Information, Stability and Dynamics in Networks under Institutional Constraints

Transcription

1 Fondazione Eni Enico Mattei Woking Papes Infomation, Stability and Dynamics in Netwoks unde Institutional Constaints Noma Olaizola Depatamento de Fundamentos del Análisis Económico I, Univesidad del País Vasco Fedeico Valenciano Depatamento de Economía Aplicada IV, Univesidad del País Vasco, Follow this and additional woks at: Recommended Citation Olaizola, Noma and Valenciano, Fedeico, "Infomation, Stability and Dynamics in Netwoks unde Institutional Constaints" (Novembe 19, 2010). Fondazione Eni Enico Mattei Woking Papes. Pape This woking pape site is hosted by bepess. Copyight 2010 by the autho(s).

2 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit Infomation, stability and dynamics in netwoks unde institutional constaints By Noma Olaizola y and Fedeico Valenciano z July 6, 2010 Abstact In this pape we study the e ects of institutional constaints on stability, e ciency and netwok fomation. Moe pecisely, an exogenous societal cove consisting of a collection of possibly ovelapping subsets that coves the whole set of playes and such that no set in this collection is contained in anothe speci es the social oganization in di eent goups o societies. It is assumed that a playe may initiate links only with playes that belong to at least one society that s/he also belongs to, thus esticting the feasible stategies and netwoks. In this way only the playes in the possibly empty societal coe, i.e., those that belong to all societies, may initiate links with all individuals. In this setting the pat of the cuent netwok within each connected component of the cove is assumed to be common knowledge to all playes in that component. Based on this two-ingedient model, netwok and societal cove, we examine the impact of societal constaints on stable/e cient achitectues and on dynamics. Jounal of Economic Liteatue Classi cation Numbes: C72 Key wods: Netwok, Non-coopeative game, Dynamics. This eseach is suppoted by the Spanish Ministeio de Ciencia e Innovación unde pojects ECO and ECO , co-funded by the ERDF. Both authos also bene t fom the Basque Govenment s funding to Gupos Consolidados GIC07/146-IT and GIC07/22-IT y Depatamento de Fundamentos del Análisis Económico I, Univesidad del País Vasco, Avenida Lehendakai Aguie, 83, E Bilbao, Spain; noma.olaizola@ehu.es. z Depatamento de Economía Aplicada IV, Univesidad del País Vasco, Avenida Lehendakai Aguie, 83, E Bilbao, Spain; fedeico.valenciano@ehu.es. Published by Bekeley Electonic Pess Sevices,

3 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] 1 Intoduction In ecent yeas the study of the economics of netwoks has attacted consideable attention fom eseaches and become one of the hottest topics of economic eseach 1. The economics of netwoks is, in Goyal s wods, an ambitious eseach pogam which combines aspects of makets (e.g., pices and competition) along with explicit pattens of connections between individual entities to explain economic phenomena (Goyal, 2007, p. 6). Seveal seminal papes study the stability and e ciency of netwoks poviding the basic models. In the simplest model links ae fomed unilateally (Goyal (1993), Bala (1996)). In this setting Bala and Goyal (2000a) study Nash stability and povide a dynamic model. A model whee links ae fomed on the basis of bilateal ageements is studied by Jackson and Wolinsky (1996), who intoduce the notion of paiwise stability. In these seminal papes it is assumed that thee is homogeneity acoss playes and also that the cuent netwok is common knowledge to all node-playes. Galeotti et al. (2006) conside heteogeneous playes, while Bloch and Dutta (2009) conside endogenous link stength. The common knowledge assumption may be unealistic in many cases, and indeed is dopped by McBide (2006), who studies the e ects of limited peception, namely, assuming that each node-playe peceives the cuent netwok only up to a cetain distance fom the node. In the seminal models netwoks povide a means fo the ow of infomation o othe bene ts though the links, but the cuent netwok is assumed to be common knowledge to all playes, who may unestictedly initiate links with any othe playes. In some cases this may be an unealistic assumption, and in geneal the lage the netwok is the moe unealistic it will be. It seems moe ealistic to assume that because they belong to the same goup (family, club, pofessional association, depatment, etc.) individuals may have a clea idea of the connections within such smalle goups and initiate links only within the goups they belong to. Moeove, an individual may belong to moe than one of these goups, shaing common knowledge of the links connecting membes of each goup. In a way this is an unothodox appoach if, as put by Goyal, the theoetical eseach on netwok e ects (..) is motivated by the idea that, within the same goup [in italics], individuals will have di eent connections and that this di eence in connections will have a beaing on thei behavio. (Goyal, 2007, p. 7). Nevetheless, this is the appoach adopted hee, and it is woth emaking that the othodox single-goup assumption is in fact a paticula case of the moe geneal setting adopted hee. In paticula, this allows Bala and Goyal s (2000a) two-way ow basic model, on which we concentate in this pape, to be integated into a wide model which sheds new light on vaious conclusions of thei model, showing which pevail and up to which point, and which do not in this wide setting. Based on this idea, in this pape we focus on the e ects of institutional and/o in- 1 Some ecent books suveying this liteatue ae Goyal (2007), Jackson (2008) and Vega-Redondo (2007)

4 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit fomational constaints on stability, e ciency and netwok fomation. Moe pecisely, an exogenous societal cove speci es social oganization in di eent goups o societies. A societal cove is a collection of possibly ovelapping subsets of the set of playes o societies that coves the whole set (i.e., each playe belongs to at least one set in this collection) such that no set in this collection is contained in anothe. It is assumed that a playe may initiate links only with playes that belong to one o moe of the societies that s/he also belongs to, thus esticting he/his feasible stategies, and as a consequence the feasible netwoks. Note that in this scenaio only the playes in the possibly empty societal coe, i.e., those that belong to all societies, may have diect access to all individuals. It is also assumed that only the pat of the cuent netwok within each component (in a sense to be speci ed late) of the societal cove is common knowledge to all playes in that component. Note also that this model collapses to Bala and Goyal s (2000a) unesticted setting fo the paticula case of the tivial societal cove consisting of a single society including all playes. Based on this two-ingedient model -netwok and societal cove- we examine the impact of the societal constaints, which ae intepeted in geneal tems as institutional constaints, on stable/e cient achitectues and on dynamics. Fo any given societal cove we constain ou attention to the admissible netwoks (i.e., those consistent with the cove) and st extend Bala and Goyal s (2000a) notion of a Nash netwok as those admissible netwoks whee no playe has an incentive to change he/his stategy, i.e., he/his choice of admissible links. We then extend thei chaacteization of Nash netwoks as those among the admissible netwoks which ae minimally connected. In this way the set of such Nash netwoks is a subset of the set of Bala and Goyal s unesticted Nash netwoks. Then Bala and Goyal s (2000a) notion of stict Nash netwok is also natually extended to this setting. Now a stict Nash netwok is a netwok consistent with the societal cove whee no playe may initiate and/o delete any admissible link(s) without loss. By contast with Nash netwoks, things tun out to be much moe complicated with stict Nash netwoks. In Bala and Goyal s setting the cente-sponsoed sta is the only (non empty) achitectue of stict Nash netwoks, while in ou setting the cente-sponsoed achitectue is feasible only when the societal coe, i.e. the set of playes belonging to all societies, is not empty. Moeove, even when the cente-sponsoed sta achitectue is feasible it is not the only possible achitectue of stict Nash netwoks. A vaiety of achitectues of stict Nash netwoks appea fo any non single-society cove, and the moe complex the societal cove the geate this vaiety is. Nevetheless, some pattens ae common to these achitectues. Moeove, a full chaacteization of all stict Nash netwoks fo a societal cove is povided by means of a condition that encapsulates synthetically the essence of the achitectue of these netwoks, embodying an implicit fom of hieachical pinciple. The main featues of thei achitectues, whee stas continue to play a pominent ole, ae studied. Paticula attention is paid to the ole of playes who belong to moe than one society, by means of whom di eent but ovelapping societies can be connected. It tuns out that stict Nash netwoks incopoate a clea hieachical stuctue: They Published by Bekeley Electonic Pess Sevices,

5 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] ae eithe oiented tees (also called aboescences in gaph theoy) o a sot of gafted oiented tees o aboescences. The latte ae possible only when thee ae hinge-playes, i.e., playes who ae the unique common membe of two societies. Finally we extend Bala and Goyal s dynamic model, whee stating fom any initial netwok each playe with some positive pobability plays a best esponse o andomizes acoss them when thee is moe than one, othewise the playe exhibits inetia, i.e., keeps his/he links unchanged. In this way a Makov chain on the state space of all netwoks is de ned. In Bala and Goyal s setting, the absobing states ae pecisely the stict Nash netwoks and they pove that stating fom any netwok the dynamic pocess conveges to a stict Nash netwok (i.e., the empty netwok o a cente-sponsoed sta) with pobability 1. While when adapted to ou setting the best esponse dynamic model does not necessaily lead to stict Nash netwoks. The eason is that in ou moe complex setting this dynamic pocess may lead to the fomation of stable incomplete stict Nash netwoks that cannot be pat of the same geneal stict Nash netwok. Thus the same logic that in thei setting leads to the absobing stict Nash netwoks, may lead in ous to stable incomplete stict Nash incompatible netwoks that block the conveging pocess. Thus, in a way, institutional constaints may hinde the way towads stict Nash netwoks. Nevetheless, best esponse dynamics lead to something vey close to a stict Nash netwok that we call quasi stict Nash netwoks, if not to a stict Nash netwok. These constitute absobing sets of minimally connected netwoks, closed with espect to best esponse dynamics, and fully connected by those dynamics in the sense that any netwok in one of these sets is eachable fom any othe in the same set by best esponse dynamics. Thus, with pobability 1, best esponse dynamics would lead eithe to a stict Nash netwok (wheneve the set of quasi stict Nash netwoks eached is a singleton) o one of these sets of quasi stict Nash netwoks whee the best esponse dynamics would oscillate fo eve. Nevetheless this is not a seious dawback because stability is eached in tems of payo s as it is poved they all quasi stict Nash netwoks within each of these sets yield the same payo s to all playes. The est of the pape is oganized as follows. In section 2 the basic model is speci ed along with the necessay notation and teminology. Section 3 studies stability and e ciency unde institutional constaints. In section 4 Bala and Goyal s dynamic model is extended to this setting. Finally, section 5 summaizes the main conclusions and points out some lines of futhe eseach. 2 The model Let N = f1; 2; ::; ng denote the set of nodes o playes. Playes may initiate o delete links with othe playes. By g ij 2 f0; 1g we denote the existence (g ij = 1) o not (g ij = 0) of a link connecting i and j initiated by i. Vecto g i = (g ij ) j2nni 2 f0; 1g Nni 3 4

6 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit speci es 2 the set of links initiated by i and will be efeed to as an (unesticted) stategy of playe i. G i := f0; 1g Nni denotes the set of i s (unesticted) stategies and G N = G 1 G 2 :: G n the set of (unesticted) stategy po les. An unesticted stategy po le g univocally detemines a diected netwok 3 that we identify with g g = f(i; j) 2 N N : g ij = 1g: Given an netwok g, and M N we denote by g j M esticting g to M, moe pecisely the netwok that esults by g j M := f(i; j) 2 M M : g ij = 1g: We now conside the following situation. An exogenous societal cove speci es a set of possibly ovelapping societies that epesent a social constaint in the following sense: Each playe in N can initiate links with any othe playe as long as they belong to the same society. Fomally, we have the following De nition 1 A societal cove of N is a collection of subsets of N (called societies ), K 2 N, such that: (i) S A = N; and (ii) fo all A; B 2 K (A 6= B), A * B: A2K Condition (i) ensues that evey playe belongs to at least one society; while condition (ii) pecludes supe uous societies: if A B; A would be supe uous given the intepetation of societies. The following notation and teminology is useful. We denote by K i the set of societies that i belongs to, and by N(K i ) the set of nodes that i may diectly access, that is: K i := fa 2 K : i 2 Ag and N(K i ) := [ A2K i A: Two nodes i; j have identical a liation if they belong to the same societies, i.e., K i = K j. Two nodes i; j have the same each if N(K i ) = N(K j ). Note that identical a liation implies the same each, but the convese is not tue. Example: If N = f1; 2; 3; 4; 5; 6; 7; 8; 9g and K := ff1; 2; 3; 4; 5; 6g; f4; 5; 6; 7; 8; 9g; f1; 2; 4; 5; 7; 8g; f2; 3; 5; 6; 8; 9gg; then 2 and 4 have the same each: N(K 2 ) = N(K 4 ) = N, but di eent a liations as K 2 6= K 4 : K 2 = ff1; 2; 3; 4; 5; 6g; f1; 2; 4; 5; 7; 8g; f2; 3; 5; 6; 8; 9gg 2 We always dop the backets f::g in expesions such as Nnfig: 3 In gaph theoy this is called a digaph without loops, i.e., edges connecting a node with itself (see, fo instance, Tutte (1984)). Published by Bekeley Electonic Pess Sevices,

7 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] and K 4 = ff1; 2; 3; 4; 5; 6g; f1; 2; 4; 5; 7; 8g; f4; 5; 6; 7; 8; 9gg: The following teminology is used. A component C of a societal cove K is a subset C K such that (i) fo all A; B 2 C thee exist A 1 ; ::; A k 2 K s.t. A 1 = A and B = A k, and A i \ A i+1 6=? fo i = 1; ::; k 1, and (ii) fo all B 2 KnC; B \ ([ A2C A) =?. The subset [ A2C A of N coveed by a component C is denoted by N(C). Fo each i, C i (K) denotes the component of K that contains K i. A societal cove is connected if it has a unique component. The societal coe of a societal cove is the set of nodes that belong to all societies coe(k) := \ A: A2K This set may be empty. Note that only the playes in the societal coe may have diect access to all individuals in N. Let K be a societal cove of N, if K 0 K we say that K 0 is a subcove of K if K 0 is a societal cove of N(K 0 ) := [ A2K 0A s.t. fo all A 2 K; A N(K 0 ) implies A 2 K 0. In paticula, a component of a a societal cove K is a (connected) subcove of K. The following de nition constains the stuctue of a netwok so as to be consistent with a given societal cove of N by uling out links connecting individuals who ae not membes of at least one society in common. De nition 2 A netwok g is consistent with a societal cove K (o is a K-netwok) if fo evey link g ij = 1 thee exists some A 2 K s.t. i; j 2 A (i.e., K i \ K j 6=?). A vecto g i = (g ij ) j2n(ki )ni 2 f0; 1g N(K i)ni speci es a set of K-feasible links initiated by i and is efeed to as a K-admissible stategy of playe i, as we assume i s capacity to choose which links to initiate in N(K i ). G i (K) := f0; 1g N(K i)ni denotes the set of i s K-admissible stategies and G K = G 1 (K) G 2 (K) :: G n (K) the set of K-admissible stategy po les. A K-admissible stategy po le g univocally detemines a K-netwok that we identify with g. Obseve that this setting is not naowe than Bala and Goyal s standad one. It is in fact moe geneal as the standad (i.e., unesticted) notions of netwok, stategy and stategy po le coespond to the paticula case of the simplest societal cove K = fng, whee a single society includes all playes and all links ae feasible. Given a netwok g, we denote g ij := maxfg ij ; g ji g. In this way a nondiected netwok g is de ned 4. g epesents the communication povided by netwok g, which is independent of who initiated the existing links accoding to the assumptions of the model. We denote by N(g) the set of non isolated nodes, that is, N(g) := fi 2 N : g ij 6= 0 fo some j 2 Ng: We say that thee is a path fom i to j in g if thee exist playes j 1 ; ::; j k, s.t. i = j 1, j = j k, and fo all l = 1; ::; k 1, g jl j l+1 = 1, and we say that such a path is i-oiented 4 In gaph theoy tems, g is the undelying gaph of digaph g (see, e.g., Tutte, 1984)

8 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit if fo all l = 1; ::; k 1, g jl j l+1 = 1. A path (diected o not) is K-feasible if all its links ae K-feasible. The set of playes with whom i initiated a link is denoted by N d (i; g), and the set of playes connected with i by a path (union fig) by N(i; g), and thei cadinalities by d i (g) := #N d (i; g) and i (g) := #N(i; g). Note that if g is a K-netwok then N d (i; g) N(K i ) and N(i; g) [ A2Ci (K)A. We say that a netwok g is an aboescence o an oiented tee if thee is a node i 0 such that fo any othe node thee is a unique i 0 -oiented path connecting it with the node oot i 0. It is assumed that each node contains valuable infomation and a link allows that infomation to ow in both diections without decay independently of who initiates it, so that each node eceives the infomation fom all nodes with which it is connected by a path. Let v ij > 0 be the payo that playe i deives fom connecting diectly (by a link) o indiectly (by a path) with playe j, and c ij > 0 the cost fo playe i of initiating a link with j. Thus the payo of playe i in g is i (g) = X X c ij : j2n(i;g) v ij j2n d (i;g) If we assume costs and bene ts to be homogeneous acoss playes (i.e., v ij = v and c ij = c; fo all i; j) and v > c, connections with new nodes ae always po table and 5 i (g) = v i (g) c d i (g): A K-netwok is e cient if it maximizes the aggegate payo unde the constaint of K-feasible payo s, that is, those that can be obtained by means of K-netwoks. As a tem of compaison we sometimes conside standad unesticted e ciency to which we efe as e cient netwoks. A component of a netwok g is a set C(g) N such that any two playes in C(g) ae connected by a path, and no playe in N n C(g) is connected by a path with a playe in C(g). We say g is connected if N is the unique component of g. A netwok is minimal if fo all i; j s.t. g ij = 1, the numbe of components of g is smalle than the numbe of components of g g ij, whee g g ij is the netwok that esults by eplacing g ij = 1 by g ij = 0 in g (similaly, when g ij = 0 we wite g +1 ij to epesent the netwok that esults by eplacing g ij = 0 by g ij = 1 in g). Remak: Note the elationship between the notions of connected component of a societal cove K of N and connected component of a K-netwok: a connected component of a K-netwok is always coveed by a connected component of the societal cove K. We denote by g i the netwok whee all links initiated by i ae deleted, and by (g i ; g 0 i) the stategy po le and netwok that esults by eplacing g i by g 0 i in g. In paticula, (g i ; g i ) = g. 5 Although the esults pesented hee can easily be extended with some slight modi cations to the case whee payo s ae, as in Bala and Goyal (2000a), given by a function ( i (g); d i (g)), whee (x; y) is stictly inceasing in x and stictly deceasing in y, we pefe this simple assumption about payo s so as to make the statements of the basic esults simple. Published by Bekeley Electonic Pess Sevices,

9 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] We next discuss some notions of stability of netwoks consistent with a given societal cove K. 3 Stability and e ciency The following de nitions ae natual extensions of the notions of Nash stability and stict Nash stability due to Bala and Goyal (2000a) fo a netwok in a scenaio whee: (i) a societal cove K allows only fo links connecting individuals belonging to the same society, and (ii) all playes in a same component C of K, i.e., in N(C), have common knowledge of the pat of the cuent netwok connecting individuals of N(C). The common knowledge assumption esticted to playes in the same component of the cove can be justi ed by assuming that infomation about the cuent netwok popagates between ovelapping societies. Note that this scenaio yields the unconstained and common-knowledge envionment of Bala and Goyal (2000a) fo the paticula case of the tivial societal cove: K = fng. De nition 3 A Nash K-netwok is a K-netwok g that is stable unde K-admissible stategies, that is, fo all i 2 N: i (g) i (g i ; g 0 i) fo all g 0 i 2 G i (K): (1) When (1) holds we say that g i is a best (admissible) esponse of i to g i. Thus, in a Nash K-netwok evey playe is playing a best K-admissible esponse to those played by the othes. Note that fo K = fng a Nash K-netwok is a Nash netwok in the standad setting. The stability notion can be e ned in the stict sense by extending Bala and Goyal s stict Nash netwoks. De nition 4 A stict Nash K-netwok is a Nash K-netwok such that fo all i 2 N: i (g) > i (g i ; g 0 i) fo all g 0 i 2 G i (K) (g 0 i 6= g i ): (2) Thus (2) means that in a stict Nash K-netwok evey playe is playing he/his unique best (admissible) esponse to those played by the othes. Also note that fo K = fng a stict Nash K-netwok is a Nash netwok in the standad setting. Given the constaints on infomation, stategies and feasible netwoks that a societal cove imposes, the set of playes N(C) in each component C of the cove, whee subcove C pescibes what links ae feasible, fom an entiely sepaate wold : No link with NnN(C) is possible and no infomation about it eaches N(C). In paticula we have the following staightfowad esult. Poposition 1 A K-netwok g is a Nash (stict Nash) K-netwok if and only if g j N(C) is a Nash (stict Nash) C-netwok fo each component C of K

10 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit Remak: Note also that although societies consisting of a single individual ae included in the model, such tivial societies ae of no inteest in this setting. Moeove, the only connected societal cove K that contains a society A s.t. #A = 1 is K = fag. Theefoe, in view of Poposition 1 and the peceding emak, in what follows we constain ou attention to connected societal coves and we always assume that all societies have at least two individuals, unless othewise speci ed. The following poposition extends Bala and Goyal s esult to this setting. Poposition 2 Given a connected societal cove K of N, a K-netwok g is a Nash K-netwok if and only if it is minimally connected. Poof. Let K be a connected societal cove of N, and g a K-netwok. Assume g is not connected. Then thee exist two nodes i; j 2 N not connected by a path in g. As cove K is connected, thee exists a nite sequence of nodes x 1 ; ::; x m, such that x 1 = i, x m = j and fo each k = 1; ::; m 1, thee is some A 2 K s.t. x k ; x k+1 2 A. Then fo at least two consecutive nodes among these m nodes, say x k and x k+1, thee is no path in g connecting them. But then it is po table fo eithe of these two nodes to initiate a link with the othe. Thus g must be connected. If it wee not minimal thee would be some edundant link that could be eliminated and that would bene t the playe that did so, and consequently g is not a Nash K-netwok. Recipocally, assume that g is minimally connected. Let i be any playe and gi 0 be any stategy gi 0 2 G i (K) (gi 0 6= g i ). We show that i (g) i (g i ; gi). 0 A new stategy gi 0 6= g i means deleting some links and initiating new ones. If g is minimally connected, then each deletion means disconnecting i with a set of nodes, and if thee is moe than one deletion any two of these sets of nodes disconnected fom i must also be disconnected fom each othe (othewise a deleted link would be edundant). Thus the numbe of links initiated should be at least equal to the numbe deleted, othewise the payo would decease. But then i s payo fo (g i ; gi) 0 cannot be geate than fo g. Theefoe if g is minimally connected no playe has an incentive to make any K-admissible change. In Bala and Goyal (2000a) the following esult is established (in ou teminology and unde the assumptions about costs and bene ts made hee 6 ): A netwok is e - cient if and only if it is minimally connected, and Nash netwoks ae those minimally connected. In view of this, we have the following Coollay 1 When the societal cove K is connected the following conditions ae equivalent fo a netwok g: (i) g is a Nash K-netwok. (ii) g is a K-consistent Nash netwok. (iii) g is an e cient K-netwok. 6 In fact, given thei weake assumptions on the payo s (see footnote 5), the empty netwok may also be Nash stable in thei setting. Published by Bekeley Electonic Pess Sevices,

11 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] H H H H H H A 1 A 2 A 1 A 2 (a) (b) Figue 1: Nash netwoks and Nash K-netwoks. Theefoe, fo any given set of nodes N and any societal cove K, the set of Nash K-netwoks is a subset of the set of standad unesticted Nash netwoks. In Figue 1 two minimally connected netwoks ae epesented 7 : (a) is a Nash K-netwok, while (b) is not a Nash K-netwok because one link connects two nodes that do not belong to the same society. We now focus on stict Nash K-netwoks. Stas of di eent types play an impotant ole in netwok stability in di eent contexts (see, Bala and Goyal (2000a, 2006), Jackson and Wolinsky (1996), Bloch and Dutta (2009)), and, as we show below, they ae also impotant in connection with stict Nash K-netwoks. In this context the following vaiant of the notion of cente-sponsoed sta poves useful. De nition 5 A set of playes M N (#M 2) is said to be connected by a centesponsoed sta s in a netwok g if g j M = s and thee is a node i 2 M s.t. N d (i; g) = Mni and g jk = 0 fo all j 2 Mni and all k 2 Mnj. Note that accoding to this de nition (i) a cente-sponsoed sta does not necessaily connect all playes in N; (ii) its cente i can be linked fom othe nodes di eent fom those in the sta; and (iii) the nodes in the peiphey, i.e., those j in M s.t. g ij = 1 can be connected with othe nodes that do not belong to the sta. Re-stated in tems of the cuent setting, notation and teminology, and adapted to it, Bala and Goyal (2000a) establish the following esult: The only stict Nash netwoks ae those consisting of a single cente-sponsoed sta that connects all playes 8. As we show below, the societal cove divesi es the stable/e cient netwoks as stict Nash K-netwoks ae not necessaily cente-sponsoed stas. A vaiety of constellations of linked stas emeges as possible stict Nash K-netwoks depending on the stuctue of the societal cove; moeove, in geneal, seveal achitectues appea as stict Nash fo a given societal cove. Ou next goal is to identify and chaacteize these netwoks. 7 As in all gues, nodes ae epesented by dots (without labels unless conveninet fo the pupose of the illustation), links by segments between them, and a lled cicle indicates the node that initiated it. 8 Given thei weake assumptions on the payo s (see footnote 5), the empty netwok may also be stict Nash in thei setting

12 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit In the chaacteization of stict Nash K-netwoks the following binay elation on N associated with a netwok g plays an impotant ole. Let g! be the tansitive closue of the binay elation L g de ned by i L g j, (i = j o g ij = 1) g That is to say, i! j if i = j o thee exists an i-oiented path fom i to j. This elation is obviously tansitive, but in geneal, fo an abitay netwok g, is not complete, antisymmetic o acyclic 9. But if g is minimally connected, then! g is cetainly antisymmetic and acyclic (othewise at least one link would be edundant). Thus, in view of Poposition 2, we have the following Lemma 1 Fo any Nash K-netwok g, the binay elation g! is a patial ode on N. Fo any Nash K-netwok g, we use the following teminology. We say that i is a pedecesso of j (o that j is a successo of i) in g if i 6= j and i g! j. We say that a node is teminal in g if it has no successos, and we say that a node is maximal in g if it has no pedecessos. The following theoem chaacteizes stict Nash K-netwoks by means of a condition that captues synthetically the essence of these netwoks, embodying an implicit fom of hieachical pinciple in thei achitectue: In such netwoks evey playe initiates links with evey node within his/he each unless it is connected (diectly o indiectly) with any of his/he pedecessos. Fomally, we have the following esult. Theoem 1 A netwok g is a stict Nash K-netwok if and only if g is a minimally connected K-netwok such that fo each node i, and all j 6= i within i s each (i.e., all j 2 N(K i )=i), g ij = 1 unless j is a pedecesso of i o thee exists a k pedecesso of i such that j 2 N(k; g). Poof. Necessity ()): Obviously, a K-netwok g that is a stict Nash K-netwok is also a Nash K-netwok, and by Poposition 2, necessaily minimally connected, and by g Lemma 1,! is a patial ode. Now let i be a node in g and assume g ij = 0; fo some j 2 N(K i )=i that is not a pedecesso of i and fo which thee is no k pedecesso of i such that j 2 N(k; g). As g is minimally connected, thee must be a path connecting i and j, that then does not contain any pedecesso of i. In paticula, on that path the st link must be a link initiated by i. But then i can delete that link and initiate a link with j without alteing i s payo, and consequently g is not a stict Nash K-netwok. Su ciency ((): Assume that g is a minimally connected K-netwok. By Poposition 2, g is a Nash K-netwok. Let i be any node and any gi 0 2 G i (K) s.t. gi 0 6= g i. We show that i (g) > i (g i ; gi) 0 if the condition in the theoem holds. Reasoning 9 A binay elation R on a set X is antisymmetic if, fo all x; y 2 X, xry and yrx, implies a = b; and R is said to be acyclic if thee is no nite chain x 1 ; x 2 ; ::; x n in X s.t. x k R k+1 fo k = 1; ::; n 1, and x 1 Rx n, unless x k = x k+1 fo k = 1; ::; n 1. Published by Bekeley Electonic Pess Sevices,

13 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] as in Poposition 2, as g is minimally connected, gi 0 6= g i involves deleting some links and initiating an at least equal numbe of new links fo (g i ; gi) 0 to be also minimally connected, othewise i s payo s would be smalle in (g i ; gi), 0 but in fact the numbe of links deleted and that of those newly initiated by i should be the same fo the same eason. Let link ii 0 be one of the fome (i.e., g ii 0 = 1 and gii 0 0 = 0) and let ij be one of the latte (i.e., g ij = 0 and gij 0 = 1). If the condition in the theoem holds, eithe j is a pedecesso of i in g o thee exists a k pedecesso of i in g such that j 2 N(k; g). But this implies a cycle in (g i ; gi). 0 The eason is this: Evidently adding link gij 0 = 1 to g means a cycle in g g ii ij, but it must be poved that this cycle is contained in (g i ; gi). 0 This is so because no link in the path in g connecting i and j can have been initiated by i (this would imply a cycle in g, which is assumed to be minimally connected). Theefoe, no matte what othe links in g i ae deleted in gi, 0 the cycle is entiely contained in (g i ; gi). 0 The same can be said about all new links in gi 0 w..t. g i, all new links ae edundant in (g i ; gi). 0 Theefoe necessaily i (g) > i (g i ; gi): 0 This chaacteization allows in paticula fo a constuctive poof of existence of stict Nash K-netwoks fo any societal cove K: Stat at any node i 0 and initiate links with all nodes in N(K i0 ), then extend the netwok by initiating new links fom those nodes, always especting the chaacteizing condition. In fact we have the following esult: Poposition 3 Fo any connected societal cove K and any node i 0 2 N thee exists an oiented tee g ooted at i 0 that is a stict Nash K-netwok. Poof. Iteate the following pocedue: - Step 0: Initially let i 0 be any playe in N, and g 0 the K-netwok that esults by i 0 initiating links with all playes in N(K i0 ). - Step fom k to k +1: If g k is the cuent K-netwok esulting fom step k, take any teminal node, say i k+1, in g k, fo which the set of nodes in N(K ik+1 )=i k+1 which ae not pedecessos of i k+1 in g k and thee is no l pedecesso of i k+1 such that i k+1 2 N(l; g k ) is not empty, and let i k+1 initiate links with all those playes. If no such node exists, stop; othewise, let g k+1 be the K-netwok that esults by adding to g k all these links initiated by i k+1. It is clea that if K is connected this iteated pocess must stop in a nite numbe of steps and the esulting netwok will be an oiented tee ooted in i 0 that foms a stict Nash K-netwok connecting all playes in N. If K wee not connected the same iteated pocedue could be applied within each component of the cove and by Poposition 2 a stict Nash K-netwok would esult. As a coollay of Theoem 1, the following popositions establish some pominent featues of the achitectue of stict Nash K-netwoks that help to fom a cleae idea about these netwoks, which we late illustate with some examples. The st shows the ole of stas in stict Nash K-netwoks. Poposition 4 In a stict Nash K-netwok g:

14 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit (i) Thee must be an i 2 N who is the cente of a cente-sponsoed sta that links with all playes in N(K i ); that is, s.t. N d (i; g) = N(K i )=i; and no othe playe in N initiated a link with i. (ii) Fo each society A 2 K, eithe no link connects two nodes of that society o all o some of the membes of that society ae connected by cente-sponsoed stas and no othe link exists connecting a pai of nodes in A. g Poof. (i) By Lemma 1, given that g is minimally connected,! is a patial ode and necessaily exists at least one maximal element, i.e., with no pedecesso. Let i 0 be a maximal element. As i 0 is maximal, by Theoem 1, necessaily N d (i 0 ; g) [ fi 0 g = N(K i0 ): (ii) Let A be a society in the cove K. Assume that fo some i; j 2 A, g ij = 1. It is enough to show that the only othe link that may exist connecting any k 2 Anfi; jg with i o j is a link initiated by i. Assume that g kj = 1. Then k can delete the link with j and initiate one with i and have the same payo. Assume that g jk = 1. Then i can delete the link with j and initiate one with k and have the same payo. Finally, assume that g ki = 1. Then k can delete the link with i and initiate one with j and have the same payo. Thus the only emaining possibility of a link connecting any k 2 Anfi; jg with i o j is a link g ik = 1. As an immediate coollay of pat (i), we have the following conclusion that yields Bala and Goyal s esult as a paticula case. Coollay 2 Thee exists a cente-sponsoed sta that is a stict Nash K-netwok if and only if the societal coe is not empty and the cente belongs to it. Obseve the similaity of the poof of pat (ii) with Bala and Goyal s poof of thei esult, and its di eences: Minimal connectedness and stict Nash-ness do not entail that all nodes ae connected by a single sta. Now the possibility of othe centesponsoed stas within a society is left open, and even the possibility of some nodes being left outside these stas (but linked though nodes belonging to societies othe than A). Thus we have in shot that in a stict Nash K-netwok g within each society eithe no pai of nodes is connected by a link o some cente-sponsoed stas connect some of the nodes in that society. But thee is at least one cente-sponsoed sta whose cente connects all nodes of all societies to which the cente belongs. The question now is: How do these stas inteconnect in g? Evidently though ovelapping societies. The following poposition answes this question moe pecisely by establishing the possible connections though ovelapping societies: Stas hand in hand, i.e., inteconnected though a fee-ide playe, ae possible only if a single playe belongs to both societies. Othewise, if moe than one playe belongs to both societies, a playe inteconnecting them necessaily initiates link(s) with playes of one o both societies. Poposition 5 Let A; B be two ovelapping societies in a societal cove K with i 2 A \ B, and g a stict Nash K-netwok. If fo some j 2 A n (A \ B) and k 2 B n (A \ B) it is _ g ij = _ g ik = 1, then g ji = g ki = 1 is possible only if A \ B = fig. Published by Bekeley Electonic Pess Sevices,

15 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] Poof. Assume that i 2 A\B and fo some j 2 An(A\B) and some k 2 B n(a\b), g ji = g ki = 1. If fig A \ B take i 0 2 A \ B, i 6= i 0. If i and i 0 wee linked then j (o k) could delete the link with i and initiate a link with i 0 without loss. Thus we should have g ii 0 = 0. As g is minimally connected eithe thee exists a path connecting i 0 and j and not containing k, o thee exists a path connecting i 0 and k and not containing j. In the st case k can delete the link with i and initiate a link with i 0, and in the second j can delete the link with i and initiate a link with i 0. In both cases this is without loss fo the playe deleting the link, theefoe poving that g is not a stict Nash K-netwok. The examples in Figue 2 illustate the chaacteization and its coollaies and convey the logic of stict Nash K-netwoks. Of couse, the chaacteizing condition holds in all cases, as the eade may check. Examples (a) and (b) epesent societal coves with a nonempty coe whee a cente-sponsoed sta is one of the possible achitectues of stict Nash K-netwoks: (d) and (c) epesent othe stict Nash K- netwoks fo the same coves. In examples (a), (b) and (d) a single cente-sponsoed sta coves (patially) each society, while two cente-sponsoed stas cove society A 3 in (c) and society A 5 in (e), and in both cases no othe link exists between pais of individuals. In all cases a maximal node exists (epesented by a white cicle ), but thee may exist moe than one, as in examples (e), (f) and (g), which illustate Poposition 5: Stas connecting hand in hand by means of a fee ide node ae possible when a single playe belongs to both societies. We have in fact the following conclusion: When no pai of societies in the societal cove K shae a single playe a stict Nash K-netwok is an oiented tee, as is poved by the following Theoem 2 Let K be a connected societal cove of N, then if fo all A; B 2 K, #(A \ B) 6= 1, then a stict Nash K-netwok is a K-netwok which necessaily foms an aboescence o oiented tee. Poof. Thee is a unique path connecting any maximal node with each node. Assume that thee ae two maximal nodes i 0 and i 1. Then thee is a path connecting i 0 and i 1, but then thee must exist thee nodes on that path i, j and k such that g ij = g kj = 1. Now if the intesection of any two societies in K is eithe empty o contains moe than a single playe, by Poposition 5, this is impossible. Theefoe thee can be only one maximal node connected with any othe node by a unique path and consequently g is an oiented tee. But note that, as examples (f) and (g) in Figue 2 show, when thee ae two o moe societies to which a single playe belongs it is possible to stat at di eent nodes at the same time, i.e., seveal maximal nodes may exist. In such cases an oiented tee does not esult. In this case two o moe gafted oiented tees may emege, so that any node is connected by an oiented tee with at least one but possibly moe maximal nodes. Finally, in the spiit of the community detection poblem (see, e.g., Jackson, 2009), we addess an issue ecipocal to that consideed so fa: Given a netwok g, can

16 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit H H b H H H H A 1 A 2 (a) A 1 @ b H B H BB B A 1 A H b H H H A 3 (b) H H H A 3 A 1 A 2 (c) (d) A 1 QQ b ( Q b Q (((( Q hhhh h Q Q Q Q A 2 A 3 A 4 A 5 (e) A 1 b H H A 2 b A 1 A 2 b H H b HH b HH A 3 A 3 (f) (g) Figue 2: Stict Nash K-netwoks. Published by Bekeley Electonic Pess Sevices,

17 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] it be intepeted as a stict Nash K-netwok fo any paticula societal cove K? Given the multiplicity of stict Nash K-netwoks fo a societal cove K, it is easy to see that this question admits many answes: In geneal, an oiented tee (o seveal gafted tees) can be seen as a stict Nash K-netwok fo di eent societal coves. Resticting attention to oiented tees, the following associated coves ae woth noting. Let g be an oiented tee ooted at i 0. The geneational cove, consisting of a minimal numbe of societies, each consisting of all nodes at the same distance fom the oot that ae not teminal along with thei o sping ; the family cove whee each node foms a society with its o sping; and the tivial binay cove whee any two diectly linked nodes fom a society. Fo all the thee societal coves the oiented tee g is a stict Nash K-netwok, and fo the latte two it is the only one with maximal node i 0. 4 Dynamics We now apply Bala and Goyal s (2000a) dynamic model adapted to this setting. Namely, stating fom any initial K-netwok g each playe i with some positive pobability esponds with a K-admissible best esponse 10 to g i o andomizes acoss them when thee is moe than one, othewise playe i exhibits inetia, i.e., keeps his/he links unchanged. In this way a Makov chain on the state space of all K-netwoks is de ned. In Bala and Goyal s setting, i.e., fo K = fng, the absobing states ae pecisely the stict Nash netwoks and they pove that stating fom any netwok the dynamic pocess conveges to a stict Nash netwok (i.e., the empty netwok o a cente-sponsoed sta) with pobability 1. The following example shows that this may not be the case fo the same dynamic model in the context of K-netwoks. Example: In Figue 3 (a) playes in A 1 have no best esponse but keep thei stategies, while playe 1 is indi eent between initiating a link with 2 o 3 o 4, and consequently the best esponse dynamic pocess would oscillate foeve. Similaly, in Figue 3 (b) all playes in A 1 and playes in A 3 keep thei stategies, while playe 1 is indi eent between initiating a link with 2 o 3, and consequently the best esponse dynamic pocess would oscillate foeve among these two netwoks. Note that in both examples the set of K-netwoks among which the best esponse dynamics oscillates ae minimally connected and yield the same payo s to all playes. The example shows an inteesting di eence with espect to Bala and Goyal s setting. It is not di cult to show that Bala and Goyal s best esponse dynamics lead to a Nash K-netwok (i.e., K-minimally connected) 11. The poblem appeas when one ties to extend the est of thei poof in seach of a stict Nash K-netwok. The eason is that in thei single-society setting, stating fom any netwok g that is not 10 Note that if g is a Nash K-netwok any stategy gi 0 of playe i such that i(g) = i (g i ; gi 0 ), is a best esponse to g i. 11 The poof is simila to that of Lemma 4.1 in Bala and Goyal (2000a), now just taking into account and especting K-feasibilty

18 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit H 4 3 A 1 A 2 A 1 A 2 A 3 (a) (b) Figue 3: Dynamic deadlock towads a stict Nash K-netwok. stict Nash one can show that with a positive pobability a stict Nash netwok may be eached by a nite sequence of best esponses. In ou moe complex setting, K- admissible best esponse dynamics lead to the fomation of achitectues consisting of inteconnected cente-sponsoed stas, but it might happen that these inteconnected cente-sponsoed stas that appea in di eent pats of the K-netwok as the dynamic pocess uns cannot be pat of the same stict Nash K-netwok, thus blocking the possibility of tansition to an absobing state. Note that in Bala and Goyal s setting with a single-society cove this last situation neve occus. Then, the same logic that in thei setting leads to the absobing stict Nash netwoks, in ous may lead to the fomation of inteconnected cente-sponsoed stas incompatible in any stict Nash K-netwok that block the conveging pocess. Nevetheless, we have a simila esult if we eplace stict Nash K-netwoks, not longe absobing states, by the following sets: De nition 6 Let K be a connected societal cove of N; a quasi stict Nash K-set is a set Q of minimally connected K-netwoks that is closed unde best esponse dynamics, and vei es full eachability by best esponse dynamics; and a quasi stict Nash K- netwok is a netwok that belongs to a quasi stict Nash K-set. By full eachability unde best esponse dynamics we mean that any netwok in one of these sets is eachable fom any othe in the same set by best esponse dynamics. Example (a) in Figue 3 shows a thee-element quasi stict Nash K-set, and example (b) a two-element quasi stict Nash K-set. Note that a stict Nash K-netwok is just a singleton quasi stict Nash K-set. We then have the following esult: Theoem 3 Stating fom any K-netwok g, best esponse dynamics each a quasi stict Nash K-set with pobability 1. Poof. Let BR be the binay elation in the set G K of all K-netwoks de ned by gbrg 0 if g 0 is one of the possible esults of a best esponse move fom g, and let BR be the tansitive closue of this elation, i.e., gbr g 0 if it is possible to each g 0 fom g by a nite sequence of best esponse moves. We now denote by H its associated equivalence elation, that is, gh g 0, gbr g 0 & g 0 BR g: Published by Bekeley Electonic Pess Sevices,

19 Fondazione Eni Enico Mattei Woking Papes, At. 516 [2010] This elation establishes a patition of the set of all K-netwoks into equivalence classes, so that in each class any netwok is eachable by a sequence of best esponse moves fom any othe in the same class. And note that a binay elation BR can be de ned on the quotient set G K =H (i.e., the set of equivalence classes): [g]br[g 0 ], gbr g 0. Note that BR is a well de ned binay elation that is e exive, antisymmetic and tansitive, i.e., BR is a patial ode in the quotient set. As the quotient set is nite, minimal classes must exist, i.e., classes [g] s.t. [g]br[g 0 ] implies [g] = [g 0 ]: Now note that the Makov chain on K-netwoks de ned by the best esponse dynamics can be extended to a Makov chain on G K =H : Fo any two di eent elated classes, i.e., [g]; [g 0 ] s.t. [g]br[g 0 ] thee is a positive pobability of eaching the latte fom the fome. Thus, thee is a positive pobability of eaching a minimal class whee the pocess will stay foeve. Now a minimal class is just a set of K-netwoks closed w..t. best esponse dynamics and such that any netwok in this set is eachable by a sequence of best esponse moves fom any othe in this set. Thus, the absobing sets of the best esponse dynamic pocess ae the quasi stict Nash K-sets, that have been intoduced in De nition 6 in best esponse dynamics tems, but it seems desiable a chaacteization of these sets, o of the quasi stict Nash K- netwoks that fom them, in tems independent of dynamics. With this pupose the following de nition is convenient. De nition 7 Let g be a minimally connected K-netwok, and let s g be a centesponsoed sta with cente i, then we say that s is K-ievesible if fo any othe centesponsoed sta s 0 g with cente j 6= i such that N(K i ) \ N(K j ) 6=?, if s 00 = s [ s 0 all nodes in N(s 00 ) \ (N(K i ) \ N(K j )) ae linked by i o all by j. That is, a cente-sponsoed sta s, that is pat of a minimally connected K-netwok is K-ievesible if fo any othe cente-sponsoed sta s 0 whose cente s each intesects with that of s, all nodes in s [ s 0 within the each of both centes ae linked fom eithe the cente of s o the cente of s 0. If this happens, spoke nodes have no best esponse and thee is no possibility of miscoodination between the centes. Then, the only feasible best esponses of the nodes (if at all any does exist) in s consists of the cente eplacing some spoke nodes with the same numbe of othe nodes within its each. In that case the cente of the sta cannot change, no the numbe of nodes that fom the sta, and we say that the sta is K-ievesible. Obseve that in Figue 3, both examples consist of minimally connected K-netwoks fomed by inteconnected K-ievesible cente-soponsoed stas. We have in fact the following chaacteization. Theoem 4 A K-netwok g is a quasi stict Nash K-netwok if and only if it is a minimally connected K-netwok consisting of inteconnected K-ievesible cente-sponsoed stas

20 Olaizola and Valenciano: Infomation, Stability and Dynamics in Netwoks unde Instit Poof. Necessity ()): By de nition, a quasi stict Nash K-netwok g is a minimally connected K-netwok. Now let g be a K-netwok not consisting of inteconnected cente-sponsoed stas. Then best esponse dynamics lead with pobability 1 to a K- netwok fomed by inteconnected cente-sponsoed stas 12. Now let g be a K-netwok consisting of inteconnected cente-sponsoed stas, and let s; s 0 g be two centesponsoed stas with centes i and j such that N(K i ) \ N(K j ) 6=? and not all nodes in N(s 00 ) \ (N(K i ) \ N(K j )) ae linked by i neithe by j, whee s 00 = s [ s 0. In that case both i and j have a best esponse among the K-feasible moves in s 00 and they can theefoe miscoodinate. Then thee is a positive pobability of tansition to a K-netwok whee eithe i o j initiate links with all nodes in N(s 00 )\(N(K i ) \ N(K j )). Applying this to evey such pais s and s 0 in sequence, the pocess leads to a K-netwok consisting of inteconnected cente-sponsoed stas that ae K-ievesible, eaching thus an equivalence class di eent fom [g], and consequently g is not a quasi stict Nash K-netwok. Su ciency ((): Assume that g is a minimally connected K-netwok consisting of inteconnected K-ievesible cente-sponsoed stas. Since fo evey cente-sponsoed stas s; s 0 g with centes i and j such that N(K i ) \ N(K j ) 6=?, all the nodes in N(s 00 ) \ (N(K i ) \ N(K j )), whee s 00 = s [ s 0, ae linked eithe fom i o j, the only K- feasible best esponse (if any does exist) that nodes in s 00 may have ae those whee just only one of the centes, say i, deletes some links with nodes in N(s 00 )\(N(K i ) \ N(K j )) and eplaces each of them by a link with anothe node in N(s 00 ) \ (N(K i ) \ N(K j )) o in some N(s 00 ) \ (N(K i ) \ N(K k )) ; whee k is the cente of a cente-sponsoed sta t g; while all othe nodes in s 00 have no best esponse, not even cente j. Since this happens fo evey such pais s and s 0, then miscoodination cannot occu and g is a quasi stict Nash K-netwok. As a coollay, we have the following esult that shows that when an absobing quasi stict Nash K-set is eached, in spite of the possibly pepetual oscillation, stability is essentially eached given that all netwoks in the same quasi stict Nash K-set yield the same payo s to all playes. Coollay 3 If Q is a quasi stict Nash K-set, fo all g; g 0 2 Q and all i 2 N, i (g) = i (g 0 ). Poof. Let Q be a quasi stict Nash K-set and g 2 Q. As g is a minimally connected K-netwok that consists of inteconnected K-ievesible cente-sponsoed stas, whose centes ae xed, any K-admissible best esponse move (if any does exist) can only involve some cente(s) changing some links keeping the netwok minimally connected. Theefoe the payo s must emain unchanged fo all playes. 12 The poof is simila to that of Theoem 4.1 in Bala and Goyal (2000a), just especting K-feasibilty. Published by Bekeley Electonic Pess Sevices,