Construction of a technology adoption decision-making model and its extension to understanding herd behavior

Knowledge-Based Systems 89 (2015) 471–486

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

Construction of a technology adoption decision-making model and its extension to understanding herd behavior Jing Gu a, Lu Li b, Zeshui Xu c,⇑, Hamido Fujita d a

School of Economics, Sichuan University, Chengdu 610064, China School of Mathematical Sciences, Peking University, Beijing 100871, China c Business School, Sichuan University, Chengdu 610064, China d Fac. of Software and Information Science, Iwate Prefectural University, 020-0193 Iwate, Japan b

a r t i c l e

i n f o

Article history: Received 10 May 2015 Received in revised form 13 July 2015 Accepted 1 August 2015 Available online 29 August 2015 Keywords: Herd behavior Information cascade Observational learning Sequential decision making Technology adoption Waiting strategy

a b s t r a c t Decision makers often face challenges during the adoption of technology. Indeed, technology adoption usually occurs sequentially, so observational learning can help the decision makers to reach reasonable decisions. In reality, the decision makers may prefer to adopt a waiting strategy when the situation is equivocal. In this study, we construct a technology adoption model called the G-WB model, where we consider a generalized waiting situation based on the Walden–Browne (WB) model. In our model, herd behavior is a difficult issue, but an information cascade occurs after herding appears. We explore the effect of different parameters on the convergence speed and extend our G-WB model. We also demonstrate that observational learning is a useful strategy during sequential decision making and our model is an optimal version of the WB model, which facilitates a better understanding of herding. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction Information technology and information system (IT/IS) are now indispensable components of our everyday lives, which are important for improving economic development, the efficiency of work, and the quality of life. However, whether IT plays critical roles depends on people actually adopting technology. Bikhchandani et al. [19], Walden and Browne [11], and Banerjee and Fudenberg [3] studied technology adoption from the perspective of social learning. In the real world, decisions are often made in order in many uncertain situations, which is known as sequential decision-making. Considering the cost of information acquisition, every decision maker may possess a small amount of information. Since the information asymmetry that is at least some relevant information is known to some but not all parties, every decision maker may possess different information. In order to make reasonable decision, decision makers usually acquire from others in the local environment, i.e., social learning. Observational learning is a typical component of social learning. Although observational learning behavior is common, simple, and easily understood, the different approaches to abstract ⇑ Corresponding author. E-mail addresses: [email protected] (J. Gu), [email protected] (Z. Xu), [email protected] (H. Fujita). http://dx.doi.org/10.1016/j.knosys.2015.08.014 0950-7051/Ó 2015 Elsevier B.V. All rights reserved.

observational learning from real situations may lead to different conclusions. Thus, observational learning models may be based on observing the immediate predecessor, random samples, or aggregation actions. In this study, we mainly consider the role of aggregation actions in observational learning. Many previous studies have addressed this issue. For example, Banerjee [2] found that observing aggregation actions can lead to herd behaviors, i.e. people will be doing what others are doing, which may result in inefficient outcomes. Bikhchandani et al. [19] claimed that human behavior is based on localized conformity (e.g., Americans act like Americans and Germans act like Germans) and the fragility of mass behavior (e.g., the attitude toward cohabiting unmarried couples has changed over time). These characteristics can also be explained by information cascades, which occur when it is optimal for an individual, having observed the actions of those ahead of him/her, to follow the behavior of the preceding individual without regard to his own information. However, Walden and Browne claimed that the concept of fragility is to some degree an artifact of the model employed by Bikhchandani et al. [19] and that other types of fragility may exist. Furthermore, Gale [7] stated that cascades may be fragile or robust depending on many factors, such as whether the signal is continuous or discrete, whether the action space is discrete or continuous, and whether the decision-making queue is endogenous or otherwise. Smith and Sorensen [16] emphasized

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the difference between information cascades and herd behaviors. In information cascades, the decision makers ignore their private information and copy their predecessors when making decisions. In herd behaviors, the subjects usually make the same decision as others but he/she might not ignore his/her private decisions. Thus, herd behavior is not necessarily the result of an information cascade [5]. Many studies have addressed the learning of aggregation actions, but there are different opinions regarding whether herd behaviors will occur, why herd behaviors might emerge, and the robustness of herd behaviors. The different assumptions regarding observational learning behavior have yielded different results. Bikhchandani et al. [19] proposed a discrete signal binary behavior model, where each decision maker has a binary signal, either H or L, and a binary decision is made about whether to adopt new technology. Walden and Browne [11] developed an observational learning model based on that described by Bikhchandani et al. [19], where they changed discrete signals into continuous signals. Doong and Ho [23] referred to this as the Walden–Browne (WB) model and they discussed it from a dynamic Bayesian network perspective. In the present study, we also consider the WB model. In all previous models, the decision makers are rational and they make once in a lifetime decisions about whether discrete signals or continuous signals are employed. However, the WB model ignores the possibility that a decision maker may prefer to adopt a waiting strategy if their private signal cannot provide sufficient information and the situation is equivocal, before making a reasonable decision after the situation becomes clearer. Faced with new options, decision makers often stick with the status quo alternative, that is, waiting, doing nothing or maintaining one’s current or previous decision [21]. For example, the development and popularization of the Internet has led to the appearance of social networking platforms such as Twitter and Renren. Initially, when the situation was not clear, many people chose to wait rather than join these networks immediately. And, after waiting for a period of time, the decision makers were likely to choose the social platform selected by most of their friends and herd behaviors may occur. Therefore, in this study, we construct a generalized WB model (G-WB model), where we consider a more general situation for technology adoption and incorporate a waiting strategy. In the GWB model, as well as selecting a specific technology that is considered to be desirable, the decision makers can wait if they are in a dilemma. We also test the G-WB model with an increased number of decisions to determine whether herd behaviors emerge, as well as addressing the controversial question of the relationship between herding and information cascades. We then investigate the effects of different parameters in our model on the convergence speed. Moreover, we show that the G-WB model is an optimal version of the WB model in terms of both accuracy and robustness. Finally, we extend the model in three respects: (a) decision makers have different preferences; (b) incomplete information is available when the decision makers observe actions in a group; and (c) the decision makers have limited rationality, where we analyze the effects of these changes on convergence. The remainder of this paper is organized as follows: In the next section, we present background material about IT adoption and social learning. Next, we describe the proposed G-WB model in Section 3. In Section 4, we analyze the G-WB model, before extending this model in Section 5. Finally, we give our conclusions in Section 6. 2. Background Given the rapid development of IT/IS, the range of services available to human society has expanded, initially for national

defense and military applications but then for industrial and commercial uses, while it now supports people in all workplaces to improve the efficiency of work, as well as enriching our daily lives, leisure pursuits, and other forms of entertainment. IT adoption can be classified as individual, group, and organizational level adoption, depending on the type of IT application that is being adopted. IT adoption can also be divided into work applications and those related to other aspects of our daily lives and entertainment. Software and other technological components are among the most complex artifacts that humans have ever built [13]. It may take many years to realize the impact of technologies [9], where the benefits of adopting useful technology can be tremendous, but the costs of adopting a failing technology can be severe. The wide scope, complexity, and high uncertainty of IT/IS applications mean that technology adoption can be a difficult issue for the decision makers. Therefore, the study of IT adoption has been a major issue since the mid-1980s. Technology adoption research in this area has focused on two main strands: studies based on classical theory and research combined with theory from other disciplines. The classical theory of IT adoption involves the research areas such as the technology adoption model (TAM), innovation diffusion theory (IDT), and the theory of planned behavior. Davis [24] proposed TAM and validated new measurement scales for the perceived usefulness and perceived ease of use, which are two distinct variables that are hypothesized to be determinants of computer usage. The IDT can be used to predict an individual’s adoption behavior, where it is proposed that beliefs affect attitudes, which then influence intentions, and thus behavior [8]. Ajzen [14] suggested that different types of behaviors can be predicted with high accuracy from attitudes toward behavior, subjective norms, and perceived behavioral control. Attitudes, subjective norms, and perceived behavioral control are also related to appropriate sets of salient behavioral, normative, and control beliefs regarding behavior, but the exact natures of these relationships are still uncertain. In addition to classical technology adoption theories, many other theories are used to analyze IT adoption from different perspectives, such as transaction cost theory in economics, cognitive theory in psychology, and change management theory in organization research. In this study, we consider technology adoption based on social learning, where word-of-mouth learning is another typical form of social learning in addition to observational learning. However, the information acquired from word-of-month learning may be unreliable because no hard evidence is provided, and opinions may not be supported by reasons, and thus it is not possible to observe the entire process based on the information provided. Moreover, it is not possible to believe everything that one is told both because hard evidence is often lacking and people often attach their personal hopes and fears to the information that they report [3]. Therefore, word-of-mouth information may sometimes be distorted, especially when it is spread sequentially among people. Indeed, ‘‘actions speak louder than words,” so the decision makers are more likely to observe the actions of others in similar situations to acquire information and make optimal decisions, i.e., observational learning. Research has shown that people use observations of others to update their own private beliefs and take actions [1], and observational learning is even more important than professional reviews for explaining technology adoption [20]. Thus, observation learning is one of the most ubiquitous and useful means available to the decision makers. Laboratory and real world studies have shown that observational learning influences the decision makers during technology adoption. With the maturation of electronic commerce, peer-topeer (P2P) internet technology now allows client machines to interact directly with each other. Thus, many people are faced with the decision of whether to adopt this new technology. Song and

J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486

Walden [15] showed that the potential adopters of P2P technology allocated considerable weight to recent adoptions but very little weight to the relative network size when making their own adoption decisions. In addition, they showed clearly that observational learning rather than network externalization explains IT adoption behaviors. Stock market reactions to electronic commerce announcements by firm X on day t were strongly predicted by electronic commerce announcements by firm Y on day t
1, because a lack of useful information about the potential value of electronic commerce leads investors to rely on the private valuations of electronic commerce made by other investors, thereby leading to the Internet bubble [10]. Studies of observational learning can be divided into observing aggregation actions, immediate predecessors, and random samples. We have already mentioned previous studies based on the observation of aggregation action but we now consider the other two aspects in the following sections. Çelen and Shachar [5] explored Bayes-rational sequential decision making in a game with pure information externalization, where each decision maker observed only his/her own predecessor’s binary action. Given imperfect information, beliefs and actions can cycle interminably but despite stochastic instability, private information is ignored over time and the decision makers become increasingly likely to imitate their predecessors, which is called an information cascade. Smith and Peter [17] assumed that everyone can only observe unordered random samples in his/her action history. In this study, we consider that herd behavior might not occur if the distant past can be sampled. 3. Proposed G-WB model In order to investigate the real process of technology decision making among people and make better understanding of herd behaviors, we consider a more generalized technology adoption situation and incorporate the strategy of waiting into a sequential decision-making model called the G-WB model. In the G-WB model, a decision maker does not have to choose the technology A or B but instead they wait for the next turn. In contrast to previous studies where the decision makers choose a specific technology once in a lifetime, our proposed G-WB model is more practical. 3.1. Assumptions In order to facilitate research into IT adoption from the perspective of sequential decision-making, it is necessary to abstract the observational learning behavior from reality and to measure the influence of observational learning. We employ several assumptions from the WB model and the assumptions in our proposed G-WB model are as follows: #1. The decision makers can choose from two new technologies called the technology A and the technology B. #2. An infinite number of decision makers must face the same problem of adopting the technology A or B in a similar situation. They make decisions in turn and at different times, i.e., sequential decision-making. The sequence of the decision makers is exogenous, where the order in which a subject makes a decision cannot be influenced by the other decision makers. #3. The benefits of the technologies A and B are the same for all of the decision makers. Specifically, if the benefit of choosing the technology A is greater than the benefit of choosing the technology B, then the technology A is better than B for all people, and vice versa.

473

#4. Every decision maker has private information about the relative merits of the technologies A and B. In particular, the decision maker receives a signal from a Gaussian distribution. The technology A and the technology B are better in two different situations, so the private signals come from different distributions. If the technology A is better than B, we denote that the signal comes from the distribution NðlA ; r2 Þ, and if the technology B is better than A, the signal comes from the distribution NðlB ; r2 Þ. Without loss of generality, lA > lB , and d ¼ lA
r lB measures the difference in the two distributions. Thus, in order to make an optimal decision, the decision maker needs to judge whether the private signal is more likely to have come from a specific distribution. #5. In addition to private information, the decision makers know the initial conditions and they can observe the actions of previous decision makers. In the initial conditions, the prior probabilities that technology A is better than B and that the technology B is better than A are denoted by PlA and PlB , respectively. We assume that PlA ¼ PlB ¼ 0:5, which indicates that there is no any prior information. People usually observe what others do, but we do not know why they do that. Thus, a decision maker knows the actions of their predecessors but he/she does not know their private signals, which are the reasons why they make specific decisions. #6. Decision makers are sufficiently rational to make decisions that depend only on the information they possess. Decision makers do not change his/her decisions simply because of his/her own preference. This is a classical rational person assumption. #7. When faced with two different new technologies and an exogenous sequence, the decision makers can wait and not choose the technology A or B if there is a dilemma. In addition, we assume that there is no cost for waiting. If the technology A and the technology B are both equally suitable, the decision maker can choose not to select the technology A or B and wait for the next turn. Let the lowercase letter a denote the choice of adopting the technology A; b denote the choice of adopting the technology B, and w denote the choice of waiting, and thus the decision maker’s decision space can be expressed as D ¼ fa; b; wg. These assumptions are all abstracted from reality and easily understood. Based on these assumptions, we can build a generalized technology adoption decision-making model. However, some of these assumptions can still be relaxed and we propose a model with the relaxed assumptions in Section 4. 3.2. Model formalization According to the model’s assumptions, the decision maker receives a continuous signal from a Gaussian distribution, where the mean value may be lA or lB , which means that the technology A is better than the technology B, and vice versa. In order to reach a reasonable decision, the decision maker must judge whether the signal is more likely to have come from a particular Gaussian distribution based on the observed actions made by the previous decision makers. Walden and Browne [11] suggested that a decision maker will believe that the technology A is better if the following decision criterion is satisfied:

pðsjlA Þ Pb pðsjlB Þ

ð1Þ

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J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486

where s is the continuous signal, and pðsjlA Þ and pðsjlB Þ are the probability density functions of NðlA ; r2 Þ and NðlB ; r2 Þ, respectively. By plugging in the Gaussian distribution density function, we can obtain a threshold for s:

rðbÞ ¼

r2

lA
lB

ln b þ

lA þ lB

ð2Þ

2

If the continuous signal s exceeds the threshold rðbÞ, then the inequality (1) is satisfied and the decision makers select the technology A; otherwise, they select B. We refer to this decision rule as the WB criterion. However, according to the assumption #7, the decision makers need not to choose one specific technology. Therefore, in the G-WB model, we add the choice of waiting, thereby improving the WB model. When s is in the vicinity of rðbÞ, the decision makers’ belief that the technology is better is equivocal, so they are unsure about whether to choose A or B. To consider the indecisive region near rðbÞ, we introduce a parameter eðe > 0Þ, which is a measure of the degree of hesitation. Denote
s ¼ rðbÞ þ e and s ¼ rðbÞ
e, and thus we obtain the following decision rule, which we refer to as the G-WB criterion:

8 > < s P
s ) choosing A s < s <
s ) waiting > : s 6 s ) choosing B

ð3Þ

which means that: (a) if s exceeds
s, then the decision maker chooses the technology A; (b) if s is less than s, then the decision maker chooses the technology B; (c) in the fuzzy region, the decision maker chooses to wait. The G-WB criterion is illustrated in Fig. 1. The optimal threshold is selected by balancing the benefits of each technology and the decision maker’s information regarding the probability that A is better than B and that B is better than A. In this case, the optimal value of b is

b¼

plB ðBenefitðbjlB Þ
BenefitðajlB ÞÞ

plA ðBenefitðajlA Þ
BenefitðbjlA ÞÞ

¼

plB plA

ð4Þ

k

where

k¼

BenefitðbjlB Þ
BenefitðajlB Þ BenefitðajlA Þ
BenefitðbjlA Þ

ð5Þ

where k represents the relative difference between the values of the two technologies in different states of the world. The numerator is

the difference between the benefit of choosing B and the benefit of choosing A in the state where the technology B is better than A. The denominator is the difference between the benefit of choosing A and the benefit of choosing B in the state where the technology A is better than B. According to the assumption #3, both the numerator and denominator are positive numbers and k is a constant. Here we denote the t-th decision maker’s belief by plA t and plB t , which depends on the sequence of decisions that they have observed. By Bayes’ theorem, it can be shown that

plA ðtþ1Þ ¼ pðlA jDt ; Dt
1 ; . . . ; D1 Þ ¼

plB ðtþ1Þ ¼ pðlB jDt ; Dt
1 ; . . . ; D1 Þ ¼

plA  pðDt ; Dt
1 ; . . . ; D1 jlA Þ pðDt ; Dt
1 ; . . . ; D1 Þ plB  pðDt ; Dt
1 ; . . . ; D1 jlB Þ pðDt ; Dt
1 ; . . . ; D1 Þ

btþ1

" # pðDt ; Dt
1 ; . . . ; D1 jlB Þ plB ¼ k¼ k pðDt ; Dt
1 ; . . . ; D1 jlA Þ plA plA ðtþ1Þ plB ðtþ1Þ

pðDt ; Dt
1 ; . . . D1 jlA Þ ¼ pðDt jAt ; lA ÞpðDt
1 ; . . . ; D1 jlA Þ

ð9Þ

pðDt ; Dt
1 ; . . . D1 jlB Þ ¼ pðDt jAt ; lB ÞpðDt
1 ; . . . ; D1 jlB Þ

ð10Þ

Thus we obtain the iterative formula:

btþ1 ¼

" #   pðDt jAt ; lB ÞpðDt
1 ; . . . ; D1 jlB Þ plB pðDt jAt ; lB Þ k ¼ b pðDt jAt ; lA ÞpðDt
1 ; . . . ; D1 jlA Þ plA pðDt jAt ; lA Þ t ð11Þ

In this manner, we can obtain the threshold of the t-th decision maker and the probability of the t-th decision maker’s choice after observing the decisions made by the previous decision makers. These possibilities are shown in Table 1. In the G-WB model, we employ the G-WB criterion where the key is the threshold rðbÞ and we find the iterative relationship between bt and btþ1 . Thus, decision makers’ actions are determined by their private signals and previous decisions in sequence. 2

N(μΒ,σ ) 2

N(μΑ,σ ) r(β)

0.8

Probability Mass

r(β)+ε

r(β)-ε

0.6 0.5

p(a|μA) p(b|μB) p(a|μB)

0.3

p(b|μA)

0.2 0.1 0.0 -5

-4

-3

-2

-1

ð8Þ

and define At ¼ fDt
1 ; Dt
2 ; . . . ; D1 g as the set of all prior decisions for all t > 1, then

0.9

0.4

ð7Þ

where Dt ðt ¼ 1; 2; . . .Þ is the tth decision maker’s choice. By combining (6) and (7), we have:

1.0

0.7

ð6Þ

0

1

2

Private signal Fig. 1. Regions with different decision outcomes.

3

4

5

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J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486 Table 1 Possible outcomes of the decision task. tth decision maker’s identification A better (a) Reality A better (lA )

pðajlA ; At Þ ¼

B better (lB )

pðajlB ; At Þ ¼

Waiting (w) R þ1

rðbÞþe

R þ1

rðbÞþe

pðsjlA Þds

pðwjlA ; At Þ ¼

pðsjlB Þds

pðwjlB ; At Þ ¼

B better (b) R rðbÞþe rðbÞ
e

R rðbÞþe rðbÞ
e

pðsjlA Þds

pðbjlA ; At Þ ¼

pðsjlB Þds

pðbjlB ; At Þ ¼

R rðbÞ
e
1

R rðbÞ
e
1

pðsjlA Þds pðsjlB Þds

1.0 0.9

r(β5)=-1

0.8

r(β4)=-0.9 r(β3)=-0.7 r(β2)=-0.3

Probability Mass

0.7 0.6

r(β1)=0.5

0.5 0.4 0.3 0.2 0.1 0.0 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Obseved Value Fig. 2. Changes in the threshold with consecutive choices of the technology A.

technology A waiting technology B

1.0 0.9 0.8

Percent Making Decision

Percent Making Decision

0.9800

0.7 0.6 0.5 0.4 0.3

0.05

0.9795

0.04

0.9790

0.03

0.02 0.9785 0.01 0.9780

0.2

91

92

93

94

95

96

97

Decision Number

0.1 0.0 5

10

15

20

25

30

35

40

45

50

55

60

65

70

Decision Number Fig. 3. Convergence of decision makers.

75

80

85

90

95

100

98

99

J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486

Percent Making Wrong / Correct Decision

476 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

WB-model G-WB-model

5

0.35

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

65

70

75

80

85

90

95

100

Decision Number

0.30 0.25 0.20 0.15 0.10 0.05 0.00 5

10

15

20

25

30

35

40

45

50

55

60

Decision Number Fig. 4. Convergence with the WB and G-WB models.

96.8%

1.0

86.7%

Percent Making Correct Decision

0.9

98%

99.4% 99.6% 99.9%

99.9%

97.9% 98.2% 98.9% 99.1% 94.7% 95.8%

WB model G-WB model

80.7% 84.9%

0.8

68.1% 0.7

68.9%

0.6 0.5 0.4 0.3

30.3% 0.2 0.1 0.0

1

5

10

50

100

500

1000

5000

10000

Decision Number Fig. 5. Convergence to the correct decision with 10,000 decision makers.

4. Analysis

If the tth decision maker selects the technology A, then by combining (12) and (11), we have:

4.1. Convergence

btþ1 < bt

First, it is necessary to differentiate between an information cascade and herd behavior. In an information cascade, the decision makers choose the same action as others while ignoring their private information. In herd behaviors, after some finite time, all of the decision makers choose the same action, but not necessarily ignore their private information. In this section, we examine whether herd behavior exists in technology adoption and we determine the relationship between herd behavior and information cascades. Based on Table 1, it is obvious that for every rðbt Þ,

and the threshold rðbt Þ increases strictly and, monotonically in regard to bt , and thus

pðajlA ; At Þ ¼

Z

þ1

rðbÞþe

pðsjlA Þds >

Z

þ1

rðbÞþe

pðsjlB Þds ¼ pðajlB ; At Þ

ð12Þ

And

pðbjlB ; At Þ ¼

Z

rðbt Þ
e


1

pðsjlB Þds >

¼ pðbjlA ; At Þ

Z

rðbt Þ
e


1

pðsjlA Þds ð13Þ

rðbtþ1 Þ < rðbt Þ

ð14Þ

ð15Þ

Similarly, if the t-th decision maker selects the technology B, then

rðbtþ1 Þ > rðbt Þ

ð16Þ

The inequality (15) means that after observing the t-th decision maker selects the technology A, the ðt þ 1Þth decision maker will relax the criterion for selecting A. The inequality (16) means that the ðt þ 1Þth decision maker will relax the criterion for selecting B after observing that the t-th decision maker chooses the technology B. If we assume that lA ¼ 1; lB ¼ 0; r2 ¼ 1; e ¼ 0:1; k ¼ 1, and all of the decision makers choose the technology A, then we obtain Fig. 2. Fig. 2 shows that the threshold value
s decreases as the number of decision makers selecting A increases consecutively. Thus, the

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J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486

ε=0.5 ε=1 ε=2

Percent Making Waiting Decision

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

65

70

75

80

85

90

95

100

Decision Number 1.0

Percent Making correct Decision

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 5

10

15

20

25

30

35

40

45

50

55

60

Decision Number Fig. 6. Effect of the degree of hesitation on convergence.

private signals become less informative and subsequent decision makers will copy the action taken by the previous decision makers, thereby leading to an information cascade. Hence, if herd behavior exists, then it is the result of an information cascade. We give the G-WB criterion for every decision maker, but every decision maker’s action is uncertain because his/her private signal is uncertain. From the viewpoint of probability theory, if the actions of decision makers converge to a certain decision with a specific probability, then herding will occur. In particular, if

lim pðDt ¼ aÞ ¼ 1

t!1

ð17Þ

is satisfied, then herding occurs. All of the decision makers choose the technology A after a finite number of decision makers select their choices. The same applies to the choice of the technology B and waiting. However, it is impossible to find an accurate mathematical expression of the probability that the t-th decision maker will choose a certain action because of the following two reasons: First, the effect of the previous decision makers’ actions on the threshold value rðbt Þ becomes very complex as t increases. Second, the expression of the probability requires the integration of a Gaussian distribution, which does not have an explicit expression. In repeated trials, as the number of tests increases, the frequency of the incident approaches a stable value, which is the

probability of the incident. Therefore, we have simulated the sequential decision-making process repeatedly and used the frequency instead of the probability to analyze herding. In Fig. 3, it is assumed that lA ¼ 1; lB ¼ 0; r2 ¼ 1; e ¼ 1; k ¼ 1, and the technology A is better than B. We performed 1000 replicates for 100 decision makers and the results represent the changes in the percentages of correct, incorrect, and waiting decisions over time. Fig. 3 shows that as the number of decision makers increases, the percentage choosing the technology A increases, and the percentages choosing the technology B and waiting decrease. The probability is 96.1% that the 100th decision maker chooses the technology A, thereby making a correct decision, whereas the probabilities of waiting and making the incorrect decision approach zero. The inset in Fig. 3, which corresponds to the 90–100th decision makers, shows that the line still fluctuates, which indicates that there are still reversals even in the interval with large numbers of decision makers. For example, the likelihood that the 98th decision maker selects the correct decision is 98% and that of the 99th decision maker is 97.9%. Thus, 0.1% of the decision makers select the technology B or wait. Therefore, there is no herding when the decision makers decide to choose the technology A after 100 decision makers have made their selections. Walden and Browne [11] indicated that convergence to the correct decision does not occur after 100 decisions in the WB model, which also applies to the G-WB model. Therefore, we have

J. Gu et al. / Knowledge-Based Systems 89 (2015) 471–486

Percent Making Waiting Decision

478

d=0.5 d=1 d=2

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compared the results obtained after simulating 1000 replicates by 100 decision makers using the WB and G-WB models, and the results are shown in Fig. 4. Fig. 4 shows that the probability of making a correct decision is higher with the WB model than the G-WB model before the 15th decision maker (including the 15-th decision maker), but the results are different after the 15-th decision maker. Furthermore, the probability of making an incorrect decision is much lower with the G-WB model than the WB model with 100 decision makers. Therefore, our results demonstrate that the G-WB model is more precise than the WB model. However, more replicates are necessary to analyze the convergence further, so we have performed simulations of 10,000 decisions and the results are shown in Fig. 5. Fig. 5 shows that the probability of making a correct decision is higher with the G-WB model than the WB model after the 16-th decision maker, and the situation does not change subsequently. Thus, herding occurs more quickly in the G-WB model than the WB model. 4.2. Factors affecting the convergence speed As shown in previous analyses, we have found that herding is a difficult issue in the G-WB model. Thus, it is necessary to pay more attention to the speed of convergence rather than the absolute level of convergence. Obviously, the parameters employed in the model have a strong impact on the speed of convergence, and thus we discuss them in the following subsection. 4.2.1. Degree of hesitation If the decision maker’s private signal falls in the interval ðr
e; r þ eÞ, then the decision maker will choose to wait. This interval is considered to be a gray area and its length is 2e; thus,

e is a measure of the degree of hesitation. The effects of different values of e on the probability of waiting and making correct decisions are shown in Fig. 6. Fig. 6 shows that the effect of the degree of hesitation on convergence is complex. The possibility of waiting is lower when the degree of hesitation (e) becomes smaller. The two curves intersect, where e is smaller and the possibility of making a correct decision is higher before the intersection, whereas the probability of making a correct decision declines subsequently when e is smaller. Thus, there must be an appropriate value of e that allows the decision makers to converge to a correct decision at the fastest speed. 4.2.2. Uncertainty of private signals If the technology A is better than B, then the private signal comes from NðlA ; r2 Þ. If the technology B is better than A, then the private signal comes from NðlB ; r2 Þ. d ¼ lA
r lB is a measure of the uncertainty of the private signal. As d decreases, the difference between the two distributions decreases, and vice versa. The effect of different values of d on the rate of convergence is shown in Fig. 7. First, the percentage of waiting decisions decreases with the increase in d. As the number of decisions increases, the percentage of waiting decisions tends toward zero with different values of d. As d increases, the decision makers converge more rapidly to the correct decision because the capacity for discrimination improves, and thus the information from their private signals and the observed actions of others are both more reliable. 4.2.3. Relative advantage in different states We have defined the parameter k in (5). If the benefit of choosing the technology B is much greater than the benefit of choosing

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the technology A in a state where B is better, and choosing A or B makes little difference in the state where A is better, then k will be high, whereas k would be small in the opposite conditions. Thus, high-risk and high-return technology leads to an extreme value for k. The effect of different values of k on convergence is shown in Fig. 8. The percentage of waiting decisions by each decision maker is zero when k is extremely large (k ¼ 100) or when k is extremely small (k ¼ 1=100). Except for extreme situations, the percentage of waiting decisions increases with k in the early stage, but the percentage of waiting decisions approaches zero as the number of decisions increases. When k is extremely small (k ¼ 1=100), the decision makers converge to the correct decision by selecting the technology A, but when k is extremely large (k ¼ 100), the decision makers are likely to choose the technology B. The percentage of correct decisions increases as k decreases. A low value for k implies

that if the technology A is better in the real world, then the correct choice A is much better than B, whereas if the technology B is better in the real world, there is no significant difference between A and B. Thus, the decision makers can relax their criteria to select the technology A when k is small. Therefore, the decision makers will converge more rapidly as k decreases if the technology A is better in the real world; otherwise, if the technology B is better in the real world, then the decision makers will converge more rapidly as k increases. Thus, we have analyzed the effect of three parameters on the convergence speed, and demonstrated the following results: (a) there is an appropriate degree of hesitation (e) that allows the decision makers to converge to the correct decision at the fastest speed; (b) as the discrimination between two Gaussian distributions (d) increases, the decision makers will converge to the correct decision more rapidly; and (c) as the relative advantage decreases

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in different stages (k), the decision makers will converge to the correct decision more rapidly if the technology A is better in the real world, and vice versa. 4.3. Robustness of the G-WB model Bikhchandani et al. [19] stated that behavioral conformity is fragile and that minor changes (even if a change does not actually occur) can disrupt an information cascade. In this paper, we are interested in the ability of the model to cope with outliers, which is the robustness of the model. Thus, robustness is used to measure the influence of extreme situations on convergence. Therefore, we investigate the impact of receiving extreme private signals of the G-WB and WB models, so we have compared the robustness of the two models. We assume that the technology A is better, so the private signal is emitted from NðlA ; r2 Þ, i.e., s  NðlA ; r2 Þ. In reality, there may be a specific period where the signal is abnormal or extreme. The

probability of the private signal falling in the interval ðlA
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Thus, we have demonstrated that herding is a difficult issue, but there must be an information cascade for herding to occur. We have also investigated the effects of different parameters on the convergence speed. Moreover, we have showed that the G-WB model is better than the WB model in terms of both accuracy and robustness. 5. Extension of the G-WB model Before constructing the G-WB model, we have defined seven assumptions in Section 3.1. However, there is always a difference between the ideal situation and reality. In order to make this study closer to reality, we relax some assumptions and extend the G-WB model in this section.

5.1. Different preferences Let us review the assumption #3. The benefits of the technology A and B are the same for all decision makers, so the benefits of the two technologies at different stages do not vary among the decision makers. This assumption is perfect, whereas people usually have their own personal preferences in reality. Thus, the two technologies A and B would have different benefits for each decision maker because of their preferences. This means that k in (5) is a variable rather than a constant. In Section 4.2.3, we have showed how the decision makers converge to the correct decision more rapidly as k decreases if the technology A is better in the real world. Now, we consider the changes in the convergence path due to the changes in k.

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Now we consider the effect of decreasing k from 1 to 0.1 for the 10th decision maker. In particular, k ¼ 1 before the 10th decision maker (including the 10th decision maker) but k ¼ 0:1 after the 10th decision maker. We have performed 1000 replicates of 100 decisions and the effects are shown in Fig. 11. Fig. 11 shows that there is a jump at the 11th decision maker. k ¼ 1 indicates that the decision maker considers that their preference is equal for the technologies A and B, whereas a small value of kð0 < k < 1Þ indicates that the decision maker prefers the technology A. The decision maker can relax the threshold value rðbÞ to choose A as k decreases and the percentage of correct decisions increases if it is assumed that A is better. Therefore, decreasing k has a positive effect on convergence if A is better than B. Furthermore, the effect of increasing k ¼ 1 before the 10th decision maker (including the 10th decision maker) to k ¼ 10 after the 11th decision maker is shown in Fig. 12. A high value of kðk > 1Þ indicates that the decision maker prefers the technology B. Thus, the decision maker can relax the threshold value rðbÞ to choose the technology B as k increases

and the percentage of correct decisions decreases when it is assumed that A is better. Therefore, increasing k has a negative effect on convergence if A is better than B, but the opposite if B is better than A in the real world. 5.2. Incomplete information According to the assumption #5, we assume that every decision maker can observe all of the previous actions made by other decision makers. In reality, the number of potential adopters may be large and thus, because of incomplete information, the number of other decision makers’ choices that each subject can observe before making a decision is limited. Therefore, it is important to investigate the effects on convergence if the decision makers are in groups, rather than in a completely open environment that can be observed by everyone else. A group is a set of potential adopters who can observe other actions. For example, if the group size is 10, then the second decision maker can observe the first decision maker’s choice, and the

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10th decision maker can observe the choices made by the decision makers 1–9; however, the 11th (the first decision maker in the next group of 10) cannot observe the behavior of other decision makers, in the same way as the first decision maker in the first group. For this scenario, the changes in the percentages of waiting and correct decisions are shown in Fig. 13. Clearly, the percentages of correct and waiting decisions fluctuate in cycles of 10, which is the group size. Due to the use of an exogenous sequence, we need to consider the probabilities of waiting and correct decisions in groups, rather than the specific probabilities for the tth decision maker. Thus, we have simulated subjects making decisions with different group sizes and 1000 replicates. Based on each replicate result, we have calculated the percentage of waiting and correct decisions in groups, thereby obtaining 1000 samples for each group size. We have treated the probabilities of waiting and correct decisions as random variables, and frequency histograms with different group sizes are shown in Figs. 14 and 15, respectively. Fig. 14 shows that the percentage of waiting decisions tends to zero as the group size increases. Fig. 15 shows that the percentage of correct decisions increases as the group size increases. These results indicate that a decision maker is more successful if their group size is larger. Next, we have calculated the averages and standard deviations for different group sizes with the WB and G-WB models, as shown in Fig. 16. The average increases as the group size increases, but the standard deviation decreases with both the WB and G-WB models. The standard deviation is smaller with the G-WB model than the WB model for every group size. The average is larger with the G-WB model than the WB model for group sizes of 50 and 100. However, the average is smaller with the G-WB model than the WB model for group sizes of 5 and 10. Overall, these results show that group size determines the accuracy and risk for decision makers.

5.3. Limited rationality According to the assumption #6, we assume that the decision makers are all rational. Now, we consider the situation if the decision makers have limited rationality. If all of the decision makers

are rational, we obtain the iterative formula in (11) by Bayes’ thepðD jl A Þ

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6. Conclusions Decision makers often face challenges related to technology adoption, but each decision maker possesses little information. The decision makers often adopt technology sequentially and they usually observe the actions of previous decision makers to update their beliefs in order to make reasonable decisions. However, the decision makers may choose to wait when the situation is equivocal. In this study, by combining observational learning and waiting, we have constructed the G-WB model based on the WB model and analyzed its performance in detail. At first, we have reviewed previous studies of technology adoption and social learning. After defining several assumptions and constructing the G-WB model, we have performed simulations to investigate the performance of the G-WB model and reached the following conclusions: (1) Herding is a difficult issue, but an information cascade occurs once when herding emerges; (2) In the G-WB model, various parameters influence the convergence speed; (3) The G-WB model performs better than the WB model in terms of both accuracy and robustness. We have also extended the model in three respects: (1) The different preferences of decision makers; (2) Incomplete information; (3) The decision makers with limited rationality, where we have analyzed their effects on convergence. Perhaps this study is most closely related to the research of Walden and Browne [10]. They construct the WB model and the results suggest that following the behavior of other similarlysituated decision makers can be a very useful strategy in adoption situations in which there is a great deal of uncertainty. However, the biggest different between the study of Walden and Browne [10] and ours is that we have considered a more generalized situation for constructing G-WB model, which allows the decision makers to wait if they are in a dilemma, and demonstrated that G-WB model is better than the WB model in terms of both accuracy and robustness. Second, we have clarified the relationship between herd behaviors and information cascades. Third we have extended our model by relaxing certain assumptions and analyzed the different influences on convergence. Technology revolution is the result of one new technology’s widely adoption and the wash out of others. Take fierce format war for example. In order to meet the demand of high-capacity data storage and the high definition entertainment, two new formats were designed to supersede the standard DVD format. One is HD DVD (short for High Definition Digital Video Disc) supported principally by Toshiba, the other is Blu-ray Disc (BD) supported by Sony. The first BD-Rom players (Samsung BD-P1000) were shipped in mid-June 2006, the first mass-market Blu-ray Disc rewritable drive for the PC was the BWU-100A, released by Sony on July 18, 2006 and on November 11, 2006, SCE (Sony Computer Entertainment Inc) released the PlayStation3 which was the first console with Blu-ray Disc player. Meanwhile, Toshiba released the HD DVD players in Japan and the Home game consoles Xbox 360 with external HD DVD player were published by Microsoft. Facing with two new technologies, there are three choices for consumers. Some of people adopted BD and a small group of people adopted the HD DVD, however the most people chose to wait. After a period of stalemate, BD hold 90% of the next generation disc market in Japan, and more than 66% of that in America. In Europe, the sales of BD kept head of the sales of HD DVD in the ratio of 3 to 1. In February 2008, Toshiba abandoned the HD DVD format, announcing it would no longer develop nor manufacture HD DVD. The HD DVD Promotion Group was dissolved on March 28, 2008. Nowadays, the Blu-ray Disc has been wildly used and developed well.

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The format war is just one example of new technology adopting process. The best explain of the stalemate are observational learning and the waiting choice. When facing two or more than two new technologies, a large group of people in practice have little private information would choose to wait. After a period of observational learning, the large group of decision makers who chose to wait at first start to make the reasonable decisions they believed in. Since the observational learning is a useful strategy, the better technology is rapidly adopted and the others would be vanished. Thus, herd behaviors occur. According to the relationship between herd behaviors and information cascades, we claim that there must be initiated information cascades. Because it’s difficult for us to get the detailed data of the two technologies (BD and HD DVD), such as parameters of lA , lB and r, so we turn to analyze the format war descriptively. Once we obtain the terminal data of two technologies, we could estimate the parameters mentioned in G-WB model, investigate the convergence of the two technologies and test the accuracy of the G-WB model. Thus, we can examine whether the G-WB model is well used in practice empirically. This paper has verified the efficiency of observational learning in technology adoption. However, we don’t know the performance of expert advising in our paper structure, since the expert advising is one of important factors in technology adoption. So we look forward to what would happen considering the expert advising and make a comparison between expert advising and observational learning in future research. What’s more, waiting has no cost in our paper, whereas the decision makers may pay to wait [22]. Therefore, how to make G-WB model more perfect is also our further research.

Acknowledgments This research was funded by the National Natural Science Foundation of China (Nos. 71401116 and 61273209), and Sichuan university (No. skgt201501).

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