ??), which means that the law of total probability is not
violated.
Hence, the classical Markov model can not explain and predict the disjunction
effect. To address it, an evidential Markov model is proposed. There are
several paradigms for studying the disjunction fallacy, in the following, a
categorization decision-making experiment is briefly introduced.

4. Categorization decision-making experiment

4.1. Experiment method
Townsend et al.[3] proposed a categorization decision-making experiment
to study the interactions between categorization and decision making. It
turned out that categorization can result in the disjunction effect on decision
making. In the experiment, pictures of faces varying along face width and lip
thickness are shown to participants. Generally, the faces can be distributed
into two different kinds: on average a narrow (N) face with a narrow width
and thick lips; on average a wide (W) face with a wide width and thin lips
(see Figure 4 for example). The The participants were informed that N
face had a 0.60 probability to come from the ”bad” guy population while
W face had a 0.60 probability to come from the ”good” guy population.
The participants were usually (probability 0.70) rewarded for attacking ”bad
guys” and they were usually (probability 0.7) rewarded for withdrawing from
”bad guys”. The primary manipulation was produced by using the following
two test conditions, presented across a series of trials, to each participant.
In the categorization-decision making (C-D) condition: participants were
asked to categorize a face as belonging to either a ”good” (G) guy or ”bad”
12

Figure 4: Example faces used in a categorization decision-making experiment

(B) guy group. Following the categorization, they were asked to make a
decision whether to ”attack” (A) or to to ”withdraw” (W). In the decisionmaking alone (D-alone) condition: participants were only asked to make an
action decision. The experiment included a total of 26 participants, but each
participant provided 51 observations for the C-D condition for a total of
26 × 51 = 1326 observations, while each person produced 17 observations for
the D condition for a total of 17 × 26 = 442 total observations.

4.2. Experiment results
The experiment results are shown in Table 1. The column labeled P (G)
Table 1: The results of Townsend et al.[3]

Type face

P (G) P (A|G) P (B)

P (A|B)

PT

P (A)

Wide

0.84

0.35

0.16

0.52

0.37

0.39

Narrow

0.17

0.41

0.83

0.63

0.59

0.69

shows the probability of categorizing the face as a ”good buy”; the column
labeled P (A|G) shows the probability of attacking given categorizing the
face as a ”good guy”; the column labeled P (B) shows the probability of
categorizing the face as a ”bad buy”; the column labeled P (A|B) shows
the probability of attacking given categorizing the face as a ”bad guy”; and
13

the column labeled PT shows the total probability of attacking in the C-D
condition computed by
P (A) = P (G) · P (A|G) + P (B) · P (A|B) ,

(12)

and the column labeled P (A) represents the probability of attacking in the
D-alone condition. Obviously, PT and P (A) are unequal for both types of
face. According to Bayesian probability theory, this violates the law of total
probability and the difference value is called the disjunction effect. In this
experiment, the disjunction effect for wide type faces is so weak that it can
be ignored and explained as the statistical error, however, the disjunction
effect for narrow type is too prominent to ignore.
The classical paradigm has been discussed in many later works. Literatures
of studying the categorization decision-making experiment and their results
are shown below in Table 2.

5. Evidential Markov model

The EM model based on D-S evidence theory and Markov modelling is proposed to model the human decision making process. In the following, we will
use the categorization decision-making experiment to illustrate our model.
The disjunction fallacy can be explained and the disjunction effect for narrow
type faces can be well predicted in our model. The flowchart of EM model
is shown in Figure ??.
Step 1: Frame of discrimination determined
In the C-D condition, the outcome will be either to attack or withdraw given
the face is categorized as a good guy or a bad guy. Thus the FOD ΘCD is
14

Table 2: Results of categorization decision-making experiments

Literature

Type

P (G)

P (A|G)

P (B)

P (A|B)

PT

P (A)

W

0.84

0.35

0.16

0.52

0.37

0.39

N

0.17

0.41

0.83

0.63

0.59

0.69

W

0.80

0.37

0.20

0.53

0.40

0.39

N

0.20

0.45

0.80

0.64

0.60

0.69

Wang & Busemeyer (2016)

W

0.78

0.39

0.22

0.52

0.42

0.42

Experiment 1[23]

N

0.21

0.41

0.79

0.58

0.54

0.59

Wang & Busemeyer (2016)

W

0.78

0.33

0.22

0.53

0.37

0.37

Experiment 2[23]

N

0.24

0.37

0.76

0.61

0.55

0.60

Wang & Busemeyer (2016)

W

0.77

0.34

0.23

0.58

0.40

0.39

Experiment 3[23]

N

0.24

0.33

0.76

0.66

0.58

0.62

W

0.79

0.36

0.21

0.54

0.39

0.39

N

0.21

0.39

0.79

0.62

0.57

0.64

Townsend et al. (2000)[3]

Busemeyer et al. (2009)[31]

Average
1

In Busemeyer et al. (2009)[31], the classical experiment was replicated.

2

In Wang and Busemeyer (2016)[23], experiment 1 used a larger data set to replicate
the classical experiment. Experiment 2 introduced a new X-D trial verse C-D trial
and only the results of C-D trial are used here. In experiment 3, the reward rate for
attacking bad people was a bit less than experiment 1 and 2.

determined as ΘCD = {AG, W G, AB, W B}, where, for example, AG represents the participant decide to attack given the face is categorized as a good
guy. In this case, as the categorization is uncertain to the participant, it is
meaningless to extend the belief states. However, in an evidential framework,
the action states should be extended. Then the states AW G (denoted UG),
AW B (denoted UB) are introduced to fill the power set of ΘCD . The state
UG represents that the participant is uncertain to attack or withdraw given
the face is categorized as G; the state UB represents that the participant
is uncertain to attack or withdraw given the face is categorized as bad. In
brief, the hesitance of a participant is shown by UG and UB.
15

In the D-alone condition, the outcome of the game will be either to attack or
withdraw without a precise categorization. Thus the FOD ΘD is determined
as ΘD = {AU, W U}, where, for example, AU represents that a participant
decide to attack given the categorization is unknown. In this case, the action
states should be extended in an evidential framework. But the state of belief
is unknown, the belief state is actually an assembly of categorizing the face
as good and bad. Then the state AW U (denoted UU) is introduced to fill
the power set of ΘD . The state UU represents that a participant is uncertain
to attack or withdraw given that the categorization is unknown, which also
shows the hesitance of a participant.
Step 2: Representation of beliefs and actions
In general, the beliefs and actions consist of the decision making process. As
the states have been extended, the initial state involves six combination of
beliefs and actions
{|BG AA i , |BG AU i , |BG AW i , |BB AA i , |BB AU i , |BC AW i} ,
where, for example, |BG AA i symbolizes the event in which the player believes
the face belongs to a good guy, but the player intends to act by attacking.
All of the possible transitions between the six states are illustrated in Figure
??.
The Markov model assumes that a person states in exactly one circle, but we
do not known which one is is. Hence, we assign probabilities of to the possible
states. At the beginning of a trail, the initial probability of starting out in
one of these six states is defined by a probability distribution represented by

16

the 6 × 1 column matrix







φ (0) = 






φAG




φU G 


φW G 
.

φAB 

φU B 

φW B

Participants’ state is a superposition of the six basis states
|ψi = ψAG · |BG AA i + ψU G · |BG AU i + ψW G · |BG AW i + ψAB · |BB AA i
+ ψU B · |BB AU i + ψW B · |BB AW i
(13)
and the initial state corresponds to an amplitude distribution represented by
the 6 × 1 column matrix







ψ (0) = 






ψAG




ψU G 


ψW G 
,

ψAB 

ψU B 

ψW B

where, for example, ψAG is the probability of observing state |BG AA i. This
probability distribution satisfies the constraint L·φ = 1, where L = [1, 1, 1, 1, 1, 1].
In this experiment, we assume that a face is shown to the participant at time
zero. The initial probability distribution is uniformly random when the participant has no time to deliberate about the categorization and the actions
to take, namely ψU G = ψU B = 0.5.
Step 3: Inferences based on prior information
17

During the decision process, new information at time t1 changes the initial
state ar time t = 0 into a new state at time t1 . In the C-D condition, if the
face is categorized as good, the state changes to

φ
 AG
 φ
 UG

 φW G
1

φ (t1 ) =
φAG + φU G + φW G 
 0

 0

0





 "
#


φ
G
=
.

0





(14)

The initial probability that the face is categorized as good equals φAG + φU G + φW G ,
and so the 3 × 1 matrix φG is the conditional probability distribution across
actions given the categorization is good.
If the face is categorized as bad, the state changes to


0


 0 
 "

#




0
0
1
=

.
φ (t1 ) =

φAB + φU B + φW B 
φB
 φAB 


 φU B 



(15)

φW B

The initial probability that the face is categorized as bad equals φAB + φU B + φW B ,
and so the 3 × 1 matrix φB is the conditional probability distribution across
actions given the categorization is bad.
In the D-alone condition, the state remains the same as the initial state,
which is a mixed state produced by a weighted average of the distribution

18

for two known conditions in C-D condition:
φ (t1 ) = φ (0)
"
#
(φAG + φU G + φW G ) · φG
=
(φAB + φU B + φW B ) · φB
"
#
"
#
φG
0
= (φAG + φU G + φW G ) ·
+ (φAB + φU B + φW B ) ·
.
0
φB
(16)
Step 4: Strategies based on payoffs
During the decision making process, the participants need to evaluate the
payoffs in order to select an appropriate action, which evolve the state at
time t1 into a new state at time t2 . The evolution of the state during this
time period corresponds the thought process leading to a action decision,
defection, cooperation or hesitance. The evolution of the state obeys a Kolmogorov forward equation (Eq. (9)) driven by a 6 × 6 intensity matrix K.
The solution is:
φ (t2 ) = eKt · φ (t1 ) .
For t = t2 − t1 , the state to state transition matrix is defined by T (t) = etK
with Tij (t) represents the probability of transiting to state i at t2 given being
in state j at time t2 . In this experiment, t is set to 2 corresponding to the
average time that a participant takes to make a decision (approximately 2
seconds)[31]. The intensity matrix K is
"
#
KG 0
H=
0 KB
with




KG = 


(17)

− 3kr 2+kw

kr

kr

kr +kw
2

−(kr + kw )

kw

kw

kr +kw
2
kr +3kw
− 2

19




,





KB = 


w
− kr +3k
2

kw

kw

kr +kw
2

−(kr + kw )

kr

kr

kr +kw
2
3kr +kw
− 2




.


The 3 × 3 intensity matrix KG applies when the face is categorized as good,
and the other matrix KB applies when the face is categorized as bad. The parameter kr represents the payoff for taking the right choice, namely attacking
the bad guy and withdrawing the good guy, and the parameter kr represents
the payoff for taking the wrong choice, namely attacking the good guy and
withdrawing the bad guy. The off-diagonal elements of K are nonnegative,
and the columns of K sum to zero, which then guarantees that the column
of T sum to one, which finally guarantees that φ (t).
In the C-D condition, if the face is categorized as good, the state changes to
# "
#
"
φG
eKG ·t
0
K·t
·
= eKG ·t · φG
(18)
φ (t2 ) = e · φ(t1 )=
KB ·t
0
0
e
If the face is categorized as bad, the state changes to
# "
#
"
KG ·t
0
e
0
·
= eKB ·t · φB
φ (t2 ) = eK·t · φ(t1 )=
φB
0
eKB ·t

(19)

In the D-alone condition, the state changes to
φ (t2 ) = eK·t · φ(0)
"
# "
#
eKG ·t
0
(φAG + φUG + φW G ) · φG
=
·
0
eKB ·t
(φAB + φUB + φW B ) · φB

(20)

= (φAG + φUG + φW G ) · eKG ·t · φG + (φAG + φUG + φW G ) · eKB ·t · φB .

Step 5: Basic probability assignments measurement
According to section 3.3, response probability of each decision can be derived.
In an evidential framework, the BPA can also be derived. Specially, when
20

the focal elements of a mass function is one, the BPA is the same with a
probability. The measurement matrix is defined as
!
MG 0
M=
.
0 MB
In the C-D condition, if the face is categorized as good, the 3 × 3 matrix MG
is set to



 







0 0 0
0 0 0
1 0 0

 
 






MG =  0 0 0  ,  0 1 0  or  0 0 0 

0 0 1
0 0 0
0 0 0

respectively to pick out the state of attacking, hesitance or defecting while
the 3 × 3 matrix MB is set to 0. If the face is categorized as bad, the 3 × 3
matrix MB is set to


1 0 0

 

0 0 0





0 0 0




 
 

 ,  0 1 0  or  0 0 0 
MB = 
0
0
0

 
 

0 0 1
0 0 0
0 0 0

respectively to pick out the state of attacking, hesitance or defecting while
the 3 × 3 matrix MG is set to 0. In the D-alone condition the 3 × 3 matrixes
MB and MG are set to


 







0 0 0
0 0 0
1 0 0

 
 






MB = MG =  0 0 0  ,  0 1 0  or  0 0 0 

0 0 1
0 0 0
0 0 0

respectively to pick out the state of attacking, hesitance or defecting.
Using L = [1, 1, 1, 1, 1, 1], the BPA of each state equals
m = L · M · etK · φ (t1 ) .

21

(21)

The derived BPAs consist body of evidences, (ℜCD , mCD ) and (ℜD , mD ) for
two conditions respectively.
(ℜCD , mCD ) = [m (AG) , m (UG) , m (W G) , m (AB) , m (UB) , m (W G)]
= (0.0264, 0.0811, 0.0625, 0.3050, 0.3960, 0.1290)
(ℜD , mD ) = [m (AU) , m (UU) , m (W U)] = (0.3314, 0.4771, 0.1915)
Step 6: Probability distribution based on uncertainty measurement
To obtain the probability distribution, the BPA of uncertain state should be
transferred. It is still an open issue to address it. The classical method is
using pignistic probability transformation. Obviously, however, the disjunction effect can not be predicted in this way. To address it, a new parameter
γ which represents the extra uncertainty degree (EUD) is determined as follows:
γ=

ED − ECD
,
ED + ECD

(22)

where ECD and ED are the information volume of (ℜCD , mCD ) and (ℜD , mD )
respectively calculated by Deng entropy as follows:
0.0264
− 0.0811log2 0.0.0811
21 −1
22 −1


0.3960
0.3050
− 0.3960log2 22 −1 −
21 −1

ED = −0.0264log2
− 0.3050log2
= 2.8715

ED = −0.3314log2
= 4.1868.



0.3314
22 −1



− 0.4771log2

0.4771
24 −1

0.0625
21 −1

0.1290
21 −1



− 0.0625log2



− 0.1915log2

0.1290log2

0.1915
22 −1



,



The cardinality of states AG, W G, AB, W B is one, the cardinality of states
UG, UB, AU, W U is two, and the cardinality of the state UU is 4.
Without a categorization, the uncertainty for the D-alone condition is obviously larger than the C-D condition. It is reasonable to assume that the
22

hesitated participants are tougher to take a final decision, especially for the
narrow type faces. According to the cognitive dissonance and social projection in psychology, people tend to be consistent with their beliefs and
actions[70–72]. The hesitated people are more tend to attack especially in
the D-alone condition. Hence, for (ℜD , mD ), the BPA of the uncertain state
should be assigned more to the attacking based on the EUD. Then the probability of attacking in the D-alone condition is calculated as


1
PD (A) = m(AU) +
+ γ m (UU ) = 0.6589.
2

(23)

As the observed experiments result shows that the disjunction effect is less
than 0.1, the information volume will not differ hugely, which guarantees
γ < 0.5. For (ℜCD , mCD ), the BPA of uncertain state is still handled by
using PPT (Eq. (4)). Then the probability of attacking in the C-D condition
is calculated as
1
1
PCD (A) = m (AG) + m (UG) + m (AB) + m (UB) = 0.57
(24)
2
2
The difference value of Eq. (23) and Eq. (24) is the predicted disjunction
effect.
Dis = PD (A) − PCD (A) = 0.0889
According to the above steps, the comprehensive process of our model is
illustrated in Figure ??.

6. Model results and comparisons

6.1. Comparison with experiment results
Applying the EM model to the categorization decision-making experiments,
the probability distributions for all the experiments can be obtained. Compare the obtained model results with the observed experiment results (for
23

narrow type face), the model results are close to the practical situation,
which verifies the correctness and effectiveness of our model. As Table 3
shows, the disjunction effect is well predicted and the average error rate is
only approximately 0.5%, which proof the correctness and effectiveness of
our model greatly.
Table 3: The results of the EM model
Literature

P (G)

P (A|G)1

P (B)

P (A|B)2

PT

P (A)

Dis

Townsend

Obs

0.17

0.41

0.83

0.63

0.59

0.69

0.1

et al. (2000)[3]

EM

0.17

0.394

0.83

0.606

0.57

0.6589

0.0889

Busemeyer

Obs

0.20

0.45

0.80

0.64

0.60

0.69

0.09

et al. (2009)[31]

EM

0.20

0.4019

0.5981

0.5588

0.80

0.6404

0.0816

Wang & Busemeyer (2016)

Obs

0.21

0.41

0.79

0.58

0.54

0.59

0.05

experiment 1[23]

EM

0.21

0.3840

0.79

0.6160

0.5673

0.6432

0.0759

Wang & Busemeyer (2016)

Obs

0.24

0.37

0.76

0.61

0.55

0.60

0.05

experiment 2[23]

EM

0.24

0.3384

0.76

0.6197

0.5622

0.63

0.0678

Wang & Busemeyer (2016)

Obs

0.24

0.33

0.76

0.66

0.58

0.62

0.04

experiment 3[23]

EM

0.24

0.3384

0.76

0.6616

0.5841

0.6436

0.0596

Obs

0.21

0.39

0.79

0.62

0.57

0.64

0.07

EM

0.21

0.3797

0.79

0.6203

0.5685

0.6432

0.0747

Average
1

Dis represents the disjunction effect.

2

Obs represents the observed experiment results.

3

EM represents the results of the Evidential Markov model.

6.2. Comparison with other uncertainty measurement method
The crucial step of the EM model to predict the disjunction effect is the
measurement of the extra uncertainty. Deng entropy is compared with many
other uncertainty measuring methods (shown in Table 4) in Deng (2016)[42].
These methods are still used to do the comparison in our method. The
comparison of the predicted disjunction effect and the standard error (SE) is
illustrated in Figure 5 and Figure 6 respectively, where the Exp 1 to 5 comes
from Townsend et al. (2000), Busemeyer et al. (2009), Wang & Busemeyer
24

Table 4: Methods of measuring uncertainty
Name

Formula

Deng entropy[42]

m (A) log2 m(A)
2|A| −1
P
m (A) log2 |A|
IDP (m) = −
A∈X
P
m (A) log2 Bel (A)
CH (m) = −
A∈X
P
m (A) log2 P l (A)
EY (m) = −
A∈X
P
P
|A∩B|
m (B) |B|
m (A) log2
DKR (m) = −
B∈X
A∈X
P
P
m (B) |A∩B|
m (A) log2
SKP (m) = −
|B|
B∈X
A∈X
h
i
P
P
|A∩B|
m (B) 1 − |A∪B|
m (A) log2
CGP (m) =
E (m) = −

P

A∈X

Dubois & Prades weighted Hartley entropy[73]
Hohles confusion measure[74]
Yagers dissonance measure[75]
Klir & Ramers discord[76]
Klir & Parvizs strife[77]
George & Pals conflict measure[78]

A∈X

B∈X

(2016) experiment 1, 2, 3 respectively. We can see that the SE of Deng
entropy is small for each experiment and the average SE of Deng entropy
is the smallest; the predicted disjunction effect of Deng entropy is close to
the observed one for each experiment and the average disjunction effect is
the most accurate. Deng entropy has a powerful ability of measuring the
uncertainty of a BOE, especially when many focal sets are multi-element.
Hence, Deng entropy is used to measure the extra uncertainty degree in the
EM model.

Figure 5: The predicted disjunction effect of different methods

25

Figure 6: The SE of different methods

6.3. Comparison with other models
As illustrated in section 3.4, the classical Markov model can not predict the
disjunction effect. However, many quantum models have been proposed to
explain the disjunction fallacy. In a quantum framework, before taking a decision, human thoughts are seen as superposed waves that can interfere with
each other, influencing the final decision. As shown in Figure ??, different
decisions can coexist at one time before the final decision is made, while the
Markov model assumes that the decision at one time must be certain. The
idea of the EM model is identified with the quantum model in some degree,
namely allowing a state to be uncertain.
The interference effect, a term of quantum mechanics, is borrowed to account
for the disjunction fallacy. One of the effective quantum models is the quantum dynamical (QD) model first proposed by Busemeyer et al. in 2006[79],
which is always used to compare with the classical Markov model. Busemeyer
et al. (2009)[31] illustrates that the QD model can predict the interference
effect while the Markov model can not in detail. Both the QD model and
the Markov model are formulated as a random walk decision process, but the
26

QD model describes the evolution of complex valued probability amplitudes
over time. In a quantum framework, the state evolves obeying a Schrödinger
equation (Eq. 25), which is driven by a Hermitian matrix. Also, two payoffs
functions are used to fill the Hermitian matrix H.
d
ψ (t) = −i · H · ψ (t)
(25)
dt
Admittedly, the disjunction effect can be predicted in the QD model, however, its problem is that the model requires a growth of parameters. For
example, in Wang & Busemeyer (2016)[23], a modified QD model, the quantum belief-action entanglement (BAE) model, is proposed. The actions and
beliefs are deemed to be entangled in some degree, which is the key to produce
the interference effect. The entanglement degree is described in a introduced
parameter whose value is set artificially. However, in our EM model, the
extra uncertainty degree is totally driven by data. And it turns out that the
uncertainty can be well measured in our model. To see the ability of predicting the disjunction effect, the model results of the probability of attacking
for the D-alone condition is compared as shown in Figure ??).
As we can see, both the QD model and the EM model can predict the disjunction effect while the Markov model can not. However, the prediction
result of our EM model is a bit more accurate. Also the free parameter of
EM model is less. Based on the above, it is reasonable to conclude that our
model is correct and efficient.

7. Conclusion
To model the decision making process and explain the disjunction fallacy,
the EM decision making model is proposed in this paper. The model combines Dempster-Shafer evidence theory with the Markov model. The classical
27

Markov model assumes that the at a certain time, people’s decision has to
be certain at one time. In an evidential framework, the action states are
extended. The hesitance of a decision maker is represented by a new introduced uncertain state. Hence, the state is allowed to be uncertain at one
time before a final decision is made. The uncertainty measurement is the
key to predict the disjunction effect. A belief entropy, named Deng entropy,
is used due to its powerful ability of uncertainty measurement. The model
results show that the EM model can well predict the disjunction effect, which
is impossible for the classical Markov model. Also, the EM model has less
free parameters than the quantum model. Hence, it is reasonable for us to
conclude that the EM model is correct and effective.

Acknowledgement

The work is partially supported by National Natural Science Foundation
of China (Grant No. 61671384), Natural Science Basic Research Plan in
Shaanxi Province of China (Program No. 2016JM6018), Aviation Science
Foundation (Program No. 20165553036), the Fund of SAST (Program No.
SAST2016083)

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