On the Windfall and Price of Friendship: Inoculation Strategies on Social Networks
Dominic Meier, Yvonne Anne Pignolet, Stefan Schmid, Roger Wattenhofer
OOn the Windfall and Price of Friendship:Inoculation Strategies on Social Networks
Dominic Meier , Yvonne Anne Pignolet ,Stefan Schmid , Roger Wattenhofer ETH Zurich, Switzerland ABB Research, Switzerland T-Labs / TU Berlin, Germany
Abstract
This article investigates selfish behavior in games where players are embeddedin a social context. A framework is presented which allows us to measure the
Windfall of Friendship , i.e., how much players benefit (compared to purelyselfish environments) if they care about the welfare of their friends in thesocial network graph. As a case study, a virus inoculation game is examined.We analyze the corresponding Nash equilibria and show that the Windfallof Friendship can never be negative. However, we find that if the valuationof a friend is independent of the total number of friends, the social welfaremay not increase monotonically with the extent to which players care foreach other; intriguingly, in the corresponding scenario where the relativeimportance of a friend declines, the Windfall is monotonic again. This articlealso studies convergence of best-response sequences. It turns out that insocial networks, convergence times are typically higher and hence constitutea price of friendship. While such phenomena may be known on an anecdotallevel, our framework allows us to quantify these effects analytically. Ourformal insights on the worst case equilibria are complemented by simulationsshedding light onto the structure of other equilibria.
Keywords:
Game Theory, Social Networks, Equilibria, Virus Propagation,Windfall of Friendship
Preprint submitted to arXiv October 24, 2018 a r X i v : . [ c s . G T ] J a n . Introduction Social networks have existed for thousands of years, but it was not untilrecently that researchers have started to gain scientific insights into phenom-ena like the small world property . The rise of the Internet has enabled peopleto connect with each other in new ways and to find friends sharing the sameinterests from all over the planet. A social network on the Internet can mani-fest itself in various forms. For instance, on
Facebook , people maintain virtualreferences to their friends. The contacts stored on mobile phones or emailclients form a social network as well. The analysis of such networks—boththeir static properties as well as their evolution over time—is an interestingendeavor, as it reveals many aspects of our society in general.A classic tool to model human behavior is game theory . It has been afruitful research field in economics and sociology for many years. Recently,computer scientists have started to use game theory methods to shed lightonto the complexities of today’s highly decentralized networks. Game the-oretic models traditionally assume that people act autonomously and aresteered by the desire to maximize their benefits (or utility). Under this as-sumption, it is possible to quantify the performance loss of a distributedsystem compared to situations where all participants collaborate perfectly.A widely studied measure which captures this loss of social welfare is the
Price of Anarchy (PoA). Even though these concepts can lead to importantinsights in many environments, we believe that in some situations, the un-derlying assumptions do not reflect reality well enough. One such exampleare social networks: most likely people act less selfishly towards their friendsthan towards complete strangers. Such altruistic behavior is typically notconsidered in game-theoretic models.In this article, we propose a game theoretic framework for social networks.Social networks are not only attractive to their participants, e.g., it is well-known that the user profiles are an interesting data source for the PR industryto provide tailored advertisements. Moreover, social network graphs can alsobe exploited for attacks, e.g., email viruses using the users’ address booksfor propagating, worms spreading on mobile phone networks and over theInternet telephony tool Skype have been reported (e.g., [12]). This articleinvestigates rational inoculation strategies against such viruses from our gametheoretic perspective, and studies the propagation of such viruses on thesocial network. 2 .1. Our Contribution
This article makes a first step to combine two active threads of research:social networks and game theory. We introduce a framework taking intoconsideration that people may care about the well-being of their friends.In particular, we define the
Windfall of Friendship (WoF) which capturesto what extent the social welfare improves in social networks compared topurely selfish systems.In order to demonstrate our framework, as a case study, we provide agame-theoretic analysis of a virus inoculation game . Concretely, we assumethat the players have the choice between inoculating by buying anti-virussoftware and risking infection. As expected, our analysis reveals that theplayers in this game always benefit from caring about the other participantsin the social network rather than being selfish. Intriguingly, however, wefind that the Windfall of Friendship may not increase monotonically withstronger relationships. Despite the phenomenon being an “ever-green” inpolitical debates, to the best of our knowledge, this is the first article toquantify this effect formally.This article derives upper and lower bounds on the Windfall of Friendshipin simple graphs. For example, we show that the Windfall of Friendshipin a complete graph is at most 4 /
3; this is tight in the sense that thereare problem instances where the situation can indeed improve this much.Moreover, we show that in star graphs, friendship can help to eliminateundesirable equilibria. Generally, we discover that even in simple graphs theWindfall of Friendship can attain a large spectrum of values, from constantratios up to Θ( n ), n being the network size, which is asymptotically maximalfor general graphs.Also an alternative friendship model is discussed in this article where therelative importance of an individual friend declines with a larger number offriends. While the Windfall of Friendship is still positive, we show that thenon-monotonicity result is no longer applicable. Moreover, it is proved that inboth models, computing the best and the worst friendship Nash equilibriumis N P -hard.The paper also initiates the discussion of implications on convergence. Wegive a potential function argument to show convergence of best-response se-quences in various models and for simple, cyclic graphs. Moreover, we reporton our simulations which indicate that the convergence times are typicallyhigher in social contexts, and hence constitute a certain price of friendship.3inally, to complement our formal analysis of the worst equilibria, simu-lation results for average case equilibria are discussed.
The remainder of this article is organized as follows. Section 2 reviewsrelated work and Section 3 formally introduces our model and framework.The Windfall of Friendship on general graphs and on special graphs is studiedin Sections 4 and 5 respectively. Section 6 discusses an alternative modelwhere the relative importance of a friend declines if the total number offriends increases. Aspects of best-response convergence and implications areconsidered in Section 7. We report on simulations in Section 8. Finally, weconclude the article in Section 9.
2. Related Work
Social networks are a fascinating topic not only in social sciences, butalso in ethnology, and psychology. The advent of social networks on theInternet, e.g.,
Facebook , LinkedIn , MySpace , Orkut , or
Xing , to name but afew, heralded a new kind of social interactions, and the mere scale of onlinenetworks and the vast amount of data constitute an unprecedented treasurefor scientific studies. The topological structure of these networks and thedynamics of the user behavior has a mathematical and algorithmic dimension,and has raised the interest of mathematicians and engineers accordingly.The famous small world experiment [29] conducted by Stanley Milgram1967 has gained attention by the algorithm community [21] and inspiredresearch on topics such as decentralized search algorithms [22, 27], routingon social networks [13, 21, 26] and the identification of communities [11, 33].The dynamics of epidemic propagation of information or diseases has beenstudied from an algorithmic perspective as well [23, 25]. Knowledge on effectsof this cascading behavior is useful to understand phenomena as diverse asword-of-mouth effects, the diffusion of innovation, the emergence of bubblesin a financial market, or the rise of a political candidate. It can also help toidentify sets of influential players in networks where marketing is particularlyefficient ( viral marketing ). For a good overview on economic aspects of socialnetworks, we refer the reader to [6], which, i.a., compares random graphtheory with game theoretic models for the formation of social networks.Recently, game theory has also received much attention by computerscientists. This is partly due to the various actors and stake-holders who4nfluence the decentralized growth of the Internet: game theory is a usefultool to gain insights into the Internet’s socio-economic complexity. Manyaspects have been studied from a game-theoretic point of view, e.g., routing [35, 36], multicast transmissions [10], or network creation [9, 31]. Moreover,computer scientists are interested in the algorithmic problems offered bygame theory, e.g., on the existence of pure equilibria [34].This article applies game theory to social networks where players are notcompletely selfish and autonomous but have friends about whose well-beingthey care to some extent. We demonstrate our mathematical frameworkwith a virus inoculation game on social graphs. There is a large body ofliterature on the propagation of viruses [4, 14, 19, 20, 37]. Miscellaneousmisuse of social networks has been reported, e.g., email viruses have usedaddress lists to propagate to the users’ friends. Similar vulnerabilities havebeen exploited to spread worms on the mobile phone network [12] and on theInternet telephony tool Skype .There already exists interesting work on game theoretic and epidemicmodels of propagation in social networks. For instance, Montanari andSaberi [30] attend to a game theoretic model for the diffusion of an innovationin a network and characterize the rate of convergence as a function of graphstructure. The authors highlight crucial differences between game theoreticand epidemic models and find that the spread of viruses, new technologies,and new political or social beliefs do not have the same viral behavior.The articles closest to ours are [2, 32]. Our model is inspired by Asp-nes et al. [2]. The authors apply a classic game-theoretic analysis and showthat selfish systems can be very inefficient, as the Price of Anarchy is Θ( n ),where n is the total number of players. They show that computing thesocial optimum is N P -hard and give a reduction to the combinatorial prob-lem sum-of-squares partition . They also present a O (log n ) approximation.Moscibroda et al. [32] have extended this model by introducing maliciousplayers in the selfish network. This extension facilitates the estimation ofthe robustness of a distributed system to malicious attacks. They also findthat in a non-oblivious model, intriguingly, the presence of malicious playersmay actually improve the social welfare. In a follow-up work [24] which gen- E.g., the Outlook worm
Worm.ExploreZip . See http://news.softpedia.com/news/Skype-Attacked-By-Fast-Spreading-Virus-52039.shtml.
Windfall of Malice has also been studied in the context of conges-tion games [3] by Babaioff et al. In contrast to these papers, our focus hereis on social graphs where players are concerned about their friends’ benefits.There is other literature on game theory where players are influenced bytheir neighbors. In graphical economics [16, 18], an undirected graph is givenwhere an edge between two players denotes that free trade is allowed betweenthe two parties, where the absence of such an edge denotes an embargo or another restricted form of direct trade. The payoff of a player is a function ofthe actions of the players in its neighborhood only. In contrast to our work,a different equilibrium concept is used and no social aspects are taken intoconsideration.Note that the nature of game theory on social networks also differs from cooperative games (e.g., [5]) where each coalition C ⊆ V of players V hasa certain characteristic cost or payoff function f : 2 V → R describing thecollective payoff the players can gain by forming the coalition. In contrastto cooperative games, the “coalitions” are fixed, and a player participates inthe “coalitions” of all its neighbors.A preliminary version of this article appeared at ACM EC 2008 [28], andthere have been several interesting results related to our work since then.For example, [8] studies auctions with spite and altruism among bidders, andpresents explicit characterizations of Nash equilibria for first-price auctionswith random valuations and arbitrary spite/altruism matrices, and for firstand second price auctions with arbitrary valuations and so-called regularsocial networks (players have same out-degree). By rounding a natural linearprogram with region-growing techniques, Chen et al. [7] present a better, O (log z )-approximation for the best vaccination strategy in the original modelof [2], where z is the support size of the outbreak distribution. Moreover,the effect of autonomy is investigated: a benevolent authority may suggestwhich players should be vaccinated, and the authors analyze the “Price ofOpting Out” under partially altruistic behavior; they show that with positivealtruism, Nash equilibria may not exist, but that the price of opting out isbounded.We extend the conference version of this article [28] in several respects.The two most important additions concern relative friendship and conver-gence . We study an additional model where the relative importance of aneighbor declines with the total number of friends and find that while friend-6hip is still always beneficial, the non-monotonicity result no longer applies:unlike in the absolute friendship model, the Windfall of Friendship can onlyincrease with stronger social ties. In addition, we initiate the study of con-vergence issues in the social network. It turns out that it takes longer untilan equilibrium is reached compared to purely selfish environments and henceconstitutes a price of friendship. We present a potential function argumentto prove convergence in some simple cyclic networks, and complement ourstudy with simulations on Kleinberg graphs. We believe that the existence ofand convergence to social equilibria are exciting questions for future research(see also the related fields of player-specific utilities [1] and local search com-plexity [39]). Finally, there are several minor changes, e.g., we improve thebound in Theorem 4.4 from n > n >
3. Model
This section introduces our framework. In order to gain insights into theWindfall of Friendship, we study a virus inoculation game on a social graph.We present the model of this game and we show how it can be extended toincorporate social aspects.
The virus inoculation game was introduced by [2]. We are given an undi-rected network graph G = ( V, E ) of n = | V | players (or nodes) p i ∈ V ,for i = 1 , . . . , n , who are connected by a set of edges (or links ) E . Everyplayer has to decide whether it wants to inoculate (e.g., purchase and installanti-virus software) which costs C , or whether it prefers saving money andfacing the risk of being infected. We assume that being infected yields adamage cost of L (e.g., a computer is out of work for L days). In otherwords, an instance I of a game consists of a graph G = ( V, E ), the inoc-ulation cost C and a damage cost L . We introduce a variable a i for everyplayer p i denoting p i ’s chosen strategy . Namely, a i = 1 describes that player p i is protected whereas for a player p j willing to take the risk, a j = 0. In thefollowing, we will assume that a j ∈ { , } , that is, we do not allow players to mix (i.e., use probabilistic distributions over) their strategies. These choicesare summarized by the strategy profile , the vector (cid:126)a = ( a , . . . , a n ). Afterthe players have made their decisions, a virus spreads in the network. Thepropagation model is as follows. First, one player p of the network is chosenuniformly at random as a starting point. If this player is inoculated, there7s no damage and the process terminates. Otherwise, the virus infects p andall unprotected neighbors of p . The virus now propagates recursively to theirunprotected neighbors. Hence, the more insecure players are connected, themore likely they are to be infected. The vulnerable region (set of players) inwhich an insecure player p i lies is referred to as p i ’s attack component .We only consider a limited region of the parameter space to avoid trivialcases. If the cost C is too large, no player will inoculate, resulting in a totallyinsecure network and therefore all players eventually will be infected. On theother hand, if C << L , the best strategy for all players is to inoculate. Thus,we will assume that C ≤ L and C > L/n in the following.In our game, a player has the following expected cost:
Definition 3.1 (Actual Individual Cost) . The actual individual cost of a player p i is defined as c a ( i, (cid:126)a ) = a i · C + (1 − a i ) L · k i n where k i denotes the size of p i ’s attack component. If p i is inoculated, k i stands for the size of the attack component that would result if p i becameinsecure. In the following, let c a ( i, (cid:126)a ) refer to the actual cost of an insecureand c a ( i, (cid:126)a ) to the actual cost of a secure player p i . The total social cost of a game is defined as the sum of the cost of allparticipants: C a ( (cid:126)a ) = (cid:80) p i ∈ V c a ( i, (cid:126)a ).Classic game theory assumes that all players act selfishly, i.e., each playerseeks to minimize its individual cost. In order to study the impact of suchselfish behavior, the solution concept of a Nash equilibrium (NE) is used. ANash equilibrium is a strategy profile where no selfish player can unilaterallyreduce its individual cost given the strategy choices of the other players. Wecan think of Nash equilibria as the stable strategy profiles of games withselfish players. We will only consider pure Nash equilibria in this article, i.e.,players cannot use random distributions over their strategies but must decidewhether they want to inoculate or not.In a pure Nash equilibrium, it must hold for each player p i that givena strategy profile (cid:126)a ∀ p i ∈ V, ∀ a i : c a ( i, (cid:126)a ) ≤ c a ( i, ( a , . . . , − a i , . . . , a n )),implying that player p i cannot decrease its cost by choosing an alternativestrategy 1 − a i . In order to quantify the performance loss due to selfishness,the (not necessarily unique) Nash equilibria are compared to the optimum8ituation where all players collaborate. To this end we consider the Price ofAnarchy (PoA), i.e., the ratio of the social cost of the worst Nash equilibriumdivided by the optimal social cost for a problem instance I . More formally, P oA ( I ) = max NE C NE ( I ) /C OP T ( I ) . Our model for social networks is as follows. We define a
Friendship Factor F which captures the extent to which players care about their friends , i.e.,about the players adjacent to them in the social network. More formally, F isthe factor by which a player p i takes the individual cost of its neighbors intoaccount when deciding for a strategy. F can assume any value between 0 and1. F = 0 implies that the players do not consider their neighbors’ cost at all,whereas F = 1 implies that a player values the well-being of its neighbors tothe same extent as its own. Let Γ( p i ) denote the set of neighbors of a player p i . Moreover, let Γ sec ( p i ) ⊆ Γ( p i ) be the set of inoculated neighbors, andΓ sec ( p i ) = Γ( p i ) \ Γ sec ( p i ) the remaining insecure neighbors.We distinguish between a player’s actual cost and a player’s perceivedcost . A player’s actual individual cost is the expected cost arising for eachplayer defined in Definition 3.1 used to compute a game’s social cost. Inour social network, the decisions of our players are steered by the players’ perceived cost . Definition 3.2 (Perceived Individual Cost) . The perceived individual cost of a player p i is defined as c p ( i, (cid:126)a ) = c a ( i, (cid:126)a ) + F · (cid:88) p j ∈ Γ( p i ) c a ( j, (cid:126)a ) . In the following, we write c p ( i, (cid:126)a ) to denote the perceived cost of an insecureplayer p i and c p ( i, (cid:126)a ) for the perceived cost of an inoculated player. This definition entails a new notion of equilibrium. We define a friendshipNash equilibrium (FNE) as a strategy profile (cid:126)a where no player can reduce its perceived cost by unilaterally changing its strategy given the strategies of theother players. Formally, ∀ p i ∈ V, ∀ a i : c p ( i, (cid:126)a ) ≤ c p ( i, ( a , . . . , − a i , . . . , a n )) . Given this equilibrium concept, we define the
Windfall of Friendship Υ. Definition 3.3 (Windfall of Friendship (WoF)) . The Windfall of Friendship Υ( F, I ) is the ratio of the social cost of the worst Nash equilibrium for I and he social cost of the worst friendship Nash equilibrium for I : Υ( F, I ) = max NE C NE ( I )max F NE C F NE ( F, I )Υ(
F, I ) > F, I ) < greater in social graphsthan in purely selfish environments.
4. General Analysis
In this section we characterize friendship Nash equilibria and derive gen-eral results on the Windfall of Friendship for the virus propagation game insocial networks. It has been shown [2] that in classic Nash equilibria ( F = 0),an attack component can never consist of more than Cn/L insecure players.A similar characteristic also holds for friendship Nash equilibria. As everyplayer cares about its neighbors, the maximal attack component size in whichan insecure player p i still does not inoculate depends on the number of p i ’sinsecure neighbors and the size of their attack components. Therefore, itdiffers from player to player. We have the following helper lemma. Lemma 4.1.
The player p i will inoculate if and only if the size of its attackcomponent is k i > Cn/L + F · (cid:80) p j ∈ Γ sec ( p i ) k j F | Γ sec ( p i ) | , where the k j s are the attack component sizes of p i ’s insecure neighbors as-suming p i is secure.Proof. Player p i will inoculate if and only if this choice lowers the perceivedcost. By Definition 3.2, the perceived individual cost of an inoculated playeris c p ( i, (cid:126)a ) = C + F | Γ sec ( p i ) | C + (cid:88) p j ∈ Γ sec ( p i ) L k j n and for an insecure player we have c p ( i, (cid:126)a ) = L k i n + F (cid:18) | Γ sec ( p i ) | C + | Γ sec ( p i ) | L k i n (cid:19) . p i to prefer to inoculate it must hold that c p ( i, (cid:126)a ) > c p ( i, (cid:126)a ) ⇔ L k i n + F · | Γ sec ( p i ) | L k i n > C + F · (cid:88) p j ∈ Γ sec ( p i ) L k j n ⇔ L k i n (1 + F | Γ sec ( p i ) | ) > C + F Ln · (cid:88) p j ∈ Γ sec ( p i ) k j ⇔ k i (1 + F | Γ sec ( p i ) | ) > Cn/L + F · (cid:88) p j ∈ Γ sec ( p i ) k j ⇔ k i > Cn/L + F · (cid:80) p j ∈ Γ sec ( p i ) k j F | Γ sec ( p i ) | . A pivotal question is of course whether social networks where playerscare about their friends yield better equilibria than selfish environments.The following theorem answers this questions affirmatively: the worst FNEcosts never more than the worst NE.
Theorem 4.2.
For all instances of the virus inoculation game and ≤ F ≤ , it holds that ≤ Υ( F, I ) ≤ PoA ( I ) Proof.
The proof idea for Υ(
F, I ) ≥ I weconsider an arbitrary FNE with F >
0. Given this equilibrium, we show theexistence of a NE with larger social cost (according to [2], our best responsestrategy always converges). Let (cid:126)a be any (e.g., the worst) FNE in the socialmodel. If (cid:126)a is also a NE in the same instance with F = 0 then we are done.Otherwise there is at least one player p i that prefers to change its strategy.Assume p i is insecure but favors inoculation. Therefore p i ’s attack componenthas on the one hand to be of size at least k (cid:48) i > Cn/L [2] and on the otherhand of size at most k (cid:48)(cid:48) i = ( Cn/L + F · (cid:80) p j ∈ Γ sec ( p i ) k j ) / (1 + F | Γ sec ( p i ) | ) ≤ ( Cn/L + F | Γ sec ( p i ) | ( k (cid:48)(cid:48) i − / (1+ F | Γ sec ( p i ) | ) ⇔ k (cid:48)(cid:48) i ≤ Cn/L − F | Γ sec ( p i ) | (cfLemma 4.1). This is impossible and yields a contradiction to the assumptionthat in the selfish network, an additional player wants to inoculate.It remains to study the case where p i is secure in the FNE but prefers to beinsecure in the NE. Observe that, since every player has the same preference11n the attack component’s size when F = 0, a newly insecure player cannottrigger other players to inoculate. Furthermore, only the players inside p i ’sattack component are affected by this change. The total cost of this attackcomponent increases by at least x = k i n L − C (cid:124) (cid:123)(cid:122) (cid:125) p i + (cid:88) p j ∈ Γ sec ( p i ) (cid:18) k i n L − k j n L (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) p i ’s insecure neighbors = k i n L − C + Ln ( | Γ sec ( p i ) | k i − (cid:88) p j ∈ Γ sec ( p i ) k j ) . Applying Lemma 4.1 guarantees that (cid:88) p j ∈ Γ sec ( p i ) k j ≤ k i (1 + F | Γ sec ( p i ) | ) − Cn/LF .
This results in x ≥ Ln (cid:18) | Γ sec ( p i ) | k i − k i (1 + F | Γ sec ( p i ) | ) − Cn/LF (cid:19) = k i Ln (1 − F ) − C (1 − F ) > , since a player only gives up its protection if C > k i Ln . If more players areunhappy with their situation and become vulnerable, the cost for the NEincreases further. In conclusion, there exists a NE for every FNE with F ≥ PoA( I ) ≥ Υ( F, I ), follows directlyfrom the definitions: while the PoA is the ratio of the NE’s social cost dividedby the social optimum, Υ(
F, I ) is the ratio between the cost of the NE andthe FNE. As the FNE’s cost must be at least as large as the social optimumcost the claim follows.
Remark 4.3.
Note that Aspnes et al. [2] proved that the Price of Anarchynever exceeds the size of the network, i.e., n ≥ PoA( I ). Consequently, theWindfall of Friendship cannot be larger than n due to Theorem 4.2. The above result leads to the question of whether the Windfall of Friend-ship grows monotonically with stronger social ties, i.e., with larger friendshipfactors F . Intriguingly, this is not the case.12 heorem 4.4. For all networks with more than three players, there existgame instances where Υ( F, I ) does not grow monotonically in F .Proof. We give a counter example for the star graph S n which has one centerplayer and n − F l and F s where F l > F s . We show that for the large friendship factor F l , there exists a FNE, F N E , where only the center player and one leaf player remain insecure.For the same setting but with a small friendship factor F s , at least two leafplayers will remain insecure, which will trigger the center player to inoculate,yielding a FNE, F N E , where only the center player is secure.Consider F N E first. Let c be the insecure center player, let l be theinsecure leaf player, and let l be a secure leaf player. In order for F N E toconstitute a Nash equilibrium, the following conditions must hold: player c : 2 Ln + 2 F l Ln < C + F l Ln player l : 2 Ln + 2 F l Ln < C + F l Ln player l : C + 2 F l Ln < Ln + 3 F l Ln For
F N E , let c be the insecure center player, let l be one of the twoinsecure leaf players, and let l be a secure leaf player. In order for the leafplayers to be happy with their situation but for the center player to preferto inoculate, it must hold that: player c : C + 2 F s Ln < Ln + 6 F s Ln player l : 3 Ln + 3 F s Ln < C + 2 F s Ln player l : C + 3 F s Ln < Ln + 4 F s Ln Now choose C := 5 L/ (2 n ) + F l L/n (note that due to our assumptionthat n > C < L ). This yields the following conditions: F l > F s + 1 / F l < F s + 3 /
2, and F l < F s + 1 /
2. These conditions are easily fulfilled,e.g., with F l = 3 / F s = 1 /
8. Observe that the social cost of the firstFNE (for F l ) is Cost ( S n , (cid:126)a F NE ) = ( n − C + 4 L/n , whereas for the secondFNE (for F s ) Cost ( S n , (cid:126)a F NE ) = C + ( n − L/n . Thus,
Cost ( S n , (cid:126)a F NE ) − Cost ( S n , (cid:126)a F NE ) = ( n − C − ( n − L/n >
C > L/ (2 n )and as, due to our assumption, n >
3. This concludes the proof.13easoning about best and worst Nash equilibria raises the question ofhow difficult it is to compute such equlibria. We can generalize the proofgiven in [2] and show that computing the most economical and the mostexpensive FNE is hard for any friendship factor.
Theorem 4.5.
Computing the best and the worst pure FNE is
N P -completefor any F ∈ [0 , .Proof. We prove this theorem by a reduction from two
N P -hard problems,
Vertex Cover [17] and
Independent Dominating Set [15]. Concretely,for the decision version of the problem, we show that answering the questionwhether there exists a FNE costing less than k , or more than k respectively,is at least as hard as solving vertex cover or independent dominating set.Note that verifying whether a proposed solution is correct can be done inpolynomial time, hence the problems are indeed in N P .Fix some graph G = ( V, E ) and set C = 1 and L = n/ .
5. We show thatthe following two conditions are necessary and sufficient for a FNE: (a) allneighbors of an insecure player are secure, and (b) every inoculated player hasat least one insecure neighbor. Due to our assumption that
C > L/n , condi-tion (b) is satisfied in all FNE. To see that condition (a) holds as well, assumethe contrary, i.e., an attack component of size at least two. An insecure player p i in this attack component bears the cost k i n L + F ( | Γ sec ( p i ) | C + | Γ sec ( p i ) | k i n L ).Changing its strategy reduces its cost by at least ∆ i = k i n L + F | Γ sec ( p i ) | k i n L − C − F | Γ sec ( p i ) | k i − n L = k i n L + F | Γ sec ( p i ) | n L − C . By our assumption that k i ≥
2, and hence | Γ sec ( p i ) | ≥
1, it holds that ∆ i >
0, resulting in p i becom-ing secure. Hence, condition (a) holds in any FNE as well. For the oppositedirection assume that an insecure player wants to change its strategy eventhough (a) and (b) are true. This is impossible because in this case (b)would be violated because this player does not have any insecure neighbors.An inoculated player would destroy (a) by adopting another strategy. Thus(a) and (b) are sufficient for a FNE.We now argue that G has a vertex cover of size k if and only if thevirus game has a FNE with k or fewer secure players, or equivalently anequilibrium with social cost at most Ck + ( n − k ) L/n , as each insecure playermust be in a component of size 1 and contributes exactly
L/n expected cost.Given a minimal vertex cover V (cid:48) ⊆ V , observe that installing the software onall players in V (cid:48) satisfies condition (a) because V (cid:48) is a vertex cover and (b)because V (cid:48) is minimal. Conversely, if V (cid:48) is the set of secure players in a FNE,then V (cid:48) is a vertex cover by condition (a) which is minimal by condition (b).14or the worst FNE, we consider an instance of the independent domi-nating set problem. Given an independent dominating set V (cid:48) , installing thesoftware on all players except the players in V (cid:48) satisfies condition (a) because V (cid:48) is independent and (b) because V (cid:48) is a dominating set. Conversely, theinsecure players in any FNE are independent by condition (a) and dominat-ing by condition (b). This shows that G has an independent dominatingset of size at most k if and only if it has a FNE with at least n − k secureplayers.
5. Windfall for Special Graphs
While the last section has presented general results on equilibria in socialnetworks and the Windfall of Friendship, we now present upper and lowerbounds on the Windfall of Friendship for concrete topologies, namely the complete graph K n and the star graph S n . In order to initiate the study of the Windfall of Friendship, we consider avery simple topology, the complete graph K n where all players are connectedto each other. First consider the classic setting where players do not careabout their neighbors ( F = 0). We have the following result: Lemma 5.1.
In the graph K n , there are two Nash equilibria with social cost NE : Cost ( K n , (cid:126)a NE1 ) = C ( n − (cid:100) Cn/L (cid:101) + 1) +
L/n ( (cid:100) Cn/L (cid:101) − , andNE : Cost ( K n , (cid:126)a NE2 ) = C ( n − (cid:98) Cn/L (cid:99) ) +
L/n ( (cid:98) Cn/L (cid:99) ) . If (cid:100) Cn/L (cid:101) − (cid:98)
Cn/L (cid:99) , there is only one Nash equilibrium.Proof.
Let (cid:126)a be a NE. Consider an inoculated player p i and an insecure player p j , and hence c a ( i, (cid:126)a ) = C and c a ( j, (cid:126)a ) = L k j n , where k j is the total numberof insecure players in K n . In order for p i to remain inoculated, it must holdthat C ≤ ( k j + 1) L/n , so k j ≥ (cid:100) Cn/L − (cid:101) ; for p j to remain insecure, it holdsthat k j L/n ≤ C , so k j ≤ (cid:98) Cn/L (cid:99) . As the total social cost in K n is given by Cost ( K n , (cid:126)a ) = ( n − k j ) C + k j L/n , the claim follows.Observe that the equilibrium size of the attack component is roughly twicethe size of the attack component of the social optimum, as
Cost ( K n , (cid:126)a ) =( n − k j ) C + k j L/n is minimized for k j = Cn/ L .15 emma 5.2. In the social optimum for K n , the size of the attack componentis either (cid:98) Cn/L (cid:99) or (cid:100) Cn/L (cid:101) , yielding a total social cost of
Cost ( K n , (cid:126)a OPT ) = ( n − (cid:98) Cn/L (cid:99) ) C + ( (cid:98) Cn/L (cid:99) ) Ln or Cost ( K n , (cid:126)a OPT ) = ( n − (cid:100) Cn/L (cid:101) ) C + ( (cid:100) Cn/L (cid:101) ) Ln .
In order to compute the Windfall of Friendship, the friendship Nash equi-libria in social networks have to be identified.
Lemma 5.3. In K n , there are two friendship Nash equilibria with social cost FNE : Cost ( K n , (cid:126)a FNE1 ) = C (cid:18) n − (cid:24) Cn/L −
11 + F (cid:25)(cid:19) + L/n (cid:18)(cid:24)
Cn/L −
11 + F (cid:25)(cid:19) , andFNE : Cost ( K n , (cid:126)a FNE2 ) = C (cid:18) n − (cid:22) Cn/L + F F (cid:23)(cid:19) + L/n (cid:18)(cid:22)
Cn/L + F F (cid:23)(cid:19) . If (cid:100) ( Cn/L − / (1 + F ) (cid:101) = (cid:98) ( Cn/L + F ) / (1 + F ) (cid:99) , there is only one FNE.Proof. According to Lemma 4.1, in a FNE, a player p i remains secure ifotherwise the component had size at least k i = k j + 1 ≥ ( Cn/L + F k j ) / (1 + F k j ) where k j is the number of insecure players. This implies that k j ≥(cid:100) ( Cn/L − / (1 + F ) (cid:101) . Dually, for an insecure player p j it holds that k j ≤ ( Cn/L + F ( k j − ) / (1+ F ( k j − k j ≤ (cid:98) ( Cn/L + F ) / (1+ F ) (cid:99) .Given these bounds on the total number of insecure players in a FNE, thesocial cost can be obtained by substituting k j in Cost ( K n , (cid:126)a ) = ( n − k j ) C + k j L/n . As the difference between the upper and the lower bound for k j is atmost 1, there are at most two equilibria and the claim follows.Given the characteristics of the different equilibria, we have the followingtheorem. Theorem 5.4. In K n , the Windfall of Friendship is at most Υ( F, I ) = 4 / for an arbitrary network size. This is tight in the sense that there are indeedinstances where the worst FNE is a factor / better than the worst NE. roof. Upper Bound. We first derive the upper bound on Υ(
F, I ).Υ(
F, I ) =
Cost ( K n , (cid:126)a NE ) Cost ( K n , (cid:126)a FNE ) ≤ Cost ( K n , (cid:126)a NE ) Cost ( K n , (cid:126)a OPT ) ≤ ( n − (cid:100) Cn/L − (cid:101) ) C + ( (cid:98) Cn/L (cid:99) ) Ln ( n − Cn/L ) C + ( Cn/L ) Ln as the optimal social cost (cf Lemma 5.2) is smaller or equal to the socialcost of any FNE. Simplifying this expression yieldsΥ( F, I ) ≤ n (1 − C/L ) C + C n/Ln (1 − C/L ) C + C n/L = 11 − C/L .
This term is maximized for L = C , implying that Υ( F, I ) ≤ /
3, for arbitrary n . Lower Bound.
We now show that the ratio between the equilibria costreaches 4 / Ln/ n/ L/n =3 nL/ n and C = L by Lemma 5.2. For F = 1 this is also theonly friendship Nash equilibrium due to Lemma 5.3. In the selfish gamehowever the Nash equilibrium has fewer inoculated players and is of cost nL (see Lemma 5.1). Since these are the only Nash equilibria they constitutethe worst equilibria and the ratio isΥ( F, I ) =
Cost ( K n , (cid:126)a NE ) Cost ( K n , (cid:126)a FNE ) = nL / nL = 4 / . To conclude our analysis of K n , observe that friendship Nash equilibriaalways exist in complete graphs, and that in environments where one playerat a time is given the chance to change its strategy in a best response mannerquickly results in such an equilibrium as all players have the same preferences. While the analysis of K n was simple, it turns out that already slightlymore sophisticated graphs are challenging. In the following, we investigate17he Windfall of Friendship in star graphs S n . Note that in S n , the socialwelfare is maximized if the center player inoculates and all other players donot. The total inoculation cost then is C and the attack components are allof size 1, yielding a total social cost of Cost ( S n , (cid:126)a OPT ) = C + ( n − L/n . Lemma 5.5.
In the social optimum of the star graph S n , only the centerplayer is inoculated. The social cost is Cost ( S n , (cid:126)a OPT ) = C + ( n − L/n.
The situation where only the center player is inoculated also constitutesa NE. However, there are more Nash equilibria.
Lemma 5.6.
In the star graph S n , there are at most three Nash equilibriawith social cost NE : Cost ( S n , (cid:126)a NE1 ) = C + ( n − L/n, NE : Cost ( S n , (cid:126)a NE2 ) = C ( n − (cid:100) Cn/L (cid:101) + 1) +
L/n ( (cid:100) Cn/L (cid:101) − , andNE : Cost ( S n , (cid:126)a NE3 ) = C ( n − (cid:98) Cn/L (cid:99) ) +
L/n ( (cid:98) Cn/L (cid:99) ) . If Cn/L / ∈ N , only two equilibria exist.Proof. If the center player is the only secure player, changing its strategycosts L but saves only C . When a leaf player becomes secure, its cost changesfrom L/n to C . These changes are unprofitable, and the social cost of thisNE is Cost ( S n , (cid:126)a NE1 ) = C + ( n − L/n .For the other Nash equilibria the center player is not inoculated. Let thenumber of insecure leaf players be n . In order for a secure player to remainsecure, it must hold that C ≤ ( n + 2) L/n , and hence n ≥ (cid:100) Cn/L − (cid:101) .For an insecure player to remain insecure, it must hold that (1 + n ) L/n ≤ C , thus n ≤ (cid:98) Cn/L − (cid:99) . Therefore, we can conclude that there are atmost two Nash equilibria, one with (cid:100) Cn/L − (cid:101) and one with (cid:98) Cn/L (cid:99) manyinsecure players. The total social cost follows by substituting n in the totalsocial cost function. Finally, observe that if Cn/L ∈ N and Cn/L >
3, allthree equilibria exist in parallel; if
Cn/L / ∈ N , NE and NE become oneequilibrium.Let us consider the social network scenario again.18 emma 5.7. In S n , there are at most three friendship Nash equilibria withsocial cost FNE : Cost ( S n , (cid:126)a FNE1 ) = C + ( n − L/n,
FNE : Cost ( S n , (cid:126)a FNE2 ) = C ( n − (cid:100) Cn/L − F (cid:101) + 1) + L/n ( (cid:100) Cn/L − F (cid:101) − , andFNE : Cost ( S n , (cid:126)a FNE3 ) = C ( n − (cid:98) Cn/L − F (cid:99) ) + L/n ( (cid:98) Cn/L − F (cid:99) ) . If Cn/L − F / ∈ N , at most 2 friendship Nash equilibria exist.Proof. First, observe that having only an inoculated center player constitutesa FNE. In order for the center player to remain inoculated, it must hold that C + F ( n − L n ≤ nL/n + F ( n − L nn = L + F ( n − L . All leaf players remaininsecure as long as L/n + F C ≤ C + F C ⇔ L/n ≤ C . These conditions arealways true, and we have Cost ( S n , (cid:126)a FNE1 ) = C + ( n − L/n .If the centerplayer is not inoculated, we have n insecure and n − n − C + F n +1 n L ≤ n +2 n L + F n +2 n L , so n ≥ (cid:100) Cn/L − F (cid:101) −
2. For aninsecure leaf player, it must hold that n +1 n L + F n +1 n L ≤ C + F n n L , so n ≤ (cid:98) Cn/L − F (cid:99) −
1. The claim follows by substitution.Note that there are instances where FNE is the only friendship Nashequilibrium. We already made use of this phenomenon in Section 4 to showthat Υ( F, I ) is not monotonically increasing in F . The next lemma statesunder which circumstances this is the case. Lemma 5.8. In S n , there is a unique FNE equivalent to the social optimumif and only if (cid:98) Cn/L − F (cid:99) − (cid:98) F ( (cid:112) − F (1 − Cn/L ) − (cid:99) − ≥ Proof. S n has only one FNE if and only if every (insecure) leaf player iscontent with its chosen strategy but the insecure center player would ratherinoculate. In order for an insecure leaf player to remain insecure we have n ≤ (cid:98) Cn/L − − F (cid:99) and the insecure center player wants to inoculate ifand only if C + F ( n − n − C + F n n L < ( n + 1) Ln + F ( n − n − C + F n n + 1 n L, which is equivalent to F n + n + 1 − Cn/L > . This implies that n ≥(cid:98) F ( (cid:112) − F (1 − Cn/L ) −
1) + 1 (cid:99) . Therefore there is only one FNE if andonly if there exists an integer n such that (cid:98) F ( (cid:112) − F (1 − Cn/L ) −
1) +1 (cid:99) ≤ n ≤ (cid:98) Cn/L − − F (cid:99) . 19iven the characterization of the various equilibria, the Windfall of Friend-ship can be computed. Theorem 5.9. If (cid:98) F ( (cid:112) − F (1 − Cn/L ) − (cid:99) + 2 − (cid:98) Cn/L − F (cid:99) ≤ ,the Windfall of Friendship is Υ( F, I ) ≥ ( n − C + L/nC + ( n − L/n , else Υ( F, I ) ≤ n + 1 n − . Proof.
According to Lemma 5.8, the friendship Nash equilibrium is uniqueand hence equivalent to the social optimum if (cid:98)
Cn/L − F (cid:99) − (cid:98) F ( (cid:112) − F (1 − Cn/L ) − (cid:99) − ≥ . On the other hand, observe that there always exist sub-optimal Nash equi-libria where the center player is not inoculated. Hence, we haveΥ(
F, I ) =
Cost ( S n , (cid:126)a NE ) Cost ( S n , (cid:126)a FNE ) =
Cost ( S n , (cid:126)a NE ) Cost ( S n , (cid:126)a OPT ) ≥ ( n − (cid:98) Cn/L − (cid:99) ) C + ( (cid:100) Cn/L (cid:101) − L/nC + ( n − L/n ≥ C ( n −
2) +
L/nC + ( n − L/n .
Otherwise, i.e., if there exist friendship Nash equilibria with an insecurecenter player, an upper bound for the WoF can be computedΥ(
F, I ) =
Cost ( S n , (cid:126)a NE ) Cost ( S n , (cid:126)a FNE ) ≤ ( n − (cid:100) Cn/L − (cid:101) ) C + ( (cid:98) Cn/L (cid:99) ) L/n ( n − (cid:98) Cn/L − F (cid:99) ) C + ( (cid:100) Cn/L − − F (cid:101) ) L/n ≤ ( n + 1) CnC + F C − C (1 + F ) + (1 + F ) L/n< ( n + 1) CC ( n + F − F )) < n + 1 n − . n , and hence indeed beasymptotically as large as the Price of Anarchy. However, if (cid:98) Cn/L − F (cid:99) −(cid:98) F ( (cid:112) − F (1 − Cn/L ) − (cid:99) − ≥ This section has focused on a small set of very simple topologies onlyand we regard the derived results as a first step towards more complex graphclasses such as Kleinberg graphs featuring the small-world property. Inter-estingly, however, our findings have implications for general topologies. Wecould show that even in simple graphs such as the star graph, the Windfall ofFriendship can assume all possible values, from constant ratios up to ratioslinear in n . This is asymptotically maximal for general graphs as well sincethe Price of Anarchy is bounded by n [2].
6. On Relative Equilibria
In the model we have studied so far, the actual cost of each friend—weighted by a factor F —is added to a player’s perceived cost. This describesa situation where friends are taken into account individually and indepen-dently of each other. However, one could imagine scenarios where the relativeimportance of a friend decreases with the total number of friends, that is, aplayer with many friends may care less about the welfare of a specific friendcompared to a player who only has one or two friends. This motivates analternative approach to describe perceived costs: Definition 6.1 (Relative Perceived Cost) . The relative perceived individual cost of a player p i is defined as c r ( i, (cid:126)a ) = c a ( i, (cid:126)a ) + F · (cid:80) p j ∈ Γ( p i ) c a ( j, (cid:126)a ) | Γ( p i ) | . In the following, we write c r ( i, (cid:126)a ) to denote the relative perceived cost of aninsecure player p i and c r ( i, (cid:126)a ) for the relative perceived cost of an inoculatedplayer.
21e will refer to an FNE equilibrium with respect to relative perceivedcosts by rFNE .It turns out that while relative equilibria have similar properties as regularfriendship equilibria and most of our techniques are still applicable, thereare some crucial differences. Let us first consider the size of the attackcomponents under rFNE.
Lemma 6.2.
The player p i will inoculate if and only if the size of its attackcomponent is k i > | Γ( p i ) | · Cn/L + F · (cid:80) p j ∈ Γ sec ( p i ) k j | Γ( p i ) | + F | Γ sec ( p i ) | , where the k j s are the attack component sizes of p i ’s insecure neighbors as-suming p i is secure.Proof. Player p i will inoculate if and only if this choice lowers the relativeperceived individual cost. By Definition 6.1, the relative perceived individualcosts of an inoculated player are c r ( i, (cid:126)a ) = C + F/ | Γ( p i ) | · | Γ sec ( p i ) | C + (cid:88) p j ∈ Γ sec ( p i ) L k j n and for an insecure player we have c p ( i, (cid:126)a ) = L k i n + F/ | Γ( p i ) | · (cid:18) | Γ sec ( p i ) | C + | Γ sec ( p i ) | L k i n (cid:19) . Thus, for p i to prefer to inoculate it must hold that c p ( i, (cid:126)a ) > c p ( i, (cid:126)a ) ⇔ k i > Cn/L + F/ | Γ( p i ) | · (cid:80) p j ∈ Γ sec ( p i ) k j F/ | Γ( p i ) | · | Γ sec ( p i ) | . Not surprisingly, we can show that friendship is always beneficial alsowith respect to relative perceived costs.
Theorem 6.3.
For all instances of the virus inoculation game and ≤ F ≤ , it holds that ≤ Υ( F, I ) ≤ PoA ( I ) also in the relative cost model. roof. Again, the upper bound for the WoF, i.e.,
PoA( I ) ≥ Υ( F, I ), followsdirectly from the definitions (see also proof of Lemma 4.2). For Υ(
F, I ) ≥ (cid:126)a (defined with relative costs) with F > (cid:126)a (cid:48) with cost C a ( (cid:126)a ) ≤ C a ( (cid:126)a (cid:48) ). If (cid:126)a is also a NE in the same instance with F = 0 then weare done. Otherwise there is at least one player p i that prefers to change itsstrategy. If p i is insecure but favors inoculation, p i ’s attack component hason the one hand to be of size at least k (cid:48) i > Cn/L [2] (otherwise there is notreason for p i to become secure) and on the other hand of size at most k (cid:48)(cid:48) i =( | Γ( p i ) | · Cn/L + F · (cid:80) p j ∈ Γ sec ( p i ) k j ) / ( | Γ( p i ) | + F · | Γ sec ( p i ) | ) ≤ ( | Γ( p i ) | · Cn/L + F ·| Γ sec ( p i ) | ( k (cid:48)(cid:48) i − / ( | Γ( p i ) | + F ·| Γ sec ( p i ) | ) so k (cid:48)(cid:48) i ≤ | Γ( p i ) |· Cn/L − F | Γ sec ( p i ) | (cf Lemma 6.2), yielding a contradiction. What if p i is secure in the rFNE butprefers to be insecure in the NE? Since every player has the same preferenceon the attack component’s size when F = 0, a newly insecure player cannottrigger other players to inoculate. Furthermore, only the players inside p i ’sattack component are affected by this change. The total cost of this attackcomponent increases by at least (see also the proof of Lemma 4.2) x = k i n L − C + Ln ( | Γ sec ( p i ) | k i − (cid:88) p j ∈ Γ sec ( p i ) k j ) . Applying Lemma 6.2 guarantees that (cid:88) p j ∈ Γ sec ( p i ) k j ≤ k i (1 + F/ | Γ( p i ) | · | Γ sec ( p i ) | ) − Cn/LF/ | Γ( p i ) | . This results in x ≥ Ln (cid:18) | Γ sec ( p i ) | k i − k i (1 + F/ | Γ( p i ) | · | Γ sec ( p i ) | ) − Cn/LF/ | Γ( p i ) |· (cid:19) = k i Ln (cid:18) − F/ | Γ( p i ) |· (cid:19) − C (cid:18) − F/ | Γ( p i ) |· (cid:19) > , since a player only gives up its protection if C > k i Ln . If more players areunhappy with their situation and become vulnerable, the cost for the NEincreases further. In conclusion, there exists a NE for every FNE with F ≥ F does no longer hold in the star graph S n . To see this,23ote that there are only at most two distinct rFNE in S n (apart from thetrivial situations where all players are either insecure or secure): the “goodequilibrium” where the center player is secure and all the leave players inse-cure, and the “bad equilibrium” where the center is insecure and a fractionof the leaves secure. The following theorem shows that the example of The-orem 4.4 for FNE is no longer true for rFNE. Theorem 6.4.
The Windfall of Friendship is monotonic in F for S n underthe relative cost model.Proof. Consider a friendship factor F . Clearly, the equilibrium where onlythe center player is secure always exists (w.l.o.g., we focus on “reasonablevalues” C and L ). When is there an equilibrium where the center is insecure?Consider such an equilibrium where x leave players are insecure. In order forthis to constitute an equilibrium, it must hold for the center player that: ( x + 1) Ln + Fn − · ( x + 1) Ln + F · C · ( n − x − n − < C + Fn − · x · Ln + F · C · ( n − x − n − ⇔ ( x + 1) Ln + Fn − · Ln < C
On the other hand, for an insecure leaf player we have:( x + 1) Ln + F L ( x + 1) n < C + F Lxn ⇔ ( x + 1) Ln + F Ln < C
Unlike in the FNE scenario, the center player is less likely to inoculate, i.e.,leaf players inoculate first. Thus, a larger F can only render the existence ofsuch an equilibrium more unlikely.Finally, note that the hardness result of Theorem 4.5 is also applicableto relative FNEs. Theorem 6.5.
Computing the best and the worst pure rFNE is
N P -completefor any F ∈ [0 , .Proof. ( Sketch ) Again, deciding the existence of a rFNE with cost less than k or more than k is at least as hard as solving the vertex cover or inde-pendent dominating set problem, respectively. Note that verifying whethera proposed solution is correct can be done in polynomial time, hence the24roblems are indeed in N P . The proof is similar to Theorem 4.5, andwe only point out the difference for condition (a): an insecure player p i in the attack component bears the cost k i /n · L + F | Γ sec ( p i ) | C + | Γ sec ( p i ) | · ( k i L/n ) / | Γ( p i ) | , and changing its strategy reduces the cost by at least ∆ i = k i L/n + F | Γ sec ( p i ) | k i L/ ( | Γ( p i ) | n ) − C − F | Γ sec ( p i ) | ( k i − L/ ( | Γ( p i ) | n ) = k i L/n − C + F L | Γ sec ( p i ) | / ( | Γ( p i ) | n ). By our assumption that k i ≥
2, andhence | Γ sec ( p i ) | ≥
1, it holds that ∆ i >
0, resulting in p i becoming secure.
7. Convergence
According to Lemma 4.2 and Lemma 6.3, the social context can only im-prove the overall welfare of the players, both in the absolute and the relativefriendship model. However, there are implications beyond the players’ wel-fare in the equilibria: in social networks, the dynamics of how the equilibriaare reached is different.In [2], Aspnes et al. have shown that best-response behavior quickly leadsto some pure Nash equilibrium, from any initial situation. Their potentialfunction argument however relies on a “symmetry” of the players in the senseinsecure players in the same attack component have the same cost. This nolonger holds in the social context where different players take into accounttheir neighborhood: a player with four insecure neighbors is more likely toinoculate than a player with just one, secure neighbor. Thus, the distinctionbetween “big” and “small” components used in [2] cannot be applied, asdifferent players require a different threshold.Nevertheless, convergence can be shown in certain scenarios. For example,the hardness proofs of Lemmas 4.5 and 6.5 imply that equilibria always existin the corresponding areas of the parameter space, and it is easy to see thatthe equilibria are also reached by best-response sequences. Similarly, in thestar and complete networks, best-response sequences converge in linear time.Linear convergence time also happens in more complex, cyclic graphs. Forexample, consider the cycle graph C n where each player is connected to oneleft and one right neighbor in a circular fashion. To prove best responseconvergence from arbitrary initial states, we distinguish between an initialphase where certain structural invariants are established, and a second phasewhere a potential function argument can be applied with respect to the viewof only one type of players. Each event when one player is given the chanceto perform a best response is called a round .25 heorem 7.1. From any initial state and in the cycle graph C n , a bestresponse round-robin sequence results in an equilibrium after O ( n ) changes,both in case of absolute and relative friendship equilibria.Proof. After two round-robin phases where each player is given the chanceto make a best response twice (at most 2 n changes or rounds), it holds thatan insecure player p which is adjacent to a secure player p cannot becomesecure: since p preferred to be insecure at some time t , the only reason tobecome secure again is the event that a player p becomes insecure in p ’sattack component at time t (cid:48) > t ; however, since p has a secure neighbor p and hence p can only have more insecure neighbors than p , p cannotprefer a larger attack component than p , which yields a contradiction tothe assumption that p becomes secure while its neighbor p is still secure.Moreover, by the same arguments, there cannot be three consecutive secureplayers.Therefore, in the best response rounds after the two initial phases, thereare the following cases. Case (A): a secure player having two insecure neigh-bors becomes insecure; Case (B): a secure player with one secure neighborbecomes insecure; and Case (C): an insecure player with two insecure neigh-bors becomes secure.In order to prove convergence, the following potential function Φ is used:Φ( (cid:126)a ) = (cid:88) A ∈S big ( (cid:126)a ) | A | − (cid:88) A ∈S small ( (cid:126)a ) | A | where the attack components A in S big contain more than t = nC/ ( F L ) − L/F + 1 players and the attack components A in S small contain at most t players in case of absolute friendship equilibria; for relative friendship equi-libria we use t = 2 Cn/ ( F L ) − L/F + 1. In other words, the threshold t to distinguish between small and big components is chosen with respect toplayers having two insecure neighbors ; in case of absolute FNEs: C + F · L · ( t − n = t · Ln + 2 F · L · tn ⇔ CnF L − LF + 1 = t and in case of relative FNEs: C + F/ · L · ( t − n = t · Ln + F · L · tn ⇔ CnF L − LF + 1 = t Note that it holds that − n ≤ Φ( (cid:126)a ) ≤ n, ∀ (cid:126)a . We now show that Case (A)and (C) reduce Φ( (cid:126)a ) by at least one unit in each best response. Moreover,26ase (B) can increase the potential by at most one. However, since wehave shown that Case (B) incurs less than n times, the claim follows by anamortization argument. Case (A):
In this case, a new insecure player p isadded to an attack component in S small . Case (B):
A new insecure player p is added to an attack component in S small or to an attack component in S big (since p is “on the edge” of the attack component, it prefers a largerattack component). Case (C):
An insecure player is removed from an attackcomponent in S big .The proof of Theorem 7.1 can be adapted to show linear convergence ingeneral 2-degree networks where players have degree at most two. In orderto gain deeper insights into the convergence behavior, we conducted severalexperiments.
8. Simulations
This section briefly reports on the simulations conducted on Kleinberggraphs (using clustering exponent α = 2). Although the existence of equi-libria and the best-response convergence time complexity for general graphsremain an open question, during the thousands of experiments, we did not en-counter a single instance which did not converge. Moreover, our experimentsindicate that the initial configuration (i.e., the set of secure and insecureplayers) as well as the relationship of L to C typically has a negligible effecton the convergence time, and hence, unless stated otherwise, the followingexperiments assume an initially completely insecure network and C = 1 and L = 4. All experiments are repeated 100 times over different Kleinberggraphs.All our experiments showed a positive Windfall of Friendship that in-creases monotonically in F , both for the relative and the absolute friendshipmodel. Figure 1 shows a typical result. Maybe surprisingly, it turns out thatthe windfall of friendship is often not due to a higher fraction of secure play-ers, but rather the fact that the secure players are located at strategicallymore beneficial locations (see also Figure 2). We can conclude that there is awindfall of friendship not only for the worst but also for “average equilibria”.The box plots in Figure 3 give a more detailed picture of the cost for F ∈ { , } . The overall cost of pure NE is typically higher than the cost ofrFNE which is in turn higher than the cost of FNE.27 .1 802.6842 403.89470.2 741.4211 399.10530.3 698.4737 400.31580.4 667.1579 405.68420.5 646.7895 407.94740.6 616.3684 4100.7 600.8947 409.21050.8 584.7895 415.84210.9 569.8421 416.73681 563.9474 419.6842F Convergence0 3000 F F Average Social Cost (n=1000)
Cost
Figure 1: Average social cost and average number of secure players as a function of F ,in the FNE resulting from round-robin best response sequences starting from an initiallycompletely insecure graph. Besides social cost, we are mainly interested in convergence times. Wefind that while the convergence time typically increases already for a small
F >
0, the magnitude of F plays a minor role. Figure 4 shows the typicalconvergence times as a function of F . Notice that the convergence timemore than doubles when changing from the selfish to the social model but isroughly constant for all values of F .
9. Conclusion
This article presented a framework to study and quantify the effects ofgame-theoretic behavior in social networks. This framework allows us toformally describe and understand phenomena which are often well-known onan anecdotal level. For instance, we find that the Windfall of Friendshipis always positive, and that players embedded in a social context may besubject to longer convergence times. Moreover, interestingly, we find that theWindfall of Friendship does not always increase monotonically with strongersocial ties.We believe that our work opens interesting directions for future research.We have focused on a virus inoculation game, and additional insights mustbe gained by studying alternative and more general games such as potentialgames, or games that do and do not exhibit a Braess paradox. Also theimplications on the games’ dynamics need to be investigated in more detail,and it will be interesting to take into consideration behavioral models beyond28 I n o c u l a t e d P l a y e r s Player s I n o c u l a t e d P l a y e r s Players I n o c u l a t e d P l a y e r s Players
F = 0 F = 1 F = 1 rel F = 0 F = 1 F = 1 rel F = 0 F = 1 F = 1 rel
Figure 2: Number of secure players in different models using L = 16: friendship often doesnot increase the number but yields better locations of the secure players. equilibria (e.g., [38]). Finally, it may be interesting to study scenarios whereplayers care not only about their friends but also, to a smaller extent, aboutfriends of friends.What about practical implications? One intuitive takeaway of our workis that in case of large benefits of social behavior, it may make sense to de-sign distributed systems where neighboring players have good relationships.However, if the resulting convergence times are large and the price of thedynamics higher than the possible gains, such connections should be dis-couraged. Our game-theoretic tools can be used to compute these benefitsand convergence times, and may hence be helpful during the design phase ofsuch a system. Acknowledgments
We would like to thank Yishay Mansour and Boaz Patt-Shamir from TelAviv University and Martina H¨ullmann and Burkhard Monien from Pader-born University for interesting discussions on relative friendship equilibriaand aspects of convergence.
References [1] V. Anantharam. On the Nash Dynamics of Congestion Games withPlayer-Specific Utility. In
Proc. 43rd IEEE Conference on Decision andControl , 2004. 29 ( F ) N E C o s t Player s ( F ) N E C o s t Player s ( F ) N E C o s t Player s F = 0 F = 1 F = 1 rel F = 0 F = 1 F = 1 rel F = 0 F = 1 F = 1 rel
Figure 3: Box plots of social cost in different scenarios. The considered equilibria resultedfrom round-robin best response sequences starting from an initially completely insecuregraph. B e s t R e s p o n s e R o und s Convergence Time (n = 1000)
SeriesSeriesSeries F Figure 4: Box plot of number of best response rounds until convergence to FNE, startingfrom an initially completely insecure graph. [2] J. Aspnes, K. Chang, and A. Yampolskiy. Inoculation Strategies forVictims of Viruses and the Sum-of-Squares Partition Problem. In
Proc.16th ACM-SIAM Annual Symposium on Discrete Algorithms (SODA) ,2005.[3] M. Babaioff, R. Kleinberg, and C. H. Papadimitriou. Congestion Gameswith Malicious Players. In
Proc. 8th ACM Conference on ElectronicCommerce (EC) , 2007. Also appeared in
Games and Economic Behavior(GEB), Volume 67, Number 1 , 2007.[4] N. T. Bailey. The Mathematical Theory of Infectious Diseases and its30pplications. Hafner Press, 1975.[5] J. M. Bilbao. Cooperative Games on Combinatorial Structures.Springer, 2000.[6] R. Blundell, W. Newey, and T. Persson (Eds). Advances in Economicsand Econometrics (Chapter 1: The Economics of Social Networks).2006.[7] P.-A. Chen, M. David, and D. Kempe. Better Vaccination Strategies forBetter People. In
Proc. 11th ACM Conference on Electronic Commerce(EC) , 2010.[8] P.-A. Chen, and D. Kempe. Bayesian Auctions with Friends and Foes.In
Proc. 2nd International Symposium on Algorithmic Game Theory(SAGT) , 2009.[9] A. Fabrikant, A. Luthra, E. Maneva, C. H. Papadimitriou, andS. Shenker. On a Network Creation Game. In
Proc. 22nd Annual Sym-posium on Principles of Distributed Computing (PODC) , 2003.[10] J. Feigenbaum, C. H. Papadimitriou, and S. Shenker. Sharing the Costof Multicast Transmissions.
Journal of Computer and System Sciences ,63(1):21–41, 2001.[11] G. W. Flake, S. Lawrence, C. L. Giles, and F. M. Coetzee. Self-Organization and Identification of Web Communities.
Computer ,35(3):66–71, 2002.[12] C. Fleizach, M. Liljenstam, P. Johansson, G. M. Voelker, and A. Mehes.Can You Infect Me Now? Malware Propagation in Mobile Phone Net-works. In
Proc. 2007 ACM Workshop on Recurring Malcode (WORM) ,2007.[13] P. Fraigniaud, C. Gavoille, and C. Paul. Eclecticism Shrinks Even SmallWorlds. In
Proc. 23rd Annual ACM Symposium on Principles of Dis-tributed Computing (PODC) , 2004.[14] J. C. Frauenthal. Mathematical Modeling in Epidemiology. Springer-Verlag, 1980. 3115] M. R. Garey and D. S. Johnson.
Computers and Intractability : A Guideto the Theory of NP-Completeness . W. H. Freeman, 1979.[16] S. M. Kakade, M. Kearns, and L. E. Ortiz. Graphical Economics. In
Proc. 17th Annual Conference on Learning Theory (COLT) , 2004.[17] R. M. Karp. Reducibility Among Combinatorial Problems.
Complexityof Computer Computations , pages 85–103, 1972.[18] M. Kearns, M. Littman, and S. Singh. Graphical Models for GameTheory. In
Proc. Conference on Uncertainty in Artificial Intelligence(UAI) , 2001.[19] J. O. Kephart and S. R. White. Measuring and Modeling ComputerVirus Prevalence. In
Proc. IEEE Symposium on Security and Privacy ,1993.[20] J. O. Kephart, S. R. White, and D. M. Chess. Computers and Epidemi-ology.
IEEE Spectrum , 30(5):20–26, 1993.[21] J. Kleinberg. The Small-World Phenomenon: An Algorithmic Perspec-tive. In
Proc. 32nd ACM Symposium on Theory of Computing (STOC) ,2000.[22] J. Kleinberg. Complex Networks and Decentralized Search Algorithms.In
Proc. International Congress of Mathematicians (ICM) , 2006.[23] J. Kleinberg.
Algorithmic Game Theory. Chapter 24 : Cascading Be-havior in Networks: Algorithmic and Economic Issues . N. Nisan, T.Roughgarden, E. Tardos, and V. V. Vazirani (Eds), Cambridge Univer-sity Press, 2007.[24] P. Kuznetsov, and S. Schmid. Towards Network Games with SocialPreferences. , 2010.[25] J. Leskovec, L. A. Adamic, and B. A. Huberman. The Dynamics ofViral Marketing.
ACM Transactions on the Web , 2007.[26] D. Liben-Nowell, J. Novak, R. Kumar, P. Raghavan, and A. Tomkins.Geographic Routing in Social Networks. In
Proc. National Academy ofSciences , number 33, pages 11623–11628. National Acad Sciences, 2005.3227] G. S. Manku, M. Naor, and U. Wieder. Know thy Neighbor’s Neighbor:the Power of Lookahead in Randomized P2P Networks. In
Proc. 36thACM Symposium on Theory of Computing (STOC) , 2004.[28] D. Meier, Y. A. Oswald, S. Schmid, and R. Wattenhofer. On the Wind-fall of Friendship: Inoculation Strategies on Social Networks. In
Proc.9th ACM Conference on Electronic Commerce (EC) , 2008.[29] S. Milgram. The Small World Problem.
Psychology Today , pages 60–67,1967.[30] A. Montanari and A. Saberi. Convergence to Equilibrium in Local In-teraction Games.
SIGecom Exch. , pages 60–67, 2009.[31] T. Moscibroda, S. Schmid, and R. Wattenhofer. On the TopologiesFormed by Selfish Peers. In
Proc. 25th Annual ACM Symposium onPrinciples of Distributed Computing (PODC) , 2006.[32] T. Moscibroda, S. Schmid, and R. Wattenhofer. When Selfish MeetsEvil: Byzantine Players in a Virus Inoculation Game. In
Proc. 25th An-nual ACM Symposium on Principles of Distributed Computing (PODC) ,2006. Also appeared in
Journal Internet Mathematics (IM), Volume 6,Number 2 , 2009.[33] M. E. J. Newman and M. Girvan. Finding and Evaluating CommunityStructure in Networks.
Physical Review E , 69, 2004.[34] Ch. Papadimitriou. Algorithms, Games, and the Internet.
Proc. 33rdAnnual ACM Symposium on Theory of Computing (STOC) , 2001.[35] T. Roughgarden.
Selfish Routing and the Price of Anarchy . MIT Press,2005.[36] T. Roughgarden and ´Eva Tardos. How Bad is Selfish Routing?
Journalsof the ACM , 49(2):236–259, 2002.[37] C. Wang, J. C. Knight, and M. C. Elder. On Computer Viral Infectionand the Effect of Immunization. In
Proc. 16th Annual Computer SecurityApplications Conference (ACSAC) , 2000.3338] J. Wright and K. Leyton-Brown. Beyond Equilibrium: Predicting Hu-man Behavior in Normal-Form Games. In
Proc. 24th AAAI Conferenceon Artificial Intelligence (AAAI) , 2010.[39] M. Yannakakis. Equilibria, Fixed Points, and Complexity Classes. In