GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.247

Competitive Targeted Marketing

Hang-Hyun Jo and Jeong-Yoo Kim

March 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Competitive Targeted Marketing
Hang-Hyun Jo
Korea Institute for Advanced Study

Jeong-Yoo Kim ∗Kyung Hee University

March 1, 2012

Abstract
In this paper, we consider two firms diffusing incompatible technologies and theirdecision of consumer targeting. The technology adoption is made in two steps. First,
once the firms sell their products to their respective targeted consumer, the technology
is diffused successively by word-of-mouth communication from the initial consumer toother consumers linked along the network. Then, in the second step, each consumer
imitates the technology the neighbors use which fares better, and through this processof imitation, the technology distribution keeps evolving until it reaches the long-run
steady state. We demonstrate that the early entrant chooses the minmax location
when firms are myopic in the sense that they do not take the imitation possibility into
account. If firms consider the possibility of imitation, the best target will tend towards
a hub, although the minmax principle in general keeps valid in the sense that it shouldbe the minmax location after considering imitation.

Keywords: Information Diffusion, Technology Adoption, Network Externalities, Compatibility, Learning, Evolution, Targeted Marketing∗

This research was begun when the second author was visiting ISER at Osaka University. W are very
grateful to Masaki Aoyagi for his hospitality during our stay. Corresponding author: Department of Economics, Kyung Hee University, 1 Hoegi-dong, Dongdaemun-gu, Seoul 130-701, Republic of Korea, Tel:
+82-2-961-0986, Fax: +82-2-966-7426, Email: jyookim@khu.ac.kr

1

1

Introduction

Technologies are evolving. An old technology soon faces competition by the emergence of a
new technology. It may be replaced by the new superior technology or they may coexist for awhile. If the competing technologies are not compatible with each other, a consumer should
be concerned about how many neighbors share the technology in deciding which technology
to adopt. The more neighbors use the technology, the higher utility a consumer usually getsby adopting it. This is called network externality.1
In the description above, special attention should be paid to the word “neighbor.” If
a consumer meets only a limited set of the whole population, the total number of users
adopting the same technology is not meaningful to him. He only needs to care about thetechnologies adopted by his neighbors, that is, those whom he interacts with. This implies
that the structure of the network among consumers in which they interact with each other is
an important factor to be considered in deciding which technology to adopt in the presenceof network externalities. However, the explicit structure of the consumer network has been
largely ignored in literature on technology adoption.
The network structure is crucial not only to consumers but also to the firms sponsoring
the technologies. As long as the technologies are diffused by word-of-mouth communication
between consumers along the consumer network, it is important for a firm to target a rightconsumer so as to maximize its market share.
In this paper, we consider two competing firms sponsoring incompatible technologies
and their decision of consumer targeting. We also consider the subsequent diffusion andevolution of technologies. The technology adoption is made in two steps. First, once thefirms sell their products to their respective targeted consumer, the technology is diffused byword-of-mouth communication from the initial consumer to other consumers linked along
the network subsequently. Then, in the second step, each consumer imitates the technology
the neighbors use which fares better, and through this process of imitation, the technology
distribution keeps evolving until it reaches the long-run steady state.
The economists’ interest in the consumer network has begun only recently. Jun and Kim(2008) seems the first paper to consider the consumer network explicitly. They consider themonopolist’s problem of choosing the referral fee to provide consumers incentives enough tomake the information about its products spread widely. Subsequently, Jun et al. (2006a,
1This concept has been actively explored in the context of technology compatibility (standardization) by
economists since David (1985), Farrell and Saloner (1985) and Katz and Shapiro (1985).

2

2006b) consider similar problems in small world networks. Also, Kim (2010) considers thetargeting decision of a monopolist whose objective is to maximize its discounted sum ofvalues generated from the information diffusion. Since a single decision maker is involved in
the paper, there is no strategic aspect to be considered. This paper is an extension of Kim(2010) into a duopoly setting.2
Recently, Duan et al. (2005) investigate a targeting strategy for the spread of opinions,diseases, new ideas, technological innovations, etc. in scale-free social networks. They aremainly interested in how to pick out a given multiple number of agents as initial targets inorder to launch a spreading process efficiently, and propose a strategy of targeting agentswith most connections (hubs) as initial adopters, so-called hub strategy. In this paper, weinvestigate the targeting strategies of competing technologies and the resulting technology
diffusion in an agent-based model by focusing more on the micro motivation of agents,3and establish the minmax principle in competitive targeting. If two firms with sequentiallydeveloped technology compete in targets, the first firm does not target naively as if he werea single player, but chooses its target by considering its effect on the second firm’s decision.
The second firm’s decision is naturally affected by the first firm’s targeting decision. Then,the optimum for the first firm is to choose the minmax node, i.e., the node to minimize
the maximum of the rival’s market share. The main insights of this paper, however, can bestraightforwardly applied to the opinion formation and others, as we will argue later.
Some could be learned from the previous works on the epidemic spreading and the efficient
immunization strategies on the complex networks.4 For example, the targeted immunization
strategy turns out to be more efficient to prevent the disease spreading than the random
immunization strategy on the scale-free networks. Duan et al. (2005) modified the strategy
by targeting the higher degree nodes with higher probability. While all those works consider
the diffusion of one kind of technology (monopoly), we take into account the competition
between two incompatible technologies, which makes the problem qualitatively different. On
the other hand, Newman (2005) studied the case in which after one pathogen spreads first on
2

We found the paper by Galeotti and Goyal (2009) which also addressed a problem of targeting in the
consumer network. However, this paper mainly considers monopoly targeting, not competitive targeting.
3
Many opinion dynamics models studied by the physicists usually simply assume the compromise process
between the agents, as in the voter model (See, for example, Liggett [1999]) and the Sznajd model See
Sznajd [2000, 2005]) without paying enough attention to the agents’ economic motivation. This might leadto misunderstanding when applied to economic phenomena.
4
For example, see Pastor-Satorras and Vespignani (2002), Havlin et al. (2003) and Jo et al. (2006).

3

a network, then the other pathogen spreads over the remaining population of that network.
Our paper is also distinguished from the work in the sense that we consider the concurrent
diffusions of two sequentially developed technologies.
The paper is organized as follows. In Section 2, we describe the model and provide its
economic rationale. In Section 3, we consider the firms’ targeting decision and the consequent technology diffusion process when the network is a tree. In Section 4, we discuss thecase of more general networks and provide some simulation results. Section 5 contains theconclusion.

2

Model and Economic Rationale

We consider only the connected network consisting of N nodes. Each node represents a consumer, who is a potential technology user. Two firms are going to introduce two technologies
which are incompatible to each other. We will call the technology introduced by firm A andB technology A and B respectively. The firms can diffuse their technology by choosing anode as the starting consumer. To avoid uncertainty due to simultaneity, we favor firm A
by giving him the first mover advantage. That is, we assume that firm A first chooses anode iA to diffuse its technology and then firm B chooses a target node iB . Considering thereality that two competing technologies are rarely introduced simultaneously, it is a realistic
assumption.
The subsequent interaction among technology users consists of two stages, each of which
again consists of multi-periods. The first stage is the technology diffusion stage. Once a
node adopts a technology α ∈ {A, B}, he passes the information on the technology to all
his neighbors in the period. Then, the neighbors in turn inform the technology to all theirneighbors in the next period, and so one. This diffusion process is proceeded until no one
remains uninformed. In other words, given the initial targets of two technologies (iA , iB ), thetechnology of node i will be αi = arg minα∈{A,B} d(i, iα ), where d(i, j) is the shortest path
length between nodes i and j. If node i receives the information on the two technologies
at the same time, i.e. d(i, iA ) = d(i, iB ), we assume that he adopts the technology A withprobability p ∈ [0, 1].5
The second stage is the technology imitation stage. Once the diffusion of the technologies
5More realistically one may assume that the probability depends positively on the number of neighbors
using the same technology. However, for the complexity reason, we do not consider the case in this paper.

4

is completed, each pair of nodes linked with each other play the 2 × 2 game in which bothusers obtain a positive payoff a (b respectively) if they choose the same technologies A(B respectively) and both obtains zero otherwise.6 The structure implies that the twotechnologies exhibit network externalities.7 After playing the 2×2 game, each user compares
the average payoff of his A-neighbors and B-neighbors including his own payoff, and imitates
the technology yielding the higher average payoff. After the transient periods the system will
be stationary, resulting in the stationary population distribution of users of each technology.
The imitation dynamics is described precisely. Let t = 0 be the period at which technology diffusion is completed and let N A (t) be the number of users adopting technology A
at period t = 0, 1, 2, · · · . Then N B (t) = N − N A (t). Let us denote the sets of a node i’s
A-neighbors and B-neighbors as NiA and NiB respectively. Then Ni = NiA ∪ NiB . FromkiA ≡ |NiA | and kiB ≡ |NiB |, the node i’s degree is ki = kiA + kiB . If the node i adopts technolkAkB
ogy A, i.e. αi = A, his payoff is given by πi = kii a. Similarly, if αi = B, then πi = kii b. Only
the node having both A-neighbors and B-neighbors has the chance to switch its technology.
If αi = A, the node i switches to B if and only if
PPl∈NiA ∪{i} πll∈NiB πl<,(1)kiA + 1kiBand similarly, if αj = B for node j, he switches to A if and only ifPPl∈NjB ∪{j} πll∈NjA πl>.
kjAkjB + 1

(2)

By applying the Bethe approximation we briefly analyze how the imitation dynamics
depends on the degrees of involved nodes. Consider a node i with αi = A is connected with6

An earlier evolutionary approach to the technology adoption game can be found in Kandori and Rob(1998). However, in their model, a pair of players playing the 2 × 2 game are randomly matched from thewhole population, not locally matched on a network. Thus, the set of players each player is expected to playthe 2 × 2 game is the same in their model, while each player plays the game with a different set of players
in our model. Another difference is that the proportion of each type evolves according to the best response
dynamics in their model, while it evolves according to the imitation dynamics. Consequently, in this model,
if the network is complete, each player will play with the same group of players as in their model, but theimitation process will be different from theirs.
7This payoff structure can be used for opinion formation. Suppose there are two conflicting views andimagine that the interaction between two individuals with similar opinions reinforces the belief of each other,while the (one-time) interaction between individuals with differing opinions yields only quarrel with no utility.
If one view, say A, is more convincing than the other, say B, then the former will have a stronger reinforcing
effect in the sense that a > b.

5

kiA A-neighbors and kiB B-neighbors. We assume for simplicity that each A-neighbor of nodei, say l, has kl neighbors and those next nearest neighbors adopt either A or B with theAprobabilities ρ = NN and 1 − ρ. Similarly we assume that each B-neighbor of node i, sayj, has kj neighbors and those next nearest neighbors adopt one of technologies in the samekA
(k −1)(1−ρ)
b, the condition (??) for switchingway. Since πi = kii a, πl = (kl −1)ρ+1
a, and πj = j kj
klfrom A to B yields
kiA πl + πi
kiB πj<,kiA + 1
kiB
i.e.,

·
¸
(kl − 1)ρ + 1
kj − 1kiA
1+a<(1 − ρ)b.
Aklkj
ki + 1 ki(3)Combined with the condition from (??), one can roughly say node i is more likely to invade
or less likely to be invaded as the number of the same type neighbors, kiA , is larger and thedegree of the opposite type neighbors, kj , is smaller. Let a link connecting i with αi = A
and j with αj = B on a tree network be the unique bridge between A-cluster and B-cluster,
implying that without that link, the tree will be split into two clusters of each own type.
Then the condition (??) reduces toki2 − 1kj − 1
a<
b.
2
ki
kj

(4)

Since ki = kiA + 1, the above analysis is clarified in this case.
The objective of each firm is to maximize the number of its technology users by targetingits starting user optimally. For the subsequent analysis, we assume that there is no discountin the profit of firms.8 This assumption implies that firms care only about the stationary
distribution, that is, cares only about the profit in the long-run future state with ignoring the
profit in the transient periods. To isolate the targeting decisions from other strategic effects,
we are also implicitly assuming that there is no price competition. If each firm can choose
its price as well as its target, the imitation decision of consumers will depend on the prices
as well. Then, the second firm may try to invade consumers using the rival’s technology bycharging a lower price than the rival. Thus, there will be a strategic interaction between
8

The assumption of no discounting might be challenged. If the evolution phase is too long, firms may careabout the profits realized before the steady state is reached. However, if we consider discounting explicitly,
the optimal targeting decision will depend on the discounting factor as well, which complicates the prediction.
See Kim (2010) for the the targeting decision of a monopolist who takes discounting into account.

6

firms in pricing. Although this is an interesting economic problem, it is beyond the scope of
this paper.

3

Consumer Targeting in a Tree

The targeting decision of the firms induces a partition of users, each of which uses the same
technology as a consequence of information diffusion. Let P and P 0 be two partitions. Notethat a partition is entirely determined by the set of users adopting technology A. Accordingly,
let A(P) be the users adopting technology A in P.
A firm’s preference over partitions is denoted by %. That is, P %A P 0 implies that firm
A prefers P to P 0 , and P ∼A P 0 implies that firm A is indifferent between P and P 0 . By theconstant-sum nature, i.e., N A + N B = N , P %A P 0 implies that P 0 %B P.
We consider two cases. One is the case that firms are myopic so that they do not consider
the possibility of technology imitation in their targeting decisions. The other is the case ofsophisticated (far-sighted) firms who make targeting decisions by considering the subsequentimitation process.
In this section, the analysis will be restricted to tree networks. Consumer targeting in
more general networks will be discussed in the next section.

3.1

Myopic Firms

Given the sequential nature of targeting, we first consider the decision of the entrant firm
B. Let i∗B (i) be the best response of firm B to A’s target i.
Lemma 1 For any iA = i, d(i, i∗B ) = 1.
This lemma implies that the targets of the firms must be minimally differentiated. The
intuition is that a firm always prefers being more closely located to the opponent in order
to secure more customers.
Proposition 1 Given any iA = i, {i∗B (i)} = arg maxj∈Ni |Sji |.
Here Sji denotes a set of node j’s successors against node i.
Now, consider incumbent A’s decision. Letting i∗A be the optimal target, we have thefollowing proposition.

7

Proposition 2 {i∗A } = arg mini∈N maxj∈Ni |Sji |.
This proposition says that it is optimal for firm A to take the minmax strategy. For
example, in a line network with N = 2m + 1 for some positive integer m illustrated in
Figure 1(a), i∗A = m + 1. Note that this target does not necessarily correspond to a hub, as
illustrated in Figure 1(b). In Figure 1(b), firm A will target node 5, although node 4 is the
hub with the highest degree.

3.2

Sophisticated Firms

If firms are sophisticated, they choose their targets not simply by maximizing the current
market share, but by taking the subsequent imitation process into account. Thus, it is notstraightforward to compare their preference between two partitions. However, the following
proposition simplifies the process of searching for the optimal target.
Proposition 3 Suppose iA = i1 , i2 induce P1 = {A1 , B1 } and P2 = {A2 , B2 } respectively.
When firms are sophisticated, A1 ⊃ A2 implies that P1 %A P2 .
Proposition 3 says that sophisticated firms prefer a larger installed base even if they take
into account the possibility that the imitation process will lead to a different market shareultimately. Due to this proposition, it is easy to see that Lemma 1 still holds in the case ofsophisticated firms. To see this, suppose d(i, i∗B (i)) ≥ 2 for some node i. Now, alternatively
take iB = j where d(i, j) = 1 and j ∈ qi,i∗B , where qi,j denotes a path from i to j. Then,clearly, the partition induced by the latter choice is preferred by the entrant B. However,
Proposition 1 does not hold any more. To see this, consider Figure 1(c). Assuming a < b,given that iA = 6, if firm B enters iB = 7, technology B cannot invade iA if a > 2b/3. If
firm B enters iB = 5 instead, it can invade node 6 if (2b/3 <) a < 8b/9. This implies thatit can be optimal for firm B to enter node 5 even if i∗B (6) = 7 in the case of myopic firms.
The main insight is that node 5 has more neighbors so as to be advantageous for invasion.
A similar intuition will be provided for the targeting decision of firm A. Take the exampleof the network in Figure 1(d) and assume that the value of a = 1 and b = 1.5 − ². If firms
are myopic, (i∗A , i∗B ) = (9, 4) as we showed. But, then, nodes 8 and 9 of type A are invaded
by type B. Ultimately, the market share of firm A will end up with 6/15. However, if firm A
targets node 4, node 4 of type A is not invaded and the resultant market share of firm A is
7/15. Thus, node 4 is preferable to firm A. The intuition is as follows. For a node of firm A
not to be invaded, it is better that the opponent node has less neighbors. Note that node 48has a large number of neighbors making him faring very well regardless of the technology hechooses. This implies that if firm A enters node 9, it will be very likely to be invaded. The
upshot is that if an early entrant expects to face a new entrant with superior technology,
it must keep the rival firm from entering the hub. Thus, it may be sometimes (not always)
good to preempt such a hub. If the incumbent has the superior technology, it may preferchoosing the hub for a different reason that it is more advantageous for invasion.
The general conclusion is that firms want their target to have more neighbors and theiropponent’s target to have less neighbors in the invasion battle. More specifically, the firmwith a superior technology prefers entering the hub to invade neighbors more easily, while
the firm with an inferior technology would rather choose the node whose rival node has lessneighbors in order to be less vulnerable to the invasion.9 However, this does not imply thatthe sophisticated firm always prefers the hub with the highest degree to the myopic optimal
targets. If the non-hub myopic targets are robust against invasion, there would be no reason
to give them up in order to preempt hub targets.
Generally, we can identify the optimal target of the two firms in the sophisticated case.
For given i, find i∗B (i) by the node j ∈ Ni that maximizes |S̃ij | where S̃ij is the set of B
technology users after the imitation is completed. The node j ∈ arg max |S̃ij | is an optimal
response of firm B. Then, an optimal target of firm A is i∗A ∈ arg mini∈N maxj∈Ni |S̃ij |.

4Discussion on Barabási-Albert Scale-Free Networks

We will briefly discuss the case of more general networks possibly containing a cycle. Firstof all, Lemma 1 is not valid any more. For example, consider a circle. Once firm A chooses
a node in a circle, any other node is optimal to firm B. In any case, the two firms split theconsumers equally in the technology diffusion stage. Suppose d(iA , iB ) = 2 in a circle graphwith N = 5. Then, by moving closer to iA , firm B may secure the node between them but
it loses other nodes in the opposite direction. In this case, note that node j is not critical in
the communication between iA and k 6= iA , iB , j. The fact that the path between two nodes
iA and iB is not unique causes Lemma 1 to fail. This observation forces us to search for firm
9

We are saying that the reason for the hub preference is either defensive (for an inferior firm) or offensive(for a superior firm), but this distinction is just an overall tendency, since the possibility of invasion depends
on the number of neighbors as well as the quality of technology. In other words, an inferior firm might prefera hub-like node for an offensive reason. This means that the two effects actually coexist. It is true, too, forthe case that the two technologies have the same quality, i.e., a = b.

9

B’s optimal response to iA not only from the neighbors but globally from the population.
More specifically, for any (nonterminal) node i and for each j ∈ Ni , one can induce theprobability distribution over the technology distribution as the result of the naive diffusion
based on p, since the existence of a cycle makes it possible for two technologies to arrive
at a node simultaneously. Then, compute the expected number of successors Ŝij . Then, an
optimal target is i∗A ∈ arg mini∈N maxj∈Ni |Ŝij |, which will be called the extended minmax
target.
We provide the simulation results in Figure 2 for N = 51. To generate networks, we
use two Barabási-Albert scale-free networks.10 Two kinds of networks are considered; a tree
which are obtained by connecting each new node to only one existing node, and a general
network with cycles which are obtained by connecting each new node to two existing nodes.
Given the generated networks, we first simulate the technology diffusion process. Takeany pair of nodes and assign A and B to them respectively. In case of trees, it suffices totake any pair of adjacent nodes. Then assign A to other nodes if they are closer to the initial
A node and assign B to nodes if they are closer to the initial B node. If the distances to theinitial A and the initial B are equal, just assign A (by assuming p = 1). Then, we simulate
the second stage technology imitation process by the random sequential updating method.
That is, at each time step of imitation process one randomly chosen node can be invaded
by its neighbors of different type. This randomness of choosing a node forces us to take theaverage of the final market share. Thus, to find the minmax strategy of sophisticated firms
we compare the averaged market shares.
In Figures 2(a) and (b), the optimal targets are (iA , iB ) = (3, 1) and (3, 7) when firms
are myopic and sophisticated, respectively. Note that k7 = 5 is larger than k1 = 3. This
supports our main insight that sophisticated firms tend to prefer nodes with higher degree.
In Figures 2(c) and (d), interestingly, the optimal targets for myopic firms are (4, 1) but the
optimal ones for sophisticated firms are reversed to (1, 4). Firm A, who is the earlier entrant,
can foresee that node 4 which is its optimal target in the myopic case will be invaded andits resulting market share will be much lower than the rival’s market share. By preempting
node 1 instead of node 4, it can increase the number of its users from 11.22 to 11.44.
In Figure 3, we illustrate the tendency of sophisticated firms towards nodes with higher
degree more dramatically for a larger population N = 501. It shows a correlation between
the degree of the extended minmax target and the market share. Note that this result should10

See Barabási and Albert (1999).

10

not be taken for granted, since it incorporates the concomitant effect of firm B’s optimal
response to firm A’s extended minmax target.
Figure 4 illustrates the correlation between the relative quality b/a and the market share
N A . It shows that the market share of technology A tends to decrease with b/a. This is not
surprising, since a technology will lose its market share more as the competing technology
has a higher quality. However, the figure also shows that the distribution of N A becomes
more dispersed as b/a increases, which is surprising. This is one consequence of the firstmover advantage. One clear aspect of the first-mover advantage is that N A is much larger
than a half of the population (250) when a = b. Because of the severe advantage, the firstmover’s market share will mostly stay above 250 when b/a ≤ 1/2. This is the main reason
why the distribution of N A is so concentrated for low values of b/a. This is because highassortativity weakens the first mover advantage.
Figure 5 shows the relationship between the degree of sophistication s and the market
share N A . By the degree of sophistication, we mean how far the firms foresee. For two
extreme cases, the myopic case corresponds to the case that s = 0 and the sophisticatedcase corresponds to the case that s = 50. The horizontal line s of Figure 5 represents
how many rounds of the imitation process firms consider in deciding their target. The
figures show that N A changes monotonically with s. Note that there are two kinds ofadvantages in this model; the quality advantage and the first-mover advantage. If a ≥ b,technology A is clearly advantageous,11 while technology B is if b À a since its quality
advantage dominates the second-mover disadvantage. The advantages will become more
stark as the firms get more sophisticated (smart). Thus, as s gets higher, N A increases
monotonically if a ≥ b and decreases monotonically if a ¿ b. This is the case for both
assortative and disassortative networks while the tendency on assortative networks is weaker
than on disassortative networks. It is again because high assortativity weakens the firstmover advantage. We also find an exceptional case in Figure 5(c), where N A increases
slightly even when b/a = 1.4 for r = 0.3. This overall result extends the result of Duan etal. showing a positive relationship between the identification ability and the contagion rate.
In this paper, the degree of sophistication is a counterpart for the identification ability andthe market share is one for the contagion rate. Figure 5 tells us that the result of Duan etal. is carried over to the case of competitive targeting in the sense that the invasion rate of11

If a = b, technology A has no quality advantage, but still has the net advantage because it is the firstmover.

11

the advantageous technology is higher as the firms are able to foresee farther.

5

Conclusion

In this paper, we examined the optimal targeting decisions of duopoly firms and obtained
the result that they must pay attention to the minmax node, not simply to a hub due tothe other competing firm’s response. The general insight provided in this paper and thesimulation method will help to better understand the evolution of competing technologies ina local interaction market with network externalities.

Appendix
Proof of Lemma 1: Let the induced technology partition be (A∗ , B ∗ ). Suppose d(i, i∗B ) ≥2. Then, there must be j ∈ qi,i∗B such that d(i, j) = 1. Now, suppose firm B chooses
j instead of i∗B as a target and let the induced partition be (A0 , B 0 ). Since j is criticalin the communication between i and k for any k ∈ B ∗ in the sense that i and k would be
disconnected without j, it follows that d(i, k) > d(j, k), implying that k ∈ B 0 for any k ∈ B ∗ .
This in turn implies that B ∗ B 0 since j ∈ B 0 but j 6∈ B ∗ . Thus, firm B prefers B 0 to B ∗ .
Contradiction.
Proof of Proposition 1: From Lemma 1, it is clear that i∗B (i) ∈ Ni . It is trivial to see that it
must generate the largest set of his successors.
Proof of Proposition 2: This is trivial, since the game is a constant-sum.
Proof of Proposition 3: Consider the set A1 \ A2 . Let Ã2 = {k ∈ A1 \ A2 | k ← A2 }. If
Ã2 ( A1 \ A2 , it is clear that A2 ∪ Ã2 ( (A1 \ A2 ) ∪ A2 = A1 . Then, it follows that P1 ÂA P2 ,since inequality (4) implies that j ∈ B1 must be invaded for a larger set A1 whenever it
is invaded for a smaller set A2 ∪ Ã2 . If Ã2 = A1 \ A2 , we have P1 ∼A P2 by the Markov
property of the population distribution.

12

References[1] Albert, R., Barabási, A.-L., 2002. Statistical mechanics of complex networks, Reviewsof Modern Physics 74, 47-97.
[2] Arenas, A., Diaz-Guilera, A., Perez, C.J., Vega-Redondo, F., 2000. Self-organized evolution in a socioeconomic environment, Physical Review E 61, 3466-3469.
[3] Barabási, A-L. and Albert, R., 1999, Emergence of Scaling in Random Networks, Science
286, 509-512.
[4] Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U., 2006. Complex
networks: Structure and dynamics, Physics Reports 424, 175-308.
[5] Castellano, C., Fortunato, S., Loreto, V., 2009. Statistical physics of social dynamics,
Reviews of Modern Physics 81, 591-646.
[6] David, P.A., 1985, Clio and the Economics of QWERTY, American Economic Review
75[7] Cohen, R., Havlin, S., Ben-Avraham, D., 2003. Efficient Immunization Strategies for
Computer Networks and Populations, Physical Review Letters 91, 247901.
[8] Duan, W., Chen, Z., Liu, Z., Jin, W., 2005. Efficient target strategies for contagion inscale-free networks, Physical Review E 72, 026133.
[9] Farrell, J. and G. Saloner, 1985, Standardization, Compatibility, and Innovation, Rand
Journal of Economics 16, 70-83[10] Galeotti, A. and S. Goyal, 2009. Influencing the Influencers: a Theory of Strategic
Diffusion, Rand Journal of Economics 40, 509-532[11] Guardiola, X., Diaz-Guilera, A., Perez, C.J., Arenas, A., Llas, M., 2002. Modeling
diffusion of innovations in a social network, Physical Review E 66, 026121.
[12] HoÃlyst, J. A., Kacperski, K., Schweitzer, F., 2000. Phase transitions in social impact
models of opinion formation, Physica A 285, 199-210.

13

[13] HoÃlyst, J. A., Kacperski, K., 2001. Social impact models of opinion dynamics. In: Stauffer, D. (Eds.), Annual Reviews of Computational Physics IX, World Scientific Publishing
Company, pp. 253-273.
[14] Jo, H.-H., Baek, S.K., Moon, H.-T., 2006. Immunization dynamics on a two-layer network model, Physica A 361, 534-542.
[15] Jun, T. and J.-Y. Kim, 2008. A Theory of Consumer Referral, International Journal ofIndustrial Organization 26, 662-278[16] Jun, T., J.-Y. Kim, B. J. Kim and M. Y. Choi, 2006a. “Consumer Referral in a Small
World Network.” Social Networks 28: 232-246.
[17] Jun, T., J.-Y. Kim, B. J. Kim and M. Y. Choi, 2006b. Network Marketing on a Small
World Network. Physica A 360, 493-504.
[18] Kandori, M. and R. Rob, 1998, Bandwagon Effects and Long Run Technology Choice,Games and Economic Behavior 22, 30-60[19] Katz, M. and C. Shapiro, 1985, Network Externalities, Competition, and Compatibility,
American Economic Review 75, 1985, 424-40[20] Kim, J.-Y., 2010, Information Diffusion and δ-Closeness-Centrality, Sociological Theory
and Methods 25, 95-106[21] Lambiotte, R., Redner, S., 2008. Dynamics of non-conservative voters, Europhysics
Letters 82, 18007.
[22] Liggett, T.M., 1999. Stochastic Interacting Systems: Contact, Voter, and Exclusion
Processes, Springer-Verlag, New York.
[23] Llas, M., Gleiser, P.M., Lopez, J.M., Diaz-Guilera, A., 2003. Nonequilibrium phase transition in a model for the propagation of innovations among economic agents, PhysicalReview E 68, 066101.
[24] Mantegna, R.N., Stanley, H.E., 2000. An Introduction to Econophysics, Cambridge
University Press, Cambridge, England.

14[25] Newman, M.E.J., 2005. Threshold Effects for Two Pathogens Spreading on a Network,
Physical Review Letters 95, 108701.
[26] Newman, M.E.J., 2003. The Structure and Function of Complex Networks, SIAM Review 45, 167-256.
[27] Newman, M.E.J., 2002. Assortative Mixing in Networks, Physical Review Letters 89,
208701.
[28] Pastor-Satorras, R., Vespignani, A., 2002. Immunization of complex networks, PhysicalReview E 65, 036104.
[29] Schweitzer, F., Behera, L., 2009. Nonlinear voter models: the transition from invasion
to coexistence, The European Physical Journal B 67, 301-318.
[30] Shao, J., Havlin, S., Stanley, H.E., 2009. Dynamic Opinion Model and Invasion Percolation, Physical Review Letters 103, 018701.
[31] Stauffer, D., 2005. Sociophysics simulations II: opinion dynamics. In: Garrido, P.L.,
Marro, J., Munoz, M.A. (Eds.), Modeling Cooperative Behavior in the Social Sciences,
AIP Conf. Proc. No. 779, AIP, Melville, NY, pp. 56-68.
[32] Sznajd-Weron, K., 2005. Sznajd model and its applications, Acta Physica Polonica B
36, 2537-2547.
[33] Sznajd-Weron, K., Sznajd, J., 2000. Opinion evolution in closed community, International Journal of Modern Physics C 11, 1157-1165.

15

(a)
1
(b)

2

3

4

5

6

7
8

1
2
4

5

6

7

3
(c)

1

9
2

5
3
(d)

6

12

7

4

5 6 7 8

4
9

10

11

12

13

14

15

1 2 3

Figure 1: Examples for (a) a line network, (b) non-hub target, (c) responses of a sophisticatedfirm B, and (d) not a highest central target.

16

(a)

(b)

24

24

40

17

16

16
21

49
41

41

19

15

18

13

28

31
3

51

7

30

29

27

43

45

22

32

25

9

30

29

27

43

45

42
37

12

34

47

7

2

38

12

26

44

46
36

1

48

50

5

23

51

22

32

3

14
11

20

33

2

38

31

8 10

25

9

15

18

13

28

44

46
36

1

48

35

26

23

33

50

5

4
19

14
11

20

8 10

21

49

4

35

40

17

34

47

42
37

6

6

39

39

(c)

(d)
51

51

39

17
46

46

40

32

25

12

10

15

40

32

25

39

17

15

16

12

10

16
2

33

27

5
14

48

33

14
48

20
1

8

27

5

35

3
23

2

3

20
1

8

23

38

41

35

38

41
11

49

31

24

49

21 18

30

45

6

24
21 18

36
42

47

28
43

7

29
37

45

6

28
43

7

26

4

30

42

31
19

13

4

36

47

11

26

19

13

29

9

37

34

50

9

34

50
22

22

44

44

Figure 2: Simulation results on the Barabási-Albert scale-free networks with N = 51 whenb/a = 1.4. Blue and red nodes denote A and B technologies, respectively. Only for thetree, blue and red arrows denote the invasion direction of A and B, respectively. (a) For
myopic firms, (iA , iB ) = (3, 1) results in (N A , N B ) = (39, 12). (b) For sophisticated firms,(iA , iB ) = (3, 7) results in (N A , N B ) = (14, 37). (c) For myopic firms, (iA , iB ) = (4, 1)
results in (N A , N B ) = (17.91, 33.09). (d) For sophisticated firms, (iA , iB ) = (1, 4) results in(N A , N B ) = (19.89, 31.11). The final market shares were averaged over 103 realizations ofimitation dynamics. The networks have been produced with the PAJEK software.

17

3

103

(d) 10

102

NA

NA

(a)

101
100

101

102

101
100

102

ki

(e)

102

NA

NA

10

0

10

1

10

103

102

101
100

1

10

2

10

(f)

NA

NA

103

102

101
100

101

101

102

ki

ki

(c)

102

ki

3

(b)

101

102

103

102

101
100

ki

101

102

ki

Figure 3: The correlations of the degree of A’s target, ki , and the A’s resulting market
share, N A , in case of sophisticated firms. We used (a)-(c) b/a = 0.6 and (d)-(f) b/a = 1.4 for
r = 0.3 (top), 0 (middle), and −0.3 (bottom). We used a scale-free network with N = 500
for each value of r.

18

500

N

A

400

300

200

r=-0.3
r=0
r=0.3

100
0.6

0.8

1

1.2

1.4

b/a
Figure 4: The dependence of N A on the ratio b/a for the sophisticated firms on the BarabásiAlbert scale-free networks with N = 500 over 20 different realizations of networks.

19

(a)

500

(c)

A

400

400

N

N

A

450

500

350

300

200

300

100
0 1 2 3 4 5 6 7 8 950

0 1 2 3 4 5 6 7 8 9

s

(b)

s

500

(d)

500

A

400

400

N

A

450

N

50

350

300

200

300

100
0 1 2 3 4 5 6 7 8 950

0 1 2 3 4 5 6 7 8 9

s

50

s

Figure 5: The dependence of N A on the degree of sophistication s when (a)-(b) b/a = 1
and (c)-(d) b/a = 1.4 for r = 0.3 (top) and −0.3 (bottom). We used the Barabási-Albertscale-free networks with N = 500 over 5 different realizations of networks (represented bydifferent colors). Each point was averaged over 50 realizations of imitation dynamics. s = 0
and s = 50 represent the myopic case and the sophisticated case, respectively.

20