University of Groningen. Collective action and network change Takacs, Karoly; Janky, Bela; Flache, Andreas. Published in: Social Networks

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1 Univesity of Goningen Collective action and netwok change Takacs, Kaoly; Janky, Bela; Flache, Andeas Published in: Social Netwoks DOI: /j.socnet IMPORTANT NOTE: You ae advised to consult the publishe's vesion (publishe's PDF) if you wish to cite fom it. Please check the document vesion below. Document Vesion Publishe's PDF, also known as Vesion of ecod Publication date: 008 Link to publication in Univesity of Goningen/UMCG eseach database Citation fo published vesion (APA): Takacs, K., Janky, B., & Flache, A. (008). Collective action and netwok change. Social Netwoks, 30(3), DOI: /j.socnet Copyight Othe than fo stictly pesonal use, it is not pemitted to download o to fowad/distibute the text o pat of it without the consent of the autho(s) and/o copyight holde(s), unless the wok is unde an open content license (like Ceative Commons). Take-down policy If you believe that this document beaches copyight please contact us poviding details, and we will emove access to the wok immediately and investigate you claim. Downloaded fom the Univesity of Goningen/UMCG eseach database (Pue): Fo technical easons the numbe of authos shown on this cove page is limited to 10 maximum. Download date:

2 Social Netwoks 30 (008) Contents lists available at ScienceDiect Social Netwoks jounal homepage: Collective action and netwok change Káoly Takács a,b,,béla Janky c, Andeas Flache b a Covinus Univesity of Budapest, Institute of Sociology and Social Policy, Hungay b Univesity of Goningen, Faculty of Behavioal and Social Sciences, Depatment of Sociology/ICS, The Nethelands c Budapest Univesity of Technology and Economics, Depatment of Sociology and Communication, Hungay aticle info abstact Keywods: Collective action Social dilemmas Social netwoks Netwok dynamics Social contol Stuctual balance Local inteaction games Netwok models of collective action commonly assume fixed social netwoks in which ties influence paticipation though social ewads. This implies that only cetain ties ae beneficial fom the view of individual actos. Accodingly, in this study we allow that actos stategically evise thei elations. Moeove, in ou model actos also take into account possible netwok consequences in thei paticipation choices. To handle the inteelatedness of netwoks and paticipation, we intoduce new equilibium concepts. Ou equilibium analysis suggests that stuctues that tend to segegate contibutos fom fee ides ae stable, but costless netwok change only pomotes all-o-nothing paticipation and complete netwoks. 008 Elsevie B.V. All ights eseved. 1. Intoduction Why and unde which social conditions ae goups successful in mobilizing collective action? Voluntay paticipation in collective actions, such as fund aising, stike movements o political upising, seems often to contadict the naowly defined self-inteest of the paticipants. Yet, empiical examples of successful mobilization abound. Students of collective action point to social netwoks as an impotant answe (see, e.g., Obeschall, 1973; Tilly, 1978; Olive, 1984; McAdam, 1986; Mawell et al., 1988; Gould, 1993a; Sandell and Sten, 1998; Chwe, 1999, 000; and fo an oveview Diani, 003a). Dense netwoks of communication and inteaction between pospective paticipants may geatly facilitate goup mobilization (Opp and Gen, 1993; Gould, 1993b; Mawell and Olive, 1993). The view that social netwoks facilitate collective action elies on the assumption that individual netwok membes have a egulatoy inteest (Heckathon, 1988; Kitts, 006) to enfoce othes contibution to the collective action. Paticulaly in dense o closed netwoks, actos can effectively employ thei social ties fo this pupose (Hechte, 1987; Coleman, 1990) both because goup membes have moe infomation about one anothe and because they have moe social means to povide ewads fo compliance o punish deviance. But it has also been agued that netwok ties have a double edge (Flache, 1996). Heckathon (1996) has pointed out how pee pessue may take the fom of oppositional contol, when net- Coesponding autho at: Covinus Univesity of Budapest, Institute of Sociology and Social Policy, H-1093 Budapest, Közaktá u. 4 6, Hungay. addess: kaoly.takacs@uni-covinus.hu (K. Takács). wok membes have stong incentives to insulate themselves fom pessues to contibute. Based on diffeent behavioal assumptions, othe theoetical studies suppoted the view that dense netwoks may sometimes undemine athe than facilitate the enfocement of contibution (Flache and Macy, 1996; Flache, 1996, 00; Kitts et al., 1999). The theoetical agument focuses on the desie of actos to obtain social ewads fom othe goup membes including those who fee ide. The desie to etain elationships with o attain behavioal confimation fom fee ides may often compomise actos willingness to exet social contol towads contibution, especially in a closely knit netwok. To claify unde what conditions netwoks ae positively o negatively elated to collective action success, models of collective action need to incopoate explicitly how individual actos make puposive decisions to use thei social elations to foste thei goals, be it to enfoce compliance o to esist pee pessue. But studies that combine positive and negative effects of social ties on coopeation (e.g., Obeschall, 1994; Heckathon, 1996; Takács, 001) have neglected a cucial implication of this pespective. Puposive action implies that netwok membes in a collective action situation not only use existing ties to attain thei goals, but they may also make o beak ties if this seves thei puposes. In geneal, most models of collective action that addess social netwok effects implicitly assume that thee is a fixed set of intepesonal elations that do not change ove time. Relational ties ae in these studies exogenously given and at most, static compaisons ae made. We ague that the elationship between collective action and social netwoks cannot be popely studied without addessing endogenous netwok change diven by individual inteests. Regulatoy inteests do not necessaily lead to enfocement, but possibly also to avoidance of unpleasant /$ see font matte 008 Elsevie B.V. All ights eseved. doi: /j.socnet

3 178 K. Takács et al. / Social Netwoks 30 (008) contol and optimization of contacts. Follustation, conside the situation of a defecto who is paticulaly sensitive to confomity pessue and who has he ties mostly with compliant goup membes. In a static netwok, it is likely that pessue to contibute bings this defecto back into line. But if netwoks can change, the defecto faces an incentive to maximize the numbe of ties she has with othe defectos and thus beak ties with coopeatos and build new ties with othe defectos. If the latte mechanism pevails, the outcome on the collective level may be a disconnected netwok with a deviant clique on which coopeatos cannot effectively impose pee pessue. But if the fist mechanism is moe impotant, the esult may be a netwok in which fee ides ae effectively sanctioned by thei compliant pees. In this example, models of collective action that neglect endogenous netwok changes might aive at questionable conclusions. In this study we emphasize that netwoks change ove time and that this is also eflected in how people behave in collective action (Kim and Beaman, 1997; Diani, 003b; Osa, 003; Gould, 003). Besides being diven by independent netwok dynamics, netwoks may also change because of the intenal deteminants of collective action. People might choose thei stuctual elations stategically in ode to maximize ewads and minimize punishments that oiginate in social contol (cf. Hasanyi, 1969). Futhemoe, in ode to facilitate paticipation, stategic establishment of communication channels and othe linkages might take place (McAdam and Paulsen, 1993; Diani, 003c). Stategic tie fomation goes vey fan the polifeation of social movement politics. Campaigning and lobbying often involves tie-fomation stategies, populaly called as netwoking (e.g., Tilly and Wood, 003). On the othe hand, anothe banch of liteatue emphasizes individual oppotunities that aise fom deleting elations and fom stuctual holes (e.g., But, 199, 005; Buskens and van de Rijt, 005; Buge and Buskens, 006). Seveing links is also consideed as a puposeful action in diffeent lines of social psychological liteatue (see Williams et al., 005 fo an oveview). Studies show that in collective action, ostacism and the theat of exclusion inceases paticipation (Olson, 198; Hishleife and Rasmusen, 1989; Ule, 005; Ouwekek et al., 005). Thee is also empiical evidence on how collective action might estuctue individual elations. Activism has cetainly a ole in changing the meaning and impact of intepesonal ties (Gould, 003). As a esult of the dynamics of dyadic elations, maco popeties of the netwok and thus pespectives of (futhe) collective action also change. Ou contibution along this line is to integate the pespective of netwok change and the esulting netwok dynamics in a theoetical model of collective action. Independently fom the liteatue we cited so fa, consideable pogess has been made ecently in modeling the dynamics of social netwoks as a esult of stategic individual action (Jackson and Watts, 00a; Dutta and Jackson, 003; Ule, 005). Pevious game theoetic models, howeve, concentated on stategic netwok fomation without collective intedependencies (Bala and Goyal, 000; Bonacich, 001). Most studies use games of stategic netwok fomation that assume values fo established ties in coopeative situations (Dutta et al., 1998; Slikke and van den Nouweland, 001) ondiect benefits fom the chain o netwok of contacts (Jackson and Wolinsky, 1996; Jackson and Watts, 00b). In this liteatue on netwok fomation, equilibium concepts have been developed that descibe equilibium netwoks in which no individuals have incentives to delete existing ties o build new ones (Jackson and Wolinsky, 1996; Watts, 001; Gilles and Saangi, 004; Ule, 005; Buskens and van de Rijt, 005; van de Rijt and Buskens, 005; Goyal and Vega-Redondo, 007). If netwok stability and behavioal stability ae embaced and adjusted to collective intedependencies, simila concepts could help us to model and undestand collective action and stability of social netwoks. The main added value of this study fo the liteatue on netwok fomation is the development and application of such equilibium concepts to n-peson games. Anothe line of wok explicitly combines collective action with netwok change (Kitts et al., 1999), but uses backwad-looking models of stuctual leaning that descibe the pocess by which individuals stengthen ties that ae beneficial fo them and abandon ties with negative expeience (Macy et al., 003). These models do not assume that decision making is stategic and puposive in the sense that individual actos weigh the costs and benefits of changing thei contibution behavio against the costs and benefits of making o beaking elationships, o a combination of both. This makes it difficult to addess effects of exogenous constaints, such as given access stuctues of netwoks o communication costs, within the stuctual leaning famewok. In this pape, we popose a game theoetical model of collective action in dynamic netwoks and aive at equilibium pedictions. Ou main innovation is that we incopoate into a collective action famewok the possibility of tie fomation and deletion, and model these decisions as intedependent with paticipation decisions. We daw on game theoetic studies of netwok evolution and model both collective action and elational change as the esult of stategic and puposive decision making that explicitly eflects the costs and benefits of the vaious decision options involved. The game theoetic appoach allows us to aive at equilibium pedictions without the need of using simulation techniques. We will use these equilibium pedictions to identify exogenous conditions that explain why sometimes netwok dynamics may undecut the effectiveness of pee pessue, while netwok influences successfully foste collective action at othe times. Examples of phenomena ou model can tackle ae the fomation of deviant cliques that emain stable despite stong social contol, o wok teams whee teamsegegation develops along a division line between wokes and shikes, even when high team pefomance would be in eveybody s inteest. Ou wok is a fist step towads addessing such phenomena in a famewok that integates collective action dynamics and netwok dynamics with the assumption of individual stategic behavio and the notion that netwok ties ae conduits of social ewads in collective action. Accodingly, the majo contibution of the pape will be theoetical. To povide the tools to tackle the substantive questions we ae inteested in, we develop an equilibium concept that embaces Nash equilibium in the n-peson collective action game and netwok stability at the same time. The concept, called stongly obust netwok equilibium, delineates stuctual conditions unde which goup outcomes ae stable both in tems of the netwok and in tems of the collective action decisions of the goup s membes. In the emainde of the pape, we poceed as follows. In Section, the basic model of social contol and collective action is lined out. In Section 3, conditions ae deived unde which stategic actos will not delete any tie o fom new ones in a given stategy pofile of collective action decisions. Section 4 contains a deivation of the conditions fo stongly obust netwok equilibia. Finally, we discuss ou esults in Section 5, whee we also fomulate conclusions and suggestions fo futhe eseach.. The model of social contol and collective action This section povides a desciption of ou model of decisions on paticipation in collective action. Collective action is modeled as a one-shot n-peson public goods game with a linea poduction function and binay decisions and it is integated with a model of dyadic intedependence between diectly elated playes. In Sec-

4 K. Takács et al. / Social Netwoks 30 (008) tion 3, we extend this model by a netwok game, in which playes can change intepesonal connections. We define N to be the set of actos, whee N contains n (n >) playes. 1 Individual actions ae binay and summaized in vecto, in which the action of i is denoted by i, whee i = 1 is contibution and i = 0 is non-contibution o defection (fo all i N). We use ( i, i ) to efe to the action pofile chosen by all goup membes, whee we distinguish between i s action and the actions chosen by othe membes of the goup, i. Each paticipation (contibution) povides a unit of public good fo all individuals. This means that the moe people paticipate, the highe the public good ewad each playe eceives. Paticipation has an individual cost c, which is highe than one unit of public good (c > ), but smalle than n units (c < n ). These payoff estictions impose a Pisone s Dilemma stuctue in the public goods game. When we neglect social netwok payoffs, univesal contibution is Paeto supeio to univesal defection, but unilateal defection is individually ational. Actos play this game not in isolation. They might have connections (links) to each othe, which define a social netwok. Fo the sake of simplicity, we conside undiected and unvalued connections, which implies that evey link is symmetic and equally impotant. The social netwok (N, R) is chaacteized by the set of actos N, and by the set of connections R R N, whee R N = {ij {i, j} N N, i j} and ij = ji. Sometimes we will efe to j as i s contact o fiend if ij R. As we conside a fixed netwok size, we will simply efe to the social netwok (N, R) asr and to the netwok fom which the elation ij is deleted as R\{ij}. The set R i = {ij ij R, j N, i j} R contains all ties of i and the set Ř i = {ij ij / R, j N, i j} denotes the null-dyads of i. The size of R i is the numbe of ties (degee) i has and is denoted by. Acto is an isolate if =0. If ij R, actos i and j influence each othe. Fist, we assume that actos pefe to act the same way as thei contacts (cf. Chong, 1991; Obeschall, 1994), which we will efe to as behavioal confimation. This assumption is suppoted by empiical studies that show that individuals even gathenfomation on-the-spot to assess whethe thei fiends will paticipate o not (Obeschall, 1993; Dixon and Roscigno, 003). When the actions of contacts match, we assume that they obtain a highe subjective payoff than when thei action is diffeent. This kind of stategic intedependence is also modeled in coodination games played on a netwok (see, e.g., Beninghaus and Schwalbe, 1996; Mois, 000; Chwe, 000; Jackson and Watts, 00a; Buskens and Snijdes, 005). Ou study, howeve, moves beyond this famewok and intoduces stategic intedependence not only in tems of the need of coodination, but also in tems of the moe challenging poblem to achieve coopeation between self-inteested actos in an n-peson Pisone s Dilemma. We assume that behavioal confimation might take two diffeent foms. Fist, we assume that the subjective payoffs i fo all i N ae inceased by the value of mass confomity b 1 fo each j if ij R and i = j. This means that an additional contact has the same influence on the acto as othe contacts, and hence the numbe of contacts with identical choice detemines the stength of this type of influence. Empiical eseach on social influence suppots that individuals ae moe likely to choose a cetain behavio when the numbe of elevant othes who behave this way inceases (see fo instance, Ploege, 1997 and Snijdes and Baeveldt, 003 fo delinquency; Blum et al., 000 fo alcohol use; Simons-Moton and Chen, 006 fo substance use). Fo example, when mass confomity opeates, an individual, who intends to paticipate in a demonstation, is pleased when thee ae many fiends in the cowd. On the othe hand, in case this individual pefes to stay at home, she is pleased to 1 Sets will be denoted by bold capitals and thei sizes by coesponding small lettes in italics. Fo example, N = n. be assued that many fiends choose to stay away fom the demonstation. As effective contol fom moe contacts pushes stonge towads paticipation, individuals face local thesholds of paticipation simila to the impact of a citical mass of paticipants on individual paticipation in othe models of collective action (Macy, 1991; Mawell and Olive, 1993; Chwe, 1999). Second, behavioal confimation is also dependent on the popotion of contacts with identical action (fom all fiends of i). We assume that the subjective payoffs i fo all i N incease by b / fo each j if i and j have a netwok tie (ij R) and thei actions ae identical ( i = j ). The paamete b scales the stength of this popotional confomity effect. Diffeent lines of aguments justify this assumption. In-degee and the popotion of elevant othes behaving the same way (e.g., mean fiendship delinquency) ae impotant pedictos of peenfluence (Haynie, 001). Futhemoe, a study using woking unit-level data found that the popotion of paticipants in the woking unit is a significant pedicto of individual stike paticipation (Dixon and Roscigno, 003). Including a popotional tem in the model is also in line with theoies on cultual change that emphasize that individuals ae influenced by elatively moe fequent cultual taits aound them (e.g., Boyd and Richeson, 1985) and with netwok models of coodination (Buskens and Snijdes, 005). In addition, stuctual elations ae the souces of positive social selective incentives (s) that ewad contibution. We assume that the subjective payoffs i fo all i N incease by i s fom each ij R connection, hence by i s in total. This implies that only contibutos eceive selective incentive ewads. If the povision of selective incentives is costly, actos would not only face a public goods poblem, but also a second ode fee-ide poblem (Olive, 1980; Heckathon, 1989). As we concentate, howeve, on selective incentives of a social chaacte like espect o status, we deem it plausible that these ae poduced without costs (Chong, 1991; Lovaglia et al., 003) and that theefoe no second ode fee-ide poblem aises (see also Coleman, 1990). This assumption eceives additional suppot by evidence that individuals punish defectos voluntaily, in spite of sanctioning costs (Feh and Gächte, 000, 00; Boyd et al., 003). We use these assumptions to expess the subjective payoff playes obtain given thei choice fo paticipation o defection. The subjective individual payoff i of i N will depend on the numbe of he contacts who paticipate, c (whee c is the size of R ic = {ij ij R i, j =1} R i ), and on the numbe of he contacts who do not paticipate (d = c ). Fomally, i fo all i N with > 0 in the given netwok R and given stategy pofile ( i, i ) is detemined as: i ( i = 0, i, R) = d b 1 + d b + n j=1,j i and n i ( i = 1, i, R) = s + c b 1 + c b + j + 1 c, i j=1,j i which can be compactly witten as ( ( i (, R) = + s + (c d ) b 1 + b ) ) c i + d ( b 1 + b ) + j n j, (1) j=1 whee j N\{i}.If =0, i (, R) = ( c) i + n j=1 j. In the main text, we assume that contibutions have positive extenalities, > 0,

5 180 K. Takács et al. / Social Netwoks 30 (008) and social contol geneates ewads (b 1 0, b 0, and s 0). Definition of payoff functions if social contol appeas in tems of punishments and the analysis of such situations is left to Appendix B. Now we can deive the conditions fondividually ational paticipation in collective action given a cetain netwok position and given a vecto of paticipation decisions of goup membes. Boadly, an acto will paticipate if the subjective payoffs fom contibution exceed the subjective payoffs fom fee iding in the given netwok and stategy pofile. This is achieved when the selective incentives and behavioal confimation ewads she eceives in he pesent netwok position ae sufficient to compensate fo the costs of contibution. Fomally, i paticipates if i ( i = 1, i, R) i ( i = 0, i, R) that is when ( + s + (c d ) b 1 + b ) c. () Expession () assets that the effect of selective incentives on paticipation incease in the numbe of ties of the given individual. Behavioal confimation pomotes contibution only when moe contacts contibute than defect. 3. Netwok changes As a next step, we include the possibility that actos might delete thei existing connections and might fom new ones. Fo the time being, we teat paticipation decisions as fixed. This allows to identify stable netwok stuctues at a given stategy pofile of the public goods game. 3 We continue to assume that social contol paametes ae ewads (b 1 0, b 0, and s 0). The costs of abandoning one tie ae denoted by a and the cost of foming one new tie is denoted by f (a 0, f 0). A connection ij is not stable if i o j pefes to delete it. 4 No compensation payments ae possible to save the connection (cf. Bala and Goyal, 000). Note that it might be beneficial fo a playe to delete a set of he links at once. Even if a deletion of ij does not esult in a highe payoff, it can be pat of the set of links that is ewading to be deleted. On the othe hand, we exclude the possibility of one-sided tie fomation, because we use non-diected elations (cf. Bala and Goyal, 000; Buskens and van de Rijt, 005). To keep ou analysis simple, we abstact fom the possibility that actos may bagain about which acto o which coalition of goup membes may bea the costs of the fomation of some subset of ties. Tie fomation is assumed to have the same costs fo both paties and no coalition possibilities beyond the dyad ae taken into account. 5 When ana- A moe efined analysis of this static model can be found in Janky and Takács (005). Main esults fo the entie netwok show that the minimum degee of the netwok is a stong deteminant of oveall collective action in case selective incentives opeate. Netwok clusteing has a stong influence when behavioal confimation mechanisms ae stong and might undemine mass collective action. Clusteed netwoks ae moe likely to have patial contibution equilibia, in which paticipants and fee ides ae segegated. The smalle the numbe of fee ides in the patial contibution equilibium, the less likely that full contibution is a payoff dominant equilibium. Moeove, the payoff dominance of full contibution equilibium is not likely in centalized stuctues when mass confomity is stong, but it is possible in case popotional confomity is pevalent. A futhenteesting esult is that not only behavioal confimation, but also selective incentives might have a non-monotonic effect on the existence of full contibution equilibia. 3 A vesion of the model allowing only fo netwok decline is also discussed in Takács and Janky (007). 4 When behavioal confimation and social selective incentives ae negative, moe links ae not stable than when behavioal confimation and selective incentives ae positive (see Appendix B). 5 Bagaining could be incopoated with a coalition analysis of netwok fomation including the use of equilibium concepts fom coopeative game theoy, such as the coe. Howeve, studies that use this appoach (e.g., Jackson and Wolinsky, lyzing netwok equilibia, we will assume that a new tie is initiated only if it is pat of an (extended) individual netwok that povides highe benefits than the pesent netwok, taken into account the costs of change and unde the assumption of no othe change in the netwok. Besides this basic and simplified cost benefit analysis, following pevious wok, we will assume that consent of the othe paty is equied to fom a new connection. We ae also concened with the possibility that playes might benefit fom the combination of deleting and building some ties. Conside, fonstance, a defecto who has only a single tie ij and this leads to a contibuto j. Since popotional confomity ewads in this case ae zeo, playe has no incentive to beak ij if a > 0. She also has no incentive to be tied with othe defectos when f > b b. Replacing ij with a connection to anothe fee ide, howeve, is beneficial fo in case b 1 + b > f + a. Such a situation is not unlikely when the cost of deleting a elation is small. Most equilibium fomulations in the eseach on games of netwok fomation, 6 such as the stable netwok concept of Watts (001) o paiwise stability of Jackson and Wolinsky (1996), posited two equiements (fo a eview, see Dutta and Jackson, 003). The fist equiement pescibes that no playentends to delete a connection and the second equiement is that no playentends to add a new elation (Watts, 001) o no new tie could be fomed fo the mutual benefit of the playes involved (Jackson and Wolinsky, 1996). A stonge vesion of the latte concept is stong paiwise stability that combines paiwise stability and stong link deletion poofness by allowing multiple ties to be deleted, but only a single tie to be built at a time (Jackson and Wolinsky, 1996; Gilles and Saangi, 004). Moe complex stability concepts, such as stong stability, allow fo a coalition of playes that is lage than two to deviate (see Dutta and Mutuswami, 1997; Jackson and van den Nouweland, 005; Jackson, 004). Fo ou puposes, we need a simple concept that is still noncoopeative in natue and unlike in stong paiwise stability, consides multiple tie fomation. The point of depatue is based on the Nash logic and states that in equilibium it should not be in anyone s inteest to make any change in he netwok. By any change we mean any combination of deleting existing and of foming new elations in which the individual is involved. This stability concept is useful fo the discussion of conditions unde which a netwok is fee of tie deletions and initiations. Applied to collective action situations, this stability concept would chaacteize netwoks in a given stategy pofile in the (collective action) game, if thee is no acto, fo whom any change in the set of he elations would esult in a bette outcome given that elational contacts outside of i ae fixed. Fomally, we would call R esistant to changes inagiven, if i (, R) i (,(R G i )\H i ) h i a g i f fo any G i Ř i,anyh i R i and fo all i N, whee the notations H i ae intoduced fo the set of ties to be deleted and G i fo the set of ties to be built by i N. The sizes of these sets ae denoted by h i and g i, espectively. It is not ou main inteest to deal with situations in which actos can impose ties on othes (as in diected netwoks). We also do not want to focus on whethenitiations of new ties take place, but we want to addess whethe new ties ae built o not, which equies consent also fom the othe paty. Accodingly, we need to move beyond the use of a puely Nash-based concept. As ties ae symmetic and tie fomation equies mutual consent, we build ou equilibium concept on the same logic as monadic stability (Gilles and Saangi, 004) and unilateal stability (Buskens and van de Rijt, 1996; Dutta et al., 1998; Slikke and van den Nouweland, 001; Jackson and van den Nouweland, 005) do not discuss collective action. 6 This line of eseach equies a chaacteistic function defined fo the netwok and an allocation ule, but its basic definitions can also be applied to ou case.

6 K. Takács et al. / Social Netwoks 30 (008) ). These concepts take into account that a new tie is only fomed if it is to the benefit of the patne, as well. Monadic stability is based on the idea that playes take it as ganted that othe playes espond affimatively to an initiation if the new link is pofitable to them, but no futhe consequences ae taken into account (Gilles and Saangi, 004). Buskens and van de Rijt (005) define a netwok unilateally stable if no acto would be bette off by changing he ties of anyone was, then at least one acto whom she poposes a new tie is wose off in the new netwok than in the oiginal netwok. We will adopt this definition and adjust it fo ou puposes when applying to collective action poblems and netwok fomation. We define a social netwok unilateally stable in a given stategy pofile in the (collective action) game, if thee is no acto, fo whom any change in the set of he elations would esult in a bette outcome of a change would esult in a bette outcome then it does not satisfy some of the new patnes given that elational contacts outside of the scope of i ae fixed. This stability concept does not allow coalition fomation beside the involved dyads, but equies consent fom new patnes involved. Definition. R is unilateally stable inagiven, if i (, R) i (,(R G i )\H i ) h i a g i f o exists such j N that ij G i : j (, R)> j (, R {ij}) f fo any G i Ř i,anyh i R i and fo all i N. Fom the fist sight, next to consideing a given stategy pofile in the collective action game, thee is a futhe diffeence compaed to the oiginal fomalization of unilateal stability. Hee, new patnes compae subjective ewads fom the oiginal netwok with subjective ewads fom the netwok to which thei new tie with i is added. Note that in ou case this is just a simplified fomalization: as stategies ae fixed and thee ae no indiect ewads fom netwok changes, j (, R {ij})= j (,(R G i )\H i ) fo all such j N that ij G i. Fo the simple case whee a = 0 and f = 0, thee is a clea-cut answe to the question which netwoks can be stable. Only a disconnected netwok with a complete component of contibutos and a complete component of defectos can sustain unilateal stability. The eason is that defectos always gain fom abandoning all ties to contibutos and contibutos always gain fom building as many as possible new ties to fellow contibutos. On the othe hand, if selective incentives ae lage enough elative to popotional confomity, then even a disconnected netwok might not be esistant to changes, because coopeatos would pefe to connect to defectos as well. We do not obtain such staightfowad esults, howeve, if thee ae costs of deleting and foming ties. In these cases, netwoks that contain ties between defectos and contibutos can also be unilateally stable. A contibuto might pofit fom a new tie to a defecto, but such a tie will neve be beneficial fo the defecto. As consent fom both paties is equied fo a new elationship, such a tie will not be ealized. In geneal, a defecto N with d > 0 is bette off by a stuctual change in which she deletes h i ties to contibutos and newly foms g i ties to defectos, if g g i b 1 + b i c + h i d ( h i + g i ) >h ia + g i f (3) holds. This condition implies that fo a defectot is always moe pofitable to abandon all elations with contibutos athe than just beaking up some of them, even when tie fomation is simultaneously possible. The eason is that fo a fee ide who has at least one contact to anothe defecto beaking contacts to actos with dissimila stategies impoves popotional confomity ewads and does not yield any loss of mass confomity o selective incentive ewads. The moe ties to coopeatos the defecto beaks, the lages the impovement of the atio of defectos to coopeatos in the defecto s pesonal netwok. This implies that any additional tie abandoned yields a geate benefit than the pevious one, wheeas the costs ae the same fo each deletion. Technically, afte substituting c fo h i, the necessay conditions of a beneficial stuctual change fo defecto with d > 0 ae given as b ic > c a + g i (f b 1 ). (4) i Similaly, fo a contibuto N with > 0 a stuctual change in which she abandons h i ties to defectos and newly foms g i ties to contibutos is beneficial, when g (g i h i )s + g i b 1 + b i d + h i c ( h i + g i ) >h ia + g i f (5) holds. Fom (5) it can be seen that fo a contibuto foming the fist new elation with anothe contibutos always at least as pofitable as futhe ones, even when consideing the simultaneous possibility of deleting ties. Again, this is caused by popotional confomity. The less ties a coopeato has (but at least one to a defecto), the lages the impovement in the atio of defectos to coopeatos fom ego s point of view if an additional tie is established with a similaly acting goup membe. Hence, the fist new tie is the most valuable fo a coopeato, in case she has at least one connection to a defecto (of she is an isolate). Technically, afte substituting 1 fo g i, the necessay conditions of a beneficial stuctual change fo contibuto with d > 0 ae given as b 1 + b id + h i c ( h i + 1) >h ia + f + (h i 1)s. (6) A netwok is unilateally stable in a given stategy pofile, if thee is no defecto fo whom Eq. (4) is satisfied with any values of g i and thee also is no contibuto fo whom Eq. (6) holds with any values of h i. Netwoks that contain ties between defectos and contibutos ae most likely to be unilateally stable when costs of link deletion (a) and link fomation (f) ae high and evey individual is tied to seveal othes. 7 Substantively, such conditions can be intepeted as high constaints on netwok change imposed by the given netwok. Besides these geneal esults, we can fomulate some illustative statements that chaacteize equilibia. Theoem I summaizes thee esults concening deleting a tie ij in a given stategy pofile (see Appendix A fo the poof). A tie ij R i will be called not stable if thee exist G i Ř i, and H i R i such that ij H i and i (,(R G i )\H i ) h i a g i f > i (, R). Theoem I. In any R and given, fo all i,j N and ij R: (a) If i = j, then i (, R) i (, R\{ij}). (b) If i =0, j =1,d >0,and b / > a, then ij is not stable. (c) If i =0, j =1, b >0 and fo sufficiently small a and f, ij is not stable. Pat (a) of Theoem I claims that no playe can incease he subjective payoff by deleting a link to anothe playe who acts the same way as she does in any netwok and given stategy pofile. Pat (b) states that if thee is a tie that connects a defecto and a contibuto, if the defecto has at least one tie to anothe defecto and 7 In case only negative social contol opeates, calculating which netwoks ae unilateally stable is easie. In Appendix B we show that in this condition no combination of deleting and building ties can be individually pofitable. An individual eithe has an incentive to delete all he ties o has an incentive to fom a new one. Hence, a stong link deletion poof netwok (see fomal definition late) in which no dyad is inteested to fom a new connection will be esistant to changes (and unilateally stable).

7 18 K. Takács et al. / Social Netwoks 30 (008) costs of tie deletion ae unde a theshold that is detemined by popotional confomity and the individual degee of the defecto, then this tie is not stable (at least the defecto wants to delete this tie). Futhemoe, pat (c) assets that elations between defectos and contibutos ae not stable, if popotional confimation is positive and tie deletion and fomation costs ae unde a specific theshold. To analyze maco-level consequences of Theoem I, we intoduce stability concepts fo the entie netwok that concen link deletions: Definition. R is link deletion poof inagiven, if i (, R) i (, R\{ij}) a fo all i N and ij R. That is, a social netwok is link deletion poof in a given stategy pofile (in the collective action game) if thee is no acto fo whom deleting a single elation would esult in highe subjective payoffs, assuming exactly the same actions and no othe change in the netwok. This concept of link deletion poofness, simila to stable netwoks (Watts, 001) and paiwise stability (Jackson and Wolinsky, 1996), concens only a single change in the netwok at once. A stability concept that allows playes to delete any set of links at once is called stong link deletion poofness (Gilles and Saangi, 004; Belleflamme and Bloch, 004). Definition. R is stong link deletion poof inagiven, if i (, R) i (, R\H i ) h i a fo any H i R i and fo all i N. That is, a social netwok is stong link deletion poof in a given stategy pofile (in the collective action game) if thee is no acto fo whom deleting any subset of he elations would esult in highe subjective payoffs, assuming exactly the same actions and no othe change in the netwok. Adopting the concept of stong link deletion poofness, we can fom two coollaies of Theoem I. Coollay I.a. Any R is stong link deletion poof if = 0 o = 1. Coollay I.b. If b / dmax > a 0(whee dmax is the highest individual degee among defectos), then in a stong link deletion poof R fo all i N with i =0holds that if ij, ik R i then j = k. The coollaies summaize typical cases unde which a netwok is stong link deletion poof. Coollay I.a states that evey netwok is stong link deletion poof in a full contibution and in a full defection stategy pofile. Coollay I.b claims that if popotional confomity ewads ae positive and costs of tie deletion ae unde a theshold value, then in a stong link deletion poof netwok in the collective action game evey defecto has ties only to defectos o only to contibutos, but not to both. The key paamete that undelies Theoem I and the coollaies is popotional confomity (b ). Popotional confomity is esponsible fo the esult that deleting all elations to contibutos is moe beneficial fo a defecto than just deleting one o some of them. The moe links to contibutos a defecto deletes, the highes the impovement of the atio of defectos to contibutos among he ties. Fo example, if this atio is 5:10, then deletion of one link impoves the atio by about , while deletion of two links to contibutos yields an impovement of about 0.15, deletion of thee yields an impovement of 0.14, etc. The theshold conditions in Coollay I.b fo the costs of tie deletion a illustate that a high level of popotional confomity makes netwoks in which defectos ae tied both to contibutos and defectos subject to link deletion. 8 The conditions imply that 8 In case of negative social contol, mass confomity and selective incentives also play a ole in stong link deletion poofness. Futhemoe, contibutos might also have an incentive to delete all thei ties to fee ides. Hence, thee ae sticte equiements fo stong link deletion poofness see Appendix B. Fig. 1. Segegation of contibutos and link deletion poofness. Notes: Example fo paamete values: s =1,b 1 =1,b =5,c = 3.5, a =. Filled nodes denote contibutos and empty nodes ae defectos. (a) A less segegated netwok and link deletions. (b) A moe segegated netwok and link deletion poofness. the highest degee among defectos who have connections to contibutos is decisive fo stong link deletion poofness of stuctues with mixed ego-netwoks. The highe the maximum degee within this subset, the moe likely it is that a stuctue with mixed egonetwoks can also be stong link deletion poof. This implies that highly centalized and vey dense netwoks that contain individuals with a high degee ae moe likely to be stong link deletion poof. 9 Coollay I.b shows that if costs of tie deletion ae low, a disconnected netwok in which defectos ae only tied to defectos and contibutos ae only tied to contibutos will be stong link deletion poof. One should note, howeve, that a bipatite netwok, in which all defectos ae only tied to contibutos and contibutos ae only tied to defectos will also be link deletion poof in this case. The level of segegation of contibutos and defectos, howeve, does not have an unambiguous impact on link deletion poofness. If the netwok is pefectly segegated, then it has a component of defectos and a component of contibutos and hence it is stong link deletion poof. But this does not imply that fo those netwoks that contain ties between contibutos and defectos, a highly segegated netwok is necessaily pone to become even moe segegated, o a modeately segegated netwok is moe likely to be stable than a highly segegated one. The examples in Fig. 1 illustate how a less segegated netwok (Fig. 1a) can be moe subject to link deletion than a moe segegated one (Fig. 1b). The two netwoks ae identical concening the sets of contibutos and defectos and have the same density. The numbes of connections of D 1 and D 3 influence netwok changes in the less segegated netwok and only D 1 s connections matten Fig. 1b. The impovement of netwok composition in tems of popotional confomity is smalle fo D 1 in netwok 1b than it is in netwok 1a. Hence, at a wide ange of paamete values, only the latte netwok is (stong) link deletion poof. Consequently, the initially less segegated netwok (in Fig. 1a) becomes moe segegated aftendividual stuctual decisions. The diving mechanism is again popotional confomity. Bidging actos with many ties do not benefit as much in tems of popotional confomity fom deleting links to individuals with dissimila choices, as compaed to the benefits that less integated bidging actos can obtain fom deleting such ties. 9 The sign of the social contol paametes does not alte ou main conclusions, although the maximum degee among contibutos with connections to defectos is also elevant fo stong link deletion poofness (see Appendix B).

8 K. Takács et al. / Social Netwoks 30 (008) We now tun to some equilibium popeties that concen tie fomation. The main esults concening foming ties ae summaized in Theoem II (see Appendix A fo the poof). A tie ij / R is initiated by i if exist G i Ř i, and H i R i such that ij G i and i (,(R G i )\H i ) h i a g i f > i (, R). Theoem II. In any R and given, fo all i,j N and ij / R: (a) If i =0and j =1,then i (, R) i (, R {ij}). (b) If i = j and i (, R) i (, R {ij}) f, then i (, R) i (, R G i ) g i f, whee ij G i fo any G i Ř i. (c) If i = j =1and s + b 1 + b d /( + i i ) >f, then ij is initiated by i. (d) If i = j =0and b 1 + b c /( + i i ) >f, then ij is initiated by i. Pat (a) of Theoem II states that new elationships ae not fomed between contibutos and defectos. The eason is that the defecto does not gain anything fom a new tie to a contibuto. Pat (b) follows fom that the maginal benefits of foming moe ties ae deceasing in the numbe of ties. Again, this is caused by popotional confomity. Hence, the fist new tie is the most valuable, in case thee is at least one connection to a defecto. If a single new tie is not beneficial, then no lage new set that contains this tie can be feasible. Similaly, assuming a coalition of defectos in which multiple ties ae fomed, the maginal benefits of foming new ties ae deceasing. 10 Pats (c) and (d) yield implications fo the chaacteistics of those netwok positions in which actos ae most likely to fom new ties. To begin with, the numbe of defecto contacts inceases the chance that a new tie is fomed between two contibutos. Futhemoe, contibutos with many connections (high ) ae less likely to fom new connections to othe contibutos, as it does not give them sufficient maginal benefits (if d > 0). Consideing two defectos that ae not tied with each othe, the likelihood of a new connection inceases with the numbe of elations to contibutos. Again, individuals with many connections ae less likely to fom new ties (if they ae tied at least to one contibuto). Individual netwok paametes ae effective because of popotional confomity. Popotional confomity benefits of a new tie ae the highest fo an individual with just one existing tie to a dissimila acto. Fo this individual, the popotion of simila actos is impoved by a half if a new tie is fomed. Stating fom moe ties o fom moe ties to simila actos mean less impovement in the composition and hence less popotional confomity gains. In a utopian setting in which contacts ae fomed feely, all contibutos would be inteested to be matched with all othe contibutos and all defectos would be happy to build elations with othe defectos to enjoy highe behavioal confimation ewads. In case selective incentives ae moe impotant than behavioal confimation, contibutos would even be inteested to get any kind of connections also including ties to defectos. A symmetic elationship equies a mutual ageement of the paties, howeve, and defectos would veto this, because additional coss goup ties may educe the benefits they enjoy fom popotional confomity. Moeove, in a full contibution stategy pofile highest benefits would come fom a netwok in which eveyone is tied to eveyone else. 11 Like fo link deletion, netwok chaacteistics influence tie fomation only within the subset of those membes who have abidging ties. Within this goup, stongly embedded, cental actos will be less likely to change thei netwoks. Unlike in the case of 10 Simila esults ae obtained also fo negative social contol (see Appendix B). 11 Much less (if any) tie fomation can be expected if social contol is expessed only as punishments (see Appendix B). link deletion, howeve, the level of segegation between contibutos and defectos has a moe clea-cut effect on netwok changes: moe segegated netwoks ae less likely to gow. Nevetheless, the impact of individual degee on changes is stonge than the impact of the numbe of abidging individual connections. To sum up, ou analysis of tie deletion and link ceation has suggested that elations in collective action tend to build up slowly and beak up easily. We found that the most pofitable stategy fo a defectos to abandon all of he ties to contibutos (assuming that she pefes to delete any tie and has at least one elation to anothe defecto). On the othe hand, we also showed that foming the fist new tie to anothe defecto has the highest maginal benefits, in case thee is at least one connection to a contibuto. Follustation, we highlight the main pedictions of ou model with netwok changes with a stylized example. Conside a wild cat stike in a factoy with a dense but not complete netwok of infomal social ties among wokes. The stike can be modeled as a one-shot public good game in which only infomal social contol fostes paticipation. When only a few wokes paticipate, tensions between stikes and goons may emege aftewads and might even esult in beaking old elations. The model assumes that stikes who ae only elated to stikes (goons only elated to goons) get positive feedback fom thei pees, and thei elationships do not come unde pessue by the event. Those who have a contact fom the opposite camp, howeve, feel shame, guilt o ae simply embaassed by the conflict with some of thei othe contacts, which geneates an incentive to beak the elationship. Ou model implies that the cleaing of such a mixed ego-netwok is moe difficult to the extent that ego has many connections. Hence, ou analysis suggests that a moe dense community is moe likely to emain cohesive even afte the heated times of the wild cat stike. Whee wokes ae less embedded, howeve, contacts between stikes and goons might dissolve and segegated factions may be fomed. The model also pedicts that in a loosely tied goup of wokes, the collective expeience of the demonstation might bing stikes close to each othe. Nonetheless, simila mechanisms opeate among those who did not paticipate in the wild cat stike. They also seek einfocement, and may fom the goup of modeates o ational egoists. In a dense community, howeve, it is less likely that such an event can contibute to the building of a lage o an even dense social netwok, because the elative impovements wokes can attain in tems of popotional confomity ae small if they ae elated to many coopeatos and many defectos at the same time. Nevetheless, those who have many abidging elations may seek new acquaintances even in a dense netwok. Nonetheless, not only stuctual chaacteistics matte. Costs of changing ties also have an impact on decisions. Lage values of a and f can be intepeted as high constaints on netwok change imposed by the given netwok. Fo example, conside the case of the stike in a poject team whose membes ae tied by a netwok of such dyadic task intedependencies that ae impotant fo thei futue wok pefomance and thus also thei caee pospects. In such a situation, team membes would not easily segegate along the lines of stikes vesus goons, because othe ewads besides selective incentives and behavioal confimation ae at stake when elations change. 4. Simultaneous social contol and stongly obust netwok equilibium In the pevious section we elaxed the taditional assumption of models of collective action that the social netwok is given and individuals cannot change thei elations. The analysis we povided is in paticula suitable fo situations in which social contol mechanisms ae delayed compaed to paticipation decisions in collective

9 184 K. Takács et al. / Social Netwoks 30 (008) action. Thee ae situations, howeve, when stuctual changes and behavion the collective action game ae simultaneous. Futhemoe, even when this is not the case, actos can anticipate stuctual changes at the time of thei paticipation decision in collective action. Unde such cicumstances, these actions ae pat of the same stategy; stuctual decisions and netwok stability should be consideed togethe with individual decisions and equilibia in collective action. Fo example, wokes who paticipate in a wild cat stike may take into account the isk of loosing fiends and the oppotunity of finding new ones. Let us conside a woke, who woks in a peipheal unit in which the majoity does not pefe to join the stike of the majo wokshop. If this woke has some fiends in the majo wokshop, then she has to decide about paticipation but also about the community she wants to be embedded in at the same time. To addess situations like this one, we need an equilibium efinement that embaces the concepts of unilateal stability and Nash equilibium in the context of games played in social netwoks. Fo this pupose, we popose the notion of stongly obust netwok equilibium. A netwok of social elations and a stategy pofile in the (collective action) game ae in stongly obust netwok equilibium, if thee is no acto, fo whom any combination of changes in he contibution decision and in he ego-netwok would esult in a bette outcome; equiing consent fo evey new elation fom patnes. Definition. We define the combination of R and stategy pofile * ( i, )astongly obust netwok equilibium, i if i (( i, i ), R) i(( i, i ), (R G i)\h i ) h i a g i f o exists such j N that ij G i : j (( i, i ), R)> j(( i, i ), R {ij}) f fo any G i Ř i,anyh i R i and fo all i N. Note that just as in unilateal stability, the concept of stongly obust netwok equilibium equies that no deviations ae individually beneficial given that stategies of othes and elations in which i is not involved ae fixed. Fom a new patne j, consent is equied only fo the fomation of ij. A new patne j would not give consent if he subjective ewads in the netwok without ij wee highe than he subjective ewads in the netwok with ij consideing the new stategy pofile, in which only the action of i might be diffeent than the oiginal choice. This fomalization is the staightfowad way to captue that stategy choices and netwok choices ae simultaneous and not independent. What ae the conditions unde which stongly obust netwok equilibia can occun collective action? It follows immediately that only a Nash equilibium stategy pofile and only unilateally stable netwoks can be in stongly obust netwok equilibium. Howeve, unilateal stability and Nash equilibium in the collective action game ae necessay but not sufficient conditions fo stongly obust netwok equilibium. Conside fonstance, a situation in which a disconnected netwok with a complete component of contibutos and a complete component of defectos is unilateally stable. Given sufficiently high benefits fom eceiving selective incentives and behavioal confimation, the stategy pofile can also be in Nash equilibium. In this situation, nobody has an incentive to abandon elations, to fom ties and eceive ageement fom the new patnes, o to change the decision in the collective action game. Restuctuing elations and changing the action in collective action, howeve, can be beneficial fo some playes. If foming new ties and abandoning existing elations ae fee, thee would always be playes fo whom such changes wee beneficial. Pat (a) of Theoem III expesses that when thee ae no costs of netwok change and social contol is expessed as ewads, the numbe of stongly obust netwok equilibia is esticted to exteme configuations: only full contibution and full defection with complete netwoks can be stongly obust netwok equilibia. Pat (b) states that when the goup is small and building and deleting costs a and f ae elatively small compaed to selective incentives and mass confomity a patial contibution pofile cannot be stongly obust netwok equilibium (see Appendix A fo the poof). Theoem III. (a) If a =0, f =0 and (b 1 >0 o s > 0), then in stongly obust netwok equilibium (R, * ): R = R N and = fo all i,j N. i j (b) If s +b 1 > nf +(n )a and b 1 f, then in stongly obust netwok equilibium (R, * ): R = R N and = fo all i,j N. 1 i j To see the intuition undelying this esult, conside a netwok in which thee ae both defectos and contibutos. Without costs fo link changes, eithe contibutos o defectos would pofit fom changing thei contibution decision, abandoning all ties to thei ex-goup and build connections to evey membe of thei new goup. Contibutos ae bette off by becoming integated defectos if in the oiginal netwok thee ae a sufficiently lage numbe of defectos. In this case, the elated gain in behavioal confimation exceeds the loss in tems of foegone povision of the collective good and lost selective incentives. Convesely, if the numbe of defectos falls below this citical level, then all defectos would gain fom tuning into contibutos and complete thei netwok with fellow contibutos. Hence, the netwok in which thee ae both defectos and contibutos is not in equilibium, because depending on the popotion of contibutos in the entie netwok eithe all defectos o all contibutos ae pulled towads an equilibium with unifom choices. One should note that if foming and deleting ties ae fee o have elatively low costs, then the assumption of simultaneous decisions about elationships and paticipation makes the initial netwok stuctue ielevant. When looking at netwok changes only, the initial netwok stuctue mattes as it constains the possible sets of contibutos and fee ides. But the initially given netwok can also be impotant fo collective action if behavioal changes ae consideed simultaneously with netwok updates. The key assumption needed fo this is that thee ae costs fo deleting o building ties that epesent stuctual constaints embodied in the existing netwok. Costly netwok changes imply that not only complete netwoks can be stongly obust netwok equilibia. When a cost of a new tie exceeds mass confomity benefits (f > b 1 ), any initial netwok without isolates is in stongly obust netwok equilibium in a full defection pofile. Futhemoe, any initial netwok with a minimum individual degee of min is in stongly obust netwok equilibium in a full coopeation pofile if the cost of a new tie exceeds selective incentives and mass confomity benefits (f > s + b 1 ) and social contol ewads exceed contibution costs ( + b + min (b 1 + s)>c to assue that switches to defection ae not beneficial). The exact condition fo pat (b) of Theoem III also detemines the cost constaint above which patial contibution with complete components of defectos and contibutos can be a stongly obust netwok equilibium. In combination with the aguments above it can be also stated that not only complete components can chaacteize patial contibution in stongly obust netwok equilibium. Fo not complete components of defectos f > b 1 should hold, fo not 1 In case of negative social contol, when deleting ties is fee, only full defection with complete isolation can be stongly obust netwok equilibium (see Appendix B).