A dynimic model of user behaviour in a social network site

A Dynamic Model of User BehaviorDae-Yong Ahn School of Marketing University of Technology Sydney dae-yong.ahn@uts.edu.au Randal Watson Department of Economics University of Texas at Aust

A Dynamic Model of User Behavior

Dae-Yong Ahn School of Marketing University of Technology Sydney dae-yong.ahn@uts.edu.au Randal Watson Department of Economics University of Texas at Austin watson@eco.utexas.edu April 16, 2010

AbstractThis paper estimates a dynamic model of user behavior in a so-cial network site using unique data on the daily login activity of asample of members of MySpace.com We view a social network as astock of capital that yields a flow of utilities over time by creatinginteractions between the site user and her friends This capital stockcan be maintained and expanded by logging in to the site and com-municating with other members The user’s login decision is thus aforward-looking one, which we model in the framework of a dynamicdiscrete-choice model We allow for two sources of persistence in mem-bers’ 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.

state transition Across types, real-time chat and messaging, features

of MySpace.com, positively affect the usage decision We use our rameter estimates to perform counterfactual simulations with the goal

pa-of providing site managers with ways to enhance firm performance.Keywords: dynamic discrete choice, online social networks, unob-served heterogeneity

of interactions with the addition of new friends to the network Since both ofthese 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 accountconsumers’ forward-looking behavior This is the task addressed here Weuse 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 tracting an increasing share of advertising expenditures Thirty-three percent

at-of adult US internet users, and 70% at-of teenagers, logged in to a social networksite 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 working by 2011’ To tap into this audience, marketers spent $1.2 billion onsocial network sites in 2007 and this figure is expected to climb to $3.6 billion

net-by 2010 worldwide (eMarketer, December 2007) Industries that advertise insocial network sites range from entertainment (25.2%), retail goods and ser-vices (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 advertisingspending in 2007 (Marketing News, July 15 2008) Media giants such asNBC and Warner Bros host sites on MySpace, while Coca-Cola, CBS, andChase promote their products on Facebook On the other hand there is also

a growing trend towards niche social network sites and marketer-sponsoredsites that attract ‘a smaller, but passionate audience’ rather than the ‘di-verse membership’ of MySpace and Facebook (CNN.com, April 16 2008).Examples include petside.com by Proctor & Gamble for its Iams pet foodsand 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.

Social network sites compete on features – instant messaging, video chat,etc – in order to attract more users and thereby bring in more advertisingrevenue.2 In this paper we assess the impact of two such features of theMySpace site – real-time chat, and a messaging or mailbox feature Weestimate a single-agent, dynamic discrete-choice model in which usage isdefined as a decision of whether or not to log in to the site This decisionrelates closely to the number of unique users that a site attracts, a benchmarkused in the industry to rank sites Our estimates allow us to propose andevaluate other features that social network sites might adopt to enhance firmperformance In particular we estimate the effects on site usage of policiesdesigned 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 fourweeks, recording three types of variables for each member: usage behavior,social interactions, and the evolution of the social network For usage be-havior we recorded whether or not one used MySpace for each member eachday For social interactions we recorded real-time chat and messaging, twofeatures of MySpace, for each member each day.3 We tracked changes inmembers’ social networks by recording the size of each member’s networkeach day

Our estimation strategy is based on the MPEC (Mathematical ming with Equilibrium Constraints) approach (Su and Judd 2008).4 This ap-proach alleviates the computational burden in estimating a dynamic discrete-choice model by formulating the NFXP (Nested Fixed Point) approach as aconstrained optimization problem We allow for two sources of persistence inmembers’ login decisions: state dependence and unobserved heterogeneity

Program-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.

that incorporates members’ demographics into segment membership bilities (Gupta and Chintagunta 1994, Erdem, Imai, and Keane 2003, Erdem,Keane, ¨Onc¨u, and Strebel 2005).

proba-We found that consumers can be classified into two distinct latent groups,according to their base rates of site usage Segment membership probabilitywas most affected by a member’s age with a 47%/53% split between the twosegments 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 thenumber of incoming messages per day, both had positive effects on usageacross segments All else equal, usage of the site was more likely on a weekdaythan on a weekend, although this effect is not statistically significant

We used our estimates to analyze the potential effects of two factual policies The first is designed to enhance the networking experienceduring online meetings An example of this type of policy is MySpace’s in-troduction of video chat using Skype in October 2007 The second policy

counter-is designed to asscounter-ist in the acqucounter-isition of new friends, for example throughthe adoption of a collaborative filtering system such as that used by onlineretailers like Amazon.com, or survey techniques used by online dating ser-vices such as Eharmony.com or Match.com We found that these policieshad varying dynamic effects on site traffic They had a similar initial effect

on usage, but subsequently a gap between the two effects appears, with thesecond 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 ing endogenous effects (social effects) in cross-sectional data from contextualand correlated effects Recent work by Graham (2008) exploits the specificnature of particular datasets to isolate social effects, separating these fromgroup-level heterogeneity by, for example, using the random assignment ofteachers and students to classes of different sizes Other studies have devel-oped structural approaches that explicitly solve network coordination gamesamong agents in static settings For example Hartmann (2008) developed

separat-a likelihood-bseparat-ased, gseparat-ame-theoretic separat-approseparat-ach to model the joint consumptiondecisions of golf players, while Bao, Gupta, and Kadiyali (2009) modelled so-cial interactions in MBA students’ choices of summer internship applications.Yet another group of studies identifies social interactions using reduced-formpanel-data approaches Examples include Trusov, Bodapati, and Bucklin(2008), who investigated the determinants of influential network users, Nair,

Manchanda, and Bhatia (2008), who examined the role of opinion leaders

in physician prescription behavior, and Mayer and Puller (2008), who ied the effects of academic and demographic factors on links between collegeusers of Facebook

stud-Our structural dynamic approach is related to that in Ryan and Tucker(2008), who use dynamic discrete-choice techniques to incorporate networkeffects into a model of new technology adoption They analyze one-timeadoption as a stopping problem, whereas we focus on an on-going usageproblem Our decision to model social effects in a single-agent frameworkalso resembles the approaches by Blume (1993) and Brock and Durlauf (2001,2002) who modeled individual choices in the presence of social interactionswhile assuming that payoffs are affected only by the aggregate behavior ofthe group

Section 2 describes the data Section 3 describes the model Section

4 describes the estimation method Section 5 discusses the results of theestimation Section 6 describes the policy experiments and discusses theresults Section 7 concludes

2 Data

We randomly selected a group of college-aged members of MySpace (aged19-23) and tracked their websites daily for four weeks from mid-January tomid-February of 2008 Our sample comprised only non-business members,excluding artists or companies who used the site for promotional or businesspurposes Privacy restrictions forced us to confine our sample to memberswhose profiles were open to the public.5

About 15% of members in the initial sample were dropped from the finalsample according to the following criteria First, we dropped members whoswitched their profiles from public to private during the data collection pe-riod Second, we dropped members who had less than 51 or more than 200friends This was due to the scarcity of observations outside this range Inaddition, this criterion allows us to reduce the size of state space Finally,

we dropped members who exhibited extreme behavior, defined as losing or

5 This may give rise to some selection issues Hence our inferences only apply to bers whose profiles are open to the public.

mem-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 ple: usage behavior, social interactions, and the evolution of the social net-work Usage behavior is defined as one’s daily decision to use a site Thisdefinition closely relates to the advertising revenue of social network sites andthus is of direct managerial relevance Since MySpace automatically updatesthe last time that a member used the site on a real-time basis, this variable iscollected with high accuracy Social interactions comprise real-time chat andmessaging, the ways through which members can interact with each other

sam-on MySpace Real-time chat is a system that allows for the back-and-forthexchange of text messages among members who are currently online Mes-saging is similar to electronic mail in that it allows members to post messages

to friends’ webpages Unlike chat, messaging does not require members to besimultaneously online in order to exchange messages Since we do not observereal-time chat directly, in our empirical work we proxy for this variable withthe number of friends online at the time of login Hence we will often usethe terms ‘real-time chat’ and ‘number of friends online’ interchangeably Tomeasure messaging activity we counted the number of incoming messages foreach 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 offriends for each member each day of the sample period The social networkmay grow or shrink over time as the member gains or loses friends.7

6 Outgoing messages of the members in our sample were posted on their friends’ pages, some of which were kept private preventing us from collecting this variable Thus

web-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.

of, and summary statistics for, our measures of site usage, social interactions,and network evolution The average rate of daily usage in our sample wasabout 52%, and the average number of friends who were online at the time ofdata collection was about four Members only received an average of about0.26 messages per day on their webpages This low number may be due

to the availability of other ways of exchanging messages such as electronicmails On average, daily changes in social networks were small: 90% of dailynetwork changes were either 0 or 1, and the mean daily change was close tozero

Table 5 shows the evolution of social networks for the members of oursample after about 4 weeks The median of monthly changes in network sizewas still zero and the mean was a gain of less than two friends However,there were substantial individual differences in network evolution across themembers About 5% of the members gained more than 10 friends after 4weeks, whereas some lost friends during that period The dispersion of net-work evolution across members demonstrates that our model must accountfor 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 Tothis 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 chatand messaging

Tables 6 and 7 show the results from Poisson regressions of the nents of per-period utility – real-time chat and messaging – on state variables

compo-In the chat regression of table 6 all effects are statistically significant at the5% 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 ber’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 anextra person to one’s network increases the expected number of friends online

mem-on any given day by just mem-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 thosetimes

Broadly similar effects are seen in the messaging regression of table 7.Messaging shows a statistically significant response to network size and lastperiod’s login While the former effect is quite small in magnitude, with anadditional friend raising the expected number of daily messages by less thanhalf a percent, the last-login effect is much larger, by a couple of orders ofmagnitude The dynamic considerations in the consumer’s choice problem,inherent in depreciation of the ‘capital stock’ if she does not log in, are thusclearly 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 othermembers of the site Enjoyment of these benefits requires the member tomake costly login efforts so as to manage existing contacts and build newones Managing contacts averts depreciation, which would be visible in ourmodel 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 beapparent in the effect of a login this period on the expected network size nextperiod, and in the flow-on effects of a bigger network on social interactions

in future periods A dynamic model allows us to incorporate expectationsabout these future effects into consumers’ login decisions

We assume a finite number of latent types for consumers as regards utility,cost, and state transition Some members may be more social than othersleading to variation in preferences towards the site Different lifestyles (e.g.,indoorsy vs outdoorsy) may affect the cost of using the site Finally, somemembers may make friends more easily than others Allowing for such het-erogeneity is important since the users in our sample show wide variations

in daily login propensities and network evolutions, which are not obviouslyexplained by their observed characteristics Furthermore our state space in-cludes the lagged action as a variable conditioning current actions As is wellknown, coefficients on such lagged actions may be incorrectly estimated ifunobserved time-constant heterogeneity is not also allowed for

3.2 Per-period utility

Let ait = 1, or 0, as consumer i does, or does not, log in at time t Byusing the site at time t she derives per-period utility uit, and incurs a logincost of cit The per-period utility from not logging in is normalized to zero,plus a random i.i.d error Let x1

it and x2

it respectively denote the amounts

of real-time chat and messaging that member i engages in at time t Weassume that, after logging in, member i of type j’s realized per-period utilitytakes the form:

uit= θj1+ x1itθj2+ x2itθj3 (1)Here θj = (θj1, θ2j, θj3) is a vector of type-specific parameters to be estimated.Realized values of x1

it and x2

it will depend on factors that are somewhatuncertain prior to login, i.e., the number of friends online and the messagingactivity This implies that a member’s login decision is based on expectations

of x1

it and x2

it, conditional on all information available prior to login Weassume that this information consists of variables that are observable tothe econometrician, collected in a finite state space S, and of unobservablecomponents of the login cost cit that are private information to the member.Since we assume that the private login costs are in fact uncorrelated withthe same-period values of x1it and x2it, only the observable states in S needcondition the member’s expectations Where sit denotes a vector of statevariables from S, we can define member i’s of type j’s expected per-periodutility at the beginning of period t as:

e

uit ≡ E[uit|sit] = θ1j + E[x1it|sit]θ2j + E[x2it|sit]θj3

= θ1j +xe1itθ2j +xe2itθj3 (2)Here ˜xit = (xe1

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.

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 numbereven if she has not logged in for a while The state space also includes theday of the week, to capture variations in the expected net utility of loginbetween 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,

s2

it is the size of her network at the start of period t, and s3

t is a vector ofdummies for days of the week

When incorporating day-of-week into the model we employ some cations in the parameterization of login costs and the transition probabilities(of chat, messaging, and network size) Specifically we assume that in thesecomponents of the model (which serve as primitives in the consumer’s choiceproblem), time-of-week enters just as a weekday-weekend dummy, ratherthan as a full set of seven dummies This abstraction adds precision to ourparameter estimates, while maintaining a reasonable approximation to thetrue transition process Note that this abstraction just applies to the modelcomponents mentioned – in solving the consumer’s overall dynamic decisionproblem (represented by the Bellman equation), we allow a different valuefunction for each day of the week, reflecting the fact that login behavior onFriday may be different to that on Monday, because of the proximity of theweekend

The cost of using a site includes associated financial costs and the costs ofany time spent in locating a computer system and socializing online Weallow this cost to differ between weekdays and weekends and formulate it as:

cit= φ1j + sWt φ2j − ε1it , (3)where φ1

j is the cost paid across all days of the week, φ2

j is a weekend premium,and sW

t = 1 on weekends, 0 otherwise Here φj = (φ1

j, φ2

j) is a vector ofparameters to be estimated, which depends on i’s unobserved time-constanttype, drawn from a finite set of such types Let cjt = φ1

j + sW

t φ2

j denote thenonrandom cost component If the member does not log in she receives azero 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., ‘multinomiallogit’)

3.5 State transition

At the beginning of period t the user observes the current state sit (and herprivate login cost cit and outside option ε0it) and forms expectations ex1

it ande

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 probabilitydistribution for states si,t+1 at the beginning of next period, at which pointthe user will face a new login decision Period-to-period transitions in theday-of-week dummy s3

t and the last period’s login status s1

it are of coursefully deterministic For s2

it, the size of the user’s network, we assume thatthe transition ∆s2it≡ s2

it reflects the valuesobserved in our data.) Furthermore we allow the parameters in this orderedlogit process to vary with a user’s unobserved time-constant type, drawnfrom a finite set of types Where γj = (γ1

j, γ2

j, γ3

j) is the user’s type-specificvector of parameters in this ordered logit, the probability of ∆s2it = +1, forexample, is thus modelled as:

pj(∆s2it = +1|sit, ait= a) =exp(κa

γj to be constant in ait

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

Here β is the discount factor and the expectation is taken with respect tofuture values of siτ, cjτ and εiτ Time paths of siτ are based on the statetransitions described in the previous subsection Those of cjτ, the determin-istic part of login costs, just depend on the day of the week By assumptionfuture 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 agentsaccording to their unobserved time-constant type j (which affectsueit, cjt andthe transitions of s2it)

Rewrite (5) in the form of Bellman’s equation:

Vj(sit, εit) = max

a it ∈A{euit− cjt+ ε1it+ βE [Vj(si,t+1, εi,t+1)|sit, ait = 1] ,

ε0it+ βE [Vj(si,t+1, εi,t+1)|sit, ait = 0]} , (6)where A ≡ {0, 1} is the action space Define gEVj(sit, ait) to be the integratedvalue function:

g

EVj(sit, ait) ≡ X

s i,t+1

pj(si,t+1|sit, ait)Eεi,t+1[Vj(si,t+1, εi,t+1)|si,t+1] (7)

Since εitis i.i.d over time, it is easy to see that the solution to (6) exhibits acutoff property Using (7), the optimal login decision is to set ait = 1 if andonly if

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

ε0i,t+1+ β gEVj(si,t+1, ai,t+1 = 0)

Con-(Nested Fixed Point) approach of Rust (1987) solves a dynamic ming problem at every iteration of the parameter estimation This makesthe NFXP approach computationally intractable for all but simple problems.The MPEC approach applies a direct optimization method to a dynamic pro-gramming problem by formulating it as a constrained optimization problem.Since the MPEC approach solves a dynamic programming problem only once

program-at the final iterprogram-ation of parameter estimprogram-ation, it achieves significant tational time-savings relative to the NFXP approach

compu-Equations (8) and (9) form the basis for estimation For a given user

i of type j, a given integrated value function gEVj and given parameters

EVj A novel aspect of our implementation is that we extend the originalspecification of SJ to allow for unobserved heterogeneity in the framework of

a finite mixture model (Erdem, Imai, and Keane 2003, Erdem, Keane, ¨Onc¨u,and Strebel 2005).10

We model segment membership probabilities as a function of demographics

as in Gupta and Chintagunta (1994) Let z1

i, z2

i and z3

i respectively denotemember i’s age, gender, and country of residence The probability for mem-ber i with demographics zi = (zi1, zi2, zi3) of being type j is

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.

4.2 Computing expectations

To form the expectations on the RHS of (9) we need transition

probabili-ties 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

There-fore we use parametric specifications as approximations As noted above,

two of the elements in sit (last login and day-of-week) evolve

deterministi-cally 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}J

j=1, {φj}J

j=1, {γj, κ}Jj=1, {αj, λj}J −1j=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{ueit− cjt+ β gEVj(sit, ait = 1) − β gEVj(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), gEVj(sit, ait) is the unique fixed point to the

contraction mapping Tθj,φj,γj,κ defined by

Therefore, we can recover structural parameters, ({θj}J

j=1, {φj}J

j=1, {γj, κ}Jj=1, {αj, λj}J −1j=1) and X = ({ait}N

i=1Tt=1,{sit}N

i=1Tt=1, {zi}N

i=1) This allows us to simplify the expression in (12) to

max

The MPEC approach alleviates the computational difficulties of the NFXP

approach by allowing endogenous regressors gEV = {gEVj}J

j=1to deviate fromthe 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, gEV ; X) explicitly expresses the

dependence of the likelihood on gEV We reformulate the maximum likelihood

estimation in (13) as the constrained optimization problem

max

(Ψ,g EV )

L(Ψ, gEV ; X)subject to EV = Tg Ψ(gEV )

(14)

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

com-pute 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.