Journal of Mathematical Sociology, 38: 233–248, 2014
Copyright # Taylor & Francis Group, LLC
ISSN: 0022-250X print=1545-5874 online
DOI: 10.1080/0022250X.2013.803100
KEY PLAYER POLICIES WHEN CONTEXTUAL
EFFECTS MATTER
Coralio Ballester
Departamento de Fundamentos del Ana´lisis Econo´mico,
Universidad de Alicante, Alicante, Spain
Yves Zenou
Department of Economics, Stockholm University, Stockholm, Stockholm,
Sweden, and Research Institute of Industrial Economics, Stockholm, Sweden
We consider a model where the criminal decision of each individual is affected by not only her
own characteristics, but also by the characteristics of her friends (contextual effects). We determine who the key player is, i.e., the criminal who once removed generates the highest reduction
in total crime in the network. We propose a new measure, the contextual intercentrality measure, that generalizes the one proposed by Ballester, Calvo´-Armengol, and Zenou (2006) by
taking into account the change in contextual effects following the removal of the key player.
We also provide an example showing that the key player can be different whether contextual
effects are taken into account or not. This means that the planner may target the wrong person
if it ignores the effect of the ‘‘context’’ when removing a criminal from a network.
Keywords: contextual effects, crime, key players, peer effects
1. INTRODUCTION
Important differences in crime rates are commonly observed across different
social groups and=or locations displaying otherwise identical economic fundamentals (Glaeser, Sacerdite, & Scheinkman, 1996). One of the main explanations put
forward to account for this phenomenon is the presence of social multiplier effects
on individual crime decisions. That is, as the fraction of individuals participating
in a criminal behavior increases, the impact on others is multiplied through social
interactions or networks (see, in particular, Sah, 1991; Glaeser et al., 1996; Rasmussen,
1996; Schrag & Scotchmer, 1997; Calvo´-Armengol & Zenou, 2004; Kleiman, 2009).
Empirically, recent research has shown the importance of peer and social multiplier
effects in crime (see e.g., Kling, Ludwig, & Katz, 2005; Bayer, Hjalmarsson, & Pozen,
2009; Patacchini & Zenou, 2012; Damm & Dustmann, 2014).
Ballester, Calvo´-Armengol, and Zenou (2006, 2010) have argued that concentrating efforts by targeting ‘‘key players,’’ that is, criminals who once removed
generate the highest possible reduction in aggregate crime level in a network, can
Address correspondence to Yves Zenou, Stockholm University, Department of Economics, 106 91
Stockholm, Sweden. E-mail: [email protected]
233
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C. BALLESTER AND Y. ZENOU
have large effects on crime because of these feedback effects or ‘‘social multipliers.’’
As a result, criminal behaviors can be magniﬁed and interventions can become more
effective. Based on a peer effect model, Ballester et al. (2006, 2010) have proposed
a centrality measure (the intercentrality measure) that determines the key player
in each network. However, in their model, contextual effects are not taken into
account; that is, only each individual’s characteristics affect her effort but not the
characteristics of her friends. In the present article, we extend this intercentrality
measure to include contextual effects and show that the formula proposed by
Ballester et al. (2006, 2010) must be adapted to a more general case.
To be more precise, we develop a network model1 where contextual effects
(which are measured here by the average characteristics of one’s friends) are taken
into account.2 We calculate the Nash equilibrium of this game and propose a new
measure, the contextual intercentrality measure, that determines the key player in
a network. This measure captures two effects. The ﬁrst effect is a pure contextual
effect, which is due to the change in the context (own and friends’ characteristics)
when the key player is removed from the network while the network is kept
unchanged. The second effect is a pure network effect, which captures the change
in crime effort due to the network structure change after the removal of the key
player. We also propose a simple example of a network with four individuals to illustrate all our results. We show that the key player is different depending on whether
or not the contextual effects are taken into account. This is important for policy
implications because this means that the planner may target the wrong person if it
ignores the effect of the ‘‘context’’ when removing a criminal from a network.
The rest of the article unfolds as follows. In the next section, we explain why
a key player analysis is important in social networks; in Section 3, we explain why
contextual effects matter in a network analysis. In Section 4, we present our network
model and determine the Nash equilibrium of this game and expose the intercentrality
measure of Ballester et al. (2006), which characterizes the key player when contextual
effects are not considered. In Section 5, we determine our new intercentrality measure
and illustrate it with a simple example. Finally, Section 6 concludes.
2. THE KEY PLAYER PROBLEM
The problem of identifying key players in a network is an old one, at least in
the sociological literature. Indeed, one of the focus of this literature is to propose different measures of network centralities and to assert the descriptive and=or prescriptive suitability of each of these measures to different situations (see, in particular,
Wasserman & Faust, 1994). Borgatti (2003, 2006) was among the ﬁrst researcher
1
There is a growing literature on networks in economics. See, in particular, Goyal (2007), Jackson
(2008), and Jackson and Zenou (2014).
2
Contextual effects are important, especially for the empirical measure of peer effects in crime. In
the standard linear-in-means models, Manski (1993, 2000) has put forward the importance of the reﬂection
problem, which is the difﬁculty of separating the contextual effect from the endogenous peer effect on own
behavior. Recent empirical papers have used the network topology to separate these two effects and to
show the importance of contextual effects in education (Calvo´-Armengol, Patacchini, & Zenou, 2009;
Lin, 2010), obesity (Cohen-Cole & Fletcher, 2008b), and crime (Patacchini & Zenou, 2012).
WHEN CONTEXTUAL EFFECTS MATTER
235
to really investigate the issue of key players, which is based on measuring explicitly
the contribution of a set of actors to the cohesion of a network. The basic strategy is
to take any whole network property, such as density or maximum ﬂow, and derive
a centrality measure by deleting nodes and measuring the change in the network
property. Measures derived in this way have been called ‘‘vitality measures’’ (see
Koschu¨tzki et al., 2005, for a review).
To be more precise, Borgatti (2003, 2006) identiﬁes two key player problems.
First, he puts forward the ‘‘key player problem=negative’’ (KPP-neg), which is
deﬁned in terms of the extent to which the network depends on its key players to
maintain its cohesiveness. It is a ‘‘negative’’ problem because it measures the amount
of reduction in cohesiveness of the network that would occur if the nodes were not
present. Borgatti gives examples of such problems. In public health, a key player
problem is whenever the planner needs to select a subset of population members
to immunize or quarantine in order to optimally contain an epidemic. In the military
or criminal justice context, the key player problem arises when a planner needs
to select a small number of players in a criminal network to neutralize (e.g., by
arresting, exposing, or discrediting) in order to maximally disrupt the network’s
ability to mount coordinated action.
The second key player problem identiﬁed by Borgatti (2003, 2006) is called
the ‘‘key player problem=Positive’’ (KPP-pos) in which the planner is looking for
a set of network nodes that are optimally positioned to quickly diffuse information,
attitudes, behaviors or goods. In a public health context, a health agency will need to
select a small set of population members to use as seeds for the diffusion of practices
or attitudes that promote health, such as using bleach to clean needles in a population
of drug addicts. In a military or criminal justice context, the planner needs to
select an efﬁcient set of actors to surveil, to turn (as into double agents), or to feed
misinformation to.
In the present article, we focus on the KPP-neg since a key player is the criminal who once removed from the network generates the highest reduction in total
crime in the network. Our approach is similar to that of Everett and Borgatti
(2010) in the sense that we decompose a node’s total centrality as the sum of its
endogenous and exogenous centrality (where exogenous is the node’s contribution
to everyone else’s centrality). This is obtained by summing all node’s endogenous
centrality, removing a node (or set of nodes) and then summing the recalculated
centralities of all remaining nodes. A major difference with the previous literature,
however, is that we explicitly model the behavior of criminals and their optimal
reaction when a key player is removed.3 Our article thus provides a rationale for
3
Researchers have also deﬁned ‘‘group’’ players so that the planner can remove more than one key
player at a time. See, in particular, Ortiz-Arroyo (2010) and Ballester et al. (2010). Ballester et al. (2010)
show that the key group problem is NP -hard from the combinatorial perspective. This means that there is
no possible sophisticated algorithm such that, given any network, will return the exact key group in
reasonable time. They show, however, that the key group problem can be approximated in
polynomial-time by the use of a greedy algorithm, where, at each step, the key player formula of Ballester
et al. (2006) is used to remove a player. They also show that the error of approximation of using a greedy
algorithm instead of solving directly the key group problem is at most 36.79%. As a result, in the present
article, we can still use our new intercentrality formula, which takes into account contextual effects, in the
greedy algorithm to tackle the issue of group players.
236
C. BALLESTER AND Y. ZENOU
the Everett-Borgatti procedure who fail to explain why the sum of centralities is
a meaningful graph invariant that is worthy of decomposition.
There are three recent papers that empirically determine key players (as we
deﬁned them in the present article) in soccer and crime. Using data from UEFA
Euro 2008 Tournament, Sarangi and Unlu (2011) evaluate a player’s contribution
to her team and relates her effort to her salaries; they show that key players regardless of their ﬁeld position have signiﬁcantly higher market values than other players.
In this paper, the authors do not take into account contextual effects because they
have no information on the characteristics of the player and thus use the standard
intercentrality measure of Ballester et al. (2006). Using data from adolescents in
the United States (AddHealth data), Liu, Patacchini, Zenou, and Lee (2012) determine the characteristics of the key players in crime. They include contextual effects
in their analysis and show that key players are more likely to be a male, have less
educated parents, are less attached to religion and feel socially more excluded than
the average criminal. They also show that contextual effects matter since the key
player may be different when the intercentrality measure used is the one given by
Ballester et al. (2006) or our formula. Using a data set of co-offenders in Sweden,
Lindquist and Zenou (2014) also include contextual effects in determining the key
player in each criminal network. They show that the key player policy outperforms
other reasonable police policies such as targeting the most active criminals or
targeting criminals who have the highest betweenness or eigenvector centrality in
the network.
3. WHY CONTEXTUAL EFFECTS MATTER?
Let us now highlight the importance of contextual effects in social networks. It
is usually easy to understand why the activities of others in one’s network would
affect one own activity level (endogenous peer effects), but it could be harder to
see why the average level of background characteristics (i.e., contextual effects) of
those in one’s network should have a direct effect on one’s activities.
To clarify this aspect, let us take the example of obesity. Obesity can easily
spread among a group of friends like a contagious disease, moving from one person
to another in an epidemic of fat. That is the ﬁnding of Christakis and Fowler (2007),
who reported that having close friends who are fat can nearly triple your risk of
becoming obese. The effect is so powerful that distance does not matter—the
inﬂuence is the same whether friends live next door or 500 miles apart.
Cohen-Cole and Fletcher (2008b) investigate the sensitivity of Christakis and
Fowler’s claim that obesity has spread through social networks by adding contextual
effects. It is indeed well known in the economics literature that failure to include contextual effects can lead to spurious inference on social network effects. Cohen-Cole
and Fletcher replicate the Christakis and Fowler (2007) results using their speciﬁcation and a complementary dataset. They ﬁnd that point estimates of the ’’ social
network effect’’ are reduced and become statistically indistinguishable from zero
once contextual effects are included. Individuals may adjust behavior because of
exposure to common inﬂuences (contextual inﬂuences). For example, the opening
of a fast food restaurant, convenience store, or gym, near a school could simultaneously affect the weight of all friends in a school’s social network. Importantly,
WHEN CONTEXTUAL EFFECTS MATTER
237
the presence of (often unmeasured) shared surroundings can lead to erroneously
implicating social network effects in individual outcomes where none exist. Finally,
individuals may alter their behavior as others in their group change theirs. In their
study, Christakis and Fowler do not include a sufﬁciently broad set of contextual
effects to account for a range of hypothesized causes of the epidemic. Also, the
Christakis and Fowler method of controlling for selection is much too a narrowing
scope. Once the ﬁrst two errors are corrected, evidence for endogenous causes
of obesity is thin. Cohen-Cole and Fletcher ﬁnd that the Christakis and Fowler
results are not robust. In fact, the econometric evidence points strongly to shared
environmental factors as the principle operative social mechanism underlying the
positive correlation in weight status within reference groups.
To show that contextual effects matter, Cohen-Cole and Fletcher (2008a)
investigate whether ‘‘network effects’’ can be detected for health outcomes that
are unlikely to be subject to network phenomena such as acne, height, and
headaches. Even if it is clear that there are no endogenous peer effects in these three
characteristics (how can the height of my friends has an impact on my own height?),
signiﬁcant network effects were observed in the acquisition of acne, headaches, and
height when contextual effects were not controlled A friend’s acne problems
increased an individual’s odds of acne problems. The likelihood that an individual
had headaches also increased with the presence of a friend with headaches, and an
individual’s height increased by 20% of his or her friend’s height. After adjustment
for environmental confounders, however, the results become uniformly smaller
and insigniﬁcant.
The problem of confounding has generally been addressed by controlling for a
rich set of individual, family, and environmental characteristics or using ﬁxed effects
at the group level. The problem, of course, is that social groups are often faced with
similar environmental characteristics. If these are neglected, one can improperly
interpret the results to imply that true ‘‘network effects’’ exist. To differentiate
between network effects of obesity and confounders, for example, one would want
to know the pattern of fast food restaurants or the caloric content of the school cafeteria available to the social network (or any other contextual effects). Inclusion of
the individual’s race, income, etc., might be reasonable proxies for some studies
but cannot distinguish two otherwise similar groups that have different environments. In their empirical models, Cohen-Cole and Fletcher (2008a) control for
school-level ﬁxed effects to control for all environmental conditions shared by
students in the same school. Their results suggest that failure to control for confounders increases the chances of attributing similarity in health outcomes between friends
to social network effects when the similarities are caused by shared environments.
4. THE MODEL WHEN THERE ARE NO CONTEXTUAL EFFECTS
4.1. The Game
A network g is a set of ex ante identical delinquents N ¼ {1, . . . , n} and a set of
links between them. We assume n 2. The n-square adjacency matrix G of a network
g keeps track of the direct connections in this network. By deﬁnition, delinquents i and
j are directly connected in g if and only if gij ¼ 1, (denoted by ij), and gij ¼ 0 otherwise.
238
C. BALLESTER AND Y. ZENOU
Links are not necessarily reciprocal, so that we may have gij 6¼ gji. We consider the
following utility function for each delinquents i who chooses effort yi:4
n
X
1
ui ðy; gÞ ¼ ai yi y2i þ /
gij yi yj ;
ð1Þ
2
j¼1
where / > 0 measures the strength of complementarities. This is a game with strategic
i ðy;gÞ
complementarities since if j is linked with i, then if yj increases, @u@y
also increases by
i
/. This is exactly the same utility function as in Ballester et al. (2006, 2010) and
Calvo´-Armengol et al. (2009) with one crucial difference: the ex ante ai of each agent
i is not given by ai ¼ xi but by:
n
1X
ai ¼ xi þ
gij xj ;
ð2Þ
gi j¼1
P
where gi ¼ nj¼1 gij is the total number of links of individual i. In other words, the
ex ante difference of individual i is not only determined by her own attribute xi
(e.g., her race, age, education) but also by the average attribute of her friends (i.e.,
the average race of her friends, the average age of her friends, the average education
of her friends). For simplicity, we assume xi 0 for all i. The ai described in (2) is
usually referred to as the contextual effect (Manski, 1993, 2000). What is crucial
for our analysis is that the structure of the network determines this ai.
4.2. The Katz-Bonacich Network Centrality Measure
½k
Let Gk be the k th power of G, with coefﬁcients gij , where k is some integer.
½k
The matrix Gk keeps track of the indirect connections in the network: gij 0
measures the number of walks of length k 1 in g from i to j. In particular,
G0 ¼ I. Given a scalar / 0 and a network g, we deﬁne the following matrix:
Mðg; /Þ ¼ ½I /G1 ¼
þ1
X
/ k Gk :
k¼0
where I is the identity matrix. These expressions are well-deﬁned for low enough
values of /.5 Let us deﬁne the Katz-Bonacich centrality (due to Katz, 1953;
Bonacich, 1987).
Definition 1. Consider a network g with adjacency n-square matrix G and a scalar /
such that M(g, /) ¼ [I /G]1 is well-deﬁned and nonnegative. Given a vector u 2 Rnþ ,
the Katz-Bonacich u-weighted centrality of parameter / in g is deﬁned as:
bu ðg; /Þ ¼
þ1
X
/k Gk u ¼ ½I /G1 u
ð3Þ
k¼0
4
Boldface lowercase letters refer to vectors while boldface capital letters refer to matrices.
Let q(G) be the spectral radius of the nonnegative matrix G, that is, the largest absolute value of its
eigenvalues. When / < q(G), the inverse [I /G]1is well-deﬁned and nonnegative (Debreu & Herstein,
1953).
5
WHEN CONTEXTUAL EFFECTS MATTER
239
The i th entry of the vector bu(g, /) is denoted by bu,i(g, /). Let 1T be the vector
of ones. Then, total centrality is deﬁned as
bu ðg; /Þ ¼
n
X
bu;i ðg; /Þ ¼ 1T Mu:
i¼1
4.3. Nash Equilibrium
Denote by q(G) the spectral radius of G. We have the following result:
Proposition 1. Consider a game where the utility function of each agent i is given by
(1) with a > 0 (i.e., ai > 0 , for all i 2 N ). Each ai is deﬁned by (2). If /q(G) < 1 , then
this game has a unique Nash equilibrium in pure strategies y , which is interior and
given by:
y ¼ ba ðg; /Þ:
ð4Þ
This proposition is a direct application of Theorem 1 in Calvo´-Armengol et al.
(2009) and says that, at the Nash equilibrium, each delinquent i ’s effort is equal to
her weighted Katz-Bonacich centrality.6
4.4. Finding the Key Player When There Are No Contextual Effects
We would like now to expose the ‘‘key player’’ policy. The planner’s objective
is to ﬁnd the key player, that is, the delinquent who once removed generates the
highest possible reduction in aggregate delinquency level. Formally,
the planner’s
P
problem is: maxfy ðgÞ y ðg½i Þji ¼ 1; . . . ; ng, where y ðgÞ ¼ i yi ðgÞ is the total
level of crime in network g, and g[i] is network g without individual i. When the
original delinquency network g is ﬁxed, this is equivalent to:
minfy ðg½i Þji ¼ 1; . . . ; ng:
ð5Þ
From Ballester et al. (2006), we can deﬁne a new network centrality measure d
(g, /, a) that solves (5). Let ba, i(g, /) be the centrality ofPdelinquent i in network g,
ba(g, /) the total centrality in network g (i.e., ba ðg; /Þ ¼ ni¼1 ba;i ðg; /Þ) and ba(g[i],
/) the total centrality in g[i].
Definition 2. Assume that ai ¼ xi for all i (no contextual effects). Then, for all
networks g and for all i , the intercentrality measure of delinquent i is:
ba;i ðg; /Þ
di ðg; /; aÞ ¼
6
n
P
mji ðg; /Þ
j¼1
mii ðg; /Þ
:
ð6Þ
In fact, uniqueness is a consequence of the spectral condition /q(G) < 1. Interiority is guaranteed
with the additional condition ai > 0.
240
C. BALLESTER AND Y. ZENOU
Figure 1 A bridge network with four deliquents.
The following result (theorem 3 in Ballester et al., 2006) establishes that when
ai ¼ xi, intercentrality captures, in an meaningful way, the two dimensions of the
removal of a delinquent from a network, namely, the direct effect on delinquency
and the indirect effect on others’ delinquency involvement.
Proposition 2. Assume that the utility function of each delinquent i is given by (1)
for ai ¼ xi . Then, a player i is the key player that solves (5) if and only if i is a delinquent with the highest intercentrality in g , that is, di ðg; /; aÞ di ðg; /; aÞ , for all
i ¼ 1, . . . , n.
Let us now illustrate Proposition 2 with a simple example when ai ¼ x. Consider the following symmetric undirected network with four individuals (i.e., n ¼ 4)
in Figure 1 and assume
0
1
0 1 0
1
x1
a1
0
B x2 C
B a2 C B 1 C
C
B C B
C
x¼B
@ x 3 A ¼ a ¼ @ a3 A ¼ @ 1 A :
1:25
x4
a4
The adjacency matrix G of this network is given by
0
0
B1
G¼B
@1
1
1
0
1
0
1
1
0
0
1
1
0C
C
0A
0
Assume / ¼ 0.4.7 It is then straightforward to see that, using Proposition 1, the
Nash equilibrium in efforts is given by
1 0
1
1 0
ba;1 ðg; /Þ
y1
5:97826
B y2 C B ba;2 ðg; /Þ C B 5:65217 C
C B
C
B C¼B
@ y A @ ba;3 ðg; /Þ A ¼ @ 5:65217 A;
3
ba;4 ðg; /Þ
y4
3:6413
0
7
ð7Þ
The spectral radius of this graph is 2.17, and thus the condition /l1(G) < 1 is satisﬁed since
2.17 0.4 ¼ 0.868 < 1.
WHEN CONTEXTUAL EFFECTS MATTER
241
so that the total equilibrium activity level is equal to
y ¼ y1 þ y2 þ y3 þ y4 ¼ ba ðg; /Þ ¼ 20:9239:
Individual 1 has the highest weighted Katz-Bonacich centrality and thus
provides the highest crime effort.
Let us now calculate the key player when there are no contextual effects. It is
easily veriﬁed that the contribution of each player is equal to
y y½1 ¼ 20:924 4:5834 ¼ 16:3406
y y½2 ¼ 20:924 4:6323 ¼ 16:2916
y y½3 ¼ 20:924 4:6323 ¼ 16:2916
y y½4 ¼ 20:924 9:9999 ¼ 10:9239:
As a result, individual 1 is also the key player because she has a central
position, intermediating the interactions taking place in the network.8 We will see
below that, when contextual effects will be taken into account, criminals 2 and 3 will
become the key players.
It is easily checked that formula (6) applies here. Indeed, for each i ¼ 1, 2, 3, 4,
we have
ba;i ðg; /Þ
y y½i ¼ di ðg; /; aÞ ¼
n
P
mji ðg; /Þ
j¼1
:
mii ðg; /Þ
For / ¼ 0.4, we have
0
3:26087
B
2:17391
M ¼ ½I /G1 ¼ B
@ 2:17391
1:30435
2:17391
2:63975
1:92547
0:869565
2:17391
1:92547
2:63975
0:869565
1
1:30435
0:869565 C
C;
0:869565 A
1:52174
ð8Þ
and it can be easily checked that y y½i ¼ di ðg; /; aÞ for all i ¼ 1, . . . , 4. For
example, for player 1, ba,1(g, /) ¼ 5.97826 (see (7)), m11(g, /) ¼ 3.26087 and
P
n
j¼1 mj1 ðg; /Þ ¼ 8:91304, so her contribution is:
d1 ðg; /; aÞ ¼
5:97826 8:91304
¼ 16:3406
3:26087
This is what we obtained above when we calculated y y½1 . The same
applies for the removal of the other players.
8
In this example, player 1 is both the most active individual and the key player. Ballester et al.
(2006) show that, in general, these notions need not to coincide.
242
C. BALLESTER AND Y. ZENOU
5. FINDING THE KEY PLAYER WHEN CONTEXTUAL EFFECTS MATTER
5.1. A Motivating Example
Consider the network described in Figure 1. If we take into account contextual
effects, then
0
1 0
1
x1
0
B x2 C B 1 C
C B
C
x¼B
@ x3 A ¼ @ 1 A
1:25
x4
and
0
1 0
1 0
1
x1 þ x2 þx33 þx4
1:08333
aC
1
B aC C B x2 þ x1 þx3 C B 1:5 C
2 C
2
B
C B
C
aC ¼ B
@ aC A ¼ @ x3 þ x1 þx2 A ¼ @ 1:5 A:
3
2
x4 þ x11
1:25
aC
4
ð9Þ
In that case, at the Nash equilibrium, using Proposition 1, we obtain9
0
1 0
1 0
1
baC ;1 ðg; /Þ
11:6848
yC
1
B yC C B baC ;2 ðg; /Þ C B 10:2899 C
B 2C C ¼ B
C B
C
@ y A @ baC ;3 ðg; /Þ A ¼ @ 10:2899 A;
3
baC ;4 ðg; /Þ
5:92391
yC
4
ð10Þ
so that the total crime level is given by
C
C
C
yC ¼ yC
1 þ y2 þ y3 þ y4 ¼ baC ðg; /Þ ¼ 38:1884:
In other words, even with contextual effects, individual 1 is the most active
individual. Observe that total crime activity is higher when contextual effects are
taken into account (from 20.9239 to 38.1884, it nearly doubles) because of the
synergies generated by them.
As before, by removing individual 1, we have now a network with three
delinquents where we have deleted the ﬁrst column and ﬁrst row in G to obtain
0
G½1
0 1
¼ @1 0
0 0
1
0
0 A:
0
What is important here is that the as also change after the removal of
delinquent 1. Denote by aC[1] the (n 1) 1 vector after the removal of delinquent
1. Then, (a2, a3, a4) are not anymore equal to (1.5, 1.5, 1.25) as in (9) but to
9
The superscripts and C indicate the Nash equilibrium values without and with contextual effects,
respectively.
WHEN CONTEXTUAL EFFECTS MATTER
0
aC ½1
1 0
1 0
1
½1
a2
x2 þ x13
2
B ½1 C @
x2 A
¼ @ a3 A ¼ x3 þ 1 ¼ @ 2 A:
½1
1:25
x4
a4
243
ð11Þ
Using the same decay factor / ¼ 0.4, we obtain at the Nash equilibrium the
following equilibrium efforts:
1 0
1 0
1
C½1
baC½1 ;2 ðg½1 ; /Þ
y2
3:33333
C
B C½1 C B
@ y3
A ¼ @ baC½1 ;3 ðg½1 ; /Þ A ¼ @ 3:33333 A;
½1
C½1
1:25
baC½1 ;4 ðg ; /Þ
y4
0
so that the total crime activity is now given by
C½1
yC½1 ¼ y2
C½1
þ y3
C½1
þ y4
¼ baC½1 ðg½1 ; /Þ ¼ 7:91667:
Thus, player 1’s actual contribution is
yC yC½1 ¼ baC ðg; /Þ baC½1 ðg½1 ; /Þ ¼ 38:1884 7:91667 ¼ 30:2717:
ð12Þ
If we perform the same procedure for the other players, we obtain the
following contextual effects when removing player 2:
0
aC½2
1 0
1 0
1
½2
4
a1
x1 þ x3 þx
1:125
2
B
C @
x3 þ x11 A ¼ @ 1 A:
¼ @ a½2
A¼
3
½2
x4 þ x11
1:25
a4
Thus, delinquent 2’s contribution is
yC yC½2 ¼ 38:1884 7:61029 ¼ 30:5781:
Similarly, if we remove player 3, we have
0
aC½3
1 0
1 0
1
½3
4
a1
x1 þ x2 þx
1:125
2
B
C @
x2 þ x11 A ¼ @ 1 A;
¼ @ a½3
A¼
2
½3
x4 þ x11
1:25
a4
so that delinquent 3’s contribution is
y yC½3 ¼ 38:1884 7:61029 ¼ 30:5781:
ð13Þ
244
C. BALLESTER AND Y. ZENOU
Finally, when removing player 4, the contextual effects become
0
aC½4
1 0
1 0
1
½4
3
a1
x1 þ x2 þx
1
2
B ½4 C @
3 A
¼ @ 1:5 A;
¼ @ a2 A ¼ x2 þ x1 þx
2
x1 þx2
½4
x3 þ 2
1:5
a3
so that delinquent 4’s contribution is given by
y yC½4 ¼ 38:1884 20 ¼ 18:1884:
Interestingly, delinquents 2 and 3 are now the key players while, when contextual effects are not taken into account, individual 1 is the key player. This shows the
importance of incorporating contextual effects in the key player analysis.
It is easily veriﬁed that, with contextual effects, formula (6) is not able to determine these contributions. This is because in (6) contextual effects are not taken into
account while they are in (13).
5.2. A New Formula for the Key Player
Let us start with some notations. Let us consider the potential removal of
player 1 from the game (it could be any player k; we focus on player 1 for the sake
of the presentation):
aC
h1i
¼
C ½1 :
x1
a
The vector aC
h1i describes the situation in which player 1 has not yet been
removed so that she has her default attribute x1 and she does not affect, or is affected
by, the other players’ contextual effects. In other words, the contextual vector aC is
computed from the network g[1].
Proposition 3. Assume
that the utility function of each delinquent i is given by (1)
Pn
1
where aC
¼
x
þ
g
i
i
j¼1 ij xj . Then, the contribution of player 1 to total effort in the
gi
game (yC yC[1]) is given by the contextual intercentrality:
d1 ðg; /; aC Þ ¼ baC ðg; /Þ baC ðg; /Þ þ
h1i
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Contextual change in g
d1 ðg; /; aC
h1i Þ
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Intercentrality of player 1 in g; for fixed aC
h 1i
baC ;1 ðg; /Þ
h 1i
¼ baC ðg; /Þ baC ðg; /Þ þ
h 1i
n
P
j¼1
m11
mj1
:
ð14Þ
Hence, 1 is the key player that solves (5) if and only if 1 is the delinquent with
the highest contextual intercentrality in g, that is, d1(g, /, aC) di(g, /, aC), for all
i ¼ 1, . . . , n.
WHEN CONTEXTUAL EFFECTS MATTER
245
The proof can be found in the Appendix. In formula (14), the ﬁrst effect is the
contextual effect, which is due to the change in the contextual effect aC (from aC
to aC
h1i ); that is, the contextual effects emanating from player 1 disappear while the
network g remains unchanged. The second effect is the network effect, which
captures the change in the network structure when the key player is removed from
the network g and the contextual vector is aC
h1i . The latter effect corresponds to the
standard inter-centrality measure of Ballester et al. (2006) (see (6)) for aC
h1i . If there
were no contextual effects so aC did not change after the removal of delinquent 1
(i.e., aC ¼ aC
ðg; /Þ ¼ 0, and we would
h1i ¼ x), we would have had baC ðg; /Þ baC
h 1i
be back to the standard formula deﬁned in (6).
5.3. Back to the Example
Let us illustrate this last result with the network described in Figure 1. As usual
and without a loss of generality, let us focus on the removal of player 1. For / ¼ 0.4
and (x1, x2, x3, x4) ¼ (0, 1, 1, 1.25), we showed that the removal of delinquent 1 led to
16:3406 ¼ d1 ðg; /; aÞyC yC½1 ¼ 30:2717:
This meant that the formula proposed in (6) was not applicable when contextual effects were taken into account. Let us now show that the new formula is valid
so that yC yC[1] ¼ d1(g, /, a).
Let us ﬁrst calculate the contextual change baC ðg; /Þ baC ðg; /Þ. We have
h 1i
baC ðg; /Þ baC ðg; /Þ ¼ 1T MaC 1T MaC
h1i :
h 1i
Recall that (see (10))
0
1 0
1
11:6848
yC
1
B yC C B 10:2899 C
B 2 C¼B
C
@ yC A @ 10:2899 A;
3
5:92391
yC
4
so that total activity level before
yC ¼ baC ðg; /Þ ¼ 38:1884. We also have
0
aC
h1i ¼
x1
B aC ½2
B 2
B C ½2
@ a3
C ½2
a4
1
the
removal
0
1
0
C B
C B 2 C
C;
C¼@
2 A
A
1:25
so that we obtain (using (8))
baC ðg; /Þ ¼ 1T MaC
h1i ¼ 36:1413:
h 1i
ð15Þ
of
any
player
is
246
C. BALLESTER AND Y. ZENOU
As a result,
baC ðg; /Þ baC ðg; /Þ ¼ 1T MaC 1T MaC
h1i ¼ 38:1884 36:1413 ¼ 2:0471:
h 1i
In other words, the direct contextual effect, baC ðg; /Þ baC ðg; /Þ, is equal to
h 1i
2.0471.
It is easily veriﬁed that baC ;1 ðg; /Þ ¼ 10:3261 (for that take the ﬁrst row of
h 1i
MaC
h1i ) and thus
d1 g; /; aC
h1i
baC ;1 ðg; /Þ
h 1i
¼
n
P
j¼1
m11
mj1
¼
10:3261 8:91304
¼ 28:2246:
3:26087
This means that the indirect contextual effect, d1(g, /, ah1i), is equal to 28.2246.
As a result, the total contextual effect is given by:
d1 g; /; aC ¼ baC ðg; /Þ baC ðg; /Þ þ d1 g; /; ah1i
h 1i
¼ 2:0471 þ 28:2246 ¼ 30:2717:
Therefore, we have d1(g, /, aC) ¼ yC yC[1] ¼ 30.2717. The same applies for
the other players.
6. CONCLUDING REMARKS
In the present article, we consider a model where the criminal decision of each
individual is affected not only by her own characteristics but also by the characteristics of her friends (contextual effects). We characterize the Nash equilibrium of this
game and determine who the key player is; that is, the criminal who once removed
generates the highest reduction in total crime in the network. We show that the formula proposed by Ballester et al. (2006) is not correct and give another one that highlights two effects. The ﬁrst effect is a pure contextual effect, which is due to the change
in the context (own and friends’ characteristics) when the key player is removed from
the network while the network is kept unchanged. The second effect is a pure network
effect, which captures the change in crime effort due to the network structure change
after the removal of the key player. We also propose a simple example of a network
with four individuals to illustrate all our results. We show that, without contextual
effects, the key player is individual 1 while, with contextual effects, the key player
is individual 2 (or 3). This is important for policy implications because this means that
the planner may target the wrong person if it ignores the effect of the ‘‘context’’ when
removing a criminal from a network. We also believe that this result is important for
empirical applications since by considering contextual effects some endogenous peer
effects may disappear as in the obesity example highlighted in Section 3.
FUNDING
Coralio Ballester acknowledges ﬁnancial support from the Fundacio´n Ramo´n
Areces and the Spanish Ministry of Education and Innovation and FEDER,
through the project SEJ–2007–62656.
WHEN CONTEXTUAL EFFECTS MATTER
247
ACKNOWLEDGMENTS
We are grateful to Phillip Bonacich as well as two anonymous referees for
helpful comments.
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APPENDIX: PROOF OF PROPOSITION 3
Let yC be the total effort before removal of player 1, and yC[1] be the total effort
after removal. Then, by Proposition 1, the change in total activity after removal
of player 1 is given by:
yC yC ½1 ¼ baC ðg; /Þ baC½1 g½1 ; /
h
i h
i
¼ baC ðg; /Þ baC ðg; /Þ þ baC ðg; /Þ baC½1 g½1 ; /
h 1i
h 1i
h
i ¼ baC ðg; /Þ baC ðg; /Þ þ baC ðg; /Þ baC½1 g½1 ; /
h 1i
h 1i
h 1i
h
i
C
¼ baC ðg; /Þ baC ðg; /Þ þ d1 g; /; ah1i
h 1i
P
h
i baC ;1 ðg; /Þ nj¼1 mj1
h 1i
¼ baC ðg; /Þ baC ðg; /Þ þ
:
h 1i
m11
&