Competitive Rumor Spread in Social Networks

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1 Compettve Rumor Spread n Socal Networks Yongwhan Lm Operatons Research Center, Massachusetts Insttute of Technology yongwhan@mt.edu Asuman Ozdaglar EECS, Massachusetts Insttute of Technology asuman@mt.edu Alexander Teytelboym Insttute for New Economc Thnkng at Oxford Martn School, Unversty of Oxford alexander.teytelboym@net.ox.ac.uk ABSTRACT We consder a settng n whch two frms compete to spread rumors n a socal network. Frms seed ther rumors smultaneously and rumors propagate accordng to the lnear threshold model. Consumers have (randomly drawn) heterogeneous thresholds for each product. Usng the concept of cascade centralty ntroduced by [6], we provde a sharp characterzaton of networks n whch games admt purestrategy Nash equlbra (PSNE). We provde tght bounds for the effcency of these equlbra and for the nequalty n frms equlbrum payoffs. When the network s a tree, the model s partcularly tractable. Keywords cascade centralty, compettve dffuson, rumor spread, threshold models. INTRODUCTION Rumors often spread quckly through socal networks and frms often compete to spread dfferent rumors. In our model an agent s decson to a spread a rumor s made accordng to the lnear threshold process [4] but the choce of rumor depends only on latest perod spreaders (as n the lnear swtchng functon of []). A good of example of ths process s Twtter whch dsplayed tweets to chronologcal order. When each frm (or poltcal party) has a unt budget.e. can only seed to exactly one ntal spreader, there may be no pure-strategy Nash equlbrum (PSNE) n general socal networks. Informally, a PSNE may fal to exst f the socal network s very symmetrc and has many cycles. However, usng cascade centralty [6], we are able to characterze PSNEs whenever they exst. Cascade centralty of an agent n a network measures the expected number of consumers who spread that rumor when the agent alone s the seed and there are no other competng rumors. In suffcently asymmetrc networks, there are certan agents those wth the hghest cascade centralty who are obvous canddates for seedng n a pure-strategy Nash equlbrum. Usng cascade centralty, we gve an explct characterzaton of PSNEs. In general, PSNEs are neffcent: a socal planner, who wants to maxmze the total number of spreaders of ether rumor Full paper avalable at: NetEcon 06 Juan-les-Pns, France Copyrght s held by author/owner(s). can outperform competng frms. We show that the prce of anarchy [5] the rato of the socally optmal number of spreaders to the number of spreaders n the worst PSNE s at most.5. The motvaton for studyng socal optmum s ths case mght be that havng no nformaton about an mportant ssue, such as an upcomng referendum, s worse from a socal pont of vew than spreadng the vew of one sde and therefore beng engaged n the poltcal process. The hghest rato of equlbrum payoffs, known as the budget multpler [], s at most. Moreover, we llustrate that both of these bounds are tght (mprovng the PoA bound of [3]). To llustrate the tractablty of our framework, we also consder the dffuson game n trees. We show that PSNEs always exst n trees and provde a complete characterzaton of ther structure and effcency propertes.. MODEL Our set up follows [3, ]. G(V, E) s a smple, undrected graph n whch nodes V represent agents and E lnks between them. Each agent n the graph s endowed wth a threshold drawn from U(0, ). There two frms, A and B, whch spread rumors a and b. An agent can be n one of three states: not spreadng a rumor, spreadng a or spreadng b. Frms seed agent smultaneously and rumors spread as n the lnear swtchng functon model of []. At tme t = 0, no agents are spreadng ether rumor and each frm smultaneously chooses one agent each as a seed for ts rumor. If frms seed the same agents, t pcks one of the rumors to spread at random. In subsequent steps, each agent checks f n the prevous perod the proporton of hs neghbours who are spreadng ether rumor s greater or equal to hs threshold. If not, he remans as a non-spreader. If hs threshold s exceeded, then the agent decdes to spread one of the rumors wth the probablty that s equal to the fracton of hs neghbors who started spreadng that rumor n the prevous perod. Eventually, the rumor spreadng cascades must stop and we can check all agents states. The frms do not know the partcular draw of agents thresholds ex ante. Defnton. The cascade centralty of node V n graph G s defned as: C (G) = + P P V \{} where s the product of the degrees of nodes (except ) along some smple path P that begns at and end at and P s the set of all smple paths that begn at and end at

2 . Snce the reference to G s usually unambguous, we refer to C (G) as C and d (G) as d. For, let us denote Ξ(, ) as the set of all paths that begn at and nclude (but do not necessarly end at). ɛ(, ) = P Ξ(,) ɛ(, ) descrbe the extent to whch node nterferes wth the cascade emanatng from node. Note that ɛ(, ) ɛ(, ) unless d = d, so a node wth a hgher degree nterferes more wth a node of a lower degree than vce versa.. Game and equlbrum The acton space of frms A and B s hence denoted Σ := Σ A Σ B := V V ; acton profle σ := (σ A, σ B). Acton profles are therefore smply (ordered) pars of nodes (agents) n the graph. A frm s payoff s smply the expected number of spreaders of ts rumor gven the acton of the other frm. For a payoff profle π := (π A(σ), π B(σ)), we can defne the game Γ := (Σ, π). Defne P (, ) P(, ) as the set of all paths from to that nclude at most one of the seeds along the path. Proposton. Consder a duopoly wth unt budgets Γ. The expected number of spreaders of rumor a (.e. frm A s payoff) s where π A(σ A, σ B) := ι = V ι P P (σ A,) { / f σ A = σ B otherwse Alternatvely, we can express frm A s payoff as π A(σ A, σ B) = { C σa / C σa ɛ(σ A, σ B) f σ A = σ B otherwse We want to focus on the pure-strategy Nash equlbra of ths game. Defnton. A profle of actons σ := (σ A, σ B) Σ s a pure-strategy Nash equlbrum of Γ f: π A(σ A, σ B) π A(σ A, σ B) for all actons σ A Σ A π B(σ A, σ B) π B(σ A, σ B) for all actons σ B Σ B Defne Σ as the set of all pure-strategy Nash equlbra of the game. 3. RESULTS In general, a game Γ does not admt a PSNE. Fgure gves an example of such a network and ts cycle best responses. We refer to smple paths as paths throughout. Snce the number of frms and the strategy spaces are fnte, a mxed-strategy Nash equlbrum always exsts. 3. Exstence and characterzaton of PSNE We now characterze both types of PSNE: n whch frms seed the same agent and n whch frms seed dfferent agents. Theorem. Consder a duopoly wth unt budgets Γ. Then Γ admts at least one PSNE f and only f at least one of the followng condtons on G s satsfed:. There exsts V such that, for any V \ {}: C ( ) ɛ(, ). There exst, V such that, C and for any k V \ {, }: C ɛ(, ) ɛ(k, ) + C k C k C ɛ(, ) ɛ(k, ) + C k C k + ɛ(, ) C ( ) ɛ(, ) If Condton s satsfed then there exsts a σ = (, ) PSNE and f Condton s satsfed, then there exsts a σ = (, ) (and σ = (, ) by symmetry) PSNE. Although the condtons set out n Theorem appear nvolved, they are rather ntutve. 3 Let us frst focus on Condton. When frms seed the same agent, ther payoff s 0 wth probablty and C wth probablty. To ensure that t s worthwhle for both frms to take that gamble, the cascade centralty of the node has to be suffcently hgh. Indeed, t s suffcent that the cascade centralty of the agent wth the second hghest cascade centralty s at most half of C as the followng corollary summarzes. Corollary. σ = (, ) s a PSNE of Γ f C any. for Condton s necessary and suffcent for the exstence of a σ = (, ) PSNE wth two dfferent seeds. The frst two parts of the condton say that the cascade centraltes of and need to suffcently hgh compared to other agents n the network. The fnal condton ensures that whle the cascade centralty of s suffcently greater than s, t s not so hgh that t ncentvzes agents to move nto the σ = (, ) equlbrum. In fact, we can summarze ths ntuton wth a clean suffcent condton. Corollary. σ = (, ) (and σ = (, ) by symmetry) s a PSNE of Γ f for any k, { } C C max, C k + ɛ(, ) { } C max, C k + ɛ(, ) 3 Note that there s a knfe-edge case n whch there are both types of equlbra.

3 Note the role that ɛ the degree of nterference between the agents plays. ɛ s small when the agents are far from each other - n terms of the lengths and numbers of paths and the degrees of agents along these paths. If nterference s small, then a suffcent condton for a PSNE wth dfferent seeds s that the cascade centralty of the agent wth the hghest cascade centralty n the network s no more than twce the cascade centralty of the second-hghest agent. Fnally, Theorem sheds lght on why there s no PSNE n the socal network n Fgure. Condton clearly cannot be satsfed: there was no agent that has a far hgher centralty than others. But Condton could not be satsfed because there was a great deal of nterference between the agents. There are several paths between and and C k s very close to C and makng t dffcult to satsfy any of the three parts of Condton. 3. Effcency of equlbra: prce of anarchy We now to turn to the effcency propertes of the PSNEs descrbed above. It s easy to fnd an example of a socal network n whch none of the PSNEs s effcent (see Fgure 4). We wll focus on the prce of anarchy, ntroduced by [5], whch measures the rato of the number of spreaders n the socally optmal outcome to the lowest expected number of spreaders among all Nash equlbra. The socal planner s obectve s to maxmze the sum of frms payoffs. Let Y (σ) := π A(σ) + π B(σ) be ths obectve. Defnton 3. Prce of Anarchy of Γ s defned as: PoA (Γ) = maxσ Σ Y (σ) mn σ Σ Y (σ) The followng theorem puts a bound on how bad equlbrum outcomes can be. Theorem. Consder a duopoly wth unt budgets Γ. For any Γ that admts at least one PSNE, PoA (Γ) <.5. Ths (upper) bound s also tght, whch s shown n the lower network n Fgure Inequalty of payoffs: budget multpler We consder the noton of the budget multpler ntroduced by []. Ths metrc measures the extent to whch the network amplfes an asymmetry n budgets nto an asymmetry n payoffs. Even though the budgets are equal n our model, equlbrum payoffs are not equal n general. Defnton 4. For arbtrary nteger budgets B A and B B, the budget multpler of game Γ s defned as: BM(Γ)= max σ Σ π A(σ)/π B(σ) B A/B B So far, we have only analyzed the case of unt budgets, where B A = B B =. Hence, the budget multpler n our π A(σ) case s smply max σ Σ π. B(σ) Theorem 3. For any Γ that admts at least one PSNE, BM(Γ) < Ths (upper) bound s also tght as shown n the upper network n Fgure. 3.4 Competton on trees If G s a tree, we can wrte P(, ) = P (, ). Denote by (G) ( (G)) the degree of the agent wth the (weakly second-) hghest degree n the network. Cascade centralty of any node n a tree s ts degree plus [6]. Defnton 5. Canddate sets of strategy profles are defned as: σ 0 = {(, ) d = }. σ = {(, ) (d, d ) {(, ), (, )}}. σ = {(, ) (d, d ) {(, ), (, ), (, ), (, )}}. For each set of strategy profles ndexed by l {, } we can defne the strategy profle that maxmzes the degree sequence product among all strategy profles n σ l as σl = arg max (,). The value of ths maxmum degree (,) σ l sequence s denoted δ l = max (,). (,) σ l Proposton. Suppose G s a tree and consder a duopoly wth unt budgets Γ. Then, Γ admts a PSNE σ, whch s characterzed as follows: σ f < If δ =, σ = σ 0 else f > σ 0 σ otherwse σ f / < Else f δ =, σ = σ 0 else f / > σ 0 σ otherwse { Otherwse, σ σ f / = σ 0 otherwse In our case, we can also solve the socal planner s problem on trees explctly. Denote a soluton by σ Y arg max σ Σ{Y (σ)}. Proposton 3. Suppose G s a tree. Then, f G s a star, σ Y = σ 0 σ. Otherwse, σ Y = { σ f δ =, δ >, σ otherwse σ Corollary 3. Suppose G s a tree. Then, there are no effcent PSNEs ff (σ = σ 0) (σ Y = σ ) are no neffcent PSNEs ff (σ = σ ) (σ Y = σ ) s an neffcent PSNE ff [(σ σ 0) (σ Y = σ)] (σ W = σ) s an effcent PSNE ff (σ σ ) (σ Y = σ ) s an effcent and an neffcent PSNE ff (σ = σ 0 σ ) (σ Y = σ ) Fgure 4 shows an example of a network wth both types of PSNEs: (, ), (.k) and (, ) (as well as symmetrc counterparts of the latter two) because the largest degree (vz. 3) s exactly equal to twce the second-largest (vz. ) mnus one. None of the PSNEs s effcent.

4 4. REFERENCES [] D. Fotaks, T. Lykours, E. Markaks, and S. Obraztsova. Influence maxmzaton n swtchng-selecton threshold models. In R. Lav, edtor, Symposum on Algorthmc Game Theory SAGT, 04. [] S. Goyal and M. Kearns. Compettve contagon n networks. In Proceedngs of the forty-fourth annual ACM symposum on Theory of computng, pages , 0. [3] X. He and D. Kempe. Prce of Anarchy for the N-player Compettve Cascade Game wth Submodular Actvaton Functons. In WINE, Lecture Notes n Computer Scence, 03. [4] D. Kempe, J. Klenberg, and É. Tardos. Maxmzng the spread of nfluence through a socal network. In SIGKDD 03 Proceedngs of the nnth ACM SIGKDD Internatonal Conference on Knowledge Dscovery and Data Mnng, pages 37 46, 003. [5] E. Koutsoupas and C. Papadmtrou. Worst-case equlbra. In STACS 99, pages Sprnger, 999. [6] Y. Lm, A. Ozdaglar, and A. Teytelboym. A smple model of cascades n networks. Techncal report, LIDS, 05. Fgure : Socal network sequence that reaches the hghest budget multpler. Fgure 3: Socal network sequence that reaches the hghest prce of anarchy. APPENDIX A. FIGURES In the graph on Fgure, the agent wth the hghest cascade centralty s. The pure-strategy best response to a frm s seedng s to seed k. But the best response to k s and, n turn, the best response to s to seed. Therefore, the only equlbra n ths game are mxed-strategy. Fgure : Example of a socal network for a game that does not admt a PSNE k that nvolves gettng a zero payoff. However, the socal planner would prefer to seed and n both cases. It s easy to show that the expected number of spreaders n equlbrum s 5, whereas the socal optmum approaches 7.5 as the length of the branch goes to nfnty. Fgure 4: effcent A tree n whch none of the PSNEs s In both graphs on Fgure, (, ) s a unque PSNE (of course, (, ) s also a PSNE by symmetry). Suppose that we create a sequence of graphs, such that at every step of the sequence we ncrease the degree of and by d and d respectvely. Ths means that along the sequence the rato d /d remans constant at and the equlbrum remans unque (up to a symmetrcal transformaton). However, the lmt payoff of the frm seedng at s d, whch s twce the lmt payoff d of the frm seedng at, as the degrees of these nodes ncrease along the sequence. In both graphs on Fgure 3, the unque PSNE s one n whch both frms seed agent. Because the cascade centralty of s so hgh, both frms are happy to take the gamble k

5 B. PROOFS Proof of Proposton. When σ A = σ B, snce the te s broken wth an equal probablty for each frm, the problem s dentcal a sngle-seed case wth a seedng probablty of a half (see [?] for detals). Hence, the payoff π A takes the prescrbed form. Suppose there are two seeds,. The probablty that node k V \ {, } spreads the rumor s + P P (,k) P P (,k) The probablty that k s nfluenced by a lve-edge path (see [4] for defnton) from seed (of frm A) s P P (,k) But P(, k) = P( va k) P (, k) (where, P (, k) s defned as all paths from to k not va the other seed as n the man text), so probablty that k s nfluenced by can be wrtten as + P P( va k) P P (,k) Summng over all k V, gves us C = + + k V k V Therefore, + k V where + k V k V P P( va k) P P (,k) P P( va k) P P (,k) = C k V P P (,k) P P( va k) π A(, ) and ɛ(, ) gvng us the desred expresson for the expected number of agents affected by seed of frm A. The expresson for seed (of frm B) s analogous. Proof of Theorem. Fx a duopoly wth unt budgets Γ. Then, there are two types of PSNE: Type : (, ) for some V. Type : (, ) for some V. By Defnton, for some V, (, ) s a type PSNE f and only f π A(, ) π A(, ) for all V, π B(, ) π B(, ) for all V. By Proposton, these condtons hold f and only f C ɛ(, ), whch s precsely the condton upon re-wrtng. Now, for some V, (, ) s a type PSNE f and only f π A(, ) π A(k, ) for all k V, π B(, ) π B(, k) for all k V. By Proposton, these condtons hold f and only f C ɛ(, ) C k ɛ(k, ) for all k, and C ɛ(, ) /, ɛ(, ) C k ɛ(k, ) for all k, and ɛ(, ) C /,. whch s precsely the condton upon re-wrtng. Proof of Corollary. We apply a condton from Theorem. Suppose C / for any. Then, C / ɛ(, )/ snce 0 ɛ(, ). Proof of Corollary. We apply a condton from Theorem. Suppose C max{/, C k } + ɛ(, ). Then, C C k + ɛ(, ) C k + ɛ(, ) ɛ(k, ). Ths s the frst condton upon dvdng by C k. The second condton can be treated smlarly. Also, C max{/, C k }+ɛ(, ) forces C /+ɛ(, ), whch s the frst half of the thrd condton. The second half s analogous. Proof of Theorem. Lemma. Suppose (, ) s a PSNE where V. Then, Y (, ) C + C. Proof of Lemma. Snce (, ) s a PSNE, n partcular, π A(, ) π A(, ) = C and π B(, ) π B(, ) = π A(, ) = C. Hence, Y (, ) = π A(, ) + π B(, ) (C + C). Lemma. For any, ɛ(, ) C. Proof of Lemma. Let δ(, ) = p Λ(,) where Λ(, ) s a set of all path startng from that excludes. Then, = ɛ(, ) + δ(, ). So, the statement amounts to showng ɛ(, ) δ(, ). Now, for each path p Ξ(, ), we may decompose t as q r where q = (q 0,, q u) Λ(, ) s a path wth q for all {0,, u} and r = (r 0,, r v) s a path wth r 0 N. Therefore, the statement follows from the followng estmate: ɛ(, ) = p Ξ(,) q Λ(,) p Λ(,) p Λ(,) = χ q d p=q r Ξ(,) C d χ r r where r 0 N {q u} (d + ) = d p Λ(,) Corollary 4. For any, V, π A(, ) C /. = δ(, ). Proof of Corollary 4. Ths follows from Lemma coupled wth the formula for the payoff.

6 Lemma 3. Suppose (, ) s a socal optmum and,, and k are parwse dstnct. Then, ɛ(, k) ɛ(, ) + ɛ(, k) ɛ(, ) < C k. Proof of Lemma 3. For a fxed n, the graph confguraton that mnmzes C k (ɛ(, k) ɛ(, ) + ɛ(, k) ɛ(, )) s a path of length n. It s easy to check for a path. Let σ be a PSNE, σ Y be a socal optmum, and ρ be the rato Y (σ Y )/Y (σ ). For any choce of σ and σ Y, t suffces to show ρ <.5. Suppose frst that σ = (, ). There are few cases to consder dependng on σ Y. Indces,, k, and l are assumed to be parwse dstnct.. σ Y = (, ). Then, ρ = <.5.. σ Y = (, ). Then, snce (, ) s a PSNE, π A(, ) < C and C / = π B(, ) π B(, ). Hence, ρ < (C + C /)/C = σ Y = (, ). Then, snce (, ) s a PSNE, by Lemma, Hence, C / = π A(, ) π A(, ) = ɛ(, ) /. ρ = C C < σ Y = (, k). Then, snce (, ) s a PSNE, for any l, π A(l, ) π A(, ) = C /. So, C l C / + ɛ(l, ). Hence, π A(, k) = ɛ(, k) C + ɛ(, ) ɛ(, k). Smlarly, π A(k, ) = C k ɛ(k, ) C + ɛ(k, ) ɛ(k, ). Usng Lemma 3, π A(, k) + π A(k, ) Therefore, ρ = πa(, k) + πa(k, ) π A(, ) + π A(, ) <.5C C =.5 ( ) C + ɛ(, ) ɛ(, k) + ( ) C + ɛ(k, ) ɛ(k, ) C + (ɛ(, ) ɛ(, k) + ɛ(k, ) ɛ(k, )) < C + C =.5C = πa(, k) + πa(k, ) C Suppose now σ = (, ). Ths tme the only hard case s σ Y = (k, l); the four other cases can be treated usng the smlar argument as before. Snce (, ) s a PSNE, for any t, π A(t, ) π A(, ). In partcular, C k π A(, ) + ɛ(k, ). Hence, π A(k, l) = C k ɛ(k, l) π A(, ) + ɛ(k, ) ɛ(k, l). Smlarly, π A(l, k) = C l ɛ(l, k) π A(, ) + ɛ(l, ) ɛ(l, k). Usng Lemma 3, π A(k, l) + π A(l, k) (π A(, ) + ɛ(k, ) ɛ(k, l)) Smlarly, + (π A(, ) + ɛ(l, ) ɛ(l, k)) (π A(, ) + π A(, )) + (ɛ(k, ) ɛ(k, l) + ɛ(l, ) ɛ(l, k)) < (π A(, ) + π A(, )) + C π A(k, l) + π A(l, k) < (π A(, ) + π A(, )) + C. So, t follows that π A(k, l) + π A(l, k) < (π A(, ) + π A(, )) + (C + C). 4 Therefore, by Lemma, ρ = = πa(k, l) + πa(l, k) (πa(, ) + πa(, )) + (C + C)/4 < π A(, ) + π A(, ) π A(, ) + π A(, ) ( ) πa(, ) + π A(, ) + ( ) π A(, ) + π A(, ) (C + )/ π A(, ) + π A(, ) + =.5 Also, the bound s sharp snce.5 bound can be acheved usng the sequence of graphs shown n Fgure 3. Proof of Theorem 3. Suppose frst that (, ) s a PSNE. Then, the budget multpler s. So, suppose (, ) s a PSNE wth <. Then, π A(, ) π A(, ) = / and π A(, ) π A(, ) = C /. Hence, So, / π A(, ) < C and C / π A(, ) <. π A(, )/π A(, ) < and π A(, )/π A(, ) <. Therefore, the budget multpler s ( ) πa(, ) πa(, ) max, <. π A(, ) π A(, ) Also, the bound s sharp snce.5 bound can be acheved usng the sequence of graphs shown n Fgure. Proof of Proposton. Suppose (, ) s a PSNE. Then, max(d, d ) =. Suppose d, d <. Then, there must be k wth d k = and π A(k, ) > π A(, ) snce π A(, ) = d + ( χ P (,)) ( ) + ( χ P (,) () ) = χ P (,) < d k + ( χ P (k,) ) () = π A(k, ). (3) Also, mn(d, d ). Wthout loss of generalty, suppose d < and d =. Then, usng the smlar argument to Equaton 3, there s k wth d k = and π A(k, ) > π A(, ). Therefore, (, ) σ 0 σ. Now, pck an element from σ 0 and σ. Suppose that the chosen elements are (, ) σ 0 and (, ) σ (wth relabelng f necessary); for (, ), suppose further that (,) = max (,) wth d = and d =. (,) σ

7 Lemma 4. Suppose. Then, (a) π A(, ) π A(, ) (b) and π B(, ) π B(, ). Proof of Lemma 4. For (a), the statement s trval f =. If then d 3. If 3 = then the choce of (, ) forces the nequalty. Otherwse, the nequalty follows from the smlar argument to Equaton (3). For (b), the statement s trval f =. If = then, = ( χ P (,) ) + π B(, ) = + χ P (,) + = π B(, ) Otherwse, d 3. So, the nequalty follows from the smlar argument that we used to prove (a). By Lemma 4, all PSNE s nvolve frms seedng at hghest and second-hghest nodes; that s, t suffces to compare π A(, ) and π A(, ) to classfy all PSNE s, where π A(, ) = + and π A(, ) = + χ P (,). If π A(, ) = π A(, ) then both (, ) and (, ) are PSNE s. Else f π A(, ) > π A(, ) then (, ) s a PSNE. Othewse, (, ) s a PSNE. When (,) =, we need to compare: or, equvalently, + and, and. If >, (, ) s a PSNE. If < then (, ) s a PSNE. If = both (, ) and (, ) are PSNE s. Now, when (,) =, we need to compare: or, equvalently, + and +, and. Fnally, when (,) >, we need to compare: + or, equvalently, and + χ P (,) and + δ where δ := χ P (,). Note that δ (0, ). In partcular, cannot be equal to + δ snce, N. Fnally, note that > +δ f and only f + snce, N. Smlarly, < + δ f and only f. Proof of Proposton 3. Suppose x σ. Then, wth relabelng f necessary, x s (, ) where d = and d =. Now, pck y = (, ) where, V and d d. Let x = + and y = d + d. Suppose Y (y) > Y (x). Then,. If = then Y (y) = d +, whch forces Y (x) = + + ( χ P (,)) + d + d + = Y (y). Suppose Y (y) = Y (x). If = then + + ( χ P (,)) = Y (x) = Y (y) = d + Snce d + N, Y (x) s ether x + or x because χ P (,) (0, ). Suppose frst that Y (x) = x +. Then, + = d forces. Snce, =. Thus, d = and = ; that s, the network has to be a star. On a star, σ Y = σ 0 σ. Now, suppose Y (x) = x. Then, + = d. Hence, 0. But, snce, ths s clearly a contradcton. Note that x y 0. Frst, suppose x y. Then, Y (y) = d + d + ( χ P (,)) < d + d + ( χ P (,) ) = Y (x) f and only f whch holds snce χ P (,) χ P (,) < x y, χ P (,) χ P (,) < χ P (,) x y. Now, suppose x y = 0. Then, necessarly d = and d =. Hence, Y (y) Y (x) by the choce of x. Fnally, for the case x y =, frst suppose / N. Then, (,), or χ P (,) /. Hence, χ P (,) χ P (,) < χ P (,) x y. Rearrangng ths nequalty yelds Y (x) = x + ( χ P (,)) > y + ( χ P (,) ) = Y (y). Fnally, suppose N. Then, Y (y) > Y (x) y + ( χ P (,) ) > x χ P (,) < ( x y) = (,) > Hence, f δ =, δ >, and σ then σ Y = σ. Otherwse, σ Y = σ unless G s a star; for a star, σ Y = σ 0 σ. Proof of Corollary 3. It suffces to check explct characterzatons of the equlbra and socal optma from Proposton and Proposton 3. Proof of Fgure. π A(, ) can be computed as: In partcular, we note that π(7, ) and π(4, ) are dfferent. We show ths by an explct calculaton usng countng formula. We assume that the seed of frm B s always fxed at. Let s perform a DFS (depth-frst search) from a seed

8 Fgure 5: Example of a socal network that does not admt a PSNE /5 /5 /30 3 /0 / /5 / /5 /5 /45 /90 3 /30 /90 6 / / /6 /30 /90 /80 3 /60 / /3 /6 8 9 Table : π(7, ) 4 3 /3 /6 3 / / /3 /6 /30 /90 / /5 / /45 / /5 /5 / /0 / /5 /30 Table : π(4, ) of frm A n a lexcographc order. There are 0 dstnct paths n the graph emanatng from 4 that avods as shown n Table. It s mmedate from here that: π(4, ) = = There are 3 dstnct paths n the graph emanatng from 7 that avods as shown n Table. It s mmedate from here that: π(4, ) = = 4 ( = 05 ) So, the dfference of the payoffs s precsely: π(7, ) π(4, ) = = 30 Now, t s clear from here that when 7 s the best response for, 4 s for, and s for 4 by symmetry. Therefore, we conclude that there cannot be a PSNE n Fgure. Proof of Fgure 4. Here are all PSNE s: σ = (, ) and σ = (, ) wth Y (σ ) = 5. σ = (, k); σ = (k, ) wth Y (σ ) = 5. σ 3 = (, ) wth Y (σ 3) = 4. Here s the unque socal optmal soluton: σ Y = (, k) and σ Y = (k, ) wth Y (σy ) = 6/3. In partcular, note that none of the PSNE s s effcent snce Y (σ ) < Y (σ Y ) for any vald par (, ).