A Heterogeneous Network Game Perspective of Fashion Cycle · Instead of providing more...

A Heterogeneous Network Game Perspective of

Fashion Cycle ∗

Zhi-Gang Cao † Cheng-Zhong Qin‡ Xiao-Guang Yang§

Bo-Yu Zhang¶

May 3, 2013

Abstract

Fashion, as a “second nature” of human being, has great economic impacts.

In this paper, we apply a heterogenous network game to analyze the existence of

a fashion cycle. There are two types of agents in the network game, conformists

and rebels. Conformists prefer to match the action taken by the majority of her

neighbors while rebels like to mismatch. Our results imply that social interaction

structures play a crucial role in the evolution of fashion, especially the emergence

of fashion cycles. For example, with a social network composed of a line or a ring,

fashion will almost always evolve into a steady state where no flux is possible

and thus can be deemed as disappeared. When the network is a star, however,

for approximately half of all the configurations of agents’ types, a fashion cycle

of length 4 will always emerge, regardless of the initial action profiles.

∗This is a sister working paper with: Bo-yu Zhang, Zhi-gang Cao, Cheng-zhongQin, Xiao-guang Yang, Fashion and Homophily (April 14, 2013). Available at SSRN:http://ssrn.com/abstract=2250898 or http://dx.doi.org/10.2139/ssrn.2250898. The research of thefirst and third authors are supported by the 973 Program (2010CB731405) and National NaturalScience Foundation of China (71101140). We gratefully acknowledge helpful discussions with XujinChen, Zhiwei Cui, Xiaodong Hu, Weidong Ma, Changjun Wang, Lin Zhao, and Wei Zhu.

†Key Laboratory of Management, Decision & Information Systems, Academy of Mathematicsand Systems Science, Chinese Academy of Sciences, Beijing, 100190, China. Email: [email protected].

‡Department of Economics, University of California, Santa Barbara, CA 93106, USA. Email:[email protected].

§Key Laboratory of Management, Decision & Information Systems, Academy of Mathematics andSystems Science, Chinese Academy of Sciences, Beijing, 100190, China. Email: [email protected].

¶Department of Mathematics, Beijing Normal University, Beijing, 100875, China. Email: [email protected].

1

KEYWORDS: fashion cycle; network games; coordination; anti-coordination;

matching pennies.

JEL Classification: A14; C62; C72; D72; D83; D85; Z13

1 Introduction

There are two dictionary definitions of fashion that are contradictory to each other.

One states that fashion is a prevailing custom, usage, or style, and the other says

that fashion is a distinctive or peculiar manner or way.1 The former definition implies

that to be fashionable is a conforming activity or practice. In contrast, under the

latter definition, to be fashionable is a rebeling activity. This may be an indication

that different people have different and even opposite opinions on what is fashionable.

Indeed, in the pursuit of fashion, many people prefer to go along with the majority,

while others like to stay in the minority. For instance, Louis Vuitton and Zara are

both regarded as fashionable brands, although the former is luxury and few people can

afford, while the latter is not luxury at all but still eagerly purchased by fashionists.2

When people say that “This year’s fashion color is black, my white skirt is not ‘in’ any

more”, they believe that conforming to the majority is fashion. In contrast, when they

exclaim with admiration that “Lady Gaga is the Godness of fashion”, they accept that

distinction is fashion.

In history, and even today, many people disgusted fashion so much that they be-

haved and claimed themselves as “anti-fashion”. However, they did not realize or

acknowledge that they were still in the influence sphere of fashion. As argued by

Simmel (1904, p. 143),

“The man who consciously pays no heed to fashion accepts its form just as

much as the dude does, only he embodies it in another category, the former

in that of exaggeration, the latter in that of negation.”

In fact, these people play very special roles as fashion leaders. But as time goes by,

more and more of them approve Simmel’s argument and are proud of being fashion

leaders.3 This change is probably not due to the diffusion of the theory of Simmel or

any other scholar, but mainly driven by the increasing influence of fashion itself.

1See, for example, Merriam-Webster: http://www.merriam-webster.com/dictionary/fashion.2Although Zara is most famously known for its “fast fashion”, “cheap fashion” fits it very well too.

Other examples of cheap fashion include H & M, Uniqlo, GAP, The Limited, and Vancl etc.3But even today, the two words “fashionist” and “fashionista” both refer only to fashion trend

followers.

2

Fashion is now a daily topic in newspapers, magazines, TVs, the Internet, etc.

Fashion is also doubtlessly a huge industry and its economic impacts are extremely

significant.4 Many people mistakenly assume that fashion is restricted only to costume

and adornment. However, the influence of fashion goes far beyond this.5 Blumer (1969)

listed the following fields where anecdotal evidences can be easily found showing the

operation of fashion: pure and applied arts (painting, sculpture, music, architecture

etc.), entertainment and amusement, medicine, business management, mortuary prac-

tice, literature, political doctrine, and even science and academic research.6 As claimed

by Svendsen (2006),

“Fashion has been one of the most influential phenomena in Western civ-

ilization since the Renaissance. It has conquered an increasing number of

modern man’s fields of activity and has become almost ‘second nature’ to

us ”.

The “second nature” viewpoint has already been implied, and eloquently proved, in

the essay of Simmel (1904), although he did not use the term explicitly. In particular,

Simmel claimed that people’s conforming and rebeling activities in fashion have a

rather stylized pattern; that is, upper class people are fashion leaders and lower class

people are fashion followers.7 As remarked in Blumer (1969):

“For him, fashion arose as a form of class differentiation in a relatively open

class society. In such a society the elite class seeks to set itself apart by

observable marks or insignia, such as distinctive forms of dress. However,

members of immediately subjacent classes adopt these insignia as a means

of satisfying their striving to identify with a superior status. They, in turn,

4For instance, the global market size of luxury industry is around $ 200 billion US dollars (Okonkwo,2007; Euromonitor International, June 2011). Only in US, $ 300 billion expenditure are driven byfashion rather than by (physical) need (Yoganarasimhan, 2012b).

5This is one of the important reasons why theoretical studies of fashion are seriously lacking, aspointed out by Blumer (1969), whose arguments on this point are still largely valid today (see An-dreozzi and Bianch, 2007). A second reason, also pointed out by Blumer (1969), is that scholars viewfashion as an “aberrant and irrational social happening”. After Gary Becker’s pioneering, convinc-ing, and influential explorations in the economic studies of addiction, crime, family, and tastes etc.,economists (and sociologists) today should have no obstruction in taking fashion as a behavior thatis rational at least to some degree. This point will be revisited in Footnote 20.

6More recent rigorous empirical studies can be found in Lieberson and Bell (1992), Schweitzer andMach (2008), Yoganarasimhan (2012b).

7Generally, there can be more than two classes, but the essence is the same: people try to imitatethe behavior of those belonging to a class that is immediately upper than them, by doing this, theydifferentiate themselves from people in the same class and the lower classes.

3

are copied by members of classes beneath them. In this way, the distin-

guishing insignia of the elite class filter down through the class pyramid. In

this process, however, the elite class loses these marks of separate identity.

It is led, accordingly, to devise new distinguishing insignia which, again,

are copied by the classes below, thus repeating the cycle. This, for Simmel,

was the nature of fashion and the mechanism of its operation.”

In the economic literature, the above logic of Simmel is largely identical to the

“conspicuous consumption” theory of Veblen (1899), which is, as its name indicates,

usually restricted to people’s consumption behavior.8 More precisely, Veblen distin-

guished between “invidious comparison” and “pecuniary emulation”, which correspond

to the rebeling behavior of the upper class and the conforming behavior of the lower

class in Simmel’s theory, respectively (see also Bagwell and Bernheim, 1996).

Subsequent literature (e.g., Pesendorfer, 1995; Bagwell and Bernheim, 1996) are

mostly based on the above theory of Simmel and Veblen.9 However, it is by no means

unchallenged. The most cogent queries on the theory of Simmel and Veblen come

from many empirical evidences where fashion is not initialized by the upper class.

The most famous (counter-)example is blue jeans, which originated from the humble

cowboys and miners (Fine and Leopold, 1993; Trigg, 2001; Benvenuto, 2000; Yoga-

narasimhan, 2012b). Other examples include slang (Benvenuto, 2000), and hip-hop

(Yoganarasimhan, 2012b), etc.. Similar to the example of slang, a prominent new ex-

ample in the era of Internet is that most fashionable expressions in the virtual world

are coined by anonymous “grassroots”. To remedy the theory of conspicuous consump-

tion, Bourdieu (1984) presents a “trickle-round” model.10 Although the usual attack

8This is also known as the Veblen effect. Strictly speaking, Veblen effect refers to the phenomenonthat as the price of a consumers’ good goes up, the demand increases rather than decreases. A similarconcept is the snob effect, which says that an individual’s demand for a consumers’ good decreases asmore people consume it. The two effects are two sides of the same coin, and their difference is that theformer is a function of price while the latter is a function of other people’s consumption. An oppositephenomenon of the snob effect is the bandwagon effect, which says that an individual’s demand fora consumers’ good increases as more people consume it. See the detailed discussions in Leibenstein(1950).

9Neither Simmel nor Veblen is the first scholar who considered fashion seriously. As reviewed byLeibenstein (1950), the first one should probably be attributed to John Rae (before 1834), who gavean extensive analysis of conspicuous consumption, which is subsequently known as the Veblen effect.See Leibenstein for more early literature.

10In literature, the fashion diffusion process of Simmel and Veblen is also summarized as “trickle-down”, and the opposite bottom-up process is also called “trickle-up” (see Trigg, 2001). In the“trickle-round” model of Bourdieu (1984), there are (at least) three classes, namely the upper class,the middle class and the working class. Besides the “trickle-down” flow of fashion diffusion, fashioncan also spread from the working class to the upper class, making a complete directed cycle.

4

loses its power on this new theory, it is still based on the differentiation of classes.

We argue in the next subsection that a “personality differentiation” theory might

be more appropriate in modern time. The fundamental reason is that the world is

becoming flatter and flatter and hence the upper class people are losing more and

more of their advantages in the game of fashion.

1.1 From class differentiation to personality differentiation

Instead of providing more counter-examples, we need first to understand the logic of

Simmel (1904) why upper class people are fashion leaders and lower class people are

followers. This is necessary in order to see why this logic worked well in the age of

Simmel, why it is failing to work today, and why it might be replaced by a classless

theory in the future.

Simmel (1904) argued that two antagonistic tendencies, imitation and differentia-

tion, union and segregation, etc., are simultaneously deeply rooted in the soul of every

individual. For instance, he observed that children “insist upon the repetition of facts”

and “they will object to the slightest variation in the telling of a story they have heard

twenty times”. He argued that “In this imitation and in exact adaption to the past

the child first rises above its momentary existence [...]” (p. 133).

The other fact about children Simmel did not point out, at least not explicitly,

is that they are also curious about new things and willing to accept challenges, al-

though this curiosity is frequently mingled with anxiety and fear.11 What’s more, even

for grown-ups, this mixture of curiosity and fear gives them a feeling of venture and

exploration and consequently, a special kind of pleasure. From the perspective of the

evolutionary theory, this subtle psychology can be explained easily as a result of natural

selection. Only possessing this state of psychology can we inherit the experience and

knowledge of our ancestors, avoid being too risky, and at the same time properly deal

with uncertainty and the unknown and create new knowledge.12 And as the children

11Hide-and-seek, a popular game loved by almost all small babies, is a good evidence. Below isanother more interesting evidence. Many new parents have the experience that when babies, aroundthe age of one year old, learn how to walk, they are fond of practicing it time and time again. Duringthis process, they enjoy the challenge a lot and usually hate the holding of adults. However, once theymaster the skills and walking is not challenging any longer (at around one and a half), they become“lazy” and tend to “conserve energy”!

12As is well known, human being, just like any other species, has two basic tendencies, namely seekpleasure and avoid pain. To make right decisions, we need to have the knowledge of which bringsus pleasure and which brings pain. Thus in a deterministic environment, we can always use purestrategies to maximize our fitness. In dealing with uncertainty, however, no gene coded pure strategy,neither “seek” nor “avoid”, is universally wise, because the former may lead to disasters and the latter

5

grow up, they are more and more used to new things, and fear becomes more and more

feeble.13 This logic on every single person can be extended to the human being as a

species, where the process of growing up is compared with the process of civilization

(see Simmel, 1904, pp. 137-138).

Also, lower class people can be compared with children and upper class people

with grow-ups, because upper class people are usually more civilized through better

education. This is the first reason why the second tendency is embodied more strongly

in the upper class than in the lower class. The second reason is that upper class people,

in general, have more access to new things. As quoted from Simmel (1904, p. 138),

“[...] the lower classes possess very few modes and those they have are seldom specific”.

The above two advantages of the upper class are becoming less and less sharp, with

the astonishingly fast development of economy and consequent consumption revolu-

tions. This trend, not surprisingly, has been noticed by many scholars in history. For

instance, Galbraith (1958, pp. 72-73) stated that “lush expenditure could be afforded

by so many that it ceased to be useful as a mark of distinction.” Canterbery (1998, p.

148) observed that, “the middle class could now emulate the rich in dress and even in

automobiles, especially as the rich downsized to Volvos”.14

Although the distinction between upper and lower classes in fashion could be blurred

by the development of the economy, the elementary logic of Simmel (1904) that the

two antagonistic tendencies are coded in every individual, however, is still largely

true and will be valid even in the expected future. Time erodes this really slowly.

Also, it is reasonable to assume that, in general, the two tendencies are not equal

for each individual. After all, diversity is universal and one of the ultimate sources

for the development of the society. For those the first tendency is stronger, we call

them conformists; and for those where the opposite is true, we call them rebels. This

distinction reflects people’s different personalities. As also noticed by Simmel (1904,

p.131), in the class-based model, “[fashion] signalizes the lack of personal freedom”.

We believe that in a world where people’s personalities are more and more freed, the

“personality differentiation” model will be more and more suitable than the “class

may miss opportunities. They are both too risky. Natural selection determines that we must takesome strategy in-between, i.e. a “mixed” strategy, when dealing with uncertainty. Thus the mingledfeeling of both fear and excitement towards uncertainty is evolved. The above viewpoint may serve asa new defence for the widely used, yet not always very clear and sometimes even debatable, conceptof mixed strategy in game theory.

13Another observation that supports this argument is that in the minds of grown-ups, “interesting”implies new and unexpected. When someone says that something is interesting, this thing is almostalways new to him/her. This is especially true in the academical world and when we talk jokes.

14The two examples are both re-quoted from Trigg (2001, p. 103).

6

differentiation” model for the study of fashion.15

1.2 What causes fashion cycle?

Fashion cycle refers to the phenomenon that the most popular choice in a specific

field periodically changes and reappears. Anecdotal observations on the existence of a

fashion cycle include fashion color and length of girls’ skirts, can be found throughout

history.16 Fashion cycle is regarded as the most prominent and the most mysterious

fact about fashion, and it is one of the key focuses in previous studies (e.g., Pesendorfer,

1995; Young, 2001; Yoganarasimhan, 2012a).

It is safe to argue that fashion cycle is caused ultimately by a combined force of

consumers and producers. However, the concrete process may be quite complicated,

and the task of theoretical as well as empirical studies is to reveal more details of

this process. In the model of Pesendorfer (1995), fashion cycle is determined and

completely controlled by a monopolist producer. The problem Pesendorfer attacked

is not on causes of fashion cycle, but how to control it optimally. Although the main

assumption that everything is decided by a powerful producer is obviously unrealistic,

because very rarely can a producer be so powerful17, this theory provides us new

understandings of the fashion cycle on the producer side. On the consumer side,

motivationally, fashion cycle may come out of the needs for class signaling (Simmel,

1904), wealth signaling (Veblen, 1899), cultural capital signaling (Bourdieu, 1984),

taste signaling (Yoganarasimhan, 2012b), or personality signaling.

1.3 Approach of this paper

In this paper, we try to analyze the causes of fashion cycle from a structural point of

view. Since fashion is generated through people’s interactions and these interactions

are usually restricted within friends (colleagues, neighbors), it is natural to analyze

fashion through a social network. After all, few people care what a stranger wears

15It should be emphasized that in the classless model, people have no absolute freedom either. Itseems that fashion will not disappear as long as the society exists, and hence people will always affecteach other and there will be no absolute freedom. However, in a classless model, people’s choices,w.r.t. chasing fashion, are not affected by their social statuses. In this sense, they are more free thanin the class-based model.

16However, as pointed out by Yoganarasimhan (2012b), “there exists no systematic empirical inves-tigation of the phenomenon of fashion. Indeed, the even more basic question of how to detect fashioncycles in data has not been studied.”

17A well-known example in history is the vain effort in 1922 by not only manufacturers but also themedia to reverse the trend towards shorter skirts. See e.g., Blumer (1969).

7

and what they do in pursuing fashion are mainly for setting a particular image or

personality in the eyes of their friends.

Our model is a fashion game adapted from Young (2001, p. 38) and Jackson (2008,

p. 271). We focus on the evolution of fashion from the perspective of consumers only.

There are two types of agents, conformists and rebels, located on a fixed social network.

Conformists like to match the majority action of her neighbors while rebels prefer to

mismatch. Each agent has two available actions, named 0 and 1, such as white or

black skirt, buying or not buying an iphone, etc. With these simplifying assumptions,

interactions between agents can be conveniently modeled by certain familiar two-player

stage games (see Section 2 for formal description).

To study fashion cycles, there must be an underling dynamic process. In Andreozzi

and Bianchi (2007), the process is determined by the log linear response and in Young

(2001), it is determined by fictitious play. In this paper, we adopt another simple

dynamic process; namely, the best response dynamic (BRD in short).18 Under this

dynamic, agents are naive and myopic, in the sense that they only have one step

memory and do not look forward.19 This dynamic is consistent with the empirical

experience that people’s behaviors in fashion are usually not so complicated.20

We consider three kinds of network structures, namely lines, rings, and stars. Our

results show that the emergence of a fashion cycle relies crucially on the network

structures. Specifically, when the social network is a line or a ring, the BRD will

almost always converge to a Nash equilibrium. The convergence is fast and thus, when

the interaction structure is a line or a ring, the system will evolve to a steady state in a

very short time where no flux or fashion cycle is possible. When the network is a star,

however, for approximately half of all the configurations of agents’ types, a fashion

cycle of length 4 will always emerge, regardless of the initial action profiles.

Our research falls into the booming field of network games (see Jackson and Wolin-

sky, 1996; Goyal, 2007; Vega-Redondo, 2007; Jackson, 2008; Jackson and Zenou, 2014).

18Both sequential BRD and simultaneous BRD are considered.19We remind the reader that in fictitious play, each agent should remember all the history that she

is involved. Actually, BRD is exactly the one-step-memory simplification of fictitious play.20In fact, irrational and even crazy are often associated with fashion, as pointed out by Blumer

(1969). We believe that there is a mixture of rationality and irrationality in the phenomenon offashion. However, we are not able to study irrationality that has no patterns. In this sense, abounded rationality model, which assumes neither absolute rationality nor patternless irrationality, isquite proper. In literature, herding models are also used to study the phenomenon of fashion. The word“herding”, historically, indicates irrational, although the most famous such models assume rationalityof agents (in fact, they are Bayesian learning models, and hence agents must be extremely smart).See Banerjee (1992) and Bikchandani et al. (1992). It should be noted that only the conforming sideof fashion can be explained in the above two models.

8

It can be taken as a new application of this field.21 In addition, the fashion game also

has its own noval theoretical features; namely, it is a mixed model of a network game

with strategic substitutes and a network game with strategic complements, the two most

extensively studied classes of network game models.22 Indeed, for conformists, it ex-

hibits a simple form of strategic complement and for rebels a simple form of strategic

substitute works. A recent model of Hernandez et al. (2013) has also strategic com-

plements and strategic substitutes. However, for a particular set of parameters, only

strategic substitute or complement but not both works. In comparison, except for two

extreme cases, both work simultaneously in our model. For fashion games with con-

formists only, Goles (1987) shows that simultaneous BRD always converges to a cycle

of length at most 2 (see also Berninghaus and Schwalbe, 1996). This result is recently

extended to more general models with arbitrary thresholds (Adam et al., 2013). Can-

nings (2009) notices that the result of Goles holds for fashion games with rebels only

too. He also observes that when the network structure is complete and the underlying

dynamic is simultaneous BRD, the only possible limit cycle lengths are 1, 2 and 4.

The computational properties of the fashion game adopted in this paper has been

recently investigated in Cao and Yang (2011). They show that testing whether a

fashion game possesses a pure Nash equilibrium is in general NP-hard23, meaning that

predicting fashion trends might be impossible.24

The rest of the paper is organized as follows. The next section introduces the model

formally. Section 3 discusses briefly the factor of heterogeneity. Section 4 presents main

results of this paper. Section 5 concludes this paper with several further discussions.

Proofs of the main results are organized in Append A and two technical lemmas used

in the proofs are presented in Appendix B.

21Other notable applications include employment (Calvo-Armengol and Jackson, 2004, 2007, 2009),education (Calvo-Armengol et. al., 2009), crime (Calvo-Armengol et. al., 2007), and R & D (Goyaland Moraga-Gonzales, 2001) etc. See also the review of Jackson and Zenou, 2014.

22See e.g. Bramoulle (2007), Bramoulle and Kranton (2007), Hernandez et al. (2013). See also thereview in Jackson (2008), and Jackson and Zenou (2014).

23Definition of this concept can be found in any textbook about computational complexity. Roughlyspeaking, a problem is NP-hard means that there seems to be no way to compute a solution in poly-nomial time. Since only polynomial time algorithms can be implemented efficiently by the computer,an NP-hard problem can be equivalently taken as unsolvable when the problem size is large enough.Thus the result of Cao and Yang (2011) tells us that when the social network is moderately large, itis impossible for us to determine whether it possess a PNE or not.

24This conjecture is supported by industrial practices from fast fashion. In Zara, for example,“leading” or even “predicting ” the fashion trends has already been given up. Hundreds of its designersfly around the world, watch the latest hot factors in people’s wearing in Paris, New York etc., andmerge these factors into their designs in extremely short times. This kind of “quick response” is oneof its core competencies.

9

0 10 1, 1 −1,−11 −1,−1 1, 1

0 10 1,−1 −1, 11 −1, 1 1,−1

0 10 −1,−1 1, 11 1, 1 −1,−1

Figure 1: Three base games. (1) (Pure) Coordination Game (left): conformist V.S.conformist; (2) (Pure) Anti-coordination Game (right): rebel V.S. rebel; (3) MatchingPennies Game (middle): conformist V.S. rebel.

2 The Fashion Game

Suppose two conformists meet. Since they both prefer to conform, i.e. match the action

of their respective opponents, they naturally play the coordination game. Similarly,

when two rebels meet, they both like to mismatch, and hence they play the anti-

coordination game.25 The most interesting case is when a conformist meets a rebel.

It is worth emphasizing that this case can be modeled by the matching pennies game.

The Row player prefers to match the Column player’s action and hence he can be

regarded as a conformist. In contrast, the Column player likes to mismatch the Row

player’s action and so he can be treated as a rebel. Throughout the rest of this paper,

we assume that those three games are all symmetric, with payoffs as shown in Figure

1.26

We assume that each agent takes the same action in all the local games she plays.

For instance, if a conformist agent has five neighbors, three of whom are conformists

and the other two are rebels, then she plays three pure coordination games and two

matching pennies games. In all the five games, however, either she always chooses

action 0 or always action 1 (we do not consider mixed strategies in this paper). This

restriction is quite common in the study of network games. Intuitively, it says that

each agent should show the same image in front of all her neighbors. The total payoff

of each agent is simply the sum of the payoffs that she receives in all the games she

plays.

An alternative way to understand our network game is that the payoff that each

25In evolutionary game theory, it is usually called the hawk-dove game. In the field of socialphysics, a closely related and extensively studied model is the “minority game” (Challet and Zhang,1997; Challet et. al., 2004). This model, also known as the El Farol Bar problem, is proposedby economist Arthur (1994). Though minority games have been applied in the study of financialmarkets early enough, Marsili (2001) noticed that people do not necessarily play any minority game instock markets, because except for a few “market fundamentalists”, most people are “trend followers”.Instead, a minority-majority game is more appropriate.

26Our main results can be easily extended to more general payoff settings.

10

agent receives is the social value of the corresponding action, where the social value of

a particular action for an agent is defined as the number of neighbors who choose an

action that she likes minus the number of neighbors whose action she dislikes (see the

companion paper for more discussions). Note that for a conformist who chooses action

0, neighbors choosing action 0 are liked by her and those choosing action 1 are disliked

by her. In contrast, for a rebel who chooses action 0, action 1 neighbors are liked and

action 0 ones are disliked.

2.1 Formal description

Throughout this paper, we use {C, R} to denote the type set. For each agent i,

Ti ∈ {C, R} is her type: Ti = C means that she is a conformist and Ti = R a rebel.

A fashion game is represented by a triple G = (N,E, T ), where

• N = {1, 2, · · · , n} is the set of agents;

• E ⊆ N ×N is the set of links;

• T = (T1, T2, · · · , Tn) ∈ {C, R}N is the configuration of agents’ types.

The underlying graph g = (N,E) is undirected. That is, ij and ji represent the same

link. We say that g is a complete graph if and only if ij ∈ E for all i, j ∈ N and i 6= j.

We say that g is a line if agents can be indexed such that E = {12, 23, · · · , (n− 1)n},a ring if E = {12, 23, · · · , (n − 1)n, n1}. g is called a star network if there exists an

agent i such that E = {ij : j ∈ N, j 6= i}, in which case i is called the central agent

and others are called peripheral ones.

Two agents i, j ∈ N are neighbors to each other if and only if ij ∈ E. We also use

Ni to denote the set of neighbors of agent i:

Ni = {j ∈ N : ij ∈ E}.

In simultaneous BRD, time elapses discretely. Players’ types do not change over

time, but their actions may change. We use

x(0) = (x1(0), x2(0), xn(0)) ∈ {0, 1}N

11

to denote the initial action profile, and define xi(t) ∈ {0, 1} as the action of agent i at

time t.27 Li(x(t)) ⊆ Ni is the subset of neighbors at time t that are liked by agent i.

Since conformists like those taking the same action, and rebels the opposite, we have

Li(x(t)) =

8<: {j ∈ Ni : xj(t) = xi(t)} if Ti = C

{j ∈ Ni : xj(t) 6= xi(t)} if Ti = R.

Set Di(x(t)) = Ni \ Li(x(t)), which is the subset of neighbors that are disliked by

agent i at time t. According to the payoff settings in Figure 1, agent i gains 1 unit from

interacting with each neighboring agent in Li(x(t)) but loses 1 unit from interacting

with each of those in Di(x(t)). Thus, given action profile x(t) = (x1(t), · · · , xn(t)) at

time t, the time-t utility of agent i is given by

ui(x(t)) = |Li(x(t))| − |Di(x(t))|.

2.2 Fashion cycle

Under simultaneous BRD, agent i switches her action at time t + 1 if and only if

ui(x(t)) < 0, because her utility will be −ui(x(t)) if she switches, assuming that her

neighbors do not change their actions. That is, simultaneous BRD evolves with time

as follows:

xi(t + 1) =

8<: 1− xi(t), if ui(x(t)) < 0

xi(t), otherwise.

Under simultaneous BRD, we use the limit cycle, a basic concept from discrete

dynamical systems, to study the phenomenon of fashion cycles. Intuitively, a limit

cycle is an ordered set of action profiles such that starting from each of these profiles

the system will evolve into its immediate successor. Since there are a total of finitely

many action profiles in the fashion game and simultaneous BRD is deterministic, after

finitely many steps some profile will reappear. From that point on, everything will be

repeated just as that state is reached for the first time. Therefore, limit cycle always

exists and it is unique for any given fashion game G = (N,E, T ) with a fixed initial

action profile x(0). In the rest of this paper, we also use I = (N,E, T, x(0)) to denote

an initialized fashion game. Below is the mathematical definition of fashion cycle.

27In this paper, agents’ strategies are simply their pure actions. Neither mixed actions nor othersophisticated strategies are allowed. And hence we use “action” and “action profile” in replace of“strategy” and “strategy profile”, respectively.

12

Figure 2: Limit cycle of the matching pennies game. Circle stands for conformist,triangle for rebel. Black color indicates that the agent chooses 1 and white for 0.

Definition 1 Let I = (N,E, T, x(0)) be an initialized fashion game, and t(I) ≥ 1 be

the first time such that

x(t(I)) ∈ {x(0), x(1), · · · , x(t(I)− 1)},

and

x(t) /∈ {x(0), x(1), · · · , x(t− 1)},∀1 ≤ t < t(I).

Let r(I) < t(I) be the time such that x(t(I)) = x(r(I)). We call the set of states

{x(r(I)), x(r(I) + 1), · · · , x(t(I)− 1)} the limit cycle of I.

In the rest of this paper, we use l(I) = t(I)− r(I) to denote the length of the limit

cycle of I. When time goes beyond r(I), we say that simultaneous BRD enters the

limit cycle.

For an initialized fashion game I, it is clear that a pure strategy Nash equilibrium

(PNE for short) will be eventually reached via simultaneous BRD if and only if l(I) = 1,

in which case we also say that simultaneous BRD converges. We show in Section 4

that if the underlying network g = (N,E) is a line or a ring, then for almost all

configurations of agents’ types T , simultaneous BRD converges for all initial action

profiles. That is, fashion will eventually stop evolving. We show that l(I) ∈ {1, 2, 4}if g = (N,E) is a line or a star network.

2.3 Several simple facts

As illustrated in Figure 2, the matching pennies game always has a fashion cycle of

length 4 under simultaneous BRD. Proposition 1(c) in Subsection 4.2 can be taken as

a generalization of the above simple fact.

Before proceeding to the next section, we introduce several more simple facts about

the fashion game. As is well-known, the matching pennies game does not have any

13

PNE, but it has a unique mixed Nash equilibrium, in which each agent plays actions

1 and 0 with equal probability. It turns out that playing actions 1 and 0 with equal

probability is also a mixed strategy equilibrium for the fashion game. Jackson (2008)

points out that the case where each agent has no less conformist neighbors than rebel

ones always has a PNE.28 If we only require that each conformist has at least as many

conformist neighbors as rebel ones, PNE can still be guaranteed (Zhang et. al., 2013).

3 The Role of Heterogeneity

Given a fashion game G = (N,E, T ), if Ti = Tj for all i, j ∈ N , i.e. either all agents are

conformists or all agents are rebels, we say that G is homogeneous. In the conformist

homogeneous case, PNE is obviously guaranteed. All agents choosing 1 and all agents

choosing 0 are the two most natural ones.29 The other homogeneous case, where all

agents are rebels, have PNEs as well.30 It can be shown that the two homogeneous

cases are both exact potential games, with the potential function given by the sum

of agents’ payoffs.31 Hence, according to the results of Monderer and Shapley (1996),

PNE will eventually be reached through a sequential BRD (with arbitrary orders).

This means that under the dynamic of sequential BRD, fashion will eventually come

to an end, unless the two types of agents both exist, implying that the heterogeneity

of consumers is a critical factor to sustain the emergence of fashion cycles.

In fact, the necessity of the heterogeneity of agents has already been noted by

Simmel. To be exact, Simmel (1904, p. 138) wrote32

“Two social tendencies are essential to the establishment of fashion, namely,

28It can be checked easily that all conformists choosing 1 and all rebels choosing 0 is a PNE.29But it should be noted that there may well be other ones. Actually, it can be proved easily that

the set of PNEs for the homogeneous fashion game with only conformists form a lattice (see Exercise9.5 of Jackson, 2008). Thus the maximum PNE and the minimum PNE are the two natural focalequilibria. The other way to refine them in previous related studies in social learning is is to usethe concept of stochastic stability instead of Nash equilibrium (Foster and Young, 1990). Since theforming of conventions can be taken as a process of coordination, this learning theory is also widelyused to study the evolution of conventions. See e.g. the classical paper of Young (1993).

30In the rebel homogeneous fashion game, its PNE corresponds to a well studied concept in graphtheory, namely the “unfriendly partition” (see Aharoni et. al., 1990).

31This can be verified easily. For the general anti-coordination game, which is an extension of thefashion game with only rebels, it is still the case. This is a well known result (see Bramoulle, 2007).

32This viewpoint is also often emphasized by subsequent scholars. For instance, Benvenuto (2000)argued that, “Undoubtedly, some of us tend more towards imitation (and thus to conformism) whileothers tend to distinction (and thus to eccentricity and dissidence), but fashion’s flux needs both ofthese contradictory tendencies in order to work.”

14

the need of union on the one hand and the need of isolation on the other.

Should one of these be absent, fashion will not be formed—-its sway will

abruptly end.”

Should the insightful observation of Simmel be verified rigorously, our understand-

ing of the two forces of conforming and rebeling would be significantly improved: On

the one hand, it is intuitive that conforming, just like positive feedback and contagion,

is likely to lead a system to extreme states. On the other hand, rebeling, just like

negative feedback, tends to lead a system to the middle, or “doctrine of the mean”.33

Each one alone will make a system eventually stable. It is the combination of the

two forces that makes a system keep in flux. In the companion paper (Zhang et. al.,

2013), we further investigate the above logic by showing that for the emergence of

a fashion cycle, these two forces may have to be “fully mixed”, namely the interac-

tions between conformists and rebels (the matching pennies games played) should be

frequent enough.

Complication comes when we consider simultaneous BRD, because neither the rebel

homogeneous case nor the conformist homogeneous case will definitely reach a PNE,

although the limit cycles can be of length at most two (Goles, 1987; Berninghaus and

Schwalbe, 1996; Cannings, 2009). It is not yet clear to us as to how likely a limit cycle of

length two can be reached, which we will carry out in a future project. However, even if

it can be shown that PNE is reached almost always under simultaneous BRD when the

fashion game is homogeneous, there is still great space to defend Simmel’s observation,

because it is very unlikely in reality for all the agents to always move simultaneously,

and our definition of fashion cycle through limit cycle is too restrictive. For instance,

when all but two agents are always satisfied in the limit cycle, it should be taken as

that fashion cycle does not occur. See Subsection 5.1 for more discussions.

4 The Role of Interaction Structures

To ease the presentation of our main results, we need several new terms.

33See Cao et al. (2011) for the exact result on a continuous learning model. A similar effect is alsofound by Galam (2004) in a voting model, where the term “contrarian” is used instead of “rebel”.

15

4.1 Three more terms

Definition 2 Given a fashion game G = (N,E, T ), if Ti 6= Tj for all j ∈ Ni and

i ∈ N , i.e. the neighbors of any conformist are all rebels and the neighbors of any rebel

are all conformists, we say that G is cross-heterogeneous.

Obviously, if G is cross-heterogeneous, then the underlying network g = (N,E)

is a bipartite graph with one side all conformists and the other side all rebels.34 If

g = (N,E) is a line or a ring network, then cross-heterogeneous means that conformists

and rebels are alternate. The concept of cross-heterogeneous, as well as homogeneous,

obviously can also be derived naturally from the homophily index. See the companion

working paper (Zhang et. al., 2013) for extensive discussions.

Definition 3 Let G = (N,E, T ) be a fashion game, and the underlying network g =

(N,E) be a line. A cluster of G is defined as any of the following components:

(i) the left most two agents that are of the same type;

(ii) the right most two agents that are of the same type;

(iii) three adjacent agents that are of the same type.

We say that G is clustered if it has at least one cluster. Otherwise, we call it

un-clustered.

For rings, we can define the concept of cluster and clusteredness similarly. The only

difference is that there are no endpoints in rings and thus the first two kinds of clusters

will never occur.

Suppose G = (N,E, T ) is a fashion game, and the underlying network g = (N,E)

is a line or a ring. Since when G is homogeneous, PNE is always guaranteed, it is

intuitive that clusters may be in favor of the convergence of BRD. This is indeed the

case. Actually, agents in a cluster are able to help each other and hence have no

incentive to deviate from the current state no matter what the other agents choose.

Clusters are like “anchors” in the evolution of fashion and their effects are “contagious”,

so that eventually all agents will stop switching and the system converges (Proposition

1(b)).

Definition 4 Given an initialized fashion game I = (N,E, T, x(0)), we say that I is

badly initialized if ui(x(0)) < 0 holds for all i ∈ N .

34A graph g = (N, E) is called bipartite, if we can partition the node set N into two parts, N1 andN2, such that the two nodes of each edge are from different parts. That is, g = (N, E) is bipartite ifand only if E ⊆ N1 ×N2. This concept can be found in any textbook about graph theory.

16

L1L1

L2L2

L3

L4

L5

L6

Figure 3: Illustration of homogeneousness, cross-heterogeneousness, clus-teredness, and bad initialization. Circles stand for conformists, and triangles forrebels. Black color indicates that the agent chooses 1, and white 0. Gray means that ini-tial action is not given. L1: homogeneous and badly initialized; L2: homogeneous butnot badly initialized; L3, L4: clustered but not homogeneous; L5: cross-heterogeneous;L6: un-clustered but not cross-heterogeneous.

Intuitively, a bad initialization is the most terrible scenario we can imagine. No

agent is satisfied initially, so they switch to the other actions simultaneously, leading to

a new state where no agent is satisfied either. In the next round, all agents switch back

to the initial state. The system keeps oscillating between the two states, all agents are

in eternal flux and no agent is ever satisfied (Proposition 1(a)). Evidently, coordination

failure is caused by simultaneous moves.

Straightforwardly, a line or a ring must be homogeneous to be badly initialized.35

For lines and rings, homogeneousness is a special case of clusteredness, cross-heterogeneousness

implies un-clusteredness, and a badly initialized fashion game must be homogeneous.

See Figure 3 for an illustration of these concepts.

4.2 Simultaneous BRD

4.2.1 Lines

Clearly, limit cycles are always of length two for badly initialized fashion games. For

line structures, the following proposition indicates that this is the only case. That is,

if the underlying network is a line, then the fashion game has a limit cycle of length

35Note that this cannot be extended to general structures.

17

two if and only if it is badly initialized.36

Proposition 1 Let G = (N,E, T ) be a fashion game with the underlying network

g = (N,E) being a line. x(0) is an initial action profile, and I = (N,E, T, x(0)) the

corresponding initialized fashion game.

(a) If I is badly initialized, then l(I) = 2;

(b) If G is clustered and I not badly initialized, then l(I) = 1;

(c) If G is cross-heterogeneous, then l(I) = 4 for all x(0);

(d) If G is un-clustered, but not cross-heterogeneous, then l(I) ∈ {1, 4} for all x(0).

Proof to the above proposition is presented in Appendix A.1. When the underlying

network g = (N,E) is a line, we know from the above proposition that to have a limit

cycle of length two, it must be that the limit cycle is entered immediately at the very

beginning of the evolution. There is no possibility to evolve for certain periods of time

and then enter a length two limit cycle.37 For general network structures, when the

system evolves into a limit cycle of length two, there is still a possibility that some, and

even many, players are always satisfied and never deviate in the limit cycle. Proposition

1 tells us that this never happens for the line structures.

Below is an immediate corollary of Proposition 1, stating that the lengths of limit

cycles of the fashion game are upper-bounded when the underlying network is a line.

Corollary 1 Let G = (N,E, T ) be a fashion game. If the underlying network g =

(N,E) is a line, then l(I) ∈ {1, 2, 4} for all x(0) ∈ {0, 1}n.

To put the above corollary another way, no matter how many agents there are in

the fashion game, the lengths of limit cycles under simultaneous BRD cannot exceed

4, as long as the interaction structure is a line. It is worth noting that we are not able

to extend this result to general structures.38

As shown in Cao and Yang (2011), when the underlying network g = (N,E) is a

line, PNE exists for G = (N,E, T ) if and only if the line is not cross-heterogeneous. The

structure of cross-heterogeneous lines is a natural extension of the matching pennies

game, a dyad composed of a conformist and a rebel. Under this structure, matching

36It seems that this may also be true for ring structures. See Proposition 2 and discussions inSubsection 5.2.

37For general networks, however, this may well happen.38Not even possible for ring structures. See Subsection 5.2 for discussions.

18

pennies is the only base game. Hence Proposition 1(c) can be taken as a generalization

of the simple fact that the matching pennies game always has a limit cycle of 4 under

simultaneous BRD.

Combining Proposition 1 and the result of Cao and Yang (2011) that is introduced

in the previous paragraph, we have an immediate interesting corollary.

Corollary 2 Let G = (N,E, T ) be a fashion game with the underlying network g =

(N,E) being a line. If G does not possess a PNE, then l(I) = 4 for all x(0).

Suppose that each agent is randomly assigned a type (without bias), then it follows

easily that the probability that a line is clustered goes to 1 as n goes to infinity.39

Since the probability that a line is homogeneous is 2/2n, Proposition 1(b) implies that

for almost all configurations of agents’ types on lines, simultaneous BRD converges to

a PNE, regardless of the initial action profiles (notice again that badly initialized is

stronger than homogeneous). We present this result formally as a corollary of Propo-

sition 1. Note that the implicit probability space is determined by unbiased coins

flipping.


(N,E) is a line, then

limn→∞Prob. [G : l(I) = 1,∀x(0) ∈ {0, 1}n] = 1,

where I = (N,E, T, x(0)) is the initialized fashion game with an initial action profile

x(0).

4.2.2 Rings

The following proposition for the ring structure is parallel to part (a) and (b) of Propo-

sition 1.


g = (N,E) being a ring. x(0) is an initial action profile, and I = (N,E, T, x(0)) the


(a) If I is badly initialized, then l(I) = 2;

(b) If G is clustered but I not badly initialized, then l(I) = 1.

39This is true because the probability that there are three or more adjacent agents that are of thesame type goes to 1, as n tends to infinity.

19

Proof to the above proposition is also presented in Appendix A.1. It is worth noting

that when g = (N,E) is a ring, we cannot derive that l(I) ∈ {1, 2, 4}. See Subsection

5.2 for more discussions. However, the following corollary is still obviously true from

Proposition 2(b).


(N,E) is a ring, then

limn→∞Prob. [G : l(I) = 1,∀x(0) ∈ {0, 1}n] = 1.

4.2.3 Stars

Combining Corollary 3 and Corollary 4, we know that when the underlying network

is a line or a ring, then almost always fashion evolves into a steady state through

simultaneous BRD. When the network is a star, however, the above statement is not

true. First of all, let’s see the following complete description for the simultaneous BRD

on star networks.


g = (N,E) being a star. x(0) is an initial action profile, and I = (N,E, T, x(0)) the


(a) If more than half of the peripheral agents are of the different type as the central

agent, and initially the central agent is unsatisfied, then l(I) = 2;

(b) If more than half of the peripheral agents are of the same type as the central

agent, then l(I) = 4;

(c) In all the other cases, l(I) = 1.

Proof to the above proposition is presented in Subsection A.2.

Parallel to Corollary 1 for the line networks, we have the following corollary for the

star networks, which is immediate from Proposition 3.


(N,E) is a star, then l(I) ∈ {1, 2, 4} for all x(0) ∈ {0, 1}n.

As before, suppose each agent is randomly assigned a type, being a conformist and

a rebel equally likely. From the above proposition, it can be calculated easily that the

20

probability that the fashion game with a star network always has a limit cycle of length

4 (case (c) and case (d)) is approximately 1/2, when n is moderately large. Whether

the limit cycle has a length of 1 or 2, however, depends on both the configuration

of types and the initial actions.40 If we assume further that the initial actions are

chosen randomly (without bias) by all agents, then it can be shown easily that with a

probability approximately 1/4 the fashion game with a star network goes into a limit

cycle of length 2, and it converges with a probability approximate to 1/4.


(N,E) is a star, then

limn→∞Prob. [G : l(I) = 4,∀x(0) ∈ {0, 1}n] = 1/2,

limn→∞Prob. [I : l(I) = 2] = 1/4,

limn→∞Prob. [I : l(I) = 1] = 1/4.

Before presenting the result parallel to Corollary 2, we need a simple lemma.

Lemma 1 Let G = (N,E, T ) be a fashion game. If the underlying network g = (N,E)

is a star, then G possesses a PNE if and only if either (i) the central agent is a

conformist and at least half of the peripheral ones are also conformists, or (ii) the

central agent is a rebel and at least half of the peripheral ones are also rebels.

Proof to the above lemma goes as follows. If condition (i) in the above lemma

holds, we let all the conformists choose 0 and all rebels choose 1. It can be checked

easily that this is a PNE. In fact, this is a special case of that considered by Jackson

(2008). If condition (ii) is valid, we let the central rebel choose 0, the peripheral rebels

choose 1, and all the conformists choose 0, then it can be seen that this is a PNE.

There are two left cases: (iii) the central agent is a conformist and more than half of

the peripheral agents are rebels, and (iv) the central agent is a rebel and more than half

of the peripheral agents are conformists. Case (iii) cannot have any PNE, because we

are not able to make both the central conformist and all the peripheral rebels happy.

Similarly, case (iv) does not admit a PNE either. This completes the proof.

40Except for the case that there are equal number of peripheral rebels and peripheral conformists,which always leads to l(I) = 1. See the proof to (c) in Subsection A.2. It is notable that this canonly happen when n is odd. And as n tends to infinity, the probability that this happens is zero.

21

Combining Lemma 1 and Proposition 3, we immediately have the following corol-

lary.


(N,E) being a star. If G does not possess a PNE, then l(I) = 4 for all x(0).

4.3 Sequential BRD

In this subsection, we discuss properties of the sequential BRD. Let I be a given

initialized fashion game. In each time step, as long as PNE is not reached, we let an

arbitrary unsatisfied agent switch her action. We say that sequential BRD converges if

and only if PNE can be definitely reached for any realized deviation order. That is, we

consider exactly the same property as for potential games, although our proofs do not

rely on explicit constructions of potential functions. We use this convergence concept

to study the fashion game on lines and rings. For the star networks, we use a slightly

weaker concept, which will be introduced soon.

Notice first that the fashion game, in general, is not a potential game for the line

network or the ring network or the star network, because in none of the three cases

can PNE be guaranteed, while a potential game always possesses a PNE. However, for

a very wide class of situations, the fashion game with one of the three kinds of network

structures is indeed a potential game.

4.3.1 Lines and rings


g = (N,E) being a line. Then sequential BRD converges for all initial action profiles

if and only if G is not cross-heterogeneous.


g = (N,E) being a ring. If G is not cross-heterogeneous, then sequential BRD converges

for all initial action profiles.

Proofs to the above two propositions are presented in Appendix A.3. Note that the

condition that G is not cross-heterogeneous is equivalent to that there are two adjacent

that are of the same type. This kind of component is weaker than a cluster. In the

previous subsection, we have seen that clusters play a critical role in the convergence

of a simultaneous BRD when the underlying network is a line or a ring. From the

22

above two propositions, it is natural to guess that sequential BRD is easier to converge

than simultaneous BRD. It is indeed the case for the cases we consider. See the next

subsection for more discussions.

The following corollary, which is similar to Corollary 3 and Corollary 4, is an

immediate result of Proposition 4 and Proposition 5.


(N,E) being a line or a ring. Then

limn→∞Prob. [G : sequential BRD converges for all initial action profiles] = 1.

We also have the following nice characterization for sequential BRD on lines.


(N,E) being a line, then the following statements are equivalent.

(a) PNE exists for G;

(b) G is not cross-heterogeneous;

(c) Sequential BRD converges for all initial action profiles;

(d) Sequential BRD converges for some initial action profile.

Proof to the above corollary is easy. In fact, the equivalence of (a) and (b) is the

result of Cao and Yang (2011) that we have used several times. The equivalence of (a)

and (c) is exactly the content of Proposition 4. That (c) implies (d) is straightforward.

When (d) holds, we must have PNE exists for G, i.e. (a) is valid, hence (c) is also true.

This completes the whole proof. When the network is a ring, unfortunately, we do not

have so nice a characterization.

4.3.2 Stars

Let I be a given initialized fashion game. In each time step, as long as PNE is

not reached, we uniformly choose an arbitrary unsatisfied agent, and let her switch

her action. We say that sequential BRD converges almost surely if and only if the

probability that PNE can be eventually reached is one. Obviously, this convergence

concept is weaker than that we used for lines and rings.

Proposition 6 Let G = (N,E, T ) be a fashion game. If the underlying network g =

(N,E) is a star, then sequential BRD converges almost surely for all initial action

profiles if and only if at least half of the peripheral agents are of the same type as the

central agent.

23

Proof to the above proposition, which is a direct application of absorbing Markov

chains, is provided in Appendix A.4. Given a star network g = (N,E), let each agent

uniformly choose a type, then the probability that at least half of the peripheral agents

are of the same type as the central agent tends to 1/2 as n goes to infinity. Combining

this fact with Proposition 6, we have the following corollary.


(N,E) being a star. Then

limn→∞Prob. [G : sequential BRD converges almost surely for all initial action profiles] = 1/2.

Similar to Corollary 9, we have the following nice characterization for sequential

BRD on star networks. The proof is trivial and thus omitted.


(N,E) being a star, then the following statements are equivalent.

(a) PNE exists for G;

(b) At least half of the peripheral agents are of the same type as the central agent;

(c) Sequential BRD converges almost surely for all initial action profiles;

(d) Sequential BRD converges almost surely for some initial action profile.

4.4 Simultaneous BRD V.S. sequential BRD

Below are the formal statements that convergence of simultaneous BRD is stronger

than that of sequential BRD when the underlying network is a line or a star.


(N,E) being a line. If simultaneous BRD converges for some initial action profile, then

sequential BRD converges for all initial action profiles.


(N,E) being a star. If simultaneous BRD converges for some initial action profile,

then sequential BRD converges almost surely for all initial action profiles.

Proofs to the above two corollaries are as follows. According to Corollary 9 and

Corollary 11, convergence of sequential BRD for lines is equivalent to the existence of

PNE, and almost sure convergence of sequential BRD for stars is also equivalent to the

existence of a PNE. However, convergence of simultaneous BRD implies the existence

of a PNE. Hence the two corollaries are valid.

24

5 Discussions

In this paper, we have investigated fashion through a heterogeneous network game

model. It is shown that the emergence of a fashion cycle relies crucially on the network

structures of social interactions.

5.1 Two weaknesses

In order to prove our results, we have defined the fashion cycle as the limit cycle in the

corresponding discrete dynamical system. This definition seems to be too restricted.

Empirically, fashion cycle describes the phenomenon that the same type of item ap-

pears as the most fashionable one again and again after regular periods of time. Only

the macro fact that “it becomes hot again” is needed, the micro requirement that

“everything comes back again” is excessive. Thus it might be more appropriate to

only look at the time series of the hottest action and define fashion cycle accordingly.

It should be noted that this new definition does not change our result that fashion

will evolve into a dead state almost surely for the line and ring structures, because

micro equilibrium always implies a macro steady state. Nevertheless, if we use this

new definition of a fashion cycle to study the impact of heterogeneity, as discussed in

Subsection 3, then Simmel’s viewpoint may well be favored.

The other weakness of the current paper is that we did not take into account errors

or mutations. That is, when people respond in the fashion dynamic, we assume that

they never make mistakes. This is obviously unrealistic, and more importantly, makes

us unable to refine the equilibria as done in standard stochastic stability analysis. Since

most of our main results (Proposition 1(b), Proposition 2(b), Proposition 4, Proposition

3 (c)(d)) do not depend on initial action profiles, once an agent makes an error during

the BRD process, at worst case we can take it as a restart of the evolution. When the

erring probability is tiny, the new evolution can reach a new limit cycle, before another

error is made, because it can be seen from our proof that the convergence is really fast.

More luckily, for the line and ring networks, the influence of an error can be blocked

by nearby clusters, and even clusters are all far away from the erring agent, it needs a

process to diffuse the impact. When errors occur after the convergence, obviously their

effect is even smaller and can be restored even faster. Hence mistakes can largely be

taken as local and are likely to have very small effect on our main convergence results.

Further studies are needed as to how errors may affect the selection of PNEs,.

25

5.2 Open problems

An obvious technical problem for further research is to give more concrete character-

izations of part (d) of Proposition 1, i.e. to distinguish between the instances with

length 1 fashion cycles and those with length 4 ones when the lines are un-clustered

but not cross-heterogeneous.

The second technical problem is to give more complete description for the simul-

taneous BRD of the fashion game on rings. The main difficulty for rings is that there

are no ending nodes, and thus we cannot expect to have a result parallel to the critial

Lemma 8 (in Appendix B). A similar “cutting” technique (see Appendix A) can still

be applied, so the only case remains unsolved is the cross-heterogeneous one. This is

also the complicated case for lines (see Appendix A). When the number of agents n

is odd, the ring cannot be cross-heterogeneous. However, when n is even, simulations

show that simultaneous BRD on a cross-heterogeneous ring behaves very differently

with that on a cross-heterogeneous line. To be specific, (1) when n/2 is also even, all

the possible lengths of fashion cycles of simultaneous BRD are 1, 2, 4 and n, (2) when

n/2 is odd, all the possible lengths of fashion cycles of simultaneous BRD are 1, 2,

4 and 2n. Figure 4 illustrates two fashion cycles with lengths n and 2n, respectively.

This shows a great difference between simultaneous BRD on lines and simultaneous

BRD on rings. Although the latter structure is but slightly modified from the former,

their properties in the extreme case may vary a lot. To be exact, lengths of limit cycles

of simultaneous BRD on rings cannot be upper-bounded as on lines. In worst cases,

they can be as long as possible, as the number of agents n goes to infinity. In that case,

it is farfetched to say that fashion cycle still exists, or even that there is any regularity

in the evolution of fashion. This may also add to the mystery of fashion.

The third technical problem is to prove or disprove that when the underlying net-

work is a star such that at least half of the peripheral agents are of the same type as

the central agent (recall from Lemma 1 that this is the sufficient and necessary condi-

tion for the existence of a PNE), then sequential BRD converges for all initial action

profiles, or equivalently, the corresponding fashion games are potential games. Note

that in this paper we have used a weaker concept, the almost sure convergence.

5.3 Other perspectives

Social learning. Our research is also closely related with the social learning theory,

where BRD and the ring structure are frequently used. Mathematically, many of the

26

Figure 4: Fashion cycles on cross-heterogeneous rings. Circles stand for con-formists, and triangles for rebels. Black color indicates that the agent chooses 1, whitefor 0. In the left example, the ring has 8 nodes, and the fashion cycle is of length 8. Inthe right example, the ring has 6 nodes, and the fashion cycle is of length 12. Unhappyfaces indicate that the corresponding nearby agents are unsatisfied.

social learning models are similar to the conforming side of the fashion game (or its

natural extensions). Accordingly, our paper contributes to the already well-studied

field.41 In addition, our model can be taken as an extension of the famous threshold

model of Granovetter (1978).42 Under the payoff settings of this paper, a conformist

prefers an action to the other if and only if it has been taken by one half or more of

her neighbors. In contrast, a rebel prefers an action if and only if at most one half of

her neighbors has chosen it. Hence the “thresholds” are 0.5 for all agents. The exact

meaning of threshold for conformists is the same as in Granovetter (1978). For rebels,

however, it is actually an “anti-threshold”.

Cellular automaton. Mathematically, simultaneous BRD of the fashion game is

a nonlinear discrete dynamical system and a boolean network, which in general are

notoriously hard to analyze. For the ring structure, the corresponding (simultaneous)

BRD is equivalent to a one-dimensional heterogeneous cellar automaton. More pre-

cisely, the evolution is governed by a combination of Rules 77 and Rule 232.43 Although

it is known to be very difficult in general to establish anything meaningful about the

41See e.g. Ellison (1993), Kandori et. al. (1993), Anderlini and Ianni (1996), Morris (2000), Young(2009), Chen et. al. (2012) for references.

42In the fields of computer science and complex systems, the network extension of the original modelof Granovetter (1978) is known as the threshold automata network, see e.g. Goles (1987), Berninghausand Schwalbe (1996).

43These rules are indexed by Wolfram (2002). The two specific ones canalso be found at the websites http://www.wolframalpha.com/input/?i=rule+77 andhttp://www.wolframalpha.com/input/?i=rule+232.

27

evolution of a cellar automaton (e.g, Ilachinski, 2001, p. 225), we have shown that the

corresponding cellar automatons in this paper behave nicely and the analysis is not

that formidably hard.

Strategic cooperation. It is widely accepted that competition and cooperation

are the two eternal topics in game theory. Zero-sum games are polar examples for com-

petition, whereas common interest games are polar examples for cooperation. Among

the three base games of the fashion game, the pure coordination game and the pure

anti-coordination game are both common interest games, while matching pennies is

a zero-sum game. For general normal form games, competition and cooperation are

both embodied. This is a kind of vertical hybrid and is the key observation of Kalai

and Kalai (2012) in their definition of the coco value, a new and promising solution for

two player strategic games. The fashion game, on the other hand, is a kind of horizon-

tal hybrid of competition and cooperation. Through simulations, Cao et. al. (2013)

find that in general people can reach an extraordinarily high level of cooperation in

small-world networks via the simple BRD. Rigorous analysis is still needed.

Matching pennies. We end this paper by remarking that our work shows a novel

value for the elementary game of matching pennies. Most other elementary two player

games, say prisoners’ dilemma, hawk and dove, and game of pig, etc., have found use-

ful interpretations and far-ranging applications, and attracted numerous attention. In

contrast, much less attention has been paid to the matching pennies game. One of the

most important reasons is that for a long time in history, there existed no convinc-

ing and fundamental interpretations for this model.44 The insightful observations of

Young (2001) and Jackson (2008) that it closely relates to the phenomenon of fashion,

hopefully can open a door for reconsidering this model. It is by no means a toy model!

For more recent work on matching pennies from the perspective of behavioral science,

we refer the reader to Palacios-Huerta (2003), and Eliaza and Rubinstein (2011).

44Another reason might be that matching pennies is not symmetric. Only symmetric two-playergames can be straightforwardly extended to multi-player and network scenarios. In evolutionary gametheory, this corresponds to single population games. To extend a two-player asymmetric game to themulti-player scenario, there must be two populations. However, the inner-population game is usuallyhard to define naturally. With the interpretation of coordination and anti-coordination, the abovedifficulty is not a barrier for the matching pennies game.

28

A Main Proofs

It is obvious that when the underlying network g = (N,E) is a line or a ring, ui(x(t))

can only possibly take five values, 2, 0, −2, 1 and −1. Precisely, when i ∈ N \ {1, n}is a conformist,

ui(x(t)) =

8>><>>: 2 if xi−1(t) = xi+1(t) = xi(t)

0 if xi−1(t) + xi+1(t) = 1

−2 if xi−1(t) = xi+1(t) = 1− xi(t)

,

and when i ∈ N \ {1, n} is a rebel,

ui(x(t)) =

8>><>>: 2 if xi−1(t) = xi+1(t) = 1− xi(t)

0 if xi−1(t) + xi+1(t) = 1

−2 if xi−1(t) = xi+1(t) = xi(t)

.

The utility of agent 1 and agent n on a line can only take a value from {1,−1}.For agents 1 and n on a ring, their utilities can also be characterized similarly. Only

notice in this case that agent 1 and agent n are neighbors of each other.

A.1 Proofs to Proposition 1 and Proposition 2

The biggest complication in this subsection comes mainly from part (c) and part (d)

of Proposition 1. Part (a) and part (b) of Proposition 1, as well as Proposition 2, are

relatively easy. So let’s prove the easier parts first. Before that, we need a new term

to ease our presentation.

Definition 5 Given an initialized fashion game I = (N,E, T, x(0)), if xi(t) keeps

unchanged when simultaneous BRD enters the fashion cycle, we say that agent i is

immune w.r.t. I.

The lemma below characterizes the convergence case.

Lemma 2 Given an initialized fashion game I = (N,E, T, x(0)), and suppose the un-

derlying structure g = (N,E) is either a line or a ring. The following three statements

are equivalent.

(a) l(I) = 1;

(b) All agents are immune w.r.t. I;

(c) There exists one agent who is immune w.r.t. I.

29

Proof. The equivalence of (a) and (b) is obvious. It is trivial that (b) implies (c), so

we are left to show that (c) implies (b). Suppose that i is immune. We claim that

j ∈ Ni, an arbitrary neighbor of i, is also immune.

Suppose not, then there must exist some time t0 ≥ r(I) such that uj(x(t0)) < 0.

So the action of i is not liked by agent j at time t0, and j will switch at step t0,

i.e. xj(t0 + 1) = 1 − xj(t0). Since i is immune, we know that she never switches, i.e.

xi(t0 + 1) = xi(t0). Therefore, the action of i is liked by agent j at time t0 + 1, and

hence uj(x(t0 + 1)) ≥ 0. Furthermore, the utility of j keeps nonnegative after time

t0 + 1, because i is immune, she never switches in the limit cycle, and her action is

always liked by j after t0. Therefore agent j will never switch any more. Thus the

action xj(t0) will never be taken again by j after t0, contradicting the fact that fashion

cycle is entered at time t0.

Notice that the above lemma relies crucially on the fact that in lines and rings,

each agent has at most two neighbors. As long as one of her two neighbors is liked by

her, she will be satisfied and will not deviate. This is not true other structures and

thus results in the above lemma cannot be extended. So are other lemmas.

The following lemma characterizes length two limit cycles.

Lemma 3 Given an initialized fashion game I = (N,E, T, x(0)), and suppose the

underlying structure g = (N,E) is either a line or a ring. The following statements

are equivalent.

(a) l(I) = 2;

(b) ∃i ∈ N , ui(x(t)) < 0 holds for all t ≥ r(I);

(c) ∃t ≥ 0, ui(x(t)) < 0 holds for all i ∈ N ;

(d) I is badly initialized.

Proof. (a) implies (b) obviously. Suppose (b) is true, then agent i keeps switching her

action in the fashion cycle. This can only happen when her neighbors keep switching

actions. Using this argument again and again, we know that no agent is satisfied in

any step of the fashion cycle, thus (c) is valid. Suppose now (c) is true, i.e. in some

time t no agent is satisfied. This can happen only when all agents are of the same type,

because for any two adjacent agents of different types, it is impossible for both of them

to be unsatisfied. Also, no agent should be satisfied initially. Suppose not, then there

exist i ∈ N s.t. ui(0) ≥ 0, then she likes at least one of her neighbors’ action initially,

and that neighbor likes her action too, because they are of the same type. Therefore,

30

the two agents are both immune, contradicting (c). (d) indicates (a) obviously, and

hence the lemma.

Notice that the equivalence of (a) and (d) in Lemma 3 has already verified part (a)

of Proposition 1 and part (a) of Proposition 2.

Intuitively, agents in a cluster are very likely to be immune, because they can help

each other. Therefore clustered fashion games are supposed to be able to reach PNEs

through BRD, unless the fashion game is badly initialized. Below is the rigorous proof

to part (b) of Proposition 1.

Proof. The condition that I is clustered says that at least one of the following must

hold, (i) T1 = T2; (ii) Tn−1 = Tn; (iii) there are three adjacent agents that are of the

same type. Due to Lemma 3, it suffices to show that l(I) ∈ {1, 2}.Suppose that T1 = T2 and they are both conformists. If at some step they take the

same action, then they will both be immune. By Lemma 2, this means that l(I) = 1

and we are done. So we suppose now they always take different actions. Since agent 1

has only one neighbor, this means that she will keep switching actions. By Lemma 3

we know that l(I) = 2. If they are both rebels, it can be shown by a similar argument.

By symmetry, it is also valid for the case that Tn−1 = Tn.

Suppose now there are three agents i, i + 1, i + 2, such that Ti = Ti+1 = Ti+2 = C.

If there exists some time t such that either xi(t) = xi+1(t) or xi+1(t) = xi+2(t), then

there will be immune agents and l(I) = 1. So we assume that xi(t) = xi+2(t) 6= xi+1(t)

holds for all time t. Therefore the middle agent i+1 will switch her action at all steps.

Therefore by Lemma 3 we know that l(I) = 2.

Suppose now there are three agents i, i + 1, i + 2, such that Ti = Ti+1 = Ti+2 = R.

If there exists some time t such that either xi(t) 6= xi+1(t) or xi+1(t) 6= xi+2(t), then

there will be immune agents and l(I) = 1. So we assume that xi(t) = xi+2(t) = xi+1(t)

holds for all time t. Therefore the middle agent i+1 will switch her action at all steps.

Therefore by Lemma 3 we know that l(I) = 2.

It can be observed from the above proof that part (b) of Proposition 2 is also true.

We are left, in this subsection, to prove part (c) and (d) of Proposition 1. We need

first to prove a property of the limit cycles for the case l(I) ≥ 3.


underlying structure g = (N,E) is either a line or a ring. If l(I) ≥ 3, then no

agent will switch in two consecutive steps, i.e. there is no time t such that ui(x(t)) =

ui(x(t + 1)) < 0.

31

Proof. We consider first the case that g = (N,E) is a line. Suppose on the contrary

that ui(x(t)) = ui(x(t + 1)) < 0. It can be shown easily that in the case i ∈ {1, n}, i

must be of the same type as her neighbor, and in the case that i /∈ {1, n}, i and her

two neighbors must have the same type. Hence I is clustered. Due to Proposition

1(a), we know that l(I) ≤ 2. A contradiction with the hypothesis that l(I) ≥ 3. The

case that (N,E) is a ring can be proved in exactly the same approach.

The key method we use to deal with part (c) and (d) of Proposition 1 is the “cutting”

technique explored by us. To ease the presentation, we need a further new term.

Given an initialized fashion game I = (N,E, T, x(0)), and suppose that g = (N,E)

is a line. For any agent i, 2 ≤ i ≤ n−1, let I ′(i) = (N ′, E ′, T ′, x′(0)) be a new (smaller)

initialized fashion game constructed by the left i agents of I in the following natural

way:

• N ′ = {1, 2, · · · , i};

• E ′ = {ij : i, j ∈ N ′, ij ∈ E};

• ∀1 ≤ j ≤ i, T ′j = Tj;

• ∀1 ≤ j ≤ i, x′j(0) = xj(0).

Definition 6 We say that I ′(i) is a complete sub-instance of I iff x′j(t) = xj(t) holds

for all t ≥ 1 and 1 ≤ j ≤ i.


underlying structure is a cross-heterogeneous line. Suppose also that l(I) ≥ 3 and

agents i and i + 1 are of the same type (2 ≤ i ≤ n − 2). Then I ′(i) is a complete

sub-instance of I.

Proof. Suppose w.l.o.g. that i and i + 1 are both conformists. In the simultaneous

BRD of I, i and i + 1 will always take the opposite actions, because otherwise they

will both be immune and the hypothesis that l(I) ≥ 3 will be violated. Therefore,

whether i will switch her action or not in step t is completely determined by the action

of agent i− 1. This is also true for the simultaneous BRD of I ′(i), where i− 1 is the

only neighbor of i. Hence the lemma.

With the assistance of two technical lemmas, Lemma 7 and Lemma 8, that are

presented in Appendix B, we are ready to prove the remaining parts of Proposition 1.

32

Suppose now I = (N,E, T, x(0)) is an initialized fashion game, and the underlying

structure is a cross-heterogeneous line. First recall that PNE does not exist in this

case. Since none of the agents has a utility of zero in the fashion cycle (Lemma 8),

and in any utility sequence 2 will never be adjacent with 2, neither will -2 be adjacent

with -2 (Lemma 7(a)), the whole sequence of utilities will consist of alternate 2 and -2

(except for the ending agents, whose utilities are 1 or -1). At any step of the fashion

cycle, either none conformist is satisfied but all the rebels are, or none rebel is satisfied

but every conformist is. Based on the above discussion, it is easy to check that the

length of any fashion cycle is exactly 4. Hence part (c) of Proposition 1 is valid.

Suppose now I = (N,E, T, x(0)) is an initialized fashion game, and the underlying

structure is an un-clustered line but not cross-heterogeneous. Since g = (N,E) is

un-clustered, by Lemma 3 we know that l(I) 6= 2 (note that un-clustered implies not

homogeneous and further not badly-configured). To prove part (d) of Proposition 1, it

suffices to show that if l(I) ≥ 3, then it is only possible that l(I) = 4.

Suppose now l(I) ≥ 3. We cut the line at an arbitrary conformist-conformist edge

or rebel-rebel edge into two pieces. Notice that un-clustered guarantees that the above

cutting will not produce isolated agents. Due to Lemma 5, they are both complete

sub-instances. If the two pieces are both cross-homogeneous, then each of them has

a fashion cycle of length 4, regardless of initial configuration of actions (part (c) of

Proposition 1). By the definition of complete sub-instance, piecing them together

gives part (d) of Proposition 1 immediately. Otherwise, we can cut them again into

even smaller pieces, and use the same argument. Since the line is finite, this process

will terminate and the proof is done.

A.2 Proof to Proposition 3

Before the main proof, we need a lemma that has a similar essence to Lemma 2 and

Lemma 3 for the line and ring structures.

Lemma 6 Let I = (N,E, T, x(0)) be an initialized fashion game and suppose the

underlying network g = (N,E) is a star. The following statements are true.

(a) At any step of the limit cycle, all the peripheral conformists choose the same

action and so do all the peripheral rebels;

(b) l(I) = 1 if and only if there is some time when all agents are satisfied, if and

only if the central agent is always satisfied in the limit cycle.

33

(c) l(I) = 2 if and only if the central agent is always unsatisfied in the limit cycle,

if and only if there is some time when no agent is satsified.

Proof. (a) We only prove for conformists, because the rebel half is analogous. Suppose

the contrary, i.e. at some step of the limit cycle, part of the peripheral conformists

choose action 1 and the others choose action 0. Since each peripheral agent has the

same single neighbor, namely the central agent, it must be true that either all the

action 1 peripheral conformists are satisfied but the action 0 ones are unsatisfied, or

the opposite. Suppose w.l.o.g. the former case occurs. Then the action 1 peripheral

conformists switch to action 0 while the action 0 ones keep their actions fixed at 0.

Hence at the next step all the peripheral conformists have an action of 0. From that

time on, either they switch simultaneously or keep unchanged simultaneously. So in

any of the following step they always share the same action, and hence the state we

assume that part of the peripheral conformists choose action 1 and the others choose

action 0 will never be reached again, contradicting our assumption that it is in the

limit cycle.

(b) The first part is straightforward, we only show the second part. When l(I) = 1,

every agent is always satisfied in the limit cycle, and thus this is true for the central

agent. On the other hand, when the central agent is always satisfied in the limit cycle,

she never switches, and it must be true that all the peripheral ones never switch either,

because if any of them ever switches, she keeps being satisfied after switching (note

again that she has only one neighbor, the central one, who never switches in the limit

cycle).

(c) When l(I) = 2, any agent, either the central one or a peripheral one, must

be either always satisfied and never switches in the limit cycle or always unsatisfied

and hence switches at any step of the limit cycle. By (b) we know that the central

agent must be always unsatisfied. Suppose now the central agent is always unsatisfied

in the limit cycle, we prove that no agent is ever satisfied in the limit cycle. This

is true because otherwise some peripheral agent will always be satisfied in the limit

cycle, which is obviously impossible since her only neighbor, the central agent, keeps

switching in the limit cycle. Suppose now there is some time such that no agent is

satisfied, then all agents will keep switching from that time on, and thus l(I) = 2.

Piecing the above three implications together, we know that (c) is correct.

Now we are ready to prove Proposition 3.

Proof. (a) Let’s first consider the case that the central agent is a rebel. We claim

that limit cycle is entered at step 2, where all the rebels choose the same action, the

34

Figure 5: Illustrations of length-4 limit cycles for the star network. Circlesstand for conformists, and triangles for rebels. Black color indicates that the agentschooses 1, white for 0.

other action is taken by the conformists, and hence no agent is satisfied and each one

keeps switching from that time on. Suppose w.l.o.g. that the central rebel chooses

action 1 initially. Since she is initially unsatisfied, at step 2 she will switch to action

0. So do the peripheral rebels who initially take action 1. For the peripheral rebels

who initially take action 0, they are initially satisfied and do not switch at the first

step. Hence at the second step, all the peripheral rebels have an action of 0. For the

peripheral conformists who take action 1 initially, they are satisfied at the first step,

and hence still have an action 1 at the second step. For the peripheral conformists who

take action 0 initially, they are unsatisfied at the first step, and hence switch to action

1 at the second step. This proves our claim.

Proof to the other case that the central agent is a conformist is quite similar. The

only difference is that in the limit cycle, all the peripheral conformists choose the same

action, which is opposite to that of the central conformist, and all the peripheral rebels

choose the same action as the central conformist.

(b) Consider the case that the central agent is a rebel. First of all, it must be

true that l(I) ≥ 2, because it is impossible for the central rebel and all the peripheral

conformists to be simultaneously satisfied. Recall that in the limit cycle, all the con-

formists choose the same action, and so do all the peripheral rebels (Lemma 6 (a)).

So it is impossible that l(I) = 2, because at any step of the limit cycle, either all the

conformists are all satisfied (when they take the same action as the central agent) or

the central rebel is satisfied (when the peripheral conformists take a different action

with the central rebel). And there must exist some time t in the limit cycle such that

the central rebel is satisfied (say she chooses action 0). Since there are more periph-

eral conformists than peripheral rebels, we know that all peripheral conformists should

35

choose action 1 at time t (hence they are satisfied), because otherwise the central rebel

would be unsatisfied. We discuss in two cases about the peripheral rebels.

• All the peripheral rebels are unsatisfied at time t.

That is, they all choose action 0 too. Then at time t+1, all the peripheral agents

switch actions, making themselves happy but the central rebel unhappy. So at

time t + 2, the central rebel switches to action 1, making herself happy and all

the peripheral agents unhappy. At time t + 3, all the conformists hold action 1,

all the peripheral rebels choose action 0, and the central rebel has action 1, thus

the only unsatisfied agent is the central rebel. By letting the central rebel switch

to action 0, we can see that the state at time t + 4 is exactly the same as at time

t. Therefore in this case we have l(I) = 4. This limit cycle is illustrated in the

left part of Figure 5.

• All the peripheral rebels are satisfied at time t.

That is, they all choose action 0. It can be checked that the state at time t+1 in

this case is exactly that at time t + 1 in the previous case. However, the time t

state in this case is none of the four states in the limit cycle of the previous case,

so it will never be reached again, contradicting our assumption that limit cycle

has been entered at time t. Therefore this case never occurs at all.

Consider the other case that the central agent is a conformist. For the same reason

as in the above proof, there must be some time t, after the entrance of limit cycle, such

that the central conformist is satisfied. Suppose w.l.o.g. that the central conformist

chooses action 0 at time t. First of all, all the peripheral rebels must choose action

0 at time t, and they are unsatisfied. We discuss in two cases about the peripheral

conformists.

• All the peripheral conformists are unsatisfied at time t.

That is, they all choose action 1. Then at time t + 1, all the peripheral agents

switch actions, making themselves happy but the central conformist unhappy. So

at time t + 2, the central conformist switches to action 1, making herself happy

and all the peripheral agents unhappy. At time t + 3, all the conformists hold

action 1, all the peripheral rebels choose action 0, and the central rebel has action

1, thus the only unsatisfied agent is the central conformist. By letting the central

conformist switch to action 0, we can see that the state at time t + 4 is exactly

36

the same as at time t. Therefore in this case we have l(I) = 4. This limit cycle

is illustrated in the right part of Figure 5.

• All the peripheral conformists are satisfied at time t.

That is, they all choose action 0. Using exactly the same logic as in the proof to

(c), we know that this case never occurs.

(c) Naturally, we discuss in four cases.

• The central agent is a rebel, more than a half of the peripheral agents are also

rebels, and initially the central rebel is satisfied.

Since the central rebel is initially satisfied, her action at the second step will be

the same as in the first step. For the peripheral rebels, the ones initially taking

the opposite action stay with this action, and the other peripheral rebels will

switch to this action too at the second step. So at the second step each of the

peripheral rebels takes an opposite action with the central rebel. Since more

than a half of the central rebel’s neighbors are rebels, we know that all the rebels

will be always satisfied from the second step on. So are the conformists. Hence

l(I) = 1.

• The central agent is a conformist, more than a half of the peripheral agents are

also conformists, and initially the central conformist is satisfied.

This case can be verified by using an analogous logic as in the previous case.

• The central agent is a rebel, and half of the peripheral agents are rebels and the

other half are conformists.

Suppose on the contrary that l(I) ≥ 2, then there is some time t in the limit

cycle that the central rebel is unsatisfied (recall Lemma ?? (b)). This must be

the case that all the agents take the same action (say action 1), because when

the peripheral conformists and the peripheral rebels take different actions, the

central rebel is satisfied. So at step t + 1, all the rebels will switch to action 0,

and the conformists still hold action 1. Therefore the central rebel is the only

satisfied agent at step t + 1. At time t + 2, the central rebel still has action 0,

the peripheral rebels have action 1, and the conformists have action 0. It can be

seen that all the agents are satisfied at step t + 2, contradicting the assumption

that l(I) ≥ 2.

37

• The central agent is a conformist, and half of the peripheral agents are rebels

and the other half are conformists.

This case can be verified by using an analogous logic as in the previous case.

This completes the proof to the whole proposition.

A.3 Proofs to Proposition 4 and Proposition 5

Proof. Recall that when the underlying network is a line, then G possesses a PNE

if and only if it is not cross-heterogeneous. So the “only if” part of Proposition 4

is straightforward. Next, we prove Proposition 5 and the “if” part of Proposition 4

simultaneously. Our proof is not based on the construction of a potential function, as

is usually done, but in a more direct way. First of all, non-cross-heterogeneous implies

that there are at least two agents, say i and i + 1, who are adjacent and are of the

same type. Then i and i + 1 can serve a similar “anchor” role as a cluster does in the

analysis of Proposition 1(b) for simultaneous BRD.

We discuss only the case that i and i+1 are both conformists. The other case that

i and i + 1 are both rebels can be proved using the same logic. We claim that there

must be some time t0, such that i and i + 1 are both satisfied from that time on, i.e.

ui(x(t)) ≥ 0, ui+1(x(t)) ≥ 0,∀t ≥ t0.

W.l.o.g., we assume that xi(0) 6= xi+1(0), because otherwise both of them would be

initially satisfied and support each other forever, that is, t0 = 0. Using the same logic,

it can be seen that as long as one of them deviates at some time, then the actions of i

and i + 1 will always be the same from that time on. So the only remaining possibility

is that none of them deviates, which implies that they are always satisfied, verifying

our claim.45

After time t0, the respective neighbors of i and i + 1, i.e. i − 1 and i + 2, will

deviate for at most one time too. So there exists some time t1, after which no agent in

45The rough logic is that according to the sequential moving rule, either deterministic or random,if one of them is ever unsatisfied, she will have a chance to deviate. If we think really carefully, thenone subtle worry may come out. That is, for certain deterministic moving orders, i may be unsatisfiedat some time, and at that time it is not her turn to move. However, when it is time for her turn, shebecomes satisfied, by the previous deviation of her neighbor i − 1. The worry is that this may cycleagain and again. A second thought tells us that this will never happen, because the hypothesis thati never deviates implies that i− 1 can only deviate for one time.

38

{i− 1, i, i + 1, i + 2} will deviate. Using this argument again and again, we know that

eventually all agents will be satisfied, and hence the sequential BRD converges.

A.4 Proof to Proposition 6

Proof. Taking all the action profiles as states, and letting the transition probabilities

be determined by the underlying dynamic (let a uniformly chosen unsatisfied agent

switch), we have a well-defined Markov chain. According to Lemma 1, the condition

that at least half of the peripheral agents are of the same type as the central agent

implies that PNE exists. In fact, it can be seen that there are exact two PNEs. What’s

more, the two PNEs correspond to the only absorbing states of the Markov chain.

What’s more, this Markov chain is absorbing: every state can reach one of the two

absorbing states. This can be shown easily from the following observation: if we let

the unsatisfied peripheral agents that are of the same type deviate first (one by one in

an arbitrary order), and the other unsatisfied peripheral agents deviate subsequently

(one by one in an arbitrary order), then we will arrive at one of the two PNEs. Since in

absorbing Markov chains, all non-absorbing states are transient, we know immediately

that PNEs will be reached with probability one, and this completes the proof.

B Two Technical Lemmas

The two lemmas provided in this appendix are used in the proofs to Proposition 1(c)(d),

the most difficult parts of that proposition. Since their proofs, mainly the latter one,

are rather tedious, we present them in this separate section.

In this appendix, we use ui(t) for short of ui(x(t)). This is reasonable, because

as long as the initial actions are set, utility of each agent is a function of the single

variable of time t.

We call [a1, a2, · · · , ak] ∈ {−2, 2, 0,−1, 1}k a utility sequence, if there exists a time t

and an agent i such that [ui(t), ui+1(t), · · · , ui+k−1(t)] = [a1, a2, · · · , ak]. To study the

cross-heterogeneous case, it is more convenient to deal with utility sequences instead

of action sequences.

Lemma 7 Given an initialized fashion game I = (N,E, T, x(0)), and suppose g =

(N,E) is a cross-heterogeneous line.

(a) The following utility sequences will never occur: [2, 2], [−2,−2], [2, 0, 2], [2, 0, · · · , 0, 2],

[2, 0, 1], [2, 0, · · · , 0, 1], [−2, 0,−2], [−2, 0, · · · , 0,−2], [−2, 0,−1], [−2, 0, · · · , 0,−1],

39

[1, 0, · · · , 0, 1], [−1, 0, · · · , 0,−1];

(b) If ui(t) = −2, then ui(t + 1) = 2;

(c) If ui(t) = 0, then ui(t + 1) ∈ {−2, 0}.

Proof. One beautiful property of the case under focus is that if one likes the action

of her neighbor, then her neighbor must hate the action of her, and vice versa. Using

this property, it is easy to check that none of the sequences in (a) will occur.

Since no pair of adjacent agents can be simultaneous unsatisfied, we know that if

agent i switches her action at some step, then none of her neighbor switches action at

the same step. Using this fact, (b) is obvious.

If ui(t) = 0, then i likes the action of exactly one of her neighbors {i − 1, i + 1},and hates the action of the other, at time t. Suppose w.l.o.g. that i− 1 ∈ Li(x(t)) and

i+1 ∈ Di(x(t)) (recall the notations defined in Section 2). Accordingly, i ∈ Hi−1(x(t))

and i ∈ Li+1(x(t)). Therefore, ui−1(t) ≤ 0 and ui+1(t) ≥ 0. Since i + 1 is satisfied at

time t, she will not switch. In the case that ui−1(t) < 0, it is evident that ui(t+1) = −2.

And in the case that ui−1(t) = 0, we have ui(t + 1) = 0. Hence the lemma.

Given an initialized fashion game I = (N,E, T, x(0)), let Z(t) be the set of agents

whose utilities are zero at time t, i.e.

Z(t) = {i ∈ N : ui(t) = 0}.

The following lemma is critical.

Lemma 8 Given an initialized fashion game I = (N,E, T, x(0)), suppose the underly-

ing structure g = (N,E) is a cross-heterogeneous line. Then Z(t) = ∅ for all t ≥ r(I).

Proof. Above all, the result of Cao and Yang (2011), as introduced in Subsection 4,

tells us that PNE does not exist for I, therefore L(I) 6= 1. Due to Lemma 3, we have

obviously that L(I) 6= 2. Hence

L(I) ≥ 3, (1)

and the results in Lemma 4, Lemma 5, Lemma 7, are all valid.

We prove in two steps.

(i) We prove first that the cardinality of Z(t) is non-increasing in simultaneous

BRD, i.e.

|Z(t + 1)| ≤ |Z(t)|. (2)

40

Let

Z+(t) = Z(t + 1) \ Z(t),

and

Z−(t) = Z(t) \ Z(t + 1),

we only need to show that

|Z+(t)| ≤ |Z−(t)|.

If Z+(t) is empty, we are done. Suppose not, take an arbitrary agent i from Z+(t),

i.e. ui(t + 1) = 0 and ui(t) 6= 0. Then by Lemma 7(b), it can only be that ui(t) = 2.

And it is obvious that i /∈ {1, n}. So agent i does not switch her action at step t, and

her utility changes from 2 at step t to 0 at step t + 1. This can only happen when

exactly one neighbors of i is unsatisfied at time t. Due to Lemma 7(a), utility sequence

[2, 2] will never occur, so the neighbor of agent i who is satisfied at time t must have a

utility of 0. Therefore, {ui−1(t), ui+1(t)} = {−2, 0} (we assume w.l.o.g. that i− 1 6= 1

and i + 1 6= n).

We consider first the case that [ui−1(t), ui(t), ui+1(t)] = [−2, 2, 0]. We claim that

Z−(t) ∩ {i + 1, i + 2, · · · , n} 6= ∅. (3)

Suppose the opposite. Then i + 1 /∈ Z−(t). Since i + 1 ∈ Z(t), this implies that

i + 1 ∈ Z(t + 1), i.e. ui+1(t + 1) = 0. As agent i + 1 does not switch her action at step

t, her left neighbor, agent i, does not switch either, and the utility of i + 1 also keeps

unchanged, we know that the right neighbor of agent i + 1, i + 2, should not switch

at step t either. Therefore ui+2(t) ≥ 0. By Lemma 7(a), utility sequence [2, 0, 2] and

[2, 0, 1] will never occur, so it is impossible that ui+2(t) ∈ {2, 1}. It can only be that

ui+2(t) = 0.

Also, Z−(t)∩{i+1, i+2, · · · , n} = ∅ implies that i+2 /∈ Z−(t). ui+2(t) = 0 tells us

that i + 2 ∈ Z(t). Using exactly the same argument as in the last paragraph, we know

that ui+3(t) = 0. But this cannot continue for ever, because for any time t, it always

holds that un(t) ∈ {1,−1}. Hence a contradiction. Therefore in this case the claim

(3) is valid. For the case that [ui−1(t), ui(t), ui+1(t)] = [0, 2,−2], (3) can be similarly

shown to be true.

Based on the above discussion, the following definition for each i ∈ Z+(t) is valid:

f(i) =

8<: min¦j : j ∈ Z−(t) ∩ {i + 1, i + 2, · · · , n}

©if [ui−1(t), ui(t), ui+1(t)] = [−2, 2, 0]

max¦j : j ∈ Z−(t) ∩ {1, 2, · · · , i− 1}

©if [ui−1(t), ui(t), ui+1(t)] = [0, 2,−2]

.

41

The facts below can also be observed from the above discussion. ∀i ∈ Z−(t),

f(i) > i implies [ui−1(t), · · · , uf(i)+1(t)] = [−2, 2, 0, · · · , 0,−2], (4)

f(i) < i implies [uf(i)−1(t), · · · , ui+1(t)] = [−2, 0, · · · , 0, 2,−2], (5)

where the “· · · ”s on the right hand sides denote both a series of 0s.

If we can show that f(i1) 6= f(i2) holds for all i1, i2 ∈ Z+(t), i1 6= i2, then the

main result we want to prove in this step that |Z+(t)| ≤ |Z−(t)| will be valid. Assume

w.l.o.g. that i1 < i2. We discuss in four cases.

• f(i1) < i1 and f(i2) > i2.

Since f(i2) > i2 > i1 > f(i1), this case is trivial;

• f(i1) > i1 and f(i2) > i2.

It can be seen from (4) that uj(t) = 0, ∀i1 < j < f(i1). i2 > i1 and ui2(t) 6= 0

imply that i2 > f(i1). Hence f(i2) > i2 > f(i1);

• f(i1) < i1 and f(i2) < i2.

By symmetry with the previous case, we have f(i1) < i1 < f(i2);

• f(i1) > i1 and f(i2) < i2.

It can be seen from (4) that uf(i1)+1(t) = −2, but from (5) we know that

uf(i2)+1(t) ∈ {0, 2}, hence f(i1) 6= f(i2).

(ii) By the monotonicity property (2) we know immediately that the cardinality of

Z(t) keeps unchanged in the fashion cycle, i.e.

|Z(t + 1)| = |Z(t)|,∀t ≥ r(I). (6)

Consequently, we know that

∀t ≥ r(I),∀j ∈ Z−(t), there exists exactly one i ∈ Z+(t) s.t. f(i) = j. (7)

Now, we are ready to prove the final result. We use the minimum counter-example

argument. Suppose the lemma is not valid, and I = (N,E, T, x(0)) is a counter-

example with the smallest number of agents. Let j∗ be the agent with the largest

42

index among all the agents having a utility of 0 for at least one time in the fashion

cycle.

First of all, it will be verified that j∗ = n−1, because otherwise we could construct

a smaller counter-example. In fact, the choice of j∗ tells us that if j∗ 6= n − 1, then

xj∗+1(t) ∈ {2,−2} holds for all t ≥ r(I). In the fashion cycle, if agent j∗ + 1 hates

the action of j∗, then her utility will be −2, because 0 is never attained, and she

will switch her action. And when agent j∗ + 1 likes the action of j∗, evidently she

will not switch. Therefore, whether j∗ + 1 switches her action at time t ≥ r(I) is

completely determined by the action of j∗. Using the same argument as in the proof

to Lemma 5, we know that I ′(j∗ + 1) = (N ′, E ′, T ′, x′(0)) is also a qualified counter-

example (fashion cycle is entered at the first step), where N ′ = {1, 2, · · · , j∗ + 1}, T ′

the corresponding sub-vector of T , E ′ the corresponding left part of E, and x′(0) =

(x1(r(I)), x2(r(I), · · · , xj∗+1(r(I)))). Since I is the smallest counter-example, this is

impossible, and thus it can only be that j∗ = n− 1.

Suppose uj∗(t′) = 0 for some t′ ≥ r(I), then there must exist t∗ ≥ t′ such that

uj∗(t∗) = 0 and uj∗(t

∗ + 1) 6= 0. Because otherwise, the utility of j∗ will keep at 0

from time t′ on, and thus she will be immune. As a result, l(I) = 1 and PNE will be

reached by simultaneous BRD. This is a contradiction with (1).

Therefore, j∗ ∈ Z−(t∗). Since t∗ ≥ t′ ≥ r(I), by (7) we know that there exists

exactly one i∗ ∈ Z+(t∗) such that f(i∗) = j∗. Because j∗ = n−1 we know that i∗ < j∗.

Therefore, by (5) we know that

[ui∗−1(t∗), · · · , uj∗(t

∗), un(t∗)] = [−2, 2, 0, · · · , 0,−1].

By Lemma 7(c), it must hold that uj∗(t∗ + 1) = −2. Therefore,

[ui∗−1(t∗ + 1), · · · , un(t∗ + 1)] = [2, 0, 0, · · · ,−2, 1]. (8)

It can be observed from (8) that j∗ − 1 ∈ Z−(t∗ + 1). Again, using (7), we know

that there is an agent k∗ ∈ Z+(t∗ + 1) such that f(k∗) = j∗ − 1. It must hold that

k∗ < j∗ − 1, because (8) tells us that the sequence [0, 2,−2] never occurs on the right

side of j∗ − 1 and (5) says that if k∗ > j∗ − 1 we would have a sequence [0,2,-2] on

the right side of j∗ − 1 (notice by definition that for any i ∈ Z+(t), we always have

f(i) 6= i). Furthermore, comparing (4) and (8) we know that k∗ = i∗ − 1, which can

only happen when ui∗−2(t∗ + 1) ∈ {−2,−1}.

ui∗−2(t + 1) = −1, i.e. i∗ − 2 = 1, is not true. Because if so, we would have

43

[ui∗−2(t∗ + 2), · · · , un(t∗ + 2)] = [1, 0, 0, 0, · · · ,−2, 2,−1], which has n − 4 0s in total.

However, it can be checked that we would further have [ui∗−2(t∗+3), · · · , un(t∗+3)] =

[1, 0, 0, 0, · · · ,−2, 2,−2, 1], which has n− 5 0s in total, a contradiction with (6).

Therefore, we have ui∗−2(t + 1) = −2 and

[ui∗−2(t∗ + 2), · · · , un(t∗ + 2)] = [2, 0, 0, 0, · · · ,−2, 2,−1]. (9)

It can be observed from (9) that j∗ − 2 ∈ Z−(t∗ + 2). Again, there is no [0, 2,−2]

on the right side of j∗ − 2, and so we can only have f(j∗ − 2) = i∗ − 2. This can only

happen when ui∗−3(t∗ + 2) ∈ {−2,−1}. Again, we can deduce that ui∗−3(t

∗ + 2) = −2

and

[ui∗−3(t∗ + 3), · · · , un(t∗ + 3)] = [2, 0, 0, 0, · · · ,−2, 2,−2, 1]. (10)

So this process will be repeated infinitely. But this is obviously impossible, because

the line is finite. Hence the lemma.

References

[1] W.B. Arthur. Inductive reasoning and bounded rationality. The American Eco-

nomic Review 84(2) (1994): 406-411.

[2] L. Andreozzi, M. Bianchi. Fashion: Why people like it and theorists do not. Ad-

vances in Austrian Economics 10 (2007): 209-229.

[3] E.M. Adam, M.A. Dahleh, A. Ozdaglar. On threshold models over finite networks.

arXiv preprint arXiv:1211.0654, 2013.

[4] R. Aharoni, E.C. Milner, et. al. Unfriendly partitions of a graph. Journal of Com-

binatorial Theory, Series B 50(1) (1990): 1-10.

[5] L. Anderlini, A. Ianni. Path dependence and learning from neighbours. Games and

Economic Behavior 13 (1996): 141-177.

[6] H. Blumer. Fashion: From class differentiation to collective selection. The Socio-

logical Quarterly 10(3) (1969): 275-291.

[7] A.V. Banerjee. A simple model of herd behavior. The Quarterly Journal of Eco-

nomics 107 (1992): 797-817.

44

[8] S. Benvenuto. Fashion: Georg Simmel. Journal of Artificial Societies and Social

Simulation 3(2), March 2000.

[9] Y. Bramoulle. Anti-coordination and social interactions. Games and Economic Be-

havior 58 (2007): 30-49.

[10] L.S. Bagwell and B.D. Bernheim. Veblen effects in a theory of conspicuous con-

sumption. The American Economic Review 86(3) (1996): 349-373.

[11] Y. Bramoulle, R.E. Kranton. Public goods in networks. Journal of Economic The-

ory 135 (2007): 478-494.

[12] W. Brock, S. Durlauf. Discrete choice with social interactions. The Review of

Economic Studies 68(2) (2001): 235-260.

[13] S. Bikchandani, B.D. Hirshleifer, I. Welch, A Theory of fads, fashion, custom

and cultural change as informational cascades. Journal of Political Economy 100(5)

(1992): 992-1026.

[14] Y. Bramoulle, D. Lopez-Pintado, S. Goyal, F. Vega-Redondo. Network formation

and anti-coordination games. International Journal of Game Theory 33(1) (2004):

1-19.

[15] S.K. Berninghaus, U. Schwalbe. Conventions, local interaction, and automata net-

works. Journal of Evolutionary Economics 6 (1996): 297-312.

[16] E.R. Canterbery. The Theory of the Leisure Class and the Theory of Demand.

In The Founding of Institutional Economics, edited by W. J. Samuels. London:

Routledge, 1998, 139-156.

[17] C. Cannings. The majority and minority models on regular and random graphs.

International Conference on Game Theory for Networks, 2009.

[18] A. Calvo-Armengol, M.O. Jackson. The effects of social networks on employment

and inequality. The American Economic Review 94 (2004): 426-454.

[19] A. Calvo-Armengol, M.O. Jackson. Networks in labor markets: Wage and employ-

ment dynamics and inequality. Journal of Economic Theory 132 (2007): 27-46.

[20] A. Calvo-Armengol, M.O. Jackson. Like father, like son: Labor market networks

and social mobility. American Economic Journal: Microeconomics 1 (2009): 124-150.

45

[21] A. Calvo-Armengol, E. Patacchini, Y. Zenou. Peer effects and social networks in

education. Review of Economic Studies 76 (2009): 1239-1267.

[22] A. Calvo-Armengol, T. Verdier, Y. Zenou. Strong ties and weak ties in employment

and crime. Journal of Public Economics 91 (2007): 203-233.

[23] H.C. Chen, Y. Chow, L.C. Wu. Imitation, local interaction, and coordination.

International Journal of Game Theory, 2012. online first, DOI 10.1007/s00182-012-

0353-7.

[24] D. Challet, M. Marsili, Y.C. Zhang YC. Minority games: interacting agents in

financial markets. Oxford Univ. Press: Oxford, 2004.

[25] Z. Cao, H. Gao, X. Qu, M. Yang, X. Yang. Fashion, cooperation, and social

interactions. PloS ONE 8(1): e49441, 2013.

[26] Z. Cao, X. Yang. The fashion game: Matching Pennies on social networks

(February 23, 2011). Available at SSRN: http://ssrn.com/abstract=1767863 or

http://dx.doi.org/10.2139/ssrn.1767863

[27] Z. Cao, M. Yang, X. Qu, X. Yang. Rebels lead to the doctrine of the mean: opinion

dynamic in a heterogeneous DeGroot model. The 6th International Conference on

Knowledge, Information and Creativity Support Systems Beijing, China, 2011: 29-

35.

[28] D. Challet, Y.C. Zhang. Emergence of cooperation and organization in an evolu-

tionary game. Physica A 246 (1997): 407-418.

[29] G. Ellison. Learning, local interaction, and coordination, Econometrica 61(5)

(1993): 1047-1071.

[30] K. Eliaza, A. Rubinstein. Edgar Allan Poe’s riddle: framing effects in repeated

matching pennies games. Games and Economic Behavior 71(1) (2011): 88-99.

[31] B. Fine, E. Leopold. The World of Consumption. London: Routlege, 1993.

[32] J.K. Galbraith. The Affluent Society. Bombay: Asia Publishing House, 1958.

[33] D. Foster, P. Young. Stochastic evolutionary game dynamics. Theoretical popula-

tion biology 38(2) (1990): 219-232.

46

[34] M. Granovetter. Threshold models of collective behavior. American Journal of

Sociology 83 (1978): 1420-1443.

[35] E. Goles. Lyapunov functions associated to automata networks. In: Fogelman-

Souli6 F, Robert Y, Tchuente M (eds) Automata networks in computer sicience.

Manchester University Press, 1987: 58-81

[36] S. Galam. Contrarian deterministic effects on opinion dynamics:“the hung elec-

tions scenario”. Physica A: Statistical and Theoretical Physics 333 (2004): 453-460.

[37] S. Goyal. Connections: An Introduction to the Economics of Networks. Princeton

Univ. Press, Princeton, 2007.

[38] A. Galeotti, S. Goyal, M.O. Jackson, F. Vega-Redondo, L. Yariv. Network games.

The Review of Economic Studies 77(1) (2010): 218-244.

[39] S. Goyal, J.L. Moraga-Gonzales. R & D networks. RAND Journal of Economics

32 (2001): 686-707.

[40] S. Goyal, F. Vega-Redondo. Network formation and social coordination. Games

and Economic Behavior 50 (2005): 178-207.

[41] P. Hernandez, M. Munoz-Herrerab, A. Sanchezc. Heterogeneous network games:

Conflicting preferences. Games and Econonic Behavavior, 2013. Online first,

http://dx.doi.org/10.1016/j.geb.2013.01.004

[42] A. Ilachinski. Cellular Automata: A Discrete Universe, World Scientific, USA,

2001.

[43] M.O. Jackson. Social and economic networks. Princeton University Press, Prince-

ton, NJ, 2008.

[44] M.O. Jackson, A. Watts. On the formation of interaction networks in social coor-

dination games, Games and Economic Behavior. 41(2) (2002): 265-291.

[45] M.O. Jackson, A. Wolinsky. A strategic model of social and economic networks,

Journal of Economic Theory 71 (1996): 44-74.

[46] M.O. Jackson, Y. Zenou. Games on Networks. Handbook of Game Theory, Vol. 4,

Peyton Young and Shmuel Zamir, eds., Elsevier Science, 2014. Available at SSRN

(August 25, 2012): http://ssrn.com/abstract=2136179

47

[47] A.T. Kalai, E. Kalai. Cooperation in strategic games revisited. The Quarterly

Journal of Economics, 2012, online first.

[48] M. Kandori, G.J. Mailath, et. al. Learning, mutation, and long run equilibria in

games. Econometrica 61 (1993): 29-29.

[49] E. Karni and D. Schmeidler. Fixed preferences and changing tastes, The American

Economic Review, 80 (1990), 262-267.

[50] H. Leibenstein. Bandwagon, snob, and Veblen effects in the theory of consumers’

demand. The Quarterly Journal of Economics 64(2) (1950): 183-207.

[51] T. Lears. Beyond Veblen: remapping consumer culture in twentieth century Amer-

ica. Marketing 2(1993): 27-40.

[52] D. Lopez-Pintado. Network formation, cost sharing and anti-coordination. Inter-

national Game Theory Review 11(1) (2009): 53-76.

[53] S. Lieberson, E.O. Bell. Children’s first names: An empirical study of social taste.

American Journal of Sociology 98(3) (1992): 511-554.

[54] D. Monderer, L.S. Shapley. Potential games. Games and Economic Behavior 14(1)

(1996): 124-143.

[55] S. Morris. Contagion, The Review of Economic Studies 67(1) (2000): 57-78.

[56] M. Marsili. Market mechanism and expectations in minority and majority games.

Physica A 299 (2001): 93-103.

[57] U. Okonkwo. Luxury Fashion Branding: Trends, Tactics, Techniques. Palgrave

Macmillan, 2007.

[58] I. Palacios-Huerta. Professionals play minimax. Review of Economic Studies 70(2)

(2003): 395-415.

[59] W. Pesendorfer. Design innovation and fashion cycles. The American Economic

Review 85(4) (1995): 771-792.

[60] G. Simmel. Fashion. International Quarterly 10 (1904): 130-155. Reprinted by

American Journal of Sociology 62(6) (1957): 541-558.

[61] L. Svendsen. Fashion: A Philosophy, Reaktion Books, 2006.

48

[62] F. Schweitzer, R. Mach. The epidemics of donations: logistic growth and power-

laws. PLoS ONE 3(1), 2008, e1458. doi:10.1371/journal.pone.0001458.

[63] Trigg, A. B. (2001). ”Veblen, Bourdieu, and conspicuous consumption.” Journal

of Economic Issues: 99-115.

[64] T. Veblen. The Theory of the Leisure Class-An Economic Study of Institutions,

London: George Allen & Unwin Ltd, 1899.

[65] F. Vega-Redondo. Complex Social Networks. Econometric Society Monograph Se-

ries. Cambridge Univ. Press, Cambridge, 2007.

[66] S. Wolfram. A New Kind of Science, Wolfram Media, 2002.

[67] H. Yoganarasimhan. Cloak or Flaunt? The Fashion Dilemma. Marketing Science

31(1) (2012a): 74-95.

[68] H. Yoganarasimhan. Identifying the Presence and Cause of Fashion Cycles in the

Choice of Given Names, UC Davis working paper, 2012b.

[69] H.P. Young. The evolution of conventions. Econometrica 61(1) (1993): 57-84.

[70] H.P. Young. Individual strategy and social structure: an evolutionary theory of

institutions. Princeton, NJ: Princeton University Press, 2001.

[71] H.P. Young. Innovation diffusion in heterogeneous populations: Contagion, social

influence, and social learning. The American Economic Review 99(5) (2009): 1899-

1924.

[72] B. Zhang, Z. Cao, C. Qin, X. Yang, Fashion and Homophily (April

14, 2013). Available at SSRN: http://ssrn.com/abstract=2250898 or

http://dx.doi.org/10.2139/ssrn.2250898

49

A Heterogeneous Network Game Perspective of Fashion Cycle · Instead of providing more...

Documents

Transcript of A Heterogeneous Network Game Perspective of Fashion Cycle · Instead of providing more...