Károly Takács* and Béla Janky **+
Collective Action and Network Dynamics
* ICS / University of Groningen, Department of
Sociology, Grote Rozenstraat 31, 9712 TG Groningen,
The Netherlands. E-mail: k.takacs@rug.nl
** Budapest University of Technology and Economics,
Department of Sociology, H-1111 Budapest, Stoczek
utca 2., Hungary. E-mail: janky@eik.bme.hu
+ We would like to thank Andreas Flache, Russell Hardin, Anthony Oberschall, Zoltán
Szántó, Benedek Kovács, and Attila Gulyás for their comments on an earlier version of the
paper. The authors acknowledge support of the Hungarian Scientific Research Fund (OTKA),
T/16, 046381; Collegium Budapest / Institute for Advanced Study; and the Netherlands
Institute for Advanced Study in the Humanities and Social Sciences (NIAS).
Short abstract
This study of collective action relaxes the critical assumption that social network relations are
fixed. Social ties influence participation as they transmit forms of social control that reward
participation or conform behavior. Social control, in turn, makes some ties more beneficial
than others and therefore actors might strategically delete and form relations. Participation
itself may depend on the prospects of network changes. We introduce equilibrium concepts
that embrace network formation and participation. Results show that when structural choices
follow collective action, segregation of contributors is paradoxically not a predictor of link
deletion, although it is important for forming new ties.
Long abstract
A critical assumption in models of collective action that count on social network effects is that
network relations are fixed. This study goes beyond the static analysis of network effects and
incorporates the opportunity of deleting and of building interpersonal ties. We analyze the
effects of collective action problems on network formation and we consider how the foresight
of structural change influences collective action outcomes.
Interpersonal ties are important mobilizing forces in collective action, as they transmit
different forms of social control such as behavioral confirmation and social selective
incentives. In order to capture the interrelations between network structure and collective
action, we incorporate social control mechanisms and consequently social network effects in
the n-person public good game. This game is extended by allowing actors to decide about
deleting or keeping social ties, and consequently allowing them to build new ties based on
mutual consent. We analyze situations in which the transmission of social control follows
collective action and situations when they are simultaneous. We introduce an equilibrium
concept that embraces equilibrium in the n-person game and network stability, called strongly
robust network equilibrium. Results show that when structural choices follow collective
action, the level of segregation of contributors and free riders is paradoxically not a predictor
of link deletion, although it plays a role in forming new ties. Perfectly segregated structures
with fully connected subsets of contributors and free riders, however, are not subject to
change. In general, density has a stronger impact on change than segregation; dense networks
are being more robust than sparse ones. At simultaneous participation and low-cost structural
decisions, only uniform action with fully connected or isolated networks can meet the
requirements of strongly robust network equilibrium.
Keywords: Collective action; Social dilemmas; Social networks; Social control; Structural
balance; Local interaction games
1
Collective Action and Network Dynamics
1. Introduction
A critical assumption that is made even by those models of collective action that count on
social network effects is that there is a fixed set of interpersonal relations that do not change
over time. Relational ties are exogenously given and at most, static comparisons are made.
Despite the interests of actors often shape the development of social ties, endogenous network
development is an issue hardly investigated in the context of collective action, especially in
single-shot encounters.
It has been argued for long in empirical studies and theoretical models that collective action is
shaped by the interpersonal relations between prospective participants (e.g., Oberschall, 1973;
1994; Tilly, 1978; Oliver, 1984; McAdam, 1986; Marwell, Oliver, and Prahl, 1988; Finkel
and Opp, 1991; Chong, 1991; McAdam and Paulsen, 1993; Marwell and Oliver, 1993; Opp
and Gern, 1993; Gould, 1993; 1995; Kim and Bearman, 1997; Sandell and Stern, 1998;
Chwe, 1999; 2000; Diani, 2003a). It is also well documented that networks change over time
and this is also reflected in how people behave in collective action (Kim and Bearman, 1997;
Diani, 2003c; Osa, 2003). Besides independent network dynamics, networks change because
of the internal determinants of collective action. People might choose their friends
strategically in order to maximize rewards and minimize punishments that originate in social
control (cf. Harsanyi, 1969). This is especially relevant when social control mechanisms are
delayed after collective action decisions. For instance, honors as selective incentives are
awarded for the heroes of a revolution well after the uprising. Even internalized selective
incentives are activated only after the collective action (“I am proud that I was part of it”).
Furthermore, in order to facilitate participation, strategic or tactical establishment of
communication channels and other linkages might take place (McAdam and Paulsen, 1993;
Diani, 2003b). Some actors might purposefully take brokerage roles and make a connection
between other individuals who are not related directly (cf. Burt, 1992). Strategic tie formation
goes very far in the proliferation of social movement politics. Campaigning and lobbying
involves institutionalized techniques as well as tie-formation strategies, popularly called as
“networking” (e.g., Tilly and Wood, 2003).
There is also empirical evidence on how collective action might restructure individual
relations. Activism has certainly a role in changing the meaning and impact of friendship ties
(Gould, 2003). Actors might reject the leadership roles or the social pressure of their network
environment by decreasing the importance of these contacts. Other contacts meanwhile might
become more important as individuals gain positive feedback from them. As a result of the
dynamics of interpersonal relations, macro properties of the network also change. As a
different development, contacts might be institutionalized, new associations or organizations
might arise as a result of successful collective action, but also in case of failure.
The perspective of network changes and the underlying network dynamics, however, has not
been captured by theoretical models of collective action. This study intends to go in this
2
direction and builds on previous models that combined game theory and structural change.
These models concentrated on long term network evolution, including theories of structural
learning (Kitts, Macy and Flache, 1999; Macy et al., 2003) and strategic network formation
without collective interdependencies (Bala and Goyal, 2000; Bonacich, 2001). Structural
learning describes the process that individuals strengthen ties that are beneficial for them and
abandon ties with negative experience. Games of strategic network formation assume values
for established ties in cooperative situations (Dutta, Nouweland, and Tijs, 1998; Slikker and
Nouweland, 2001) or indirect benefits from the chain of contacts (Jackson and Wolinsky,
1996). Individual benefits of tie formation were emphasized earlier also by sociological
approaches in obtaining social capital and receiving information on job opportunities. This
can be achieved by strategic building of bridging ties, brokerage relations, or by creating
structural holes (e.g., Granovetter, 1973; 1974; Rees, 1966; Burt, 1992). More psychological
approaches emphasize the reduction of tensions in structural relations. An example is the
drive towards balance in triads: “the friend of my friend becomes my friend”; “the enemy of
my friend becomes my enemy” (Heider, 1946; Cartwright and Harary, 1956). Another well
documented psychological mechanism is homophily, the attraction towards similar others
(Mark, 1998; McPherson, Smith-Lovin, and Cook, 2001) and its mirror image, distancing
oneself from dissimilar others (Mark, 2003). In addition, new ties naturally emerge while
people perform new activities, while they are at new locations, and at coincidental encounters.
Common in these approaches is that there is much more emphasis on building ties than on
abandoning ties and all processes that are taken into account are independent from collective
interdependencies. Everyday experience tells us, however, that relationships emerge and fade
away as a consequence of collective interdependencies, and expectations about possible
changes affect individual behavior (e.g., Burt, 2002). A theoretical model proposed in this
study, therefore, will incorporate the possibility of strategic tie formation and deletion, and
these decisions will be handled as interdependent with participation decisions in collective
action. This includes both the possibility that interpersonal relations change due to
expectations about collective action and the possibility that network relations are transformed
as a result of collective interdependencies. Hence, this study aims to open the floor for
considering relational changes endogenously and it will be considered how the foresight of
structural change influences collective action outcomes in single encounters. That is, the study
aims to analyze the impact of collective action on network dynamics and the consequences of
simultaneous considerations of collective action and relational changes on the outcome of
collective action and on the stability of the social network.
In short, the study provides a theoretical framework for the analysis of collective action
situations and network effects. As a new development, the traditional assumption of fixed
networks is relaxed. The theoretical model will represent collective action as a public goods
game with interpersonal influence in which players can abandon existing relationships or
build new ones in case it is in their rational interest. The study also tries to delineate the
structural conditions under which collective action can be achieved in a way that it does not
destroy personal relations. In this way, stable structures can be determined in which nobody
3
has an incentive to change behavior or contacts. This means that this study also contributes to
the foundation of equilibrium concepts for interdependent collective and network dynamics.
2. The modeling framework
This section provides a description of the fundamental elements of the modeling framework.
Collective action is modeled as an n-person public goods game and it is integrated with a
model of dyadic interdependence between connected players. This fundamental model will be
extended in the subsequent section by a network game, in which players can change their
interpersonal connections.
We model collective action as a standard n-person public goods game with a final set of
players (N={1, …, i, …, n}, where n>2) and with a linear production function and binary
decisions. Participation (contribution) means a provision of a unit of a public good and
defection (no participation) means no additional provision. The action taken by the individual
iN is denoted by i, where i=1 is contribution and i=0 is defection. Contribution has a
cost c, and the value of a unit of the public good provided by any player is  for all
individuals (c>). Although the cost of contribution is higher than the gain of the provision of
one unit of the public good, there is a threshold number of contributors n* (1<n*n), for
which n*>c.
To incorporate network effects in the model, we assume that players might have connections
to others. For a general model, we do not specify the type (e.g., friendship, discussion) and the
strength of these relationships. We only assume that actors who are connected influence each
other and this influence is inevitable once there is a connection. To have a parsimonious
model, we model the web of social contacts by an undirected and unvalued graph, in which
nodes are individuals and edges are relationships. This means that we assume that every tie is
symmetric and equally important. We will denote the existence of a direct relationship
between individuals i and j by rij (i,jN, and i  j), where rij = rji = 1 if there is a direct
relationship between them, and rij = rji = 0 if they are not directly related. For the sake of
n
simplicity, we will denote the total number of i’s ties (degree) by ri (  rij  ri , where i  j).
j 1
The dyadic interdependence between connected individuals will be modeled as follows. These
behavioral assumptions are largely based on previous research that connected collective
action with social influence and social control (e.g., Olson, 1965; Oliver, 1984; Marwell,
Oliver, and Prahl, 1988; Heckathorn 1989; Coleman, 1990; Chong, 1991; Gould 1993;
Marwell and Oliver, 1993; Oberschall 1994; Flache and Macy 1996; Sandell and Stern, 1998;
Takács, 2001), especially on Janky and Takács (2004).
The choice of a related person influences the actor in different ways. Individuals might prefer
to do the same that related actors do (cf. Chong, 1991; Oberschall, 1994). In this case, the
individual’s deviation from a related player's choice implies lower payoff than the outcome
4
where they behave in the same way. Both the absolute number and the proportion of deviators
among the related actors might matter. That is, we assume that behavioral confirmation might
take two different forms. The first form of behavioral confirmation is a linear function of the
number of friends with the same choice and we call it mass conformity. Formally, we will
assume that all related actors with the choice equivalent to i’s decision change i's payoff with
b1. For example, when mass conformity operates, an individual, who intends to participate in
a demonstration, likes when there are many friends in the crowd. On the other hand, in case
this individual prefers to stay at home, she or he likes to be assured that many friends choose
the same option.
The second form of behavioral confirmation is received as a linear function of the proportion
of friends with the same choice, and we refer to it as proportional conformity. Proportional
conformity is independent of the number of ties the given individual has. The coefficient of
the proportional type of conformity is denoted by b2.
Moreover, we suppose that every player receives rewards (punishments) for contribution
(defection) from each of his or her friends. The amount of these social selective incentives
from a single tie is denoted by s. One should note that in our model, social selective incentives
are elements of the rational calculation of the receiver, but are not of the one of the provider
as they are free to produce. In case of costly provision of selective incentives, the model
should take account of the second order free rider problem (Oliver, 1980; Heckathorn, 1989).
As we concentrate, however, on selective incentives of a social character like respect or
status, we often see that these are produced without costs. Besides, there is also a documented
tendency that individuals punish defectors voluntarily (Fehr and Gächter 2000; 2002; Boyd et
al. 2003).
Denote C and D two disjoint sets of the group, such sets that C={N\D}. Moreover, let us
denote ric and rid the numbers of i's connections who are elements of sets C and D,
respectively (ric+rid=ri). If every member of C participates and every member of D defects,
then the payoffs of defection and participation for i are the following:
 i  i  0  rid b1 
 i  i  1  ri s  ricb1 
n
rid
b2    j
ri
j 1
(1)
n
ric
b2   ( j  1)  c
ri
j 1
where j  N\{i}. In the main text, we assume that >0 and social control parameters are
rewards (b10, b20, and s0). The analysis of situations in which social control appears in
terms of punishments is left to the Appendix. From (1) it follows that participation of i is
rational if
5
  ri s  ( ric  rid )( b1 
b2
)c.
ri
(2)
From (2) it follows that selective incentives foster participation relative to the number of
connections of the given individual. Behavioral confirmation promotes contribution only
when there are more contributing friends than defectors. In case the number of free riding
friends exceeds the number of participating friends, behavioral confirmation drives towards
defection. In case the cost of contribution is too high we cannot expect any provision of the
public good. When there are strong incentives for contribution, however, collective action can
be established.
A more refined analysis of this static model can be found in Janky and Takács (2004). Main
results for the entire network show that the minimum degree of the network is a strong
determinant of overall collective action in case selective incentives operate. Network
clustering has a strong influence when behavioral confirmation mechanisms are strong and
might undermine mass collective action. Clustered networks are more likely to have partial
contribution equilibria, in which participants and free riders are segregated. The smaller the
number of free riders in the partial contribution equilibrium, the less likely that full
contribution is a payoff dominant equilibrium. Moreover, the payoff dominance of full
contribution equilibrium is not likely in centralized structures when mass conformity is
strong, but it is possible in case proportional conformity is prevalent. A further interesting
result is that not only behavioral confirmation, but also selective incentives might have a nonmonotonic effect on the occurrence of full contribution.
3. Collective action and link deletion proofness
The model presented in the previous section will be extended with the perspective of network
change below. As we prefer stepwise model building (cf. Lindenberg, 1992), first the
possibility that actors might abandon relations will be considered. This leads us towards the
investigation of stable network structures that are link deletion proof.
Link deletion proofness or robustness of networks is an issue that has been investigated
previously in different contexts (see Albert and Barabási, 2002; Gilles and Sarangi, 2004).
Structures, in which an incremental change triggers a chain of dissolution of contacts, are
considered as highly vulnerable networks. The logic of investigation in these studies
implicitly assumes that dissolution of ties is a “backward development”. Other studies
emphasize the individual opportunities that arise from abandoning relations. In this line of
research, Burt (1992) claims that social capital can be obtained by optimizing the ego-network
structure, for instance by creating structural holes, in which information does not circulate as
it does in dense ego-networks. Abandoning relations is assumed to serve for the benefit of the
group also in different lines of social psychological literature and hence it is considered as a
6
purposeful action (see Williams, Forgas, and von Hippel, forthcoming for an overview).
Ostracism and the threat of exclusion is documented to have a significant relevance in
establishing collective action (Olson, 1982; Hirshleifer and Rasmusen, 1989; Ouwerkerk et
al., forthcoming).
As a first step, we consider the n-person public goods game with social control and with the
extension that actors in the single-shot game also decide about breaking one of their ties or
keeping their ties. A dyadic connection is not preserved if one of the actors prefers to abandon
it. We assume that the other actor cannot pay compensation in order to save the relation. In
this way, we remain in the realm of non-cooperative games (cf. Bala and Goyal 2000).
We define a social network as link deletion proof in a given strategy profile in the (collective
action) game, if there is no iN, for whom an abandonment of a single relation (from ri)
would result in a better outcome, assuming exactly the same strategy profile and no other
change in the network.
This concept is an equilibrium concept in network terms. A network is link deletion proof, assuming everything else is given, - if nobody has an incentive to change his or her network
strategy from keeping his or her relations to breaking one of them. The following statements
hold for non-negative social control. When social control is expressed in the form of
punishments, networks are consequently subject to more tie deletion, which is discussed in the
Appendix.
Theorem I.: If deleting relations is free then a) only a relation between a defector and a
contributor can be subject to link deletion; b) such a relation is always subject to link deletion
if b2>0 and the defector has at least one tie to another defector (at least the defector wants to
abandon the relation under any conditions).1
Proof: a) Deleting a relation between two defectors decreases their payoff with some
proportion of behavioral confirmation and there is no source of compensation. Even when
behavioral confirmation incentives are zero, there is no improvement by breaking relations.
Deleting a relation between two contributors decreases their payoff with some proportion of b
and also with s and there is no source of compensation. b) The payoff of a defector in any
strategy profile can consist only the following elements: c, nc, ridb1, and (rid/ri)b2. The first
three elements do not change, if he abandons a relation to a contributor. The last element
always increases in case b2>0 and rid>0, which completes the proof.
Corollary I.1.: If breaking relations is free and behavioral confirmation rewards are positive,
then a link deletion proof network in the collective action game exists only when the network
is fully segregated (contributors have ties to contributors and defectors are connected to
defectors) except that there might be defectors who are only connected to contributors.
Corollary I.2.: Every network is link deletion proof in a full contribution and in a full
defection strategy profile.
1
If the defector has no ties to other defectors, the contributor j wants to abandon this relation if b2rjc/rj(rj-1)>a.
7
The corollaries point to a single network property (perfect segregation)2 that is relevant for
link deletion proofness in the collective action game with dyadic social control and no costs of
structural change. This result is comparable to the structural balance theorem (Heider 1946;
Cartwright and Harary 1956), but with the important distinction that relations in this model
are filled with content and there are both local and global interdependencies in the network
structure. The link deletion proofness of defector subgroups is backed by the assumption that
selective incentives are expressed as rewards and not as punishments (cf. Appendix).
When there are costs of deleting a relation, we still assume that a tie can be broken
unilaterally and the other side cannot apply compensation mechanisms to save the relation. In
this case, part a) of Theorem I. still holds; that is, a relation between two contributors and in
case of positive selective incentives a relation between two defectors cannot be subject to link
deletion. Part b) of Theorem I., however, might not always hold even when b2>0 and when a
defector has at least one tie to another defector. If we denote the cost of abandoning one
relation by a (a0), then for a defector i D it is more beneficial to break a relationship with a
contributor, if
b2
rid
a
ri ri 1
(3)
holds.3 This follows from the proof of part b) of the theorem, since benefits of abandoning a
tie to a contributor are expressed on the left side of equation (3). In case of positive social
control, the decision of a free rider about deleting a tie with a cooperator is independent of
selective incentives and mass conformity. The neutrality of selective incentives and mass
conformity is due to the assumption that they are rewards and not punishments. When they
are expressed as punishments, they make link deletion more likely (see Appendix). Stronger
proportional conformity increases instability, since a contributor friend causes more
frustration for a free rider when there is more emphasis on the distribution of contributing and
free riding friends. Furthermore, as network parameters in equation (3) show, it is less likely
to delete a tie when the individual's ego-network is larger.
The proposed concept of link deletion proofness, similar to stable networks (Watts, 2001) and
pairwise stability (Jackson and Wolinsky, 1996), concerns only a single change in the egonetwork at once. A stability concept that allows players to sever any set of links at once will
be called strong link deletion proofness (Gilles and Sarangi, 2004; Belleflamme and Bloch,
2004; Goyal and Joshi 2003).
We should consider the conditions of abandoning multiple relations in case a free rider iD
has ties to more contributors. Assuming that breaking each tie has a cost of a, how many
relations should the defector abandon? If we denote this number with x (ricx0), individual i
2
Segregation refers to the segregation of contributors and defectors in the network.
8
should maximize the net benefits of his or her action. Based on (3) this means that we should
find the maximum value of
b2
xrid
 xa,
ri ri  x 
(4)
which is the difference between the benefits and the costs of abandoning x relations. From the
derivative of (4) it follows that as x increases, the net benefits are also increasing (rid>0). This
means that the most beneficial for a free rider is to abandon all his or her ties to contributors,
which holds also for the case of negative social control (see Appendix). It might well be that
meanwhile it is not profitable to delete one or only few ties, it is profitable to abandon all
links to contributors.
This result has the corollary that a network will be strong link deletion proof in a strategy
profile, if there is no defector who would be better off by abandoning every relation to
contributors. Substituting to (4) it means that there is no defector i D, for whom
b2
ric rid
 r a,
ri ri  ric  ic
holds, which can be simplified to
b2
 a.
ri
(5)
This result shows that high proportional behavioral confirmation makes networks vulnerable.4
One should also note that the minimum degree of the subset of defectors with connections to
contributors influences the link deletion proofness of the network. The higher the minimum
degree within this subset, the more likely is that a non-segregated structure is strong link
deletion proof. This also means that density is positively correlated with link deletion
proofness. More cohesive networks are less vulnerable since the expected value of minimum
degree is higher in denser networks. It also follows from (5) that at a given level of density
and degree variance, the likelihood of link deletion proofness depends on the relative network
positions of defectors with connection to contributors. The network is less vulnerable if
defectors with “abridging” ties are highly embedded, central actors.
The level of segregation of contributors and defectors does not have an unambiguous impact.
If the network is perfectly segregated, then it is strong link deletion proof. If there are ties
3
The contributor j prefers to delete this relation if b2rjc/rj(rj-1)>s+a.
In case of negative social control, mass conformity and selective incentives also play a role in strong link
deletion proofness. Furthermore, contributors might also have an incentive to delete all their ties to free riders.
Hence, there are stronger requirements for strong link deletion proofness - see Appendix.
4
9
between members of C and D, however, then segregation that would appear as the relative
number of ties to contributors (ric/ri) in the micro conditions, is not relevant for link deletion
proofness. Except extremely dense and sparse networks, the transformation towards a higher
segregation of contributors and defectors is as likely in a less segregated network than in a
more fragmented one.5
The examples in Figure 1 illustrate how can a less segregated network (Figure 1a) be more
vulnerable than a more segregated one (Figure 1b). The two networks are identical concerning
the sets of contributors and defectors and have the same density. The numbers of connections
of D1 and D3 influence network changes in the less segregated network and only D1‘s
connections matter in Figure 1b. At a wide range of parameter values, only the latter network
is link deletion proof. Consequently, the initially less segregated network (in Figure 1a)
becomes more segregated after individual structural decisions.
Figure 1 around here
Let us take a stylized example that can highlight the main predictions of the model. Consider
a wild cat strike in a factory with a rich but not complete network of informal social ties
among workers. The strike can be modeled as a one-shot public good game in which only
informal social control fosters participation. When few workers participate only, tensions
between strikers and goons may emerge afterwards. These tensions can be captured as
unpleasant forms of social control that might even result in breaking old friendship ties. The
model assumes that strikers surrounded only by strikers (goons surrounded only by goons) get
positive feedback from their peers, and their relationships are reinforced by the event. Those
who have friends from the opposite camp, however, feel shame, guilt or are simply
embarrassed by the conflict with some of their friends. The model predicts that the ‘clearing’
of a ‘mixed’ ego-network is more difficult, if the ego has many connections. As a
consequence, a more interconnected community is more likely to remain cohesive even after
these heated times. Where workers are less embedded, however, contacts between strikers and
goons might dissolve and segregated fractions may be formed. A more interconnected
community might be too valuable for its members to ruin it for one conflict.
4. Collective action and building ties
As a natural subsequent development, in this section we examine the possibility that actors
can build ties in order to enjoy higher payoffs from the structurally embedded game. As we
consider non-directed relations, we exclude the possibility of one-sided tie formation (cf. Bala
and Goyal, 2000). Two individuals have to agree in order to form a new connection, which
5
Our results are robust to various kinds of modifications in the link-deletion cost function. The sign of the social
control parameters does not alter our main conclusions, although the minimum degree among contributors with
connections to defectors is also relevant for strong link deletion proofness (see Appendix).
10
would normally lead us out from the realm of non-cooperative analysis. Since mutual consent
is required, a coalitional description and equilibrium concept is necessary or the protocol of
proposing and accepting links has to be determined (Jackson, 2004).6 We concentrate on the
structural determinants and consequences of collective action, and for this objective it is
sufficient to present a basic and simplified cost-benefit analysis of dyadic tie formation. We
simply assume that there is no bargaining problem about who should bear the costs and sidepayments are not possible. Tie formation has the same costs for both parties and a tie is
formed only if rewards for both parties are higher than the costs. This also means that if only
one party benefited from tie formation, this would not result in a change in the network.
In a utopian setting in which contacts are formed freely, all contributors would be interested
to be matched with all other contributors and all free riders would be happy to build relations
with other free riders to enjoy higher behavioral confirmation rewards. In case selective
incentives are more important than behavioral confirmation, contributors would even be
interested to get any kind of connections including also ties to free riders. A symmetric
relationship requires a mutual agreement of the parties, however, and free riders are not likely
to give in. Moreover, in a full contribution strategy profile highest benefits would come from
a network in which everyone is tied to everyone else. Much less (if any) tie formation can be
expected if social control is expressed only as punishments (see Appendix).
When there is a network structure given in advance and there are costs of forming new ties,
new relationships are most likely formed between two contributors or between two defectors.
Two unconnected contributors i, jC both have an incentive to be matched, assuming a cost
of forming a tie f, if
s  b1  b2
rkd
 f , k=i, j,
rk rk  1
(6)
which follows from a comparison of marginal benefits and costs. Hence the number of free
rider friends increases the chance that a new tie is formed between two contributors.
Furthermore, individuals with many connections (high ri) are less likely to form new
connections, as it does not give them sufficient marginal benefits (if rid>0).
Two unconnected free riders i, jD both have an incentive to be matched, assuming a cost of
forming a tie f, if
b1  b2
rkc
 f , k=i, j,
rk rk  1
6
(7)
For the sake of simplicity, we do not go into the details of cooperative analysis that is offered by a growing
body of studies (e.g., Jackson and Wolinski, 1996; Dutta, van den Nouweland, and Tijs, 1998; Slikker and van
den Nouweland, 2001; Jackson and van den Nouweland, 2005).
11
considering a similar reasoning as in the case of two contributors. Here the likelihood of a
new connection increases with the number of relations to contributors. Again, individuals
with many connections are less likely to form new ties (if they are connected at least to one
defector).
One can see from (6) and (7) that when mass conformity (b1) is larger than the costs of
forming a new tie (f), only networks in which the subsets of contributors and of defectors are
fully connected will not grow. If b1 is not as large, however, network characteristics play an
important role in shaping the chances of new tie formation.
Concerning the building of multiple new relations, for all individuals involved, the set of new
ties should provide higher benefits than the cost of tie formations. Consider a contributor who
has the option to form multiple ties at the same time. The cost of forming one tie (f) is the
same for every new relation and for both sides. Denoting the number of new ties of individual
iC by yi, this structural change is beneficial for i, if
s  b1  b2
rid
 f
ri ri  yi 
(8)
is satisfied. This shows that the marginal benefits of forming more ties are decreasing. The
first new tie is the most valuable, in case there is at least one connection to a defector. If new
dyads are not beneficial, no larger coalition that contains a new set of ties can be feasible.
Similarly, assuming a coalition of defectors in which multiple ties are formed, the marginal
benefits of forming new ties are decreasing. Similar results are obtained also for negative
social control (see Appendix).
If we consider the example of strike, the model predicts that in a loosely tied group of
workers, the collective experience of the demonstration might bring strikers closer to each
other. The collective action might be a symbolic event around which a new community is
formed. The rationale behind this process is the participants’ urge to find reinforcement for
their decision. Nonetheless, similar mechanisms operate among those who did not participate
in the wild cat strike. They also seek reinforcement, and may form the group of “moderates”
or “rational egoists”. In a more interconnected community, however, workers receive
sufficient feedback from their peers; thus, it is less likely that such an event can contribute to
the building of a larger or an even more dense social network. Nevertheless, those who have
many abridging relations may seek new acquaintances even in a dense network.
One should note that the outcome of the collective action game in the initial network structure
might not be Nash-equilibrium, but still can happen that nobody will build new relations.
There might be a lack of partners for a contributor (defector) intending to form a new tie
within C (D). Thus, a network is subject to growth, if there are at least two contributors for
whom (6) holds, or there are at least two defectors for whom (7) holds.
Similarly to the case of link deletion, network characteristics matter only within the subset of
those who have abridging ties. Another similarity is that strongly embedded, central actors
12
will be less likely to change their networks. Unlike in the case of link deletion, however, the
degree of segregation between contributors and defectors has a more clear-cut effect on
network changes: more segregated networks are less likely to grow. Nevertheless, the impact
of individual degree on changes is much stronger than the impact of the number of abridging
individual connections.
5. Strongly robust networks and collective action
In the previous sections we showed that the most profitable for a free rider is to delete all of
her or his ties to contributors (assuming that she or he prefers to delete any tie and has at least
one relation to another free rider). On the other hand, we also showed that forming the first
new tie to another free rider has the highest marginal benefits, in case there is at least one
connection to a contributor. These results suggest that in collective interdependence, relations
build up slowly and break up easily. As a subsequent step, we consider that relations can be
deleted and new ties can be formed at the same time.
Most equilibrium formulations in the research on games of network formation,7 such as the
stable network concept of Watts (2001) or pairwise stability of Jackson and Wolinsky (1996),
posited two requirements (for a review, see Dutta and Jackson, 2003). The first prescribed that
no player intends to delete a connection and the second required that no player intends to add
a new relation (Watts, 2001) or no new tie could be formed for the mutual benefit of the
players involved (Jackson and Wolinsky, 1996). A stronger version of the latter concept is
strong pairwise stability that combines pairwise stability and strong link deletion proofness by
allowing multiple ties to be deleted, but only a single tie to be built at a time (Jackson and
Wolinsky, 1996; Gilles and Sarangi, 2004). More complex stability concepts, such as strong
stability, allow for a coalition of players that is larger than two to deviate (see Dutta and
Mutuswami, 1997; Jackson and van den Nouwenland, 2005; Jackson, 2004). Subsequently,
we will investigate the conditions for strong robustness of networks in collective action, in
which no player intends to change any combination of his relations to others, including
deleting and building multiple ties at the same time. This follows more closely the logic of
Nash equilibrium analysis and does not allow coalition formation beside the involved dyads,
but requires consent from new partners involved.
When equation (5) is satisfied, defectors prefer to delete all their ties with contributors. As
this happens, building one new tie to another defector has exactly the same marginal benefit
as building multiple connections. This marginal benefit is greater than zero when mass
conformity is larger than the cost of tie formation (b1>f). When this condition is satisfied,
defectors are motivated to establish relations to every defector. The situation is similar for
contributors, except that they are less motivated to abandon relations to defectors because of
7
This line of research requires a characteristic function defined for the network and an allocation rule, but its
basic definitions can also be applied to our case.
13
selective incentives, but once segregated, they are more motivated to form new ties to other
contributors (s+b1>f should hold).
Conditions (5) and (7) are sufficient conditions for defectors for structural change, but they
are not necessary conditions. There might be individuals who do not benefit from abandoning
relations and do not benefit from building new ties, but they benefit from the combination of
the two. Consider, for instance, a defector who has a single tie to a contributor. This defector
has no incentive to break this relation when a>0. He or she has no incentive to be tied with
other defectors when f>b1+0.5b2. Replacing the existing tie with a connection to another free
rider, however, is beneficial for this individual in case b1+b2>f+a. This is not at all unlikely
when the cost of deleting a relation is small.
In general, a defector iD is better off by a structural change in which he or she deletes x ties
to contributors and newly forms y ties to defectors, when
yb1  b2
yric  xrid
 xa  yf
ri ri  x  y 
(9)
holds. From (9) it can be seen that for a free rider deleting all relations with contributors is
always more profitable than just breaking up some of them, even when considering the
simultaneous possibility of tie formation. Hence, after substituting ric for x, the necessary
conditions of a beneficial structural change for iD are given as
b2
ric
 ric a  y f  b1 .
ri
(10)
Similarly, for a contributor jC a structural change in which he or she abandons x ties to
defectors and newly forms y ties to contributors is beneficial, when
 y  x s  yb1  b2
yr jd  xrjc
r j r j  x  y 
 xa  yf
(11)
holds. From (11) it can be seen that for a contributor forming the first new relation with
another contributor is always at least as profitable as further ones, even when considering the
simultaneous possibility of deleting ties. Hence, after substituting 1 for y, the necessary
conditions of a beneficial structural change for jC are given as
b1  b2
r jd  xrjc
r j r j  x  1
 xa  f   x  1s.
(12)
After looking at individual benefits and costs of deleting and forming ties, we would like to
say more about the conditions of the stability of the entire network. In this perspective, we
14
have to extend the concept of network stability. We define a social network strongly robust in
a given strategy profile in the (collective action) game, if there is no iN, for whom any
change in his or her relations (in his or her ego-network) would result in a better outcome in
the game given that relational contacts outside of i are fixed. By “any change” we mean any
combination of deleting existing and of forming new relations in which i is involved.
If there are no costs of deleting and forming ties, only a perfect segregation of contributors
and defectors with fully connected networks within the subsets C and D will be a strongly
robust network. We do not obtain such a straightforward result, however, if there are costs of
deleting and forming ties. For these cases, in which integrated networks can also be strongly
robust, there should not be any iD for whom equation (10) is satisfied with any values of y
and there should not be any jC for whom equation (12) holds with any values of x. This is
most likely at high costs a and f, and with larger ego-network sizes.
In case negative social control operates only, calculating which networks are strongly robust
goes easier. In the Appendix we show that no combination of deleting and building ties can be
individually profitable. An individual either has an incentive to delete all his or her ties or has
an incentive to form a new one. Hence, a strong link deletion proof network in which no dyad
is interested to form a new connection will be strongly robust.
6. Simultaneous social control and strongly robust network equilibrium
In the previous section we relaxed the traditional assumption of models of collective action
that the social network is given and individuals cannot change their relations among each
other. The analysis we provided is in particular suitable for situations in which social control
mechanisms are delayed compared to participation decisions in collective action. There are
situations, however, when structural changes and behavior in the collective action game are
simultaneous. Furthermore, even when this is not the case, actors can anticipate structural
changes at the time of their participation decision in collective action. Under such
circumstances, these actions are part of the same strategy; structural decisions and network
stability should be considered together with individual decisions and equilibria in collective
action. For example, workers who participate in a wild cat strike may take into account the
risk of loosing friends and the opportunity of finding new ones. Let us consider a worker, who
works in a peripheral unit in which the majority does not prefer to join the strike of the major
workshop. If this worker has some friends in the major workshop, then she or he has to decide
about participation but also about the community she or he wants to be embedded in at the
same time.
In this perspective, we can formulate an equilibrium refinement that embraces the concepts of
strongly robust network and Nash equilibrium in the context of games played in social
networks. The equilibrium defined in this way integrates the main ideas that have been put
forward in this paper.
15
We define strongly robust network equilibrium as a network of social relations and a strategy
profile in the (collective action) game, in which there is no iN, for whom any combination
of changes in his or her action and in his or her ego-network would result in a better outcome.
It is clear that only a Nash equilibrium strategy profile and only strongly robust networks can
be in strongly robust network equilibrium. Strong network robustness and Nash equilibrium
in the collective action game, however, are necessary but not sufficient conditions for strongly
robust network equilibrium. Consider for instance, the case of a fully segregated strongly
robust network in partial contribution equilibrium. Contributors are connected to every other
contributor and free riders are connected to every other free rider. In this situation, nobody has
an incentive to abandon relations, to form new ties, or to change his or her strategy in the
collective action game. Restructuring relations and changing the action in collective action,
however, can be beneficial for some players. If forming new ties and abandoning existing
relations are free, there would always be players for whom such changes were beneficial. This
has the consequence that only full contribution and full defection with complete networks can
be strongly robust network equilibria (assuming no structural costs and positive selective
incentives or positive behavioral confirmation).
For the proof of this statement, we demonstrate that in partial contribution equilibria, either
contributors or free riders would have the incentive to abandon all their existing ties, form
new ties with every member of the other camp and change their action in the collective action
game. Contributors are better off by remaining in the fully connected camp of contributors if
  nc  1s  nc  nd  1b1  c.
(13)
On the other hand, defectors have no incentive to become “integrated” contributors, if
  nc s  nc  nd  1b1  c.
(14)
Equations (13) and (14) cannot be simultaneously satisfied when selective incentives or mass
conformity are positive, which completes the proof.
In case of negative social control, when deleting ties are free, only full defection with
complete isolation can be strongly robust network equilibrium (see Appendix).
One should note that if forming and deleting ties are free, then the assumption of
simultaneous decision about relationships and participation makes the initial network
structure irrelevant. In the model settings presented before, however, initial network structure
played a role by constraining the possible sets of contributors and free riders. Nevertheless,
one could bring back the importance of initially given network by introducing costs for
deleting or building ties. More detailed analysis of the simultaneous decision model could
show that the effects of the initial network are mostly similar to those derived from the
16
simpler modeling setup. However, the assumption of simultaneous choice results in smaller
sets of stable network structures. That is, conclusions about the impacts of various network
characteristics are similar to but more restrictive than those of the simpler models.
7. Discussion
Previous research on social network effects in collective action considered social relations as
given. Individuals change their network ties, however, and changes are partly consequences of
collective interdependence. This paper investigated the impact of collective action on network
structure and the effect of the prospect of network dynamics on equilibria in single-shot nperson public good games.
For these objectives, this study proposed a new, integrated framework of analysis combining
insights of game theory and social network analysis. In the proposed model, network effects
were incorporated in the standard n-person public goods game through different social control
mechanisms that are transmitted through interpersonal relations. Relationships and individuals
were considered anonymous; there were no leaders, privileged actors, or binding coalitions.
Social control mechanisms, namely selective incentives and forms of behavioral confirmation
were modeled as rewards (and punishments) that influence individual decisions through
actors’ relationships to relevant others and make collective action possible.
Social control that is delayed after collective action is partly also responsible for breaking
relations in the network and for building new ones. In order to avoid unpleasant influence and
to enjoy more rewards of social control, individuals might strategically revise their network
relations. This study demonstrated that it does not necessarily mean an inflating segregation
dynamics of contributors and free riders. If it is possible to delete individual ties after
collective action, then initial segregation will not be a predictor of link deletion. Hence
initially sparser sub-networks of contributors and free riders may end up in more densely knit
communities than some originally denser ones. This can be the case even if the relative
measures of segregation and degree variance are the same in the original networks compared.
Another model implication is that denser networks are more robust than sparse networks as
they are less likely to be subject to link deletion as well as to tie formation. Furthermore,
individual level analysis of the possibility of building new ties demonstrated that from a
partial contribution equilibrium those contributors can improve on their situation who have
only few connections and relatively many of them leads to free riders, meanwhile those free
riders are keen to build new ties who have few ties and relatively many goes to contributors
among them. This means that there is an effect of initial segregation on tie formation,
although it is not as strong as the effect of individual degree.
For a synthesis of analyses, we introduced a new equilibrium concept that combines
equilibrium in collective action and network stability. In this strict sense, in case of positive
social control and no costs for structural change, only full contribution and full defection with
complete networks can be strongly robust network equilibria. In case of negative social
17
control and no costs of deleting relations, full defection and isolation is the only strongly
robust network equilibrium.
The model and its predictions have a direct relevance for empirical research on collective
action. Although empirical studies do not often find clear-cut tie formation strategies and
equilibria, micro foundations of this model and the results about the interrelation of collective
action and network dynamics can help explaining real world situations and can pinpoint to
optimal strategies and outcomes.
Our aim was to provide a foundation for subsequent research that recognizes the interrelation
of collective action and network structure. The presented framework can be extended to
similar situations with different collective structure of interdependence, including public good
provision with a different production function, sustaining a public bad (cf. Kuran, 1995) and
other n-person games. Another assumption we could relax is the binary character of social
relations (two individuals are either friends or not). We could assume that there are good
friends and also mere acquaintances in the network by ordering weights to each tie. In such a
framework, ties are not abandoned or built, but weights are reconsidered. This model
extension would also allow considering asymmetric ties (for a similar dynamic analysis see
Kitts, Macy and Flache, 1999).
We adopted a model of forward-looking, strategically rational individuals. This could be
regarded as a serious shortcoming, although we are convinced that there are well-founded
theoretical reasons for taking this type of actor-model as given.8 A possible way of relaxing
the strict assumptions of the model is to consider boundedly rational actors (cf. Macy, 1993;
Macy and Flache, 2002) or stochastic decision making.
A further step would be an analysis of the iterated public goods game and consequently a
dynamic interrelated analysis of repeated collective action problems and structural dynamics.
For the complexity of this problem, however, agent-based simulation techniques would be
more appropriate than analytical methods. Model predictions can be tested in laboratory
experiments using artificial networks or in field experiments.
8
The emphasis here is both on this type and on taking it as given. See Granovetter (1985) and Raub and Weesie
(1990) on this issue in the context of network models.
18
Appendix
Main results in case social control appears in form of punishments
We assume in the following calculations that selective incentives and behavioral confirmation are punishments
that decrease individual payoff. These calculations can be contrasted with those in the main text that hold for
positive rewards of social control. The payoffs of free riding and contribution for i are:
n
 i  i  0    j ri s  ricb1 
j 1
n
ric
b2
ri
 i  i  1   ( j  1)  c  rid b1 
j 1
(1b)
rid
b2
ri
where j  N\{i}. From these equations it follows that the conditions for participation of individual i to be rational
are exactly the same as in equation (2).
Theorem Ib.: If deleting relations is free, then a) free riders prefer to delete a link to a contributor and they
prefer to delete all their links to contributors. Once they deleted all their links to contributors, they prefer to
delete all their other links, as well. b) Contributors prefer to delete their links to defectors. Theorem Ib. follows
from equations (1b).
Corollary I.1b: A network is link deletion proof if there are links only between contributors.
Corollary I.2b: Every network is link deletion proof in a full contribution strategy profile.
Conditions for deleting a link in case it has a cost:
A free rider iD prefers to delete a link with a contributor, if
s  b1  b2
rid
a.
ri ri  1
(3b-1)
A free rider prefers to sever all links with contributors, if
s  b1 
b2
 a.
ri
(5b-1)
The left side of equation (5b-1) is never smaller than the left side of equation (3b-1), hence the most beneficial
for a free rider is to delete all his or her ties to contributors.
When behavioral confirmation has an effect, deleting a tie to a contributor always gives higher benefits for the
free rider than deleting a tie to another free rider. This has the consequence that a free rider might have an
incentive to delete a tie to another free rider only once all ties to contributors have already been deleted. When
this is the case, a free rider will delete all relations, if s>a holds; and will delete no relations to other free riders,
if s<a.
Furthermore, a contributor jC might also prefer to delete a link with a free rider, if
b1  b2
r jc
r j r j  1
a
19
(3b-2)
holds.
A contributor prefers to delete all links with free riders, if
b1 
b2
 a.
rj
(5b-2)
Hence, the most beneficial for a contributor is to delete all his or her ties to free riders. A contributor always
prefers to keep the links with other contributors.
From these equations it follows that in case of negative social control, the network will be strong link deletion
proof in a strategy profile, if there is no iD for whom (5b-1) holds, and there is no jC for whom (5b-2) holds.
Conditions for building a link in case it has a cost:
Two unconnected contributors i, jC both have an incentive to be matched, assuming a cost of forming a tie f, if
b2
rkd
 f , k=i, j,
rk rk 1
(6b)
which follows from a comparison of marginal benefits and costs.
Two unconnected free riders i, jD both have an incentive to be matched, assuming a cost of forming a tie f, if
b2
rkc
 f  s , k=i, j.
rk rk 1
(7b)
Assume now that contributors can form multiple ties at the same time. The cost of forming one tie f is the same
for every new relation and for both sides. Denoting the number of new ties of individual iC by yi, this structural
change is beneficial for i, if
b2
rid
 f
ri ri  yi 
(8b)
is satisfied. This shows that the marginal benefits of forming more ties are decreasing. Similarly, the marginal
benefits of forming new ties are decreasing also for defectors.
Conditions for structural change when deleting and building ties can be combined:
As it was discussed, when s>a holds, a free rider prefers to delete all of his or her relations. When s<a and (5b1) hold, a free rider deletes all relations to contributors and builds no ties. When (5b-1) is not satisfied, but (7b)
holds (high costs of link deletion and cheap tie formation), then the free rider is interested to build a new tie to
another free rider, but keeps all his or her ties to contributors.
20
A similar result can be obtained for contributors. When (5b-2) holds, a contributor prefers to delete all ties to free
riders and has no interest to form new ties. When (5b-2) is not satisfied and (6b) holds, a contributor would like
to form a new tie to another contributor, but is not interested to delete any of his or her ties.
Hence, a combination of deleting and forming ties is never profitable in case of negative social control. This has
the corollary that a strongly robust network is a strong link deletion proof network in which no dyad is interested
to form a new connection.
Simultaneous social control and strongly robust network equilibrium
As there are no rewards of social control, when deleting relations is free, only full defection with no ties can be
strongly robust network equilibrium, irrespective of the costs of forming new ties. This follows directly from
(1b).
21
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Figure 1: Segregation of contributors and link deletion proofness
Notes: Example for parameter values: s=1, b1=1, b2=5, c-α=3.5, a=2. Green nodes denote
contributors and blue nodes are defectors.
C1
C3
C2
D1
D2
D3
C4
Figure 1a. A less segregated network and network vulnerability
C1
C3
C2
D1
D2
D3
C4
Figure 1b. A more segregated network and link deletion proofness
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