A dynimic model of user behaviour in a social network site

A Dynamic Model of User Behavior
in a Social Network Site∗
Dae-Yong Ahn
School of Marketing
University of Technology Sydney

Randal Watson
Department of Economics
University of Texas at Austin

April 16, 2010

Abstract
This paper estimates a dynamic model of user behavior in a social network site using unique data on the daily login activity of a
sample of members of MySpace.com. We view a social network as a
stock of capital that yields a flow of utilities over time by creating
interactions between the site user and her friends. This capital stock
can be maintained and expanded by logging in to the site and communicating with other members. The user’s login decision is thus a
forward-looking one, which we model in the framework of a dynamic
discrete-choice model. We allow for two sources of persistence in members’ login decisions: state dependence and unobserved heterogeneity.
We found two distinct types of consumers as regards utility, cost, and
∗

We are grateful to Susan Broniarczyk, Romana Khan, Vijay Mahajan, Om
Narasimhan, Raghunath Rao, and Garrett Sonnier for their comments, suggestions, and
corrections.

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state transition. Across types, real-time chat and messaging, features
of MySpace.com, positively affect the usage decision. We use our parameter estimates to perform counterfactual simulations with the goal
of providing site managers with ways to enhance firm performance.
Keywords: dynamic discrete choice, online social networks, unobserved heterogeneity

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1

Introduction

This paper structurally estimates a dynamic model of user behavior in a
social network site using data from MySpace.com. We view a social network
as a stock of capital that yields a flow of utilities over time by creating social
interactions between the owner and her friends. Using a social network site
may have two effects on this stock of capital: (1) maintaining the quality of
interactions with an existing group of friends, and (2) expanding the quantity
of interactions with the addition of new friends to the network. Since both of
these effects will yield a higher flow of utilities in the future, it is appropriate
to analyze usage of a social network site with a model that takes into account
consumers’ forward-looking behavior. This is the task addressed here. We
use our model to measure how features of a social network site affect usage,
and to evaluate counterfactual policies that are managerially relevant.
As their popularity grows among consumers, social network sites are attracting an increasing share of advertising expenditures. Thirty-three percent

of adult US internet users, and 70% of teenagers, logged in to a social network
site at least monthly in 2007. A market research firm eMarketer projects that
‘50% of online adults and 84% of online teens in the US will use social networking by 2011’. To tap into this audience, marketers spent $1.2 billion on
social network sites in 2007 and this figure is expected to climb to $3.6 billion
by 2010 worldwide (eMarketer, December 2007). Industries that advertise in
social network sites range from entertainment (25.2%), retail goods and services (17.6%), and telecommunications (16.2%) to financial services (6.3%)
and automotive (5.1%) (Nielsen/NetRatings, September 2006).1
eMarketer reports that MySpace.com and Facebook.com, the two largest
US social network sites, accounted for 72% of social network advertising
spending in 2007 (Marketing News, July 15 2008). Media giants such as
NBC and Warner Bros. host sites on MySpace, while Coca-Cola, CBS, and
Chase promote their products on Facebook. On the other hand there is also
a growing trend towards niche social network sites and marketer-sponsored
sites that attract ‘a smaller, but passionate audience’ rather than the ‘diverse membership’ of MySpace and Facebook (CNN.com, April 16 2008).
Examples include petside.com by Proctor & Gamble for its Iams pet foods
and artofcookie.com by Campbell Soup Company for its Pepperidge Farm
1
The numbers in parentheses are industries’ shares of the total advertising dollars spent
on social network sites.

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cookies.
Social network sites compete on features – instant messaging, video chat,
etc. – in order to attract more users and thereby bring in more advertising
revenue.2 In this paper we assess the impact of two such features of the
MySpace site – real-time chat, and a messaging or mailbox feature. We
estimate a single-agent, dynamic discrete-choice model in which usage is

defined as a decision of whether or not to log in to the site. This decision
relates closely to the number of unique users that a site attracts, a benchmark
used in the industry to rank sites. Our estimates allow us to propose and
evaluate other features that social network sites might adopt to enhance firm
performance. In particular we estimate the effects on site usage of policies
designed to enhance the networking experience while online, and to expand
a user’s network.
Our data are observations on a random sample of college-aged members
of MySpace. We collected data from their webpages on a daily basis for four
weeks, recording three types of variables for each member: usage behavior,
social interactions, and the evolution of the social network. For usage behavior we recorded whether or not one used MySpace for each member each
day. For social interactions we recorded real-time chat and messaging, two
features of MySpace, for each member each day.3 We tracked changes in
members’ social networks by recording the size of each member’s network
each day.
Our estimation strategy is based on the MPEC (Mathematical Programming with Equilibrium Constraints) approach (Su and Judd 2008).4 This approach alleviates the computational burden in estimating a dynamic discretechoice model by formulating the NFXP (Nested Fixed Point) approach as a
constrained optimization problem. We allow for two sources of persistence in
members’ login decisions: state dependence and unobserved heterogeneity.
To accommodate unobserved heterogeneity, we use a finite mixture model
2
Advertising rates on these sites are set as fees per quantity of page views. For example
Merrill Lynch reports a rate of $1.83 per thousand views (BusinessWeek, November 7
2007).
3
For real-time chat we actually measure a proxy, namely the number of friends online
at the same time of each day.
4
See Rust (1987), Hotz and Miller (1993), Hotz, Miller, Sanders, and Smith (1994),
Keane and Wolpin (1994), Magnac and Thesmar (2002), Aguirregabiria and Mira (2007),
Bajari, Benkard, and Levin (2007), Pakes, Ostrovsky, and Berry (2007), Pesendorfer and

Schmidt-Dengler (2007), Arcidiacono and Miller (2008), Imai, Jain, and Ching (2009) for
estimation of dynamic discrete-choice models.

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that incorporates members’ demographics into segment membership probabilities (Gupta and Chintagunta 1994, Erdem, Imai, and Keane 2003, Erdem,
ă u, and Strebel 2005).
Keane, Oncă
We found that consumers can be classified into two distinct latent groups,
according to their base rates of site usage. Segment membership probability
was most affected by a member’s age with a 47%/53% split between the two
segments averaged across demographics – older members have lower rates
of site usage on average. Expected quantities of real-time chat, measured
by the number of friends who are online, and messaging, measured by the
number of incoming messages per day, both had positive effects on usage
across segments. All else equal, usage of the site was more likely on a weekday
than on a weekend, although this effect is not statistically significant.
We used our estimates to analyze the potential effects of two counterfactual policies. The first is designed to enhance the networking experience
during online meetings. An example of this type of policy is MySpace’s introduction of video chat using Skype in October 2007. The second policy
is designed to assist in the acquisition of new friends, for example through
the adoption of a collaborative filtering system such as that used by online
retailers like Amazon.com, or survey techniques used by online dating services such as Eharmony.com or Match.com. We found that these policies
had varying dynamic effects on site traffic. They had a similar initial effect
on usage, but subsequently a gap between the two effects appears, with the
second policy resulting in larger gains in site usage over time.
There is a growing empirical literature on the economics of social networks
or social network websites. Manski (1993) illustrated the difficulty of separating endogenous effects (social effects) in cross-sectional data from contextual
and correlated effects. Recent work by Graham (2008) exploits the specific

nature of particular datasets to isolate social effects, separating these from
group-level heterogeneity by, for example, using the random assignment of
teachers and students to classes of different sizes. Other studies have developed structural approaches that explicitly solve network coordination games
among agents in static settings. For example Hartmann (2008) developed
a likelihood-based, game-theoretic approach to model the joint consumption
decisions of golf players, while Bao, Gupta, and Kadiyali (2009) modelled social interactions in MBA students’ choices of summer internship applications.
Yet another group of studies identifies social interactions using reduced-form
panel-data approaches. Examples include Trusov, Bodapati, and Bucklin
(2008), who investigated the determinants of influential network users, Nair,
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Manchanda, and Bhatia (2008), who examined the role of opinion leaders
in physician prescription behavior, and Mayer and Puller (2008), who studied the effects of academic and demographic factors on links between college
users of Facebook.
Our structural dynamic approach is related to that in Ryan and Tucker
(2008), who use dynamic discrete-choice techniques to incorporate network
effects into a model of new technology adoption. They analyze one-time
adoption as a stopping problem, whereas we focus on an on-going usage
problem. Our decision to model social effects in a single-agent framework
also resembles the approaches by Blume (1993) and Brock and Durlauf (2001,
2002) who modeled individual choices in the presence of social interactions
while assuming that payoffs are affected only by the aggregate behavior of
the group.
Section 2 describes the data. Section 3 describes the model. Section
4 describes the estimation method. Section 5 discusses the results of the
estimation. Section 6 describes the policy experiments and discusses the
results. Section 7 concludes.

2

Data

2.1

Data description

We randomly selected a group of college-aged members of MySpace (aged
19-23) and tracked their websites daily for four weeks from mid-January to
mid-February of 2008. Our sample comprised only non-business members,
excluding artists or companies who used the site for promotional or business
purposes. Privacy restrictions forced us to confine our sample to members
whose profiles were open to the public.5
About 15% of members in the initial sample were dropped from the final
sample according to the following criteria. First, we dropped members who
switched their profiles from public to private during the data collection period. Second, we dropped members who had less than 51 or more than 200
friends. This was due to the scarcity of observations outside this range. In
addition, this criterion allows us to reduce the size of state space. Finally,
we dropped members who exhibited extreme behavior, defined as losing or
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This may give rise to some selection issues. Hence our inferences only apply to members whose profiles are open to the public.

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gaining more than 10 friends on any given day. The final data set consists of
111 members.

Three types of variables were recorded daily for each member of our sample: usage behavior, social interactions, and the evolution of the social network. Usage behavior is defined as one’s daily decision to use a site. This
definition closely relates to the advertising revenue of social network sites and
thus is of direct managerial relevance. Since MySpace automatically updates
the last time that a member used the site on a real-time basis, this variable is
collected with high accuracy. Social interactions comprise real-time chat and
messaging, the ways through which members can interact with each other
on MySpace. Real-time chat is a system that allows for the back-and-forth
exchange of text messages among members who are currently online. Messaging is similar to electronic mail in that it allows members to post messages
to friends’ webpages. Unlike chat, messaging does not require members to be
simultaneously online in order to exchange messages. Since we do not observe
real-time chat directly, in our empirical work we proxy for this variable with
the number of friends online at the time of login. Hence we will often use
the terms ‘real-time chat’ and ‘number of friends online’ interchangeably. To
measure messaging activity we counted the number of incoming messages for
each member, which serves as a proxy for messaging activities as a whole.6
We tracked the evolution of the social network by recording the number of
friends for each member each day of the sample period. The social network
may grow or shrink over time as the member gains or loses friends.7

2.2

Descriptive statistics

Tables 1 and 2 give definitions of, and summary statistics for, demographic
and state variables used in our paper. Over the time period of our sample,
an average member was 21 years of age, and had about a 50% probability of
being female, or of coming from outside the US.8 On average, a member of
our sample had about 120 friends on MySpace. Tables 3 and 4 give definitions
6
Outgoing messages of the members in our sample were posted on their friends’ webpages, some of which were kept private preventing us from collecting this variable. Thus

we implicitly assume that the number of incoming messages is proportional to that of
outgoing messages.
7
All data were recorded at the same time of each day so as to minimize any noise from
time-of-day effects.
8
Most non-US users in our sample are from Australia and the UK.

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of, and summary statistics for, our measures of site usage, social interactions,
and network evolution. The average rate of daily usage in our sample was
about 52%, and the average number of friends who were online at the time of
data collection was about four. Members only received an average of about
0.26 messages per day on their webpages. This low number may be due
to the availability of other ways of exchanging messages such as electronic
mails. On average, daily changes in social networks were small: 90% of daily
network changes were either 0 or 1, and the mean daily change was close to
zero.
Table 5 shows the evolution of social networks for the members of our
sample after about 4 weeks. The median of monthly changes in network size
was still zero and the mean was a gain of less than two friends. However,
there were substantial individual differences in network evolution across the
members. About 5% of the members gained more than 10 friends after 4
weeks, whereas some lost friends during that period. The dispersion of network evolution across members demonstrates that our model must account
for idiosyncratic time paths of individuals’ social networks.
The central notion of our paper is that members can maintain and invest
in social networks now so that they yield larger social interactions in future.

Members perceive social networks as a stock of social capital, the ‘dividends’
of which are the expectation of chat and messaging activities upon login. To
this end we test whether members’ login decisions last period and the sizes
of social networks – state variables that serve as proxies for maintenance of,
and cumulative investment in, social capital – positively affect real-time chat
and messaging.
Tables 6 and 7 show the results from Poisson regressions of the components of per-period utility – real-time chat and messaging – on state variables.
In the chat regression of table 6 all effects are statistically significant at the
5% level or better. Chat is increasing in the member’s network size – this is
to be expected since chat itself is measured as the number of friends online.
It is also increasing in last period’s login decision, suggesting that a member’s stock of social capital depreciates if he or she does not log in frequently.
Moreover this effect is quite strong in a relative sense – whereas adding an
extra person to one’s network increases the expected number of friends online
on any given day by just one percent, a login last period increases the mean
by around 14%. Finally, the number of friends online is lower on weekends
– perhaps reflecting the alternative activities available to members at those
times.
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Broadly similar effects are seen in the messaging regression of table 7.
Messaging shows a statistically significant response to network size and last
period’s login. While the former effect is quite small in magnitude, with an
additional friend raising the expected number of daily messages by less than
half a percent, the last-login effect is much larger, by a couple of orders of
magnitude. The dynamic considerations in the consumer’s choice problem,
inherent in depreciation of the ‘capital stock’ if she does not log in, are thus
clearly illustrated.

3

Model

We propose a dynamic model of a member’s usage of a social network site
(henceforth, a ‘site’). Consumers derive utility from interacting with other
members of the site. Enjoyment of these benefits requires the member to
make costly login efforts so as to manage existing contacts and build new
ones. Managing contacts averts depreciation, which would be visible in our
model as the negative effect on the expected amounts of chat and messaging
if the member failed to log in last period. Acquisition of new capital will be
apparent in the effect of a login this period on the expected network size next
period, and in the flow-on effects of a bigger network on social interactions
in future periods. A dynamic model allows us to incorporate expectations
about these future effects into consumers’ login decisions.

3.1

Member types

We assume a finite number of latent types for consumers as regards utility,
cost, and state transition. Some members may be more social than others
leading to variation in preferences towards the site. Different lifestyles (e.g.,
indoorsy vs. outdoorsy) may affect the cost of using the site. Finally, some
members may make friends more easily than others. Allowing for such heterogeneity is important since the users in our sample show wide variations
in daily login propensities and network evolutions, which are not obviously
explained by their observed characteristics. Furthermore our state space includes the lagged action as a variable conditioning current actions. As is well
known, coefficients on such lagged actions may be incorrectly estimated if
unobserved time-constant heterogeneity is not also allowed for.

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3.2

Per-period utility

Let ait = 1, or 0, as consumer i does, or does not, log in at time t. By
using the site at time t she derives per-period utility uit , and incurs a login
cost of cit . The per-period utility from not logging in is normalized to zero,
plus a random i.i.d. error. Let x1it and x2it respectively denote the amounts
of real-time chat and messaging that member i engages in at time t. We
assume that, after logging in, member i of type j’s realized per-period utility
takes the form:
uit = θj1 + x1it θj2 + x2it θj3 .
(1)
Here θj = (θj1 , θj2 , θj3 ) is a vector of type-specific parameters to be estimated.
Realized values of x1it and x2it will depend on factors that are somewhat
uncertain prior to login, i.e., the number of friends online and the messaging
activity. This implies that a member’s login decision is based on expectations
of x1it and x2it , conditional on all information available prior to login. We
assume that this information consists of variables that are observable to
the econometrician, collected in a finite state space S, and of unobservable
components of the login cost cit that are private information to the member.
Since we assume that the private login costs are in fact uncorrelated with
the same-period values of x1it and x2it , only the observable states in S need
condition the member’s expectations. Where sit denotes a vector of state
variables from S, we can define member i’s of type j’s expected per-period
utility at the beginning of period t as:

uit ≡ E[uit |sit ] = θj1 + E[x1it |sit ]θj2 + E[x2it |sit ]θj3
= θj1 + x1it θj2 + x2it θj3 .

(2)

Here x˜it = (x1it , x2it ) denotes a vector of expected chat and messaging activities
upon login conditional on sit .

3.3

State space

Our specification of S includes as state variables the members’ histories of
login activity and size of social network. The member’s login history is
represented by a binary indicator of whether or not she logged in last period.9
9

The members engage in other activities such as sending messages to friends during
login sessions. We do not have this type of information in our data. To the extent that
other activities are conditional on the member’s login decision, the login histories serve as
a gross proxy for the activities after login.

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We measure network size by the member’s actual number of friends observed
in period t, presuming that the member is able to keep track of this number
even if she has not logged in for a while. The state space also includes the
day of the week, to capture variations in the expected net utility of login

between weekdays and the weekend. Overall an element of S is a vector
sit = (s1it , s2it , s3t ), where s1it denotes member i’s login status at period t − 1,
s2it is the size of her network at the start of period t, and s3t is a vector of
dummies for days of the week.
When incorporating day-of-week into the model we employ some simplifications in the parameterization of login costs and the transition probabilities
(of chat, messaging, and network size). Specifically we assume that in these
components of the model (which serve as primitives in the consumer’s choice
problem), time-of-week enters just as a weekday-weekend dummy, rather
than as a full set of seven dummies. This abstraction adds precision to our
parameter estimates, while maintaining a reasonable approximation to the
true transition process. Note that this abstraction just applies to the model
components mentioned – in solving the consumer’s overall dynamic decision
problem (represented by the Bellman equation), we allow a different value
function for each day of the week, reflecting the fact that login behavior on
Friday may be different to that on Monday, because of the proximity of the
weekend.

3.4

Cost

The cost of using a site includes associated financial costs and the costs of
any time spent in locating a computer system and socializing online. We
allow this cost to differ between weekdays and weekends and formulate it as:
2
cit = φ1j + sW
t φj − ε1it ,

(3)

where φ1j is the cost paid across all days of the week, φ2j is a weekend premium,
and sW
= 1 on weekends, 0 otherwise. Here φj = (φ1j , φ2j ) is a vector of
t
parameters to be estimated, which depends on i’s unobserved time-constant
2
type, drawn from a finite set of such types. Let cjt = φ1j + sW
t φj denote the
nonrandom cost component. If the member does not log in she receives a
zero mean utility plus a random return ε0it . Let the pair of random errors
εit = (ε0it , ε1it ) be jointly distributed i.i.d. extreme-value (i.e., ‘multinomial
logit’).
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3.5

State transition

At the beginning of period t the user observes the current state sit (and her
private login cost cit and outside option ε0it ) and forms expectations x1it and
x2it about the levels of chat and messaging that would eventuate were she
to log in. This login decision, along with sit , will determine a probability
distribution for states si,t+1 at the beginning of next period, at which point
the user will face a new login decision. Period-to-period transitions in the
day-of-week dummy s3t and the last period’s login status s1it are of course
fully deterministic. For s2it , the size of the user’s network, we assume that
the transition ∆s2it ≡ s2i,t+1 − s2it ∈ {−2, −1, 0, +1, +2} arises from an ordered
logit process, with yit ≡ (s1it , log(s2it /10), sW

t ) as regressors, conditional on
the login decision ait . (The set of possible values for ∆s2it reflects the values
observed in our data.) Furthermore we allow the parameters in this ordered
logit process to vary with a user’s unobserved time-constant type, drawn
from a finite set of types. Where γj = (γj1 , γj2 , γj3 ) is the user’s type-specific
vector of parameters in this ordered logit, the probability of ∆s2it = +1, for
example, is thus modelled as:
pj (∆s2it = +1|sit , ait = a) =
exp(κa+1 − γj yit )
exp(κa0 − γj yit )
−
.
1 + exp(κa+1 − γj yit ) 1 + exp(κa0 − γj yit )

(4)

Here a = 0 or 1 and κa−2 , κa−1 , . . . denote the cutoff parameters in the ordered
logit. The subscript j on p recognizes the fact that transition probabilities for
the network-size variable are type-specific. We allow these to vary with the
login decision ait , reflecting the conditionality of future expectations on current ‘investment’ decisions. For parsimony we restrict the other parameters
γj to be constant in ait .

3.6

Decision rule

Member i of type j at time t chooses a sequence of actions {ait } over the
infinite horizon to maximize the discounted sum of expected utilities minus
costs. Define the value function by
∞

Vj (sit , εit ) =

sup
{ait ,ai,t+1 ,...}

β τ −t [aiτ (uiτ − cjτ + ε1iτ ) + (1 − aiτ )ε0iτ )] |sit

E
τ =t

(5)
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.


Here β is the discount factor and the expectation is taken with respect to
future values of siτ , cjτ and εiτ . Time paths of siτ are based on the state
transitions described in the previous subsection. Those of cjτ , the deterministic part of login costs, just depend on the day of the week. By assumption
future values of εiτ are distributed independently of the current value of εit ,
so εit need not condition the expectation on the RHS of (5). The j subscript
on V allows for the fact that the choice problem may differ across agents
according to their unobserved time-constant type j (which affects uit , cjt and
the transitions of s2it ).
Rewrite (5) in the form of Bellman’s equation:
Vj (sit , εit ) = max {uit − cjt + ε1it + βE [Vj (si,t+1 , εi,t+1 )|sit , ait = 1] ,
ait ∈A

ε0it + βE [Vj (si,t+1 , εi,t+1 )|sit , ait = 0]} ,

(6)

where A ≡ {0, 1} is the action space. Define EV j (sit , ait ) to be the integrated
value function:
pj (si,t+1 |sit , ait )Eεi,t+1 [Vj (si,t+1 , εi,t+1 )|si,t+1 ] .

EV j (sit , ait ) ≡

(7)

si,t+1

Since εit is i.i.d. over time, it is easy to see that the solution to (6) exhibits a
cutoff property. Using (7), the optimal login decision is to set ait = 1 if and
only if
uit − cjt + β EV j (sit , ait = 1) − β EV j (sit , ait = 0) ≥ ε0it − ε1it .

(8)

We can then integrate (6) over εit , and combine with (7) to get the following
modified Bellman equation:
pj (si,t+1 |sit , ait )×

EV j (sit , ait ) =
si,t+1

Eεi,t+1 max

ai,t+1 ∈A

ui,t+1 − cj,t+1 + ε1i,t+1 + β EV j (si,t+1 , ai,t+1 = 1),

ε0i,t+1 + β EV j (si,t+1 , ai,t+1 = 0)

4

.

(9)

Estimation

We adopt the MPEC (Mathematical Programming with Equilibrium Constraints) approach outlined by Su and Judd (2008; henceforth SJ). The NFXP
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(Nested Fixed Point) approach of Rust (1987) solves a dynamic programming problem at every iteration of the parameter estimation. This makes
the NFXP approach computationally intractable for all but simple problems.
The MPEC approach applies a direct optimization method to a dynamic programming problem by formulating it as a constrained optimization problem.
Since the MPEC approach solves a dynamic programming problem only once
at the final iteration of parameter estimation, it achieves significant computational time-savings relative to the NFXP approach.
Equations (8) and (9) form the basis for estimation. For a given user
i of type j, a given integrated value function EV j and given parameters
θj1 , θj2 , θj3 , φ1j , φ2j , equation (8) implies logit probabilities with which to form a
likelihood function. We maximize this likelihood subject to the constraints in
equation (9), which comprises a finite set of equalities that implicitly defines
EV j . A novel aspect of our implementation is that we extend the original

specification of SJ to allow for unobserved heterogeneity in the framework of
ă u,
a finite mixture model (Erdem, Imai, and Keane 2003, Erdem, Keane, Oncă
10
and Strebel 2005).

4.1

Segment membership probabilities

We model segment membership probabilities as a function of demographics
as in Gupta and Chintagunta (1994). Let zi1 , zi2 and zi3 respectively denote
member i’s age, gender, and country of residence. The probability for member i with demographics zi = (zi1 , zi2 , zi3 ) of being type j is
q(j|zi , αj , λj ) =

exp(αj + zi λj )
J
j =1

exp(αj + zi λj )

.

(10)

Here parameters αj and λj = (λj1 , λj2 , λj3 ) are specific to each type j =
1, . . . , J. We normalize αJ and λJ to zero for identification.
10

Ackerberg (2001) proposes a method based on importance sampling that allows for

individual-level unobserved heterogeneity in a dynamic discrete-choice model. Our decision
to adopt a finite mixture model is motivated by both computational and data-related
reasons. Allowing for individual-level unobserved heterogeneity is infeasible even with the
MPEC approach. Also, we do not have enough data to identify individual-level differences.
Note that the Ackerberg’s approach solves a dynamic programming problem once at the
first iteration of parameter estimation, whereas the SJ’s approach does so at the last
iteration.

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4.2

Computing expectations

To form the expectations on the RHS of (9) we need transition probabilities between states sit , and also state-conditional distributions for post-login
‘chat’ and ‘messaging’ activities. Ideally we would like to use non-parametric
estimates of these distributions. In practice such non-parametric estimates
impose a heavy data requirement that is not met in our application. Therefore we use parametric specifications as approximations. As noted above,
two of the elements in sit (last login and day-of-week) evolve deterministically. For the third element, network size, we use the ordered logit with
type-conditional thresholds described in section 3.5. Pre-login expectations
of ‘chat’ and ‘messaging’ are generated from the Poisson regressions in tables
6 and 7. For simplicity (unlike for the network-size ordered logit) we restrict
the parameters in these Poisson regressions to be the same across member
types.

4.3

Recovering {θj }Jj=1 , {φj }Jj=1 , {γj , κ}Jj=1 , {αj , λj }J−1

j=1

Since we assume that εit follows an extreme value distribution, i.i.d. over
members and time, the probability of member i of type j to log in at time
t given states sit and parameters {θj , φj , γj , κ} has the multinomial logit
formula:
P (ait = 0|sit , θj , φj , γj , κ) =

1
1 + exp{uit − cjt + β EV j (sit , ait = 1) − β EV j (sit , ait = 0)}

and
P (ait = 1|sit , θj , φj , γj , κ) = 1 − P (ait = 0|sit , θj , φj , γj , κ) .
As in, e.g., SJ (equation 14), EV j (sit , ait ) is the unique fixed point to the
contraction mapping Tθj ,φj ,γj ,κ defined by
pj (si,t+1 |sit , ait ) ×

EV j (sit , ait ) = Tθj ,φj ,γj ,κ (EV j )(sit , ait ) =
si,t+1 ∈S

log exp[ui,t+1 − cj,t+1 + β EV j (si,t+1 , ai,t+1 = 1)] + exp[β EV j (si,t+1 , ai,t+1 = 0)]

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. (11)


Therefore, we can recover structural parameters, ({θj }Jj=1 , {φj }Jj=1 , {γj , κ}Jj=1 , {αj , λj }J−1
j=1 ),

by maximizing the likelihood function
N

J

i=1

j=1

L=

T

P (ait |sit , θj , φj , γj , κ)pj (sit |si,t−1 , ai,t−1 , γj , κ) .

q(j|zi , αj , λj )
t=1

(12)
N T
Let Ψ = ({θj }Jj=1 , {φj }Jj=1 , {γj , κ}Jj=1 , {αj , λj }J−1
j=1 ) and X = ({ait }i=1 t=1 ,
N
T
{sit }N
i=1 t=1 , {zi }i=1 ). This allows us to simplify the expression in (12) to

max L(Ψ; X) .
Ψ

(13)

The MPEC approach alleviates the computational difficulties of the NFXP
approach by allowing endogenous regressors EV = {EV j }Jj=1 to deviate from
the constraints in (11) during parameter estimation. It assigns a penalty
function for such deviations which must tend to zero at the final iteration of
parameter estimation.
An augmented likelihood function, L(Ψ, EV ; X) explicitly expresses the
dependence of the likelihood on EV . We reformulate the maximum likelihood
estimation in (13) as the constrained optimization problem
max

L(Ψ, EV ; X)
(14)

(Ψ,EV )

subject to EV = TΨ (EV ) .
SJ show the mathematical equivalence between (13) and (14) and recommend
use of software with high-level interfaces with state-of-the-art commercial
solvers. For our application, we use AMPL with a nonlinear solver Knitro.11
We use nonparametric bootstrapping to compute the standard errors of
our parameter estimates. Finite mixture models suffer from label switching,
which makes it difficult to implement nonparametric bootstrapping to compute standard errors. We adopt labeling restrictions by Geweke and Keane
(1997) that prevent the components of the mixture from interchanging. An
example of such restrictions is αj ≥ αj+1 for j = 1, . . . , J − 1 (with αJ = 0)
using our notation.
11

Readers are referred to Luo, Pang, and Ralph (1996) and Nocedal and Wright (2006)

for the convergence properties of the MPEC estimator.

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4.4

Identification issues

Our data do not record the identity and individual login behavior of all the
friends of any given user – hence we cannot model the full coordination game
among all players on a network. Such a model would in any case present
a major computational challenge because of the extensive overlaps between
different groups of friends on MySpace. Instead we draw a relatively small
sample from the millions of MySpace users and in essence adopt a singleagent framework. To obtain identification of the parameters we then need
certain assumptions on the random errors in the model. First, the errors in
the login cost (εit ), in the messaging and chat regressions, and in the ordered
logit for network evolution are all independent over time and independent of
each other. Second, each of these errors is independent of the variables in sit ,
in particular the last login s1it and the current network size s2it . Third, the
errors in the ordered logit and the Poisson regressions are also independent
of the current-period action ait .
User i’s latent type controls for a time-constant unobservable which might
otherwise introduce a correlation between these random errors and the last
login, and between the errors and the network size. (Correlation of the latter
type would reflect a tendency for rather ‘chatty’ people to log in a lot and
also accumulate a lot of friends.) Note that this time-constant unobservable
could be something personal to user i, or more generally some characteristic
shared by all of her friends. To maintain the above assumptions on the errors

we require in addition that there be no time-varying shared unobservables
affecting logins, which would otherwise confound the causal inferences in our
model. Suppose for example that, over the period of our sample, occasional
sporting or cultural events cause many site members to simultaneously use
their PC’s (to check scores, perhaps), leading to more observed MySpace
logins and perhaps more messaging about the event(s). If interest in the
event carries over more than one day, such behavior would show up in our
data as a correlation between last login and current login, and a correlation
between last login and current messaging and chat. The latter correlation
would imply that the user expects more messaging and chat this period if she
logged in last period. The structure imposed in our model would then infer
from the former correlation that expectations of messaging and chat cause
logins. Of course this may be a false causality, since it is really the shared
unobservable (the sporting event) that is driving the observed behavior.
A couple of features of our model may mitigate the possibility of such
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false causality. Since we include day of the week in the state space, any
time-varying shared unobservables that follow a regular cycle (e.g., sporting
fixtures, popular TV shows, etc.) are already accounted for. That is, since
the value function Vk may vary over the week, consumer choice behavior may
also vary over this cycle, independently of expected chat and messaging.
Another factor working in our favor is that we measure responses to both
messaging and number of friends online. A positive response to messaging,
for a given number of friends online, suggests that this factor induces more
logins even after controlling for any shared unobservables that cause more
friends to use their PC’s.
Furthermore our framework assumes that user i’s action responds to

friends’ expected current-period activity, not to actual activity (which is not
known prior to login). In turn these expectations are conditioned on state
variables (last login and network size) that are determined by actions in
previous periods. With respect to user i’s action the potential endogeneity problem is thus one of serial correlation between εi and previous values
of any shared unobservables. The hope is that by incorporating the latent
time-constant user types we have controlled for most sources of such serial
correlation.12
Our counterfactual simulations inherit the assumptions of our model, and
therefore suppose that the systematic part of login behavior is all driven by
time-of-week, personal characteristics, and expectations of chat, messaging,
and network evolution. We suppose that a site manager’s strategic tools (the
introduction of new site features and so on) affect the latter expectations.
Even if some shared unobservables (not on a weekly schedule) remain in the
model, our estimates and methodology might still be of interest as providing
an upper bound to what managers might achieve with enhancements to the
networking experience. If the manager believes that in fact some common
unobservables are affecting login behavior then he may want to scale back
the simulated impacts derived from a model of the present type.
Under the above assumptions the model infers the parameters governing
the dynamics of consumer behavior from observed login responses to changes
in network size and changes in presumed expectations of chat and messag12

Our construction of expected chat and messaging effectively assumes that friends do
not observe i’s contemporaneous private costs εit when making their own login decisions.
For if they did, then i would presumably be aware of this, in which case εit would become
part of the ‘public’ state and would need to condition i’s expectations of current-period
chat and messaging activity.

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ing activity. In general the discount factor β is not identified in dynamic
discrete-choice models (Rust 1987, 1994). Hence we set β = 0.98. The utility
parameters (including the constant and coefficients on chat and messaging)
and the weekend premium are identified by the variation in the member’s
login propensity across states, conditional on the transition probabilities and
associated future expectations. State transition parameters (including cutoff
points) are identified by the variation in network evolution across actions and
states. Segment membership probabilities are identified by the correlation
across consumers between time-averaged login propensities and demographic
variables.

5

Results

We estimated our model for up to 3 segments and chose a 2-segment model
based on Bayesian Information Criterion (BIC). Table 8 presents the estimates of per-period utility. The coefficients for real-time chat and weekend
premium were restricted to be the same across the two segments; allowing
them to be different does not improve the model fit. The most pronounced
difference between the two segments appears in the base rates of login with
estimated constants of −2.1520 and −0.1149 for segments 1 and 2 respectively. This translates into an average difference across states in predicted
login probabilities of about 55% between the two segments; the minimum
predicted login probabilities are 14.2% and 63.1% for segments 1 and 2 respectively. Based on this substantial difference the need for a model that
allows for unobserved latent login propensities is clear.
Both segments show per-period utility significantly increasing in chat (the
expected number of friends online) and messaging. Interestingly the response
to messaging is considerably higher in segment 2, the group with the higher
latent login propensity, than in segment 1. It is also interesting to note that

the estimated effect on per-period utility of one extra incoming message is
much larger than that of an extra online friend. This suggests that, although
a member may have friends online when she logs in, on average they do not
necessarily chat all that much. Instead, social network sites are used as online scrapbooks, where users share personal lives with friends by exchanging
messages. The login cost is estimated to be higher on weekends, although it
is not statistically significant at the 10% level. The negative sign of the weekend premium implies that some users may have limited internet access on the
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weekend. Alternatively, site usage may be an inferior good with respect to
the time constraint, leading users to substitute toward other activities when
they have more leisure.
Table 9 presents the estimated network evolution or state transition probabilities. We allowed the cutoffs of the ordered logit probabilities to differ
between ait = 1 and ait = 0. The magnitudes of the cutoffs seem quite
intuitive. The probability of gaining friends increases by nearly two percentage points averaged across states and segments if a member logs in than not.
Not many parameters are statistically significant, except for the network-size
parameter in segment 2. Relative to segment 1, members of segment 2 seem
to be ‘social butterflies’ who attract more and more friends as they engage
in social networking.
Table 10 presents the estimation results for segment membership probabilities. The constant for segment 1 is estimated to be negative and statistically significant at the 10% level, implying that all else equal segment 1
is smaller in size than segment 2. Across all demographics, the membership
probabilities are 47.3% and 52.7% for segments 1 and 2 respectively. Age
has a positive and statistically significant effect on a member’s probability of
belonging to segment 1: older members are less active in site usage and make
fewer friends than younger ones. Supposing that most of our sample are college students, this suggests that it is on average harder for a site manager to
get upper-year students to log in, perhaps because they are busy looking for
jobs and preparing for graduation.

6

Policy experiment

Parameter estimates are used to perform two counterfactual simulations, with
the goal of providing managers with policies that increase site traffic. The
first policy is designed to enhance the intensity of a user’s online experience
with a given set of friends. We exogenously raise the coefficient on real-time
chat and simulate the effects on the user’s login behavior, and on the size
of the user’s social network. The second policy fixes the coefficients in perperiod utility at the existing levels, and exogenously raises the propensity to
acquire new friends. This may not have an immediate effect on the member’s
login behavior, but bigger networks may feed back into enhanced utility from
social networking.
The first policy might be implemented via technology aimed at enhancing
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the networking experience during online meetings. An example of this is
MySpace’s decision to allow for video chat by incorporating Skype into its
internet messenger system in October of 2007. We simulate such a policy
by exogenously raising by 10% the coefficient on real-time chat in per-period
utility, which is common to both segments.
For our second policy simulation we increase the propensity to acquire
new friends by 5% for both segments. A manager seeking to implement
such a network expansion could for example adopt collaborative filtering
mechanisms or survey techniques to match members based on common traits.
Many online retailers like Amazon.com use collaborative filtering systems
to recommend products to customers based on their purchase histories, and
online dating services like Match.com and Eharmony.com use detailed surveys
to match members.

Both policy experiments were performed by simulating paths of actions
and states at the observed demographics-states combinations. We simulated
ten 365-day sequences per demographics-states combination and then averaged the results over all time paths. Note that our policy experiments are
inherently partial equilibrium in nature. After adjusting the relevant parameter (for utility of online meetings, or for network evolution) we do not solve
for a new equilibrium of the whole coordination game between user i and
her friends. Instead we hold friends’ strategies (their state-conditional action distributions) fixed, at the levels estimated for the original equilibrium
(which is that observed in the data), and just allow i to adjust her own best
response. Were we to solve for a new equilibrium (or, more likely, equilibria), it is certainly possible that the friends would raise their own login and
messaging activity, creating feedback effects that lead to network sizes still
larger than those seen below.
We use two metrics to compare the effectiveness of the two policies with
that of the baseline policy – the status quo of MySpace. First, we divide
52 weeks into four quarters and compute the average daily site traffic for
each quarter. Second, we compute average individual network size for each
quarter. The first metric is directly related to the benchmark that ranks the
sizes of social network sites – number of unique users per month. The second
metric measures an aspect of members’ well-being, because social network
sites are in part used to make new friends.
Table 11 shows average daily usage rates for the three policies – status
quo, enhanced utility for real-time chat, and bigger network. Not surprisingly
the two counterfactuals both do better than the status quo in daily site usage
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after 52 weeks. However the usage trends over time show some differences
between the two counterfactual policies. Whereas the two counterfactual
policies have a similar effect on the usage rate upon adoption, the second has
the higher rate of increase in site usage after one year. One can observe a
widening gap in site usage between the first and second policies over time.

Table 12 shows average network sizes under the two policies, with the
counterfactuals naturally leading to larger networks than the status quo.
Note that the effect of the first policy on network sizes is purely indirect:
enhanced utility in real-time chat with friends leads a member to spend
more time online, which in turn causes them to make new friends. The two
counterfactual policies have markedly different effects on network size over
time. Whereas the average network size of the first policy is nearly identical
to that of status quo over time, the second policy has about an 8% bigger
network size than the status quo by the fourth quarter.

7

Conclusions

This paper structurally estimates a dynamic model of usage behavior and
network effects in social network sites using data from MySpace. Our goal
is to estimate the effects of site features – real-time chat and messaging –
on the usage behavior of members and to provide managers with business
policies that may enhance firm performance. The framework could easily be
extended to incorporate other site features as data on these become available.
We found that (1) there are distinct types of consumers as regards utility,
cost, and state transition, but (2) features of MySpace positively affect site
traffic across all segments. Using the parameter estimates, we performed
policy experiments to simulate the effects of two counterfactual policies –
enhanced utility for real-time chat at any given network size, and increased
propensity to expand one’s network. The two policies yield distinct time
paths of usage rates and network sizes.
In addition to studying other site features, future research might focus
on linking the dynamics of login behavior to other current topics in social
networks. One such topic is the question of how consumers in social networks

share information about products they have purchased. To study this question rigorously it is important to know the fundamental incentives that drive
social networking – the model presented here might be one ingredient in such
an analysis. Another current topic relates to the role of influential members
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in social networks. To the extent that such members think about their roles
in a forward-looking manner, the model here might again contribute to a
richer empirical understanding of the structure of social networks and how
managers can best exploit these structures.

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