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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
Diffusion Dynamics of Network Technologies With
Bounded Rational Users: Aspiration-Based Learning
Youngmi Jin, Member, IEEE, George Kesidis, Senior Member, IEEE, and Ju Wook Jang, Member, IEEE
Abstract—Recently, economic models have been proposed to
study adoption dynamics of entrant and incumbent technologies
motivated by the need for new network architectures to complement the current Internet. We propose new models of adoption
dynamics of entrant and incumbent technologies among bounded
rational users who choose a satisfying strategy rather than an optimal strategy based on aspiration-based learning. Two models of
adoption dynamics are proposed according to the characteristics
of aspiration level. The impacts of switching cost, the benefit from
entrant and incumbent technologies, and the initial aspiration
level on the adoption dynamics are investigated.
Index Terms—Aspiration, bounded rationality, evolutionary
game theory, technology adoption, technology diffusion dynamics.
I. INTRODUCTION
I
N SPITE of its great success, the current Internet has
generic deficiencies such as security. Recently, there have
been calls for new network architectures that can complement
the features lacking in the current Internet architecture. As the
need for new and viable network technologies arises, the adoption dynamics of a new technology has drawn the attention of
the networking research community [6], [12], [13], [20], [26].
The goal of such studies is to develop models assessing the
viability of a new network technology and explain the diffusion
process of the entrant.
The various models for diffusion of new technologies assume
that users make rational decisions with given information. In
general, users may not have the perfect or complete information necessary for optimal decision making, or they may be
misinformed. It is also possible that users have cognitive limitations or that users just act randomly. That is, users may not
be fully rational, but bounded rational. Many existing works
studying the diffusion of new technologies consider bounded rationality due to limited information, for example, the number of
Manuscript received June 23, 2010; revised February 22, 2011 and January
09, 2012; accepted February 21, 2012; approved by IEEE/ACM TRANSACTIONS
ON NETWORKING Editor D. M. Chiu. Date of publication March 22, 2012;
date of current version February 12, 2013. This work was supported by the
Basic Science Research Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Science and Technology,
under Grant 2010-0006611 and National Science Foundation under CNS Grant
1152320.
Y. Jin is with the Department of Electrical Engineering, KAIST, Daejon 305701, Korea (e-mail: youngmi_jin@kaist.ac.kr).
G. Kesidis is with Pennsylvania State University, University Park, PA 16802
USA (e-mail:kesidis@engr.psu.edu).
J. W. Jang is with the Department of Electronic Engineering, Sogang University, Seoul 121-742, Korea (e-mail: jjang@sogang.ac.kr).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNET.2012.2189891
users adopting a technology [13], [18] or when only local information is available such as that of neighboring nodes without
the information of global network structure) [11], [21], [22].
Among many aspects of bounded rationality, we are interested
here in users’ decision-making behaviors, such as the choice of
a satisfying strategy (an inferior strategy) instead of an optimal
strategy [7], [25].
The theory of satisfying behavior says that players are more
apt to satisfy rather than to optimize [5], [27]: Instead of
choosing a strategy giving the highest payoff, a player sets a
standard representing the payoff level she wants, called aspiration level, and searches a strategy giving her payoff higher
than the aspiration level. Once she finds such a strategy, she
sticks to it even though it does not give her the highest payoff.
The decision-making process itself is called aspiration-based
learning [7], [16].
We start with a basic game model, a coordination game,
that is widely used in the analysis of technology diffusion [9], [11], [14], [21] and propose two aggregate diffusion
models with bounded rational users using common aspiration
level. Most economic literature studies 2 2 (two users and
two strategies) game models in which each user adjusts her own
standard level (aspiration) based on the payoff she has received
users, where
is suffiin the past. Our model considers
ciently large and all users have a common aspiration level as
in [25]. Two aggregate diffusion models with bounded rational
users are proposed: constant aspiration level and time-varying
aspiration level. With a fixed common aspiration, the diffusion
dynamics of an entrant technology can be modeled as a continuous-time Markov process on finite state space. Since there
is uncertainty in decision making of bounded rational users, it
is important to learn from past experience. In aspiration-based
learning, learning from past experience is incorporated into
aspiration level through adaptation [4], [7], [16], [25]. When
the common aspiration level is time-varying, users evaluate
the technologies and adjust the aspiration level in the direction
toward the average payoff as the technology diffusion goes
on. We examine the dynamics between one incumbent and
one entrant and the dynamics among one incumbent and two
entrants, respectively.
The main contribution of this paper is the formulation
and analysis of models of aggregate diffusion dynamics of
bounded rational users. Considering bounded rationality in a
user’s strategy choice, we introduce a new modeling approach
adopted from the economics literature into networking research. The rest of the paper is organized as follows. Section II
summarizes related literature and describes how our work
herein is related to the existing work. In Section III, we formu-
1063-6692/$31.00 © 2012 IEEE
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
late a Markov process model using a fixed common aspiration
level and analyze the user’s adoption behavior through the
Markov process model. In Section IV, we formulate mean
field dynamics between one entrant and one incumbent with
time-varying aspiration update. The equilibrium points of the
dynamics are found, and their stability properties are investigated. Section V studies the mean field diffusion dynamics
when two entrants are introduced to the market with an incumbent. Section VI concludes with summary of the results of the
paper. The proofs of the theorems are placed in the Appendix.
II. RELATED WORK
There are two streams of work relevant to our study: adoption dynamics of technologies and aspiration-based modeling
in economics. Both are briefly discussed in this section.
A. Adoption Dynamics of Technologies
The diffusion dynamics of a new technology are an active
line of research in economics and management science. This
topic has recently received attention from the network research
community for new network technologies for the future Internet.
Chan et al. discussed in [6] why newly proposed secure BGP
protocols are not adopted by ISP in spite of existing security
problems of current BGP. The authors explained the nonadoption of innovative secure BGP protocols by weak adoptability,
the strength of a protocol’s properties in driving the adoption
process. Their methodology assessing the adoptability is based
on users’ incentive-compatible choice of technologies; users
choose a new protocol if the benefit of adopting it is higher than
the transition cost (or switching cost). One important factor in
user benefit is network externality, i.e., how many users adopt a
given technology. In [1], network externality is recognized as a
major factor hindering the deployment of security technologies
as well as information asymmetry. The impact of network externalities is studied in various analytic models. Joseph et al. proposed a “static” economic model in which every user chooses
either a new technology or an old one [13]. A user adopts the
new technology or adopts a converter that enables the user to
use new technology with partial benefit while still using an old
technology (if the new technology gives higher benefit to the
user than the old one). In [12] and [26], the authors formulate dynamic diffusion models among heterogeneous users who
choose a technology giving them highest positive payoff. In
their models, a user chooses no technology if none gives her positive utility. No choice of technologies and the heterogeneous
user assumption result in various equilibrium points—for example, full market penetration of one technology, only one technology survival with partial penetration level, and coexistence
of two technology of full adoption or partial adoption.
Immorlica et al. [11] studied how the entrant can spread to
most of the users, when there exist users who employ “converters” that enable simultaneous use of both technologies.
They identified the payoff structure and the cost of converters
for the pervasive adoption of the entrant in -regular graphs.
Their results depend on the underlying network graph structure, which means neighboring nodes of an individual have
more influence upon her payoff. The impact of an underlying
network graph structure on technology diffusion dynamics is
also studied in [9], [21], and [22].
29
In [18] and [19], security investment is studied. Users on a
network decide to deploy security protection at a positive cost.
Adopting security lowers the probability of loss due to malware.
One interesting result is that if the security technology provides
strong protection against risk, then the incentive to invest in
security reduces as many users adopt security.1 Also, if security
provides weak protection, then critical mass is observed. Two
stable equilibria and one unstable equilibrium are possible.
This paper does not consider the impact of a network graph
structure. We assume that users are fully connected and well
mixed.
B. Aspiration-Based Learning
Bounded rational user behaviors selecting an inferior strategy
instead of an optimal one have been studied to model learning,
experimentation, and evolutionary processes [4], [14], [16].
This approach has been used to select an equilibrium among
multiple Nash equilibria and describe explicit dynamics to
reach the equilibrium through trial and error [14]. Aspiration-based learning is one particular form of such bounded
rational behavior modeling [4], [7], [16], [25]. In the aspiration-based learning, each user chooses her strategy at each
time-step. Each user sets a standard representing the payoff
she hopes to get and compares her choice to the standard at
each time-step. The standard is called aspiration. If the payoff
from her current choice exceeds the standard, then she keeps
the choice at the next step. Otherwise, she drops the choice and
chooses another alternative with some positive probability at
the next time-step. Aspiration-based learning has been studied
extensively to explain the convergence to the Pareto-optimal
point in prisoner’s dilemma game. In the economic literature,
many assume a 2 2 game where each user adjusts her own
aspiration level. Our work is motivated by [25], which studies
the emergence of cooperation when two randomly chosen
players among a large population repeatedly match and play a
prisoner’s dilemma game with time-varying common aspiration
level. We assume that all users have common aspiration level
as in [25] and model the cases when the common aspiration
level is constant or time-varying.
III. MARKOV PROCESS MODEL WITH FIXED ASPIRATION
A. Model
Consider a fixed population of users and two network technologies, labeled
and , representing a new entrant technology and an incumbent, respectively. We assume that all users
adopt technology at the initial time, and technology
has
technical features that technology
does not provide for the
users. Once the entrant technology is introduced to the market,
each user adopts either technology or technology .
Let
be user ’s technology choice and
be the other users’ choice.
, for given : If is , then
,
Let
1A similar phenomenon has been observed for inoculation uptake in studies
of the epidemiology of infectious disease. The probability of side effects of inoculation may have greater associated risk than that of infection when a critical
mass of the population is already inoculated against the disease. Therefore, inoculation uptake is disincentivized when a critical mass of the population is not
infectious.
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
PAYOFF MATRIX IN A 2
TABLE I
2 COORDINATION GAME:
and vice versa. The payoff of user ’s choice
is
users’ choice
under other
where
is a payoff in 2 2 coordination game of which
payoffs are given by Table I. We assume that
in the sense
that new technology provides more desirable technical features
for users than technology .
We further assume that every user repeatedly makes a choice
between and . Let
be the number of users adopting ,
the number of users choosing at time . Each user
and
independently makes her decision with rate 1. We will denote
the th decision time of user by
. The subscript will be
omitted when is obvious.
Now, fix a user and consider how she makes a decision.
As in the introduction section, the user does not seek the best
strategy, but a strategy satisfying her. An aspiration level is the
criterion to determine whether she is satisfied. If her payoff with
the current choice is above or equal to the aspiration level, then
she is satisfied with her decision. Hence, she does not want
to change her choice at the next opportunity, even though she
might get a higher payoff by switching her choice. It is not difficult to find such user behavior in everyday life. Even if a new
product or technology is available, users often do not try the new
one if they are satisfied by the present one they use.
Let and
be the strategy and aspiration level of user at
time , respectively. At time
, a user compares her utility at
with the aspiration level
. She keeps
if
. Otherwise, she switches her choice to the other technology
. More precisely, the strategy of
with probability ,
user at time
is
with prob.
with prob.
if
if
if
where
. In economics literature,
is called inertia,
and switching probability [7]. If the switching cost is high,
users will not tend to switch easily.
In this section, we will assume that all users have common
aspiration level and that it is constant:
for all and
. The assumption of constant aspiration level might be strong
since, in general, users reevaluate their expectation of the available technologies after using them. However, this assumption
of constant aspiration level explains a widespread (but not always true) belief that a new and better technology will eventually dominate. Moreover, it reveals the difference between the
modeling of bounded rational behavior and that of fully rational
behavior: We will see a counterintuitive thought that higher aspiration level takes a longer time for full adoption of the entrant
than lower aspiration level, which comes from bounded rationality and lack of memory.
With the assumption of common constant aspiration level,
becomes a continuous-time Markov process on the state
space
as follows. Suppose that
at
time . Then, there are users choosing
and
users
gets
choosing . Therefore, at time , any user choosing
utility
. Let
. If
, then a user adopting is disappointed by and
tends to switch from technology to technology with probability . Hence, the system transits from state to state
with transition rate . If
, users adopting
are
satisfied by
and are not willing to change their technology.
Hence, the system stays in state . The transition rate from
state
to state
is
if
otherwise
Similarly, users choosing
transition rate from state
(1)
have
and the
to state
if
otherwise.
is
(2)
B. Analysis
With the model in Section III-A, we examine the limiting
behavior of
and the average time to reach state starting
from state 0.
If
, then
: There is no transition if all users
adopt technology at the initial time. Since all users are satisfied with current technology, they do not need to switch to the
entrant, even though new technology is available. Network externality (sometimes called network effect) is one of the essential parameters to explain of nonadoption or a failure of wide deployment of a new and superior technology [6], [12], [13], [26].
Besides network externality and the size of installed base, our
model provides an explanation to nonadoption of a new and superior technology, the degree of user disappointment (or satisfaction): If users are satisfied with the current incumbent, they
do not want to try something new.
Suppose that
. First consider the case that
for all . From (1), this happens only if
,
. If
, then
which is equivalent to
for all
and
for all
(since
). The resulting
, is irreducible, and
transition rate matrix,
the invariant distribution such that
for this case is
(see [15])
for
The invariant distributions are depicted in Fig. 1.
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
31
The average hitting time can be obtained by solving the system
of linear equations (see [23, Theorem 3.3.3])
(4)
for
(5)
for
Fig. 1. Invariant distribution
when
(6)
.
Since
for all , changing states continually occur
from one state to either
or
without absorbing
into a particular state. The birth–death dynamics are observed
because neither of the two technologies satisfies the users and
the users are bounded rational without memory. If users are
fully rational (i.e., they choose the optimal strategy), they will
choose the entrant giving them the higher utility when neither
technology satisfies them, under the circumstance when they
should choose one. If users have memory of their strategy and
payoff history, such cyclic behavior might not be observed. Incorporation of prior experience of an individual into her decision making can be achieved through the adaptation of aspiration level [4], [7], [16]. That is, a user observes her payoff and
(or) the payoff of others resulting from current decision making
and sets a new standard (new adjusted aspiration level) based
on her experience. Adaptation of aspiration level and diffusion
dynamics with time-varying aspiration level will be studied in
Section IV.
Suppose that
. Let
Solving the system of linear equations (4)–(6), we have the
following result that tells us how much time is taken for the
entrant to take full adoption of it.
Theorem 2: The average hitting time to state starting from
state 0 is
(7)
and
, the average hitting time to state
, is given by
from state , for
for
for
where
. Note that this is a unique integer such that
. The transition rates are now
for
for
for
and the transition rate diagram is shown in Fig. 2. It can be easily
and
checked that the invariant distributions are
for
. Thus, we have the following result.
Theorem 1: If
, then
.
can be interpreted as follows. The aspiration
That
level higher than implies that there is real market demand
for the entrant, and the fact
means that the users do
not want a new technology better than technology . Hence,
technology
is the one the market seeks, and ultimately all
users adopt it.
Our next point of interest is the average time to reach the
state
from state 0. The hitting time to the state
starting
from state for a given sample path is defined as
Theorem 2 tells us that it takes an exponentially proportional
and a linearly proporamount of time from state 0 to state
tional amount of time from state to state . As in Theorem 2,
the average hitting time to state from state 0 depends on the
value of that in turn depends on the aspiration level (or ,
when is fixed). A question arises: How does the average hitting time vary with (or )? Fig. 3 shows that
is an in. Since increases
creasing function of for the case
as the common aspiration level increases, high aspiration level
results that it takes longer to reach to the state where all users
adopt . The rationale is as follows. From (7), we can express
where
(3)
where
is the number of users adopting
average hitting time is
at
and the
Once the system reaches state , it stochastically moves toward
the absorbing state without returning to state 0. However, the
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
Fig. 2. Transition rate diagram for
.
level, taking into account the payoff of her own and (or) payoff
of others in evolutionary game theory [4], [7], [14], [16]. In a
lot of prior work, the aspiration level of a single user (call her )
is updated in discrete time as
Fig. 3. Average hitting time with
,
.
system moves back and forth to reach state
exponential increase (note the term in
, which results in
) in time.
IV. MEAN FIELD MODEL
A fixed aspiration level may not be a realistic a standard for
decision making, as we have observed in Section III. It may be
set too low or too high when users have no experience using a
new technology. Therefore, it is necessary to adapt aspiration
levels that take into account the learning from the past decision making as real-world experiences accumulate with time.
This section studies technology diffusion dynamics with timevarying aspiration level.
As the aspiration adapts with time, the degree of disappointment, expressed by the difference between the aspiration level
and the received utility, also becomes time-varying. In general,
users tend to switch their choice with high probability when the
disappointment is big as Axelrod wrote, “… what works well for
a player is more likely to be used again, whereas what turns out
poorly is more likely to be discarded,” in [3]. This behavior is
incorporated into the switching probability in aspiration-based
learning [7], [16], [25]. We consider a switching probability
“function” depending on the degree of disappointment rather
than a constant switching probability: the bigger disappointed,
the higher switching probability. This section investigates the
adoption dynamics when the aspiration level is evolutionary but
all users have the common time-varying aspiration level. With
time-varying common aspiration level, the diffusion dynamics
of a new technology are modeled as a dynamical system consisting of two differential equations when the population size is
big.
A. Time-Varying Aspiration and Switching Probability
Function
To incorporate prior experience of an individual into her decision making, a user sets a new standard, i.e., new aspiration
(8)
where
and
denote her payoff and aspiration level
at time , respectively, and
a weight function
(see [7] and [16]). Note that current aspiration level
depends on the past aspiration level
and the past utility
in (8). The aspiration level
represents a payoff
level that a player hopes to get based on her past history of utility
up to time .
In the environment considered in this paper, users observe
experiences of other users and share their aspiration levels. We
assume that they imitate each other’s aspiration level with the
same weight on each individual. These dynamics are possible
owing to our assumption that users are located on a fully connected network. “Moreover,” we assume that the average aspiration level is delivered to every user as soon as there is a change
on the value of the current aspiration level, without delay. To
consider the evolution of common aspiration level, we assume
that
is constant and every user simultaneously
adapts her aspiration level at the time
(in discrete time unit)
with the adaptation rule (8). Then, by summing (8) over and
dividing the summation by , the average of updated new aspiration level at time
is
where
and
, the total
sum of payoffs, and
is a constant . Based on the above
reasoning, we assume that the change on the common aspiration level in a short time interval is proportional to the quantity
. An update of common aspiration levels according
to some population statistics was proposed in [25]. In theoretic
biology [8], [24] where the emergence of cooperation among
selfish genes is actively studied, average payoff, an example of
population statistics, plays the role of standard for rational decision making. In this case, a common aspiration level is average payoff itself: That is, there is no adaptation procedure for
aspiration.
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
33
The first term is the amount of decrease in the number of
users choosing the entrant, i.e., the number of users who switch
from technology to technology , and the second term is the
number of users who switch from the incumbent to the entrant.
Recall from Section IV-A that the change on the common aspiration level in a short time interval is proportional to
where
is the total sum of payoffs at time
Fig. 4. h(z).
Note that
A switching probability function
continuously differentiable such that
is nondecreasing and
if
otherwise
, which is used in [16] and [25].
with
Note that
for
. Also,
is bounded for
all
since
and
is continuous.
Under high switching cost, users are reluctant to change their
past decision. The upper bound of
can be interpreted in
this context, and switching cost is incorporated into the value of
[7]. A general -shaped
is depicted in Fig. 4.
We assume that a user changes her technology choice with
probability proportional to her degree of disappointment, i.e.,
(aspiration—utility). The value of represents how much a user
is disappointed,
where
is a utility and
is an aspiration of a user. In this section
is an average
aggregate utility and
is a common aspiration level to study
aggregate dynamics.
is a Markov process defined on
. Using Kurtz’s Theorem [17], we
will see the dynamics of
can be approximated as a set of deterministic differential equations when
is sufficiently large.
First, note that if we let
,
,
and
and assume that is sufficiently large that
, then by (9)
Since the change of the common aspiration level in a short
time period is proportional to
,
the evolution of the common aspiration level can be modeled as
The average payoff
is
Thus, the dynamics can be modeled as
B. Modeling
Let
be the number of users who adopt the entrant and
be the common aspiration level at time , when there are
total
users. We assume that the common aspiration level is
bounded, i.e.,
for all .
Consider how the number of users adopting
changes. A
user choosing the entrant
gets the utility
A user choosing the incumbent
has the utility
(10)
on
and
.
Indeed the solutions of the above differential equations are
the deterministic approximation of
as we will
see. We assume that
takes discrete values on the space
such that
.
However, even if
is real (continuum) valued, the following
results still hold.
Let
be the utility of a user adopting the entrant,
be the utility of a user adopting the incumbent, and
be total
sum of the payoffs when users choose the entrant; that is
at time . Note that user utility linearly increases as the number
of adopters of a given technology increases. At time , a user
choosing
changes her technology choice with probability
and a user choosing changes her technology choice with probability
. Therefore, the change in
in a short time interval is
(9)
Let
process (since
. Then,
is a Markov
is a Markov process) defined on
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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
the state space
and
.
Since a user choosing changes her technology choice (to
the entrant) with probability
with rate
Since a user choosing changes her technology choice (to the
incumbent) with probability
with rate
Recall that the change on the common aspiration level in a
short time interval is proportional to
. More
is positive, then the updating
specifically, if
probability of is proportional to
. If
is negative, then the updating probability is proportional
to
. Therefore
with rate
with rate
By Kurtz’s theorem [17] (see also the Appendix), we have the
following result.
Theorem 3: For a fixed time interval
and
,
the Markov chain
converges almost surely to
, which is the solution of the (deterministic) differential (10).
We study the dynamics of the dynamical system defined by
(10). From now on, in (10) is replaced by . Note that
is a
stochastic process, but
is the deterministic approximation
of common aspiration
in the above theorem. An
equilibrium
is the point at which
and
. At an equilibrium point, the number of users who are
satisfied with their chosen technology equals with the number
of adopters of the technology with the aspiration level equal to
the average aggregate payoff.
C. Analysis
Clearly,
and
are the equilibria of the dynamical
system of (10). We call
and
trivial (boundary) equilibria and
with
a nontrivial (interior)
equilibrium.
Theorem 4: The equilibria of (10) are
,
, and
.
By the above theorem, the dynamical system has three equilibrium points,
,
,
: full market penetration of the incumbent, full market penetration of the entrant,
and coexistence of both technologies. When the two technologies share the market, the portion of the market share taken by
the entrant is
since is bigger than . Hence, the incumbent has larger share than the entrant. Note that
,
the aspiration level at the nontrivial equilibrium, is less than the
payoff of the incumbent.
Theorem 5: The equilibrium point
is unstable
while
and
are locally stable.
Since
is unstable, the adoption dynamics will
reach either a pure win of the entrant or a pure win of the
incumbent with probability 1.
A similar result is also observed in [18] and [19], which
studies the impact of network externality on security investment
in networks. In their models, users on a Erdös–Renyi graph
decide to deploy security protection at positive cost to reduce
probability of loss. When the security protection is not strong,
there are three deployment levels as equilibria: 0, ,
with
, where 1 means full deployment. No deployment and deployment level
are stable, but
is unstable.
Note that , the Pareto-optimal equilibrium, is strictly less than
one, while full deployment of the new (better) technology is the
Pareto-optimal point in our case. This difference comes from
the unique characteristics of network externality in security. In
a security deployment case, when many users invest in security,
the incentive for security investment decreases, and the benefit
(network externality) from security deployment is greater to the
security nonadopters than the benefit to the security adopters,
which results in partial security deployment.
Regarding Theorem 5, a different model of bounded rational
users may show different dynamics. For example, the -person
coordination game model in [14] exhibits a different long-run
equilibrium. In that model, at a discrete time unit
,
a user chooses the best response strategy with probability
, according to the information at time , and chooses
the other strategy with probability
. It is shown that if
the total population size is greater than 2, then the long-run
equilibrium (the equilibrium as
is the pure win of the
entrant, the better technology, with probability 1 as goes to 0
(see [14, Corollary 2]).
The mean field model results in a diffusion dynamics different from that of Markov process model in Section III. When
, in the Markov process model, eventually all the
users adopt the entrant technology X. However, in the mean field
model, the asymptotic behavior does not always converge to the
state of full adoption of technology , even
. The initial aspiration level and switching probability, where switching
probability depends on switching cost and the degree of disappointment, determine the asymptotic state
. When
, the Markov process model exhibits a cyclic behavior
without absorbing into one state, but repeatedly moving from
one state to another. The mean field model does not show such
a cyclic behavior because the aspiration level is adjusted when
the initial aspiration level is too high so that no technology satisfies users. In the Markov process model with fixed aspiration
level, when
, the hitting time to full adoption of the
entrant is increasing with the fixed aspiration level. However,
in the mean field model, the time needed to reach full adoption of the entrant is decreasing if the initial aspiration level is
high, as we will see in Section IV-D This phenomenological difference comes from the fact that the switching probability depends on the degree of disappointment and the aspiration level
is adapted according to the past users’ decision making. The
proof of Theorem 5 shows that there is no closed orbit, which
means that every trajectory of the dynamical system governed
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
35
by (10) converges to a limit. Note that the limit of a trajectory
is not unique, but depends on the initial value of
and .
D. Numerical Examples
We conducted numerical experiments to investigate how the
values of , , and
affect the adoption dynamics. In our
scenario, all users adopt the incumbent technology at the initial
time,
, and the entrant technology is introduced with
: When
, clearly no one has the need to try the
new technology. We used
and the switching probability
function
if
otherwise
in all experiments. This function was used in [25].
Figs. 5–7 show the impact of , the payoff from the entrant, on
the limiting behavior of diffusion dynamics. We used
and
. In Fig. 5, where
, the entrant does not survive with any initial aspiration level
. Recall that
in the Markov process model,
guarantees the elimination
of the incumbent. The failure of the entrant superior to the incumbent is explained by the network externality (or the size of
installed base) of the incumbent as in other work [6], [12], [13].
As increases, initial aspiration level determines the asymptotic
behavior of dynamics like Fig. 6: The entrant technology dies
out for
, but it defeats the incumbent if
.
If further increases as in Fig. 7, most of the trajectories with
converge to the state of the entrant’s full market
penetration. We see that high enables the entrant to survive
with high probability, as expected.
To examine the impact of the switching cost, the upper
bound of
, on the adoption dynamics, we do the same
experiments with the same parameters used in Figs. 5–7 except
the value of . Recall that depends on switching cost and high
switching cost means low value of . The numerical experiments with
are presented in Figs. 8–10. In Fig. 8, the
entrant with
perishes in the market for all
,
while the entrant wipes out the incumbent for
when
as in Fig. 6. In Fig. 10, when
, the trajectories
with
converge to the state of full adoption of the
incumbent . Note that most of trajectories converge to the
state of full adoption of the entrant if
in Fig. 7. The
entrant should provide higher benefit in a market with high
switching cost than with low switching cost in a market.
Fig. 11 shows the smallest number of iterations necessary to
reach the state of full adoption of the entrant from the initial state
where there is no user choosing the entrant, for various initial
aspiration levels. Since the number of iterations is proportional
to time in this case, Fig. 11 shows that the time to reach the state
of the entrant’s dominance is decreasing in the initial aspiration
level. This is different from the observation in Section III-B in
which the average hitting time to the state of full dominance
of the entrant is increasing (see Fig. 3). The difference mainly
comes from the switching probability function (bigger disappointment, higher rate switching) and the evolving aspiration
(learning from the past).
From these numerical examples, we see that the entrant can
take the full adoption with high probability if
is high. High
Fig. 5. Diffusion dynamics:
Diamond: interior equilibrium.
and
. Circle: limit of a trajectory.
Fig. 6. Diffusion dynamics:
Diamond: interior equilibrium.
and
. Circle: limit of a trajectory.
Fig. 7. Diffusion dynamics:
Diamond: interior equilibrium.
and
. Circle: limit of a trajectory.
causes bigger disappointment with the incumbent than
low
, and big disappointment expedites the diffusion of the
entrant.
V. TWO COMPETING EMERGING TECHNOLOGIES AND ONE
INCUMBENT
This section studies the adoption dynamics of two entrant
technologies and one incumbent one among users.
We denote the entrants
and
with
and
. The incumbent is denoted by
with
. We assume that
for
and
.
is the number of users choosing
technology
and
at time and is sufficiently
large.
36
Fig. 8. Diffusion dynamics:
Diamond: interior equilibrium.
Fig. 9. Diffusion dynamics:
Diamond: interior equilibrium.
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
and
and
. Circle: limit of a trajectory.
. Circle: limit of a trajectory.
Fig. 12. Diffusion dynamics with two entrants
switching cost (circles are limits of trajectories).
and
with low
Fig. 13. Diffusion dynamics with two entrants
switching cost (circles are limits of trajectories).
and
with high
Each user adopts only one technology of the three
’s.
. At time , any user who adopts
Hence,
receives the utility
for
.
A user choosing
switches to
,
with switching
probability
if she is not satisfied with
where
and
. The change in
in
a short time interval for
is similarly modeled as
The average payoff function is
Fig. 10. Diffusion dynamics:
Diamond: interior equilibrium.
and
. Circle: limit of a trajectory.
If we assume
for all and , as in Section IV, we have
for
Fig. 11. Hitting time (number of iterations) to reach the state
.
Clearly, there are three equilibrium points,
,
and
, for the dynamical system.
Figs. 12 and 13 show the adoption dynamics of an incumbent
and two entrants when
,
. If switching cost is low
as in Fig. 12, then either
or the incumbent is adopted by all
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
Fig. 14.
dominates for
(circles are limits of trajectories).
37
and
increase together until
. Once the superior entrant gets bigger adoption than 0.5, then the inferior entrant loses
its market and technology
eventually becomes the winner
of the market. It is worthwhile to note the adoption dynamics
around
. Since trajectories for
in Fig. 14 are close to
, small perturbation can drastically change the limiting behavior. In our experiment, we increased
to 0.51 when
. Our intentional perturbation is shown in Fig. 16, showing that the inferior entrant
eventually gets full adoption. What this implies is that near the
point (0.5, 0.5), the inferior entrant can effectively employ free
seeding (providing a technology or sample products at a low
price, discounted rate, or free of charge) to encourage users to
adopt it with the purpose of winning the market. As in Fig. 16,
the winner of the market changes after the free seeding strategy
of the inferior entrant. In this case, the two entrants will experience an intensive competition to win the market. We had same
observation with high switching cost.
VI. CONCLUSION
Fig. 15.
dominates for
(circles are limits of trajectories).
Fig. 16.
dominates for
(circles are limits of trajectories).
users depending on the initial aspiration level: If the initial aspiration is high, then the superior entrant takes full adoption. If
the initial aspiration is low, then the incumbent takes full adoption and the entrants perish. Note that
does not survive in
either case. Fig. 13 shows that if the switching cost is high, the
entrants are defeated by the incumbent.
Comparing the case of Fig. 13 to that of Fig. 9, we learn
the impact of two entrants on the market: The competition of
two entrants can hinder the adoption of both new technologies.
In Fig. 9, the entrant can survive if the aspiration level is high
enough
. However, in Fig. 13 (when there are two
entrants), no new technology defeats the incumbent even for the
case
with
and
. This shows
that if switching cost is high, the inferior entrant can hinder the
adoption of the superior one.
Figs. 14–16 illustrate the adoption dynamics when
is small. In Fig. 14, with sufficiently high initial aspiration,
In this paper, we proposed models of diffusion dynamics
among bounded rational users on a fully connected network
graph. In particular, we used aspiration-based learning in
which a user chooses a strategy that satisfies her rather than
an optimizing strategy. We suggested a birth–death Markov
process model and a mean field model depending on the characteristics of the assumed common aspiration. Our Markov
process model, with assumed constant aspiration, explained
how the incremental deployment guarantees the success of an
entrant technology. In the mean field model, where common
aspiration level adapts with time toward averaged payoff,
the diffusion dynamics show various behaviors depending on
switching cost and initial aspiration level as well as and .
The model shows clearly the impact of switching cost, the
payoffs of two technologies, and the initial aspiration level on
the diffusion dynamics. Through numerical experiments, we
showed that high switching cost prevents users from adopting
the new technology, and initial high aspiration has the users
more dissatisfied by the incumbent, hence encourage the users
to switch to the entrant.
We also numerically examined the diffusion dynamics with
one incumbent and two entrants. If there is little difference between the payoffs from the two entrants, then the competition
between two entrants can be intensive. If the difference in payoffs from the two entrants is notable, then either the superior
entrant or the incumbent is adopted by all users. Under high
switching cost, the competition between two entrants causes
them to perish in the market.
One interesting research direction of this work is to investigate how the underlying network graph structures affect the diffusion dynamics when aspiration-based learning is employed.
Users put more weight on the payoffs resulting from the interactions with their neighbors and less or no weight on the payoff
from the interactions with non-neighbors. In this case, common
aspiration level may not be available since users can observe
only their neighbors.
38
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
APPENDIX
Proof of Theorem 2: From (6) and
Note that there are two equations for
, one from (11) and
the other from (12). They give
in the theorem statement.
, we have
(11)
for
.
Using induction, we can show that for
Proof of Theorem 4: The proof consists of two parts: We
will show that
is an equilibrium point, and then there
is no equilibrium point other than
,
,
.
Suppose that
is an equilibrium point where
and
. Letting
, we have
(16)
(12)
By (4), we have
. Hence, (12) holds for
Note that
. Suppose that (12) holds with . For
substituting (12) into (5), we have
if we let
as in (12).
From
for
. Therefore,
.
,
Letting
, we have two cases.
— Case 1:
.
— Case 2:
.
Case 1: Suppose that
.
Note that
holds if and only
since
is a nonnegative
function. Hence, we have
and
by the definition of
. From
and (16), we have
(13)
can be expressed
(17)
From
in (13)
and (16), we have
(18)
(14)
for
.
We will show that
(15)
The only possible value simultaneously satisfying (17) and (18)
is
. Therefore,
is the equilibrium point for
Case 1.
Case 2: Suppose that
is an equilibrium
point where
,
and
. Respectively from
and
,
we have
from
, (15) holds
by induction. Since
for
. Suppose that this holds for . Using the expression
of
of (13) and (14),
becomes
Since
, we have
. From
and
, we have
or
From
(19)
, we have
or
From (14), for
which gives
, we have
(20)
However, there is no satisfying (19) and (20) simultaneously.
From the results of Cases 1 and 2,
is a
unique “interior” nontrivial equilibrium point of the dynamical
system.
To prove that there are only three equilibrium points,
,
, and
, we examine the possibility of a
“boundary” equilibrium point that is neither
nor
.
If
, then
should be . Hence, there is no equilibrium of the form
except
. With the same argument, there is no equilibrium of the form
except
.
The only possible equilibrium on the boundary takes the form
. However, it is impossible since the value of
for nonnegative .
JIN et al.: DIFFUSION DYNAMICS OF NETWORK TECHNOLOGIES WITH BOUNDED RATIONAL USERS
Proof of Theorem 5: To investigate the stability property at
an equilibrium point, we examine the eigenvalues of the Jacobian matrix
of the dynamical system governed by (10) at
the equilibrium points (by Hartman–Grobman Theorem). The
Jacobian matrix at is
where
in (10).
At equilibrium point
and
for
. The eigenvalues of
are
and
. Therefore,
is locally stable.
At equilibrium point
, the Jacobian matrix
has
eigenvalues
. Hence,
is locally
stable.
At the equilibrium point
,
has
eigenvalues
and -1. Hence, the eigenvalues
of the Jacobian matrix
do not determine the stability at
. To show that
is unstable, we
first show that there is no orbit encircling
using
Liouville’s theorem in the theory of ordinary differential equations (see [2, p. 198]).
Liouville’s Theorem: Let
for
be a
dynamical system defined on an open set
where
is
differentiable with respect to
. Let
and
and
denote the volume of
. Then
39
Since there are two locally stable equilibrium points (0, 1) and
, there should be a separatrix that demarcates the basins of
attraction of (0, 1) and
. The separatrix should be either a
closed orbit encircling or a path that passes through an unstable
equilibrium (see [10]). However, as we have seen, there is no
closed orbit. Hence, the separatrix demarcating the basins of
attraction should pass through some unstable equilibrium point.
Since we have only one equilibrium
, the separatrix
should pass it, and it must be an unstable one.
Kurt’z Theorem (See [17, Sec. 3]): Let
be the
-dimensional integer lattice. For positive integer
,
is a Markov chain with
state space
. Suppose that
there exist nonnegative functions
and positive
constants , such that
since
for
state
Let
where
If
,
where
to state
is the transition rate from
.
.
then
almost surely
where
.
Since the trace of
is negative, the
dynamical system defined by (10) cannot have a closed orbit that
encircles the equilibrium
since any closed orbit has
positive volume (area).
Moreover, the dynamical system does not have any closed
orbit in the
space,
. To show that
there is no closed orbit in
, we need the following result: Index Theorem: Any closed orbit enclosing
an open set
should have a singular (equilibrium) point in
(see [2, pp. 254–256]).
If there is a closed orbit induced by the dynamical system,
then the index or winding number of the closed orbit in
should be positive and the orbit should have at least
one equilibrium by the above theorem. However, there is no
equilibrium except
, and there is no orbit enclosing
.
for every
.
Proof of Theorem 3: Let
easily checked that
,
. It can be
40
IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 21, NO. 1, FEBRUARY 2013
where
and
. Indeed,
has its maximum on
since
and
is continuous.
Note that
if
,
which results in
. Note that
,
, which
satisfies the Lipschitz continuity condition
for any and in bounded open set over . Since
, the Lipschitz condition holds for any
,
.
ACKNOWLEDGMENT
The authors would like to thank the associate editor, the
anonymous referees, and Prof. R. Guérin for their valuable
comments.
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Youngmi Jin (M’05) received the B.S. and M.S.
degrees in mathematics from the Korea Advanced
Institute of Science and Technology (KAIST),
Daejeon, Korea, in 1991 and 1993, respectively, and
the M.S. and Ph.D. degrees in electrical engineering
from Pennsylvania State University, University
Park, in 2002 and 2005, respectively.
She was a Member of Technical Staff with KT,
Seoul, Korea, where she did research on traffic
engineering and broadband wireless access (IEEE
802.16) systems. From 2007 to 2008, she was a
Postdoctoral Researcher with the University of Pennsylvania, Philadephia.
She is currently with the Department of Electrical Engineering, KAIST, as a
Research Professor. Her research interests include Internet economics, Internet
pricing, incentive engineering, P2P networks, wireless networks, and social
networks.
George Kesidis (M’90–SM’93) received the Ph.D.
degree in electrical engineering and computer science
from the University of California, Berkeley, in 1992.
He was a Professor with the Electrical and
Computer Engineering Department, University
of Waterloo, Waterloo, ON, Canada, from 1992
to 2000. Since 2000, he has taught with both the
Computer Science and Engineering and Electrical
Engineering Departments of Pennsylvania State
University, University Park. His research experience
includes several areas of communication networking
and machine learning.
Dr. Kesidis served as the Technical Program Committee Co-Chair of IEEE
INFOCOM 2007 and is now serving on the Editorial Boards of the ACM Transactions on Modeling and Computer Simulation and the IEEE Communications
Surveys and Tutorials.
Ju Wook Jang (M’11) received the B.S. degree in
electronic engineering from Seoul National University, Seoul, Korea, the M.S. degree in electrical
engineering from the Korea Advanced Institute of
Science and Technology (KAIST), Daejeon, Korea,
and the Ph.D. degree in electrical engineering from
the University of Southern California (USC), Los
Angeles.
From 1985 to 1988 and 1993 to 1994, he was with
Samsung Electronics, Suwon, Korea, where he was
involved in the development of a 1.5-Mb/s video
codec and a parallel computer. Since 1995, he has been with Sogang University,
Seoul, Korea, where he is currently a Professor. His current research interests
include WiMAX protocols, mobile networks, and next-generation networks.