Friend of My Friend: Network Formation with Two-Hop Benefit

Transcription

1 Friend of My Friend: Network Formation with Two-Hop Benefit Elliot Ansheleich, Onkar Bhardwaj, and Michael Usher Rensselaer Polytechnic Institte, Troy NY, USA Abstract. How and why people form ties is a critical isse for nderstanding the fabric of social networks. In contrast to most existing work, we are interested in settings where agents are neither so myopic as to consider only the benefit they derie from their immediate neighbors, nor do they consider the effects on the entire network when forming connections. Instead, we consider games on networks where a node tries to maximize its tility taking into accont the benefit it gets from the nodes it is directly connected to (called direct benefit), as well as the benefit it gets from the nodes it is acqainted with ia a two-hop connection (called two-hop benefit). We call sch games Two-Hop Games. The decision to consider only two hops stems from the obseration that hman agents rarely consider contacts of a contact of a contact (3-hop contacts) or frther while forming their relationships. We consider seeral ersions of Two-Hop games which are extensions of well-stdied games. While the addition of two-hop benefit changes the properties of these games significantly, we proe that in many important cases good eqilibrim soltions still exist, and bond the change in the price of anarchy de to two-hop benefit both theoretically and in simlation. 1 Introdction How and why people form ties is a critical isse for nderstanding the fabric of social networks. In arios models, inclding pblic good games (see e.g., [8, 12, 17] and the references therein), stable matching (see e.g., [6, 16]), and others [5], it is often assmed that people make strategic decisions or form friendships/partnerships based on the benefit they derie from their immediate neighbors, independent of the rest of the network. On the opposite end of the spectrm, many game-theoretic models sch as [15] and its many extensions (see [7] and references therein) consider players that form a network with the goal of maximizing their inflence oer nodes that can be far away from them, i.e., caring not jst abot their local neighborhood bt abot their position in the entire network. In many settings, howeer, agents are neither so myopic as to consider only the benefit they get from their immediate connections alone, nor do they form relations considering the effects on the whole network. For example, one of the aspects people consider when forming a relationship is the two-hop benefit they can get from the friends of sch a friend. This is especially important in the world of bsiness, bt also occrs natrally when forming eeryday friendships and collaborations: we jdge people by the company they keep, and become better friends with those whose friends we like as well. Inspired by sch settings, we consider games on networks where a node tries to maximize its tility taking into accont the benefit it gets from the nodes it is directly connected to (called direct benefit), as well as the benefit it gets from the nodes it is acqainted with ia a two-hop connection (called two-hop benefit). We will call sch games Two Hop Games. Before formally defining Two Hop games, we point ot a difference between two concepts - one being the ability to form a relationship with someone, and another being the ability to extract benefit ot of a direct or two-hop acqaintance. The ability to form a relationship indicates whether two agents can interact directly with each other (de to geographical proximity, etc.) The ability to extract benefit ot of a direct or two-hop acqaintance instead tells s abot how compatible the agents are with each other. We distingish between these two concepts by haing two arbitrary ndirected graphs: Connection Graph(G C ): The edges in this graph denote which pairs of agents are able to form connections/relationships with each other. This work was spported in part by NSF awards CCF , CCF , and CNS Preliminary ersion of this paper appeared in Symposim on Algorithmic Game Theory, 2013

2 Friendship Graph(G F ): The edges in this graph indicate whether two agents are compatible with each other. If they are compatible, then they can derie benefit if they are connected either directly or ia a two-hop connection. Ths G F goerns the tility extracted from acqaintances (the formation of which is goerned by G C ). Two Hop Games: Now we will formally define Two-Hop games. Each Two-Hop game is defined by the following: We hae a Connection Graph (G C ) and a Friendship Graph (G F ) as described aboe, both haing the same node set. These nodes are the players of the game. We want to model the case where different friendships and relationships can be of different strength. Ths the strategy of a node, say, consists 1. The total contribtion that a node can make is limited by a constant k. The contribtion represents the effort pts into its relationship with. Note that we restrict the contribtions of to edges adjoining in G C, as those are the nodes that can connect to directly. We can represent the strategy of in a compact way sing a ector x =( ) with nmber of components eqal to the degree of in G C.As sal, x =(x 1,, x 1, x +1,, x n ) denotes the strategies of all other nodes except. We restrict () x = min(k, d )whered is the degree of in G C. The limit of k represents the fact that any person has only finite time/resorces at his disposal to form acqaintances, and ths can contribte at most k effort in total. The objectie of a node is to maximize its tility gien by U (x, x )= r w (x w,x w w )+ s w (,x,xw,xw w ) (1) (w) G F G C of choosing to contribte to each of its adjacent edges () ing C,withanamont0 (w) G F (),(w) G C The fnction r w (x w,xw w ) represents the strength of the direct relationship between and w: this depends only on the effort that and w pt into the relationship. The fnction s w (,,x w,x w w )represents the strength of a bond between and w formed de to a mtal friend. The strength of sch a two-hop acqaintance can potentially depend on all the intermediate efforts on the 2-link path. Ths the tility of a node is the total strength of its (direct and 2-hop) relationships with all of its neighbors in G F, i.e., the nodes who actally benefit node. Note that the two-hop benefit oer all two-hop paths between and w adds p: a larger nmber of mtal friends increases how mch people can inflence each other, a larger nmber of internal referrals increases the chances that a job-seeker gets an interiew, etc. We are interested in the following two types of Two-Hop games. Sm Two Hop Games (S2H Games): s w (,x r (, )=,xw,xw w )=α (x + (2) xw + x w w x ) (3) We call 0 α 1 the two-hop benefit factor. It represents the intitie notion that a two-hop acqaintance between and w ia shold yield less benefit than a direct acqaintance. Eqation 2 defines the strength of a relationship as the addition of strengths in each direction: strength in the direction is gien by, and in the reerse direction gien by x. Similarly the term x xw in Eqation 3 represents the strength of the two-hop acqaintance between and w ia in the direction w (e.g., how likely information is to pass from to w ia ). The term x w w is the strength of this indirect relationship in the other direction. If not for the 2-hop effect, S2H Games wold be a simple ariation of network contribtion games [5] (see Section 2 for frther details) and it is not difficlt to show that in sch a game, a Nash Eqilibrim always exists and its Price of Anarchy is 1. As we show in this paper, howeer, the addition of 2-hop benefit changes the properties of this game. Min Two Hop Games (M2H Games): s w (,x r (,xw,,xw w )=min(, )=α min(x ) (4) ) min(xw,xw w ) (5),x In M2H games, a relationship is only strong if both participants contribte a lot of effort. As before, the strength of a 2 hop effect is the prodct of the strengths of the two relationships in the 2 link path, attenated by a factor α [0, 1]. 2

3 Withot the 2-hop effect, this game is essentially a fractional ersion of k-stable matching (see Section 2 for details). As discssed in Section 2, existing work on stable matching immediately implies arios reslts abot the existence and qality of eqilibrim for sch a game. Howeer, jst as with S2H games, the addition of 2-hop benefit greatly changes the properties of this game. To assess the qality of a soltion M in S2H and M2H games, we will se social welfare, gien by U(M) = U (x, x ). For S2H games, we will focs on the existence and the qality of Nash Eqilibria (NEs). For M2H games, howeer, sing the concept of 2-strong Nash Eqilibrim, also called pairwise eqilibrim [21], makes more sense to consider than the concept of Nash eqilibrim. A pairwise eqilibrim (PE) is a soltion stable with respect to deiations by any pair of players, as well as any single player. This is consistent with preios work on sch games: if we think of integral ersions of these games (where is constrained to be in {0, 1}) as network formation games, then S2H corresponds to a game where a node can nilaterally form a link and reap the benefits of this link, while M2H corresponds to a game in which both endpoints of a link are needed to form this link. Traditionally pairwise eqilibria hae been sed to analyze the latter types of games [5,13] simply de to the fact that any single-player deiation wold not be able to create a new link. Similarly, in or fractional ersion of M2H, it is reasonable to expect for a pair of people (, ) to increase the leel of their friendship at the same time, ths increasing min(, ). Ths for M2H games, we stdy pairwise eqilibria and inestigate their qality compared to the optimal soltion. We call the ratio between the qality of the worst pairwise eqilibrim and the optimal soltion 2-PoA to differentiate it from the PoA (price of anarchy) with respect to Nash Eqilibria. Or Contribtion: We define Two-Hop games, which are natral generalizations of well-stdied games. As mentioned aboe, S2H games withot any two-hop benefit redce to simple network contribtion games; ths they are potential games, an integral NE always exists for them, and they hae Price of Anarchy (PoA) of 1. As we show in Section 3, despite the introdction of two-hop benefit, a NE always exists for general S2H games. Howeer, an integral NE may no longer exist, and S2H games are not potential games (except for some special cases: see Theorem 4). The introdction of two-hop benefit also changes the behaior of PoA. For the important special cases of G F G C (I can connect to all of my friends) and G C G F (I can only connect to friends), we show a tight PoA bond of 1 + αk, and in the ery nice case when G F and G C are complete graphs, the PoA is 1+αk 1+α(k 1). As we show in Theorem 5, in general for S2H games PoA decreases as the oerlap between G F and G C increases, i.e., PoA decreases as nodes get more opportnities to form acqaintances with nodes they are compatible with. For example, if eery node has at least k/2 nodes which are its neighbors in both G F and G C, then the PoA is at most 1 + 2αk. Note that for the most reasonable ales of α the PoA bonds aboe are rather small. For example, we can often assme that a single direct friendship brings more benefit than connecting to someone solely becase of the 2-hop contacts being made; this is qantified by assming that α 1 k since any node can hae at most k friends. For this range of α, the aboe PoA bonds become merely 2 and 3. We frther consider weighted S2H games (See Section 3.2) in which different acqaintances can potentially yield different intrinsic benefit, and show that the reslts obtained for S2H games also hold for weighted games when G F G C. Becase of its connection to many-to-many stable matching, it is not difficlt to show that for M2H games withot 2-hop benefit an integral pairwise eqilibrim (PE) always exists, and 2-PoA is at most 2. For general M2H games with 2-hop benefit, howeer, we show that an integral PE may not exist (existence of a fractional PE for this and related games is an important open qestion). For the cases when PE does exist, or main reslt for M2H games proes that 2-PoA for the important case of G C G F is at most 2+2αk, which for the reasonable range of α [0, 1/k] mentioned aboe ealates to at most 4. For weighted ersions, we also carried ot simlations by scattering nodes niformly in a nit sqare and experimented with different classes of weight fnctions which depend on the distance between the nodes. We fond that althogh the worst-case PoA bonds cold be qite high, the aerage qality of eqilibria was ery close to the optimm. We also fond that althogh integral NE may not exist for S2H games, in or simlations for majority of the instances it did exist, and simple dynamics conerged to it extremely qickly in almost all instances. Or simlations also showed that as two-hop benefit decreases, nodes transition from forming small interconnected clsters to forming more of a backbone tree-like network. Remark: Instead of assming a limit of k on the contribtions that each node can make, we can instead consider the case when the limits are different for each node, say k for node. In this case all of or reslts 3

4 hold with k replaced by max k, except the cases 1 and 3 of Theorem 5, where we will mention the difference in the bonds in the discssion following the proof. 2 Related Work and Impact of Two-Hop Benefit Network formation games, and games on networks more generally, hae been stdied extensiely. In many network formation games, nodes connect to each other with the goal of maximizing their tility, which depends on their position in the whole network. For example, it may depend on the aerage distance to the rest of the nodes, or on arios other notions of centrality and node importance. [15] stdies the setting where nodes try to minimize their aerage distance to the other nodes by bilding network links, howeer the fixed link cost for each link they decide to bild acts as a damping factor. [15] goes on to analyze how the existence and the qality (Price of Anarchy) of Nash eqilibria aries with respect to the connection cost of establishing each new link. The athors in [13] analyzed the bilateral connection ersion (i.e., bilding a link reqires both endpoints to cooperate) of the aboe model in [15] and bonded the efficiency of pairwise (2-strong Nash) eqilibria along with the range of link costs to garantee the existence of sch eqilibria. The athors in [22] consider the scenario where each node has a bdget for prchasing links and different preferences for connecting to different nodes, with nodes trying to prchase adjacent edges to connect to the nodes that matter the most. The ersion considered in [7] is similar to [22] except that the nodes aim for maximizing their betweenness centrality. For frther examples of the approach where nodes hae the goal of maximizing their tility based on their position in the whole network, see [3, 9, 14, 19] and the references in [7]. On the other hand, in many models of the network formation games, the agents are concerned only abot the direct benefit they derie from their immediate neighbors. One sch class of models is stable matching and its ariants. Stable matching was introdced in [16] nder the basic setting of nodes partitioned into two sets (i.e., a bipartite graph), with each node haing preferences oer the nodes from the other set, and each node desiring to get matched (i.e., get assigned) to at most one node from the other set as high as possible in its preference list. The stable matching problem inestigates the possibility of stable otcomes (called stable matchings) in this setting, where stability corresponds to no two nodes prefering each other oer their present partners. Different ariants of stable matching problems hae been stdied intensiely oer the last few decades. On the algorithmic side, existence, efficient algorithms, and improement dynamics for stable matchings oer bipartite graphs hae been of interest (for references, see standard textbooks sch as [18, 23, 25]). Many-to-one and many-to-many ersions of stable matching hae also been stdied in the literatre and it has been shown that stable matchings exist when the nderlying graph is bipartite [6,26,27]. Another example of network formation games where the agents are concerned only abot the benefit deried from their immediate neightbors are network contribtion games introdced in [5]. Here athors stdy 2-strong Nash eqilibria nder the setting of each node haing a constraint on the total bdget it can inest on adjoining edges in order to maximize its tility. For more examples, see [2, 4, 8, 11, 12, 17]. Howeer, in this paper we are interested in settings where agents may neither be concerned abot the actions of remote nodes nor be so myopic as to consider only the benefit they derie from their immediate neighbors. Two-Hop games fall nder this category. The decision to consider only two hops stems from the obseration that hman agents rarely consider contacts of a contact of a contact (3-hop contacts) or frther while forming their relationships. As mentioned aboe, we distingish between the ability of a pair of agents to interact directly (represented by the connection graph G C ) and their capability of being able to derie benefit if they are connected directly or ia a two-hop connection (represented by the friendship graph G F ). The friendship graph G F can be seen as a social context which dictates the benefits obtained by the nodes by playing a game on the connection graph G C. Some other work which explores different forms of social context are [10] and [20]. In this work, the cost of a node in a resorce-sharing game depends on its own cost and the costs of its friends, where friend nodes are its neighbors in an nderlying social network. Let s describe the impact that introdcing two-hop benefit with social context can hae. Withot two-hop benefit, the tility of a node in S2H games becomes U (x, x )= ( () () G C G F + x ) (6) 4

5 This is most closely related to one of the models for network contribtion games considered in [5] where the athors also consider that the tility obtained by a node from contribting an edge is gien by c ( + ) where c > 0 is an edge-specific constant. S2H games withot two-hop benefit can be seen as a ariation of this game where we hae c = 1 if () G C G F and 0 otherwise, howeer we place an additional pper bond of 1 on s along with a limit () x = k. In [5], when the tility fnction of a node is gien by () c ( + ), the athors proe that a (integral) Nash Eqilibrim always exists and that the PoA is 1. Here by integral NE we mean that s are restricted to {0, 1}. Althogh we place an additional constraint of 1, sing similar techniqes it is easy to show that for S2H games with α = 0 (i.e., no two-hop benefit) an integral NE always exists and PoA is 1. With addition of two-hop benefit, we can iew S2H games as an extension of sch network contribtion games: we show that a NE still exists (althogh it may not be integral), and that the PoA does not increase by too mch. For M2H games, if we do not hae any two-hop benefit then the tility of a node becomes U (x, x )= = min(,x ) (7) () G F G C () G F G C For k = 1, the ersion of this with edge weights becomes eqialent to stable matching. More precisely, it becomes the correlated ersion of stable matching [1] for which a stable matching is known to exist for arbitrary graphs, and the 2-PoA (qality of stable matching compared to the optimm one) is bonded by 2 [4]. For k>1, this becomes a many-to-many ersion of stable matching, where each node is allowed to match with k partners. Many-to-one and many-to-many ersions of stable matching hae been stdied in the literatre and it has been shown that stable matchings exist when the nderlying graph is bipartite [6,26,27]. In fact, many bilateral network formation games can also be interpreted in the stable matching framework. Howeer, to the best of or knowledge the correlated ersion of many-to-many stable matching has not been stdied before. Neertheless, it is easy to show sing the techniqes from [4] that existence of integral stable matching and the same bond on 2-PoA still holds. This een holds for fractional stable matching, by which we mean that a node is allowed to choose an adjacent edge with a fractional amont 0 1sch that () x = k, and an edge () is present in a matching with fraction min(, ). This is a precise generalization of the sal notions of stable matching and pairwise eqilibrim, since a single node can destroy or weaken an edge, bt both endpoints are reqired to form or strengthen an edge. With the introdction of two-hop benefit, an integral pairwise eqilibrim may no longer exist; howeer we show that the qality of pairwise eqilibrim remains good in the instances when they do exist. 3 Sm Two Hop Games Recall that in S2H games, the tility U (x, x ) of a node is obtained by sbstitting Eqation (2) and (3) into Eqation (1) which gies s U (x, x )= ( () G C G F + x )+α (),(w) G C s.t. (w) G F ( xw + x w w x ) (8) We introdce some more notation which will proe sefl later. Define U ot (x, x )andu in (x, x )as: Note that U ot (x, x )= U in (x, x )= () G F G C () G F G C + α (w) G F (),(w) G C + α (w) G F (),(w) G C x w w x w (9) x (10) U (x, x )=U ot (x, x )+U in (x, x ) (11) 5

6 U ot(x, x ) can be interpreted as the share of the tility of node that gets by irte of its own contribtions, i.e., this is the share of tility U (x, x ) in which the contribtions made by play a role. Similarly, U in(x, x ) can be interpreted as the share of the tility U (x, x ) independent of the contribtions made by. Let s define U ot (M) for any soltion M as U ot (x, x )andu in (M) as U in (x, x ). Frthermore, it can be obsered that U(M) =2 U ot (M) =2 U in (M) (12) Withot any two-hop benefit, there always exists an integral pre NE for S2H games (i.e., a NE in which all the contribtions are either 0 or 1) and they are exact potential games. Integral NE s represent an important combinatorial case where the contribtions of the agents take fixed ales, i.e., either I am or I am not a friend of someone. Sch soltions make sense in some applications in which a relationship either exists or does not (e.g., haing a physical network link between two parties), bt for less tangible relationships sch as friendships, fractional soltions make sense as well. For or game, een after introdcing two-hop benefit (i.e., α>0), we can proe that a NE always exists for S2H games sing Proposition 20.3 of [24]. Howeer, all NE may be fractional, and this ceases to be a potential game. Theorem 1 For S2H games, a pre Nash Eqilibrim always exists. Proof. We begin the proof by defining some notation. Let d denote the degree of node in G C. If 1, 2,..., d are neighbors of in G C then the strategy space of is a set of d -dimensional ectors gien by: S = {x : 0 x 1 and ( i) i =min{k, d }} (13) where by 0 and 1 we denote the ectors with all components zero and one respectiely. Let S denote S 1 S 2... S n. Define a preference relation for a node on the set S as follows: For x, y S, we hae y x wheneer the tility of the node in the otcome y is at least as mch as its tility in the otcome x. We will proe that a NE exists for S2H games sing Proposition 20.3 of [24]. Proposition 20.3 of [24] states that we hae a (pre) Nash Eqilibrim wheneer the strategy space S of each node is a non-empty, compact, conex set and the preference relation continos and qasi-concae on S for each node. We will explain the terms as we will enconter them in the proof. We will proe the existence of NE by showing that each of these conditions hold for S2H games. S is a non-empty, compact and conex set for each node : If there are no isolated nodes in G C, as we hae assmed already, then the set S is not empty as seen from Eqation (13). Since S is also a closed and bonded set, it is also a compact set. From Eqation (13), we can also see that it is a conex set. The preference relation is continos for each node : Let {x k } k=0 denote a seqence of strategy profiles x1, x 2, with each strategy profile in the seqence {x k } k=0 belonging to set S. Now let s define by what is meant by preference relation for a node being continos on S (definition taken from Section 1.7 of [24]). Sppose there are two strategy profile seqences {x k } k=0 and {yk } k=0 conerging to x S and y S respectiely sch that xk y k for all k. If this also implies x y for any two sch seqences of strategy profiles then the preference relation is said to be continos. Recall that x k y k means U (x k ) U (y k ). Now for S2H games, the tility fnction U (x) is continos in x. This continity implies x y where x, y S are limits of two seqences {x k } k=0 and {yk } k=0 sch that xk y k for all k. Ths for any node, the preference relation is continos. The preference relation is qasi-concae on S : Let s define B (x, x )={y S :(y, x ) x}. Informally,ifwefixx then the set B (x, x )isthesetofstrategiesfor for which gets at least as mch tility as it gets by playing x. The preference relation is said to be qasi-concae on S if B (x, x ) is conex for each x S. Thstoproethat is qasi-concae on S,wehaeto 6

7 proe that if (y, x ) x and (z, x ) x hold tre then (λy +(1 λ)z, x ) x holds tre (where 0 λ 1). This follows from the linearity of U (x, x )inx gien x, i.e., U (λy +(1 λ)z, x ) = (λ(y () G C G F + (),(w) G C s.t. (w) G F [αλ(y + )+(1 λ)(z + )) + x w x w w )+α(1 λ)(z + x w x w w )] = λ U (y, x )+(1 λ) U (z, x ) λ U (x)+(1 λ) U (x) U (x) (14) Hence the preference relation is qasi-concae on S for each node. Ths we hae shown that for each node, its strategy space S is a non-empty, compact, conex set and the preference relation is continos and qasiconcae on S. Ths by proposition 20.3 of [24], a (pre) NE exists for S2H games. Theorem 2 There are instances of the general S2H game which do not admit any integral pre Nash eqilibrim. Proof. We will describe the constrction of an instance of S2H games for arbitrary k sch that an integral NE does not exist for sch instances. Figre (1) shows sch an instance bt for k = 2. To constrct an instance for arbitrary k, consider three nodes,, w sch that the edges (), (w) and(w) existing C bt not in G F, i.e. (), (w), (w) G C \ G F. We will call the triangle formed by (), (w), (w) asthe central triangle. The node is frther connected to k nodes otside the central triangle in G C,withthese edges also belonging to G F. We will call these nodes as satellite nodes of (e.g., in Figre 1, nodes 1 and a are satellite nodes of ). Similarly and w will hae their own k satellite nodes. Among these satellite nodes, we select nodes 1,, k 1 and connect them frther to k nodes sing edges in G C G F. We will call these k nodes as sbordinate nodes of the satellite node nder consideration (e.g., Figre 1, satellite node 1 has y and z as its sbordinate nodes). Each node of the central triangle is connected in G F to all the sbordinate nodes of its satellite nodes, howeer these edges do not belong to G C. Note that there exists a special satellite node for each node of the central triangle sch that it has no sbordinate nodes. For example satellite node a is sch a node corresponding to. Each ertex of the central triangle is frther connected in G F to satellite nodes of the ertex of the central triangle in clockwise direction. These edges belong to G F bt not in G C. For example, node is connected to satellite nodes of (only in G F bt not in G C ). To smmarize the notation for Figre (1), the solid edges belong to G C G F, semi-solid edges belong to G C and dotted edges belong to G F.Wechoose1/k < α < 1/(k 1). Now let s proe that an integral NE does not exist for sch a constrction. In this proof, by = 1. Since we are considering only integral contribtions, a satellite node, say 1,hasto choose k edges for its contribtions from k + 1 adjoining edges. Ths 1 makes a fll contribtion to at least k 1 sbordinate nodes of 1. Now we claim that a ertex of the central triangle, say, always contribtes 1 to the edges leading to all its satellite nodes, except a, in any candidate soltion for NE. To proe this we will mean =0then can always = 1 by making its contribtion to some other edge 0. To see it, notice claim, we show that if there is any satellite node, say 1 (except a), sch that x 1 increase its tility by making x 1 that: Node cannot contribte to (a) and not to ( 1 ) becase in sch a case, by making x a = 0 node can lose tility by 1 and it can more than compensate for this loss by making x 1 = 1 as it increases its tility by at least 1 + α(k 1).Thisisbecasenow obtains an additional direct benefit of 1 by contribting to ( 1 ) and also gets the two-hop benefit of α(k 1) obtained from the two-hop paths of natre 1 p where p is a sbordinate node of 1 as 1 contribtes 1 to at least k 1ofits sbordinate nodes. Node wold rather prefer contribting to ( 1 ) instead of (w) becase node does not obtain any tility by contribting to (w) as there is no direct or two-hop benefit to be obtained by contribting 7

8 y 1 z a Edges in G F G C Edges in G C \ G F Edges in G F \ G C c w b Fig. 1: An integral NE does not exist for S2H game. Example shown for k =2.Note1/k < α < 1/(k 1). to (w). There is no benefit for in contribting to (w) becase (w) / G F and no node connected to w in G F (solid or dotted edges) has a two-hop path from in G C.Ths can always make x 1 =1 instead of contribting to (w) and get the additional tility of 1 + α(k 1) as explained in the preios case. Sppose node contribtes 1 to edge () and not to ( 1 ). Then in this case by redcing contribtion to () to0,node can lose only two-hop benefit as () G C \ G F, ths there being no direct benefit obtained by contribting to (). The two-hop benefit that node can lose is at most αk becase of destroying paths of the natre p since there are at most k sch paths. Ths, if remoes its contribtion from () and instead contribted to ( 1 ) then it gets a tility of 1 + α(k 1) (as explained in the preios case) which more than compensates for its loss of at most αk tility de to or assmption that 1/k < α < 1/(k 1). Ths node makes a contribtion of 1 to the edges leading to all of its satellite nodes except a in any candidate soltion for NE. Similarly, node (and node w) makes a contribtion of 1 to the edges leading to all of its satellite nodes except b (except c) in any candidate soltion for NE. Ths to examine the possibility of NE, we only need to examine where the ertices of the central triangle make their remaining one contribtion. In fact, the only choices for the remaining contribtion for node are edges (a) and() (and not edge (w)) becase it can always choose to contribte 1 to edge (a) to get a direct benefit of 1 instead of contribting to edge (w) from which it gets no tility as arged aboe. Similarly the only choices for the remaining contribtion for node (node w) areedges(b), (w) (edges(w), (wc)). Now we will show that no combination of these choices is possible in a candidate soltion for NE. For conenience we will call edges (), (w), (w) as the edges of the central triangle. Consider the following cases: chooses (a), chooses (b)andw chooses (wc). In sch a case, wold prefer to remoe its contribtion from (a) tocontribteto() instead. This is becase in this process gains the two-hop benefit of αk > 1 from the satellite nodes of which is more than the tility of 1 that it loses by remoing its contribtion to (a). Sppose none of the edges (a), (b), (wc) were chosen for the remaining contribtions, i.e. chooses (), chooses (w), w chooses (w). In this sitation, by contribting to (w), node gets an tility (only two-hop benefit) of α(k 1) by irte of establishing two-hop paths to (k 1) satellite nodes of w where w makes its (k 1) contribtions (i.e., nodes w 1,,w k 1 ). Howeer, by remoing its contribtion to (w) and instead choosing to contribte to (b) gets an tility of 1 which is more than the tility of α(k 1) it loses by remoing its contribtion to (w).thisisbecasewehaeassmedα<1/(k 1). 8

9 Other then the aboe two cases, at least one and at most two edges from (a), (b), (wc) arechosenfor the remaining contribtions by nodes,, w. To coer both these cases, withot the loss of generality assme that (a) is chosen and (wc) is not chosen whereas (b) may or may not be chosen. This means that chooses (a) andw chooses (w) to contribte. Now if has chosen (b) then as explained in the first case aboe, can remoe its contribtion to (a) and choose to contribte to (w) instead, increasing its tility in this process. Otherwise, if has chosen (w) to contribte then can remoe its contribtion from (w) to contribte to (b) instead to increase its tility as explained in the second case aboe. Ths we hae shown that none of the combinations of choices for the remaining contribtions of nodes,,w can lead to an integral NE, which was the only step remaining to be proen for non-existence of integral NE in this constrction. Hence we hae proen that there may not exist an integral NE for S2H games. Theorem 3 The general S2H game is not a potential game. Proof. We gie an instance where better-response dynamics can cycle, ths establishing that this is not a potential game. Figre 2 shows sch an instance of S2H games. In this instance, we hae k = 2. Now we will describe the constrction and later proe that sch a cycle of states exists for this instance. 1 2 b 2 w w 1 w 2 1 a Edges in G C \ G F Edges in G F \ G C q p Fig. 2: Example showing that the S2H game is not a potential game. Example shown for k = 2. Note that this is a partially drawn figre: not all the adjoining edges of the nodes of the central pentagon are shown except for the node. Also, both the nodes p and q hae a sbgraph attached to them otside the central pentagon, analogos to sbgraph formed by the nodes 1, 2, a, b arond the node.the complete example is a symmetric figre. In this example, it is possible for the nodes to exhibit a cycle of better responses, ths leading to the conclsion that the S2H game is not a potential game. Figre 2 consists a cycle of ertices p q w p. WecallthiscycleC 1. The total nmber of ertices in C 1 is 5 (althogh the proof works with C 1 ha+ing any odd nmber of ertices greater than 3.) We frther stitch one more cycle C 2 throgh these ertices by connecting each ertex to a ertex two positions ahead of it in C 1. Since we hae chosen the total nmber of ertices in C 1 as an odd nmber greater than 3, this process ensres that connecting this way indeed creates another cycle. All the edges of C 1 and C 2 are in G C \ G F. Each of the ertices of C 1,say, is connected in G C \ G F to two special nodes 1, 2 which we call as satellite nodes of. Similarly other ertices of C 1 hae their own satellite nodes with analogos notation (not all of these are pictred in the figre). Each satellite node is frther connected in G C \ G F to another 9

10 node (for example 1 is connected to a) which we will call sbordinate nodes of the satellite node nder consideration. So, the total nmber of nodes in the example is 5 5 = 25. Now let s describe the edges in G F.EachertexofC 1 is connected in G F \ G C to sbordinate nodes of its satellite nodes. For example, is connected to a and b. Inaddition,eachertexofC 1 is connected in G F \ G C to satellite nodes of two ertices of C 1 in clockwise direction, for example, node q is connected to 1, 2, 1, 2 (again, not all sch edges are shown in the figre). In Figre 2, the edges that are in G C \ G F are shown in solid pattern and the edges that are in G F \ G C are shown in dotted pattern. Note that since G F G C =, no node can obtain any direct benefit from connecting to a neighbor. Let s choose the initial strategies of the nodes. We choose the strategies of satellite and sbordinate nodes as follows: For the lack of choice, 1 contribtes 1 to both ( 1 )and( 1 a). Let eery satellite node follow analogos strategies. Consider a sbordinate node, say a. Again for the lack of choice, a contribtes 1to( 1 a). Let other sbordinate nodes follow analogos strategies. We choose the initial strategies of the ertices of C 1 as follows: let all the ertices of C 1 except p and q contribte 1 to the edges leading to their satellite nodes. Howeer, let node q contribte 1 to edges (q), (q) andletnodep contribte to edges (pp 1 ), (p). Let s call this initial state as S 0. Beginning with S 0, we will show that better-response dynamics can lead to a cycle of states. In the first state transition, let remoe its contribtions from edges ( 1 ), ( 2 ) and contribte 1 to (), (w) instead. Obsere that if completely remoes its contribtion from the edge ( 1 )thenit loses an tility of α. This is becase it loses the two-hop benefit obtained from the path 1 a. Howeer if choosestocontribteto() then it increases its tility by 2α becase of the the two-hop benefit obtained from the paths 1 and 2. Using this, we conclde that if remoes its contribtions from edges ( 1 ), ( 2 ) and contribte 1 to (), (w) instead, then increases its tility by 2(2α α) =2α. Let s call this state S 1. Howeer, the transition from S 0 to S 1 destroys all the benefit node p had been getting in S 0 by contribting to edge (p). This is becase in state S 1 node p does not obtain any direct benefit by contribting to (p) as(p) G C \ G F and it also does not obtain any two-hop benefit in S 2 by contribting to (p) as the node does not contribte to any edge (t) s.t.(t) G F (See Eqation (8) to see how two-hop benefit is compted). Rather than getting no benefit by contribting to (p), node p wold moe this contribtion to edge (pp 2 ). This is becase in this process p increases its tility by α de to a two-hop benefit of α obtained by establishing two-hop path directed to the sbordinate node of p 2 ia p 2.Forthe similar reason, node q wold prefer moing its contribtion from edge (q) toedge(qq 1 ). Ths in the next transition, nodes p and q moe their contribtions from the edges leading from them to to edges (pp 2 )and(qq 1 ) respectiely. Let s call this state S 2. Notice that the state S 2 is isomorphic to S 0 in the sense that in S 2 nodes q and play the roles that p and q respectiely played in S 0. Ths we hae shown that better-response dynamics can cycle in S2H games, and ths S2H games are not potential games. Howeer, we now gie a family of instances for which S2H games are exact potential games and an integral NE exists. Proing that the total tility changes in a fixed proportion to the change in the tility of anode when changes its strategy is the crcial component of proing that a game has an exact potential fnction. Ths to proe that a family of S2H games has an exact potential fnction, it wold be sfficient to ensre that when a node changes its strategy, the change in the tility of neighbors is proportional to the change in the tility of node. This occrs, for example, if G F is the complete graph, since then all nodes are affected by all others, and ths the change in tility for me is similar to the change in tility for the other nodes in the graph. In fact, as we show in the following Theorem 4, it is een enogh that G F always joins all pairs of nodes which hae a potential two-hop path in G C. In other words, if we let d C (, ) denote the distance between and in G C, then the following holds. Theorem 4 If d k for all nodes and if the condition d C (, ) 2 implies () G F for all the pairs of nodes then the S2H game is an exact potential game and an integral NE exists. 10

11 Proof. First we will proe the potential game part of the theorem. The condition d C (, ) 2 implies () G F tells s that () G C () G F (15) (), (w) G C (w) G F (16) Ths the tility of node gien by Eqation (8) takes the following simpler form where the smmations jst depend on G C : U (x, x )= ( + )+α ( x w + x w w ) (17) () G C (),(w) G C w We hae introdced the ersions of U ot(x, x )andu in(x, x ) in Eqation 9 and 10. The ersions of these qantities after applying the constraints gien by Eqation (15) and (16) are: U ot (x, x )= () G C + α U in (x, x )= = min(k, d )+α () G C + α (),(w) G C w (),(w) G C w (),(w) G C w x w (18) xw (19) x w w (20) We hae also defined U ot (M) = U ot (x, x ). Note that Eqation (18) represents that part of the tility of (gien by Eqation (17)) which depends on the contribtions made by, whereas Eqation (20) is the remaining part which does not depend on contribtions made by. ThsinaNE,anode mst choose a strategy that maximizes U ot(x, x ) gien the contribtions of other nodes. Also, sing Eqation 12 we hae U(M) =2 U ot (M). Ths in order to look at the behaior of U(M) after a node changes its strategy, it is sfficient to look at the behaior of U ot (M) andintrnatu ot (x, x ) for eery node. Notice that U ot(x, x ) depends only on the contribtions of and its neighbors. Ths when the node p changes its strategy, U ot(x, x ) changes only for p and for those nodes which are neighbors of p. Let set N(p) denote the set of neighbors of p. Now let s inestigate how U ot (x, x ) changes for p and the nodes in N(p): First let s consider node p. Sppose p changes its strategy from x p to y p. Using Eqation (19), we get the following: Up ot (y p, x p ) Up ot (x p, x p )= αyp p x w αx p p x w (p),(w) G C w p = N(p) = N(p) = N(p) αy p p α(y p p α(y p p w N() w p x w (p),(w) G C w p N(p) αx p p x p p ) (min(k, d ) x p ) xp p ) min(k, d )+ N(p) w N() w p x w α(x p p yp p )xp Also note that the theorem statement assmes that min(k, d ) = k for eery node. Ths the first smmation in the aboe eqation redces to zero giing s Up ot (y p, x p ) Up ot (x p, x p )= α(x p p yp p )xp (21) N(p) 11

12 Now consider the nodes in N(p). We se Eqation (19) to sm the U ot(x, x )oeralltheneighbors of before and after p changes its strategy. The terms that change are the terms which depend on the contribtions of p. Hence the cmlatie change in the tilities of node in N(p) afterp changes its strategy is gien by: Howeer z N(p),z ypz p N(p) αx p = k y p p z N(p),z y pz p N(p) αx p and similarly z N(p),z xpz p eqation, the change in the ale of cmlatie U ot N(p) αx p (k yp p ) N(p) z N(p),z x pz p = k x p p (x, x ) of the neighbors of p is gien by αx p (k xpp )= N(p). Using this in the aboe αx p (xp p yp p ) (22) As arged before, U ot (x, x ) changes only for p and its neighbors. Ths sing Eqations 21 and 22 we conclde that U ot (M) increasesby2 (Up ot (y p, x p ) Up ot (x p, x p )) and ths U(M) increasesby 4 (Up ot (y p, x p ) Up ot (x p, x p )) sing Eqation 12. This proes that if d k for all nodes and if the condition d C (, ) 2 implies () G F for all the pairs of nodes then the S2H game is a potential game and U(M) andφ(m) =U(M)/4 is an exact potential fnction. Now we will proe how the existence of potential fnction implies the existence of an integral NE in this case. Notice that the aboe analysis also applies erbatim if we restrict the contribtion ariables to take the ales only from the set {0, 1}. Hence a potential fnction exists and implies the existence of an integral NE for the integral ersion of the game. Let s denote this integral NE by M. We will now show that this integral NE is also a NE if we allow the contribtions to be fractional. To proe this, we show that in the fractional case, gien x, there exists at least one strategy x for node sch that the contribtions of are integral and choosing x maximizes the tility of node. ThsinM, if cold improe its tility when fractional contribtions are allowed then in fact cold hae also improed its tility by choosing a strategy in which contribtions are integral. This wold contradict or initial assmption of M being an NE for the integral ersion of the game and in trn proe that M is also an NE when fractional contribtions are allowed. Ths now the only part that is to be proed is to show that in the fractional case, gien x, there exists at least one strategy x for node sch that the contribtions of are integral and choosing x maximizes the tility of node. As mentioned before, in a NE M, a node mst choose a strategy that maximizes U ot(x, x )gien x. From Eqation (18), U ot (x, x ) can alternatiely be expressed as U ot (x, x )= (1 + x w ) (23) () G C (w) G C w Let s denote (w) G C,w xw by c. Notice that since x are fixed, the terms c s are constants. Ths applying the constraints of S2H games, the problem of maximizing U ot(x, x )for becomes a problem of choosing a strategy x which soles the following optimization problem maximize (),() G C s.t. 0 1and () c = min(k, d ) It is easy to see that if we sort the elements in {c :(), () G C } in descending order and choose the top min(k, d ) terms from the sorted array and set the contribtions s of node corresponding to these terms as 1, with other contribtions set to 0 then this strategy soles the aboe problem. Note that this strategy is integral, i.e., the contribtions are chosen from the set {0, 1}. As arged before, gien fixed x, the strategies that maximize U (x, x ) are exactly the strategies that maximize U ot(x, x ), and ths we hae shown that there exists a strategy x that maximizes U (x, x ) sch that all the contribtions of are integral. As discssed before, this was the remaining step to be proen in order to show that M is also an NE when fractional contribtions are allowed, and hence we hae proed or claim. 12

13 3.1 Price of Anarchy To begin with, we gie a qick oeriew of the reslts of this section. We know that withot any two-hop benefit, the PoA of S2H games is 1. We will first show that with two-hop benefit, PoA can become nbonded for arbitrary G F and G C if G F G C =. Howeer we will later proe that as the oerlap between G F and G C increases then the PoA for S2H games decreases and for the interesting cases of G F G C and G C G F, PoA becomes 1 + αk. Increasing the oerlap between G F and G C can be interpreted as nodes getting more opportnities to become directly acqainted with the nodes they are compatible with. Claim: For S2H games, PoA can be infinite if G F G C =. z Edges in G C \ G F Edges in G F \ G C w Fig. 3: The example showing that the PoA for S2H games can be infinite when G C G F = φ. Proof. Figre 3 shows an instance of S2H games sch that G F G C = and the PoA is infinite. In this instance, we hae k =1,andthreenodes,, w are connected to a central node z in G C. Howeer, G F consists of only (). Ths G F G C =. For analyzing this example, we will take p q to mean x pq p =1. It can be erified that wheneer we hae z w, z and z then it is a NE and has zero tility. Howeer an optimm soltion is z,z, w z which has tility 2α. Ths the PoA is infinite for this instance. Now we will show that as the oerlap between G F and G C increases then the PoA for S2H games decreases and for the interesting cases of G F G C and G C G F, PoA becomes 1 + αk. First, we formally qantify what we mean by oerlap between G F and G C.LetF denote the degree of in G F G C. We define oerlap between G F and G C as ρ(g F,G C )=min F. We now gie PoA bonds for seeral interesting cases: Theorem 5 For the S2H game, k 1. For arbitrary G F and G C,PoA 1+αk min(k,ρ(g F,G C)). Ths when there is a large oerlap between G F and G C,sayρ(G F,G C ) k/2 then we hae PoA 1+2αk. 2. Frthermore, if G C G F or G F G C then PoA 1+αk. 3. For the special case of G F = G C = K n, we hae PoA = 1+αk 1+α(k 1). First, we gie a brief otline of the proof of Theorem 5. For arbitrary G F and G C, the PoA bond will follow by simple obserations on the minimm tility obtained by a node in a NE and its maximm attainable tility. This PoA bond can be large for a small oerlap becase een if a node is capable of getting little direct benefit becase of its small degree in G F G C (which is a lower bond on minimm tility obtained by it in a NE), it can still get a large two-hop benefit (hence large maximm attainable tility) by connecting to a lot of its friends ia G C \ G F. Howeer this changes in G C G F becase G C \ G F =. This also changes with G F G C becase here if a node can get a large two-hop benefit by connecting to a lot of friends in G C then it mst hae a lot of frinds, and therefore its degree in G F G C mst be high. Ths these cases reslt in a mch improed bond on PoA regardless of the oerlap size. Now we will proceed to proe each of the aboe cases in details. 13

14 Proof (Theorem 5). Using Eqation 12, PoA can be expressed as PoA = U ot (OPT ) max Mis a NE U ot (M) (24) Recall that we hae defined F as {() :() G F G C }, i.e. F qantifies the degree of in G F G C. Let s define q as min(k, F ). Since F is at least as mch as the oerlap ρ(g F,G C ), we hae: Lemma 1 In any NE, U ot (x, x ) q. q = min(k, F ) min(k, ρ(g F,G C )) (25) Proof (Lemma 1). Notice from Eqation (10) that U in(x, x ) is independent of the strategy chosen by. Ths a node in a NE mst choose a strategy which maximizes U ot.nowifu ot (x, x ) <q in some NE M then can simply make =1foranyq edges from the set {() :() G F G C } to increase U ot (x, x ) and in trn increasing its tility. This contradicts M being an NE, hence proing that U ot (x, x ) q in any NE. Haing proed Lemma 1, now let s contine the proof of Theorem 5. Let s consider the first case, i.e., sppose G F and G C are arbitrary. Consider a NE M. We already hae U ot (x, x ) q in any NE M. Now let s calclate the maximm ale that U ot(x, x ) can take in any soltion. An pper bond on the ale of the two-hop benefit component of U ot(x, x ) in any soltion is gien by α x w α [ x w ] α k αk 2 (26) () G C (w) G C () G C (w) G F (),(w) G C Ths the two-hop benefit component of U ot (x, x )inanysoltioncanbeatmostαk 2. We also know that the direct benefit component of U ot(x, x )inanysoltioncanbeatmostq sing the definition of q.thswegetu ot(x, x ) q + αk 2 in any soltion. We already know that U ot(x, x ) q in a NE sing Lemma 1. The ratio of these two bonds is 1 + αk k/q. Using Eqation 25, we get that this ratioislessthan1+αk k/ min(k, ρ(g F,G C )). Using this in Eqation 24, we get PoA 1+αk k min(k, ρ(g F,G C )) (27) NowspposewehaeG C G F. Let s compte an pper bond on the ale of U ot(x, x )inany soltion. When G C G F, an pper bond on the two-hop benefit component of U ot (x, x )inanysoltion is gien by α (w) G F (),(w) G C x w α [ () G C (w) G C x w ] αk () G C Bt since G C G F, eery adjoining edge of in G C is also in G F.Ths () G C (28) q. Using this in (x, x )inanysoltion (x, x ) q sing the (x, x ) q (1 + αk) in any soltion (in particlar an optimm soltion). We (x, x ) q in a NE from Lemma 1. Combining these obserations, we hae Eqation (28), we conclde that the ale of two-hop benefit component of U ot can be at most αk q. We already know that direct benefit component of U ot definition of q.thsu ot also know that U ot (w) G F PoA 1+αk (29) Now sppose G F G C. Let s compte an pper bond on the ale of U ot(x, x ) in any soltion. Now the ale of the two-hop benefit component of U ot (x, x ) in any soltion can be bonded as follows: α x w α α k = αf k (30) (),(w) G C (),(w) G C (w) G F (w) G F 14

15 Combining this with Eqation (26), we get that when G F G C the two-hop benefit component of U ot(x, x ) in any soltion is pper bonded by min(αk 2,αkF ) which can also be expressed as αkq sing Eqation (25). (x, x ) q in any soltion sing the definition (x, x ) q (1 + αk) in any soltion (in particlar an optimm soltion). We (x, x ) q in a NE from Lemma 1. Combining these obserations, we hae We already know that direct benefit component of U ot of q.thswehaeu ot also know that U ot PoA 1+αk (31) When G F = K n and G C = K n,wehaeq = min(k, n 1). The ale of U ot (x, x )inanecanbe lower bonded by U ot (x, x )= () G F G C q + α q + α + α () G C () G C (w) G F (),(w) G C (w) G C,w (k 1) x w x w q + αq (k 1) (32) When G F = G C = K n, for an pper bond on U ot(x, x ) in any soltion (in particlar an optimm soltion), we can se the pper bond of q (1 + αk) wehadderiedonu ot (x, x )forg F G C,since G F = G C = K n is a special case of G F G C. Combining this pper bond with Eqation (32), we get PoA 1+αk 1+α(k 1) (33) Note that if we hae different bdget constraints k on the total contribtions of different nodes, instead of a single bdget k for eeryone, then almost all of or reslts hold with k replaced by k max =max k. The only exceptions are that the bond for S2H games with arbitrary G F and G C becomes 1 + αk max k max min(k min,ρ(g F,G C)) and the bond when both of them are complete graphs changes to (1 + αk min)/(1 + α(k min 1)), where and k min =min k. Theorem 6 The bonds on the price of anarchy in Theorem 5 are asymptotically tight. Proof. To begin, we will inestigate the case when G F and G C can be arbitrary. For this case, the PoA is pper bonded by 1 + αk k/ min(k, ρ(g F,G C ), see Theorem 5. We hae already discssed before that PoA can be infinite when ρ(g F,G C ) = 0 sing Figre 3. Now we will describe how to constrct an instance to demonstrate the tightness for any non-zero ale of ρ(g F,G C ). Consider Figre 4 where nodes hae been diided into fie sets A, B, C, D 1, D 2.LetsetB contain q k nodes. Let set D 1, D 2 contain k nodes each. Let set C contain k q nodes. Let set A contain the remaining n 3k nodes. A solid bidirectional arrow between two sets denotes that all the nodes in one set are connected with all the nodes in the other set in G F G C. In Figre 4, sch solid bidirectional arrows exist between set A and B, setb C and set D 1 D 2. The semi-solid bidirectional arrow between set A and D 1 denotes that all the nodes from set A are connected with all the nodes in set D 1 only in G F bt not in G C. The dotted bidirectional arrow between A and C denotes that all the nodes from A are connected with all the nodes in C in G C bt not in G F. Note that there exist no edges in either G F or G C between the nodes belonging to the same set. From Eqation (8), in S2H games a node has an opportnity to obtain two-hop benefit only if it is a ertex of a triangle w sch that (), (w) G C and (w) G F bt the two-hop benefit it actally obtains depends on the contribtions made by the nodes,, w. Ths in Figre 4, the only nodes that hae an opportnity to obtain two-hop benefit are the nodes from set A and D 1. Recollect that for an instance, the oerlap ρ(g F,G C ) is the minimm nmber of neighbors a node has in G F G C in that instance. By the constrction of the instance in Figre 4, the oerlap ρ(g F,G C )for 15

16 A (n 3k nodes) B (q nodes) C (k q nodes) D 1 (k nodes) D 2 (k nodes) Fig. 4: Tight example for PoA of S2H games for arbitrary G F and G C. A solid, semi-solid, and dotted bidirectional arrow between two sets means these two sets form a complete bipartite graph with edges in G F G C, G F \ G C, and G C \ G F respectiely. The oerlap is gien by ρ(g F,G C)=q and is dictated by the nmber of nodes in set B. As described in the proof of Theorem 6, in the worst Nash Eqilibrim, the nodes in set B end p contribting to the edges leading to the nodes in set D 2 and no node obtains any two-hop benefit. This key obseration leads to an asymptotically tight bond on the PoA of S2H games with G F and G C. A 1 B 1 A 2 B 2 k nodes k nodes k nodes k nodes C1 k nodes C 2 k nodes Fig. 5: Tight example for PoA for the S2H game when G F =G C. Solid bidirectional arrow between two sets means these two sets form a complete bipartite graph. The key obseration for this example is that in the worst Nash Eqilibrim, all the nodes in set A 1 contribte to the edges leading to set A 2 and ice ersa with analogos contribtions for the nodes in the sets B 1, B 2, C 1, C 2. Ths in the worst NE, no node obtains any two-hop benefit. This obseration leads to the bond of 1 + αk being tight. 16

17 this instance is gien by the degree of a node in set A in G C G F, and ths ρ(g F,G C ) = q for this instance. To analyze the PoA for this instance, we will se the alternate definition of PoA for S2H games gien by Eqation (24). Recollect that U ot (M) isgienby U ot(x, x )whereu ot (x, x )isgien by Eqation (9). We will now constrct two soltions: M and M, sch that M is an optimal soltion and M is a NE. We will show that the qantity U ot (M )/U ot (M) can be taken arbitrarily close to the PoA bond in Theorem 5, as desired. Let soltion M be as follows: Any node in set A, D 1, D 2 has exactly k adjoining edges in G C,thsit contribtes 1 to all these edges. Let each node in B and C contribte 1 to the k adjoining edges that lead to nodes in D 1. Now let s compte U ot (M ). Notice that in M each node in B, C, D 1, D 2 contribtes to the edges that are in G C as well as G F. Hence in M, the expression () G F G C ealates to k for any node that belongs to B C D 1 D 2. Howeer, in M any node A contribtes to exactly q edges that are in G F G C, i.e., the edges connecting to the nodes in set B. ThsinM the expression () G F G C ealates to q for any node A. Thsifwesmtheexpression () G F G C oer all the nodes, we get (n 3k)q +3k k = nq +3k(k q). We already mentioned that the only nodes that hae an opportnity to obtain two-hop benefit are the nodes in A and D 1. Now recall that in Figre 4, for eery node A and w D 1,wehae(w) G F.InthesoltionM jst described aboe, we hae that xw =1and(w) G F A, B C, w D 1 (34) Hence in M,foranode A we hae (w) G F (),(w) G C xw = k 2 (35) Using the aboe eqation, we get that in M the following holds: x w =(n 3k)k 2 (36) Ths in total, A D 1 (w) G F (),(w) G C U ot (M )=nq +3k(k q)+α(n 3k)k 2 (n 3k) (q + αk 2 ) (37) Now we constrct a NE M as follows: Let all the nodes hae the same strategy they follow in M except that let the nodes in B and C contribte 1 to the edges leading to the nodes in D 2.NotethatinM, for any node the expression () G F G C (the direct benefit of ) takes the same ale as it has in M. Howeer notice that in M no node obtains two-hop benefit, ths we hae U ot (M) =nq +3k(k q) =(n 3k)q +3k 2 (38) Now we will show that M is indeed a NE: none of the nodes can change their strategies to increase their tility. Consider a node in A D 1 D 2. While analyzing soltion M, we arged that sch a node does not hae a choice for strategy in any soltion bt to contribte 1 to all the adjoining edges in G C.Thsa node belonging to sets A D 1 D 2 cannot change its strategy. A node B C cannot receie any 2-hop benefit, and so it simply tries to maximize the direct benefit that depends on its contribtions, i.e., U ot. Since it cannot get any 2-hop benefit, from this node s point of iew contribting to any k edges gies the same tility, and so M is a NE. Using Eqations (37) and (38), we get U ot (M ) U ot (M) 1+αk k/q 1+ 3k2 (n 3k)q Note that in this instance we had q = min(k, ρ(g F,G C )). Ths from Eqation (39), by increasing n, we can constrct instances to reach within arbitrary precision of the pper bond on PoA gien by 1 + αk k/min(k, ρ(g F,G C ). (39) 17

18 Now let s proe that the pper bond of 1 + αk on PoA for G F G C and G C G F is tight. To proe this, we will describe an instance with G F = G C, ths coering both cases G F G C and G C G F.The scheme of sch an instance is depicted in Figre 5. This instance consists of two complete tripartite graphs, with each partition consisting of k nodes. Sets A 1, B 1, C 1 constittes one of the tripartite graphs and sets A 2, B 2, C 2 constittes another tripartite graphs. Sets A 1 and A 2 also constitte a complete bipartite graph. Similarly sets B 1, B 2 (and sets C 1, C 2 ) constitte a complete bipartite graph. To analyze the PoA for this instance, we will se the alternate definition of PoA for S2H games gien by Eqation (24). Recollect that U ot (M) isgienby U ot(x, x )whereu ot(x, x ) is gien by Eqation (9). The direct benefit component of U ot (x, x )isgienby () G C G F which can be at most k in any soltion. From Eqation (26) the two-hop benefit component has an pper bond of αk 2 in any soltion. Combining these obserations, U ot (x, x ) k(1+αk) in any soltion (in particlar an optimm soltion). This implies that for or instance if we cold find a soltion in which eery node obtains an tility of exactly k(1 + αk) then it is an optimm soltion. Now we will constrct sch a soltion. Let all the nodes in set A 1 contribte 1 to all the edges leading to the nodes in set B 1. Note that this does not iolate the = k as each partition consists of exactly k nodes. Also, let all the nodes in set B 1 (set C 1 ) contribte 1 to all the edges leading to the nodes in set C 1 (set A 1 ). Let the nodes in sets A 2, B 2, C 2 make analogos contribtions. Let s denote this soltion by M.SinceG F = G C, the two-hop benefit component of a node in M is gien by constraint of S2H games that () G c α (),(w) G C w x = α () (w) G C,w x w = α () k = αk2 In the aboe series of eqalities, the second eqality follows from the first one sing the fact that in the soltion M, at most one endpoint of an edge contribtes to it. It is straightforward to see that the directbenefit component of U ot (x, x ) which is gien by () G C G F is exactly k for each node in soltion M.ThsU ot (x, x ) for each node is k(1 + αk) inthesoltionm. As arged before, this proes that M is an optimm soltion. Ths U ot (M )=nk(1 + αk). Now we will constrct a NE, denoted by M, in which each node obtains U ot(x, x ) exactly eqal to k. Ths we will hae U ot (M) =nk. Combining it with U ot (M )=nk(1 + αk) proes the tightness of PoA bond of 1 + αk. Now let s constrct sch a NE M. ToconstrctM, let all the nodes in A 1 contribte 1toalltheedgesleadingtonodesinA 2 and ice ersa. Let the nodes in all other partitions make analogos contribtions. Now let s proe that M is a NE. Since the network is symmetric, to proe that M is a NE, it is sfficient to show changing its strategy cannot increase tility for any one node. Hence withot loss of generalization, consider a node A 1. The only part of the tility U (x, x ) (See Eqation 8) which depends on the contribtions of is expressed by U ot (x, x ). Hence to proe that the tility of cannot increase by changing its strategy, it is sfficient to proe that U ot(x, x ) cannot increase by changing its strategy. As arged before, the direct benefit component of U ot(x, x ) can be at most k in any soltion. In M, the direct benefit component of U ot (x, x )isexactlyk. Hence if can at all improe its tility by changing its strategy, then after changing the strategy, the two-hop benefit component of U ot (x, x ) mst go positie from its ale of 0 in M. Now we will show that this cannot happen, which will proe that cannot change its strategy. To see this, notice that wheneer changes its strategy in M, itmsthappen that it decreases its contribtions on some of the edges leading to nodes in A 2 and increases its contribtions on some of the edges leading to nodes in B 1 or C 1.Let() be among sch edges where increases its contribtion. Withot loss of generalization, let B 1. Howeer its contribtion to () cannot make the two-hop benefit component of U ot(x, x ) positie becase in M the node B 1 does not make any contribtions to edges of the type (w) sch that (w) G F (See Eqation 9). This holds irrespectie of s other contribtions. Ths any change in the strategy of cannot make the two-hop benefit component of U ot(x, x ) positie which was the only remaining part while proing the tightness of PoA bond of 1+αk. Ths we hae proed that the PoA bond of 1 + αk is tight for G F G C and G C G F. 18

19 3.2 Weighted S2H Games Sometimes a person can hae different leels of intrinsic interest in different acqaintances. We incorporate this scenario into S2H games by haing a positie weight f on each edge () G F.Wecallthisextension as Weighted S2H Games. The tility of a node U (x, x ) in Weighted S2H games is gien by: U (x, x )= ( () G C G F + )f + α (w) G F (),(w) G C ( x w + x w w )f w (40) It is not difficlt to see that the argment for the existence of NE for Weighted S2H games is the same as the argment for the existence of NE of S2H games. Also, despite haing arbitrary weights on the edges of G F, wheneer we hae G F G C the PoA proes to be at most 1 + αk as it was in the absence of weights. Here by G F G C, we mean that the nweighted ersion of G F is a sbset of G C. Becase of haing arbitrary positie weights on the edges of G F we do not treat the case of G F = G C = K n (i.e., the nweighted G F is eqal to K n ) separately bt iew it as a special case of G F G C. Ths we get the following reslts: Theorem 7 For Weighted S2H games, a Nash Eqilibrim always exists. Theorem 8 For Weighted S2H games, wheneer G F G C we hae PoA 1+αk. Proof (Theorem 8). We first introdce some notation. Analogos to Eqations (9) and (10) we define U ot (x, x )andu in (x, x ) for Weighted S2H games: U ot (x, x )= U in (x, x )= f + α () G F G C f + α () G F G C (w) G F (),(w) G C (w) G F (),(w) G C f w f w x w w xw (41) (42) The definitions of U ot (M), U in (M) are the same as the ones in Section 3.1. We also hae U (x, x )= U ot (x, x )+U in (x, x ). Lemma 12 also holds for Weighted S2H games. Let d be the degree of node in G C.LetE denote the edge set adjoining in G F G C. Define W as W = max S E s.t. S min(k,d ) () S f (43) In other words, if we compte the cmlatie weight of the member edges for each of the sbsets of E with at most min(k, d ) edges then W is the maximm ale of cmlatie weight of member edges for any sch set. This is the maximm direct benefit that node cold possibly attain from its incident edges. Now we will proe Lemma 2 and Lemma 3 which will help s proe Theorem 8. Lemma 2 For Weighted S2H games, U ot(x, x ) W in any NE. Proof (Lemma 2). Notice from Eqations (41) and (42) that U ot(x, x ) is the only part of the tility obtained by that depends on the contribtions made by whereas U in (x, x ) is independent of the contribtions of. Hence a node in a NE M mst choose a strategy that maximizes U ot (x, x )gien the strategies played by the other nodes. If in M, wehaeu ot(x, x ) <W then the node can increase U ot (x, x ) (and in trn its tility as U in (x, x ) doesn t change) by simply making a contribtion of 1 to the edges that add p to W. This is a contradiction hence we hae proed or claim. Lemma 3 For Weighted S2H games, wheneer G F G C, U ot(x, x ) W (1 + αk). 19

20 Proof (Lemma 3). Recall that () x min(k, d ). Using this constraint and the definition of W,we hae that for any soltion () G F G C f W. Now we claim that for any soltion α (w) G F (),(w) G C f w x w αkw (44) Note that the aboe ineqality combined with the obseration () G F G C f lemma. To show that this ineqality always holds, consider the qantity c w defined as: c w = W proes the :(),(w) G C x w (45) Since the total contribtions of any node are at most k and each x w 1, then clearly c w () for eery, w. Now consider fixing node, and smming c w oer all w sch that (, w) G F.Thiseqals () [x (w) xw ], which is at most k2. Now let s proe that Eqation (44) holds. The left hand side of Eqation (44) can be rewritten as α f w x w = α f w x w (w) G F (w) G F (),(w) G C = α (w) G F f w c w (),(w) G C w Since each c w is bonded by k and their sm is bonded by k 2, the aboe qality is maximized when c w is set to k for the k edges (, w) G F with maximm ale f w, and set to 0 otherwise. Since G F G C,then these are exactly the edges considered in W, and so the aboe qality is bonded by αkw, as desired. From Lemma 3, we hae that in an optimm soltion, U ot(x, x ) W (1+αk) for any node. Combining this with Lemma 2 we get that PoA for Weighted S2H games is at most 1 + αk, proing Theorem 8. Althogh we proed the pper bond of 1 + αk on the PoA for the weighted S2H games with G F G C,the same reslt does not easily extend to the case of G C G F. This is becase when G C G F, the weights on the edges in G F \ G C make it difficlt to bond the two-hop benefit obtained by the nodes along the lines of Eqation 30, nlike the nweighted ersion. k (46) 4 Min Two-Hop (M2H) Games Recall that M2H games are a natral extension of fractional stable matching games obtained by introdcing two-hop benefit. Denoting min(,x )byx, the tility U (x, x ) of a node in M2H games can be written as: U (x, x )= + α x w (47) () G C G F (),(w) G C (w) G F We will call the first smmation in Eqation 47 as direct benefit of node and the term with the coefficient of α in Eqation (47) as two-hop benefit of node. Recall from Section 1 that we se the concept of pairwise eqilibria (PE s) to assess the qality of a soltion in M2H games, denoting the ratio between the worst PE and the optimal soltion as 2-PoA. Recall that in an integral PE all the contribtions are either 0 or 1. An integral PE exists for M2H games withot two-hop benefit (see Related Work). With two-hop benefit, we constrct an instance of M2H games which does not admit any integral pre NE by adapting Example 1 from [11] which is an instance of stable roommates problem sch that no stable matchings exist. We also gie 2-PoA bonds in Thm 10 for some important cases to assess the qality of PE s wheneer they exist. 20

21 Theorem 9 There exist instances of M2H games that do not admit any integral pairwise eqilibrim. Proof. To constrct an instance of M2H games where an integral PE does not exist, consider a pentagon constitted by nodes,, w, y, z (see Figre 6). We will refer to this pentagon as the central pentagon. All the edges of the central pentagon are in G C G F. These and the other edges that are in G C G F are shown in solid pattern in Figre 6. Each ertex of the central pentagon has three other nodes connected to it in G F G C which we will refer as the satellite nodes of the ertex nder consideration. For example a, b and c are the satellite nodes of the ertex. Each satellite node is connected to 3 frther nodes which we will refer as sbordinate nodes of the satellite node nder consideration. In Figre 6, for conenience we hae shown the sbordinate nodes a1, a2, a3 of only one satellite node a, see the expanded iew of the balloon pointed by the dark arrow. All other balloons consist of analogos connections networks. A ertex of the central pentagon is connected in G F \ G C to all the sbordinate nodes of its satellite nodes. These and the other edges that are in G F \ G C are shown in dotted pattern. In addition, a ertex of the central pentagon is also connected in G F \ G C to exactly one satellite node of the ertex of the central pentagon in clockwise direction. For example, ertex is connected in G F \ G C to a,ertex is connected in G F \ G C to w a and so on. We set k = 4 and choose 0 <α<1. Now we will show that an integral PE does not exist for this instance. a a1 a2 a3 a z b c y w Edges in G F G C Edges in G F \ G C Fig. 6: Example showing that an integral PE may not exist for M2H game. Example shown for k =4.Each petal in the figre (e.g., the ones arond a, b,and c represent a gadget which is illstrated on the top right of the figre for node a. In this example, it is possible to show that the only candidates for an integral PE are the soltions where the node contribtes 1 to the edges ( a), ( b c), ( c) (and analogos obserations hold for the nodes, w, y, z). Now each node of the central pentagon has a remaining bdget of 1 for which they face a choice between the two adjoining edges of the central pentagon. The proof of Theorem 9 ses this obseration along with the strctre of the example to proe that no integral PE can exist for this example. To begin with, notice that for the lack of choice, all the satellite nodes and sbordinate nodes contribte 1 to all the adjoining edges in G C. De to these nodes haing a fixed strategy in any soltion, we need not analyze these nodes for changing the strategy. Ths we will be concerned abot the strategies of only the nodes that are the ertices of the central pentagon. Now we claim that in any candidate soltion for an integral PE, node will contribte 1 to edges ( a ), ( b )and( c ). This is for the following reason: its contribtion to each of these edges, say ( a ), fetches it a direct benefit of 1 and a two-hop benefit of 3α as x a x aa1 = x a x aa2 = x a x aa3 = 1 (refer to Eqation (47) to see how two-hop benefit is calclated). Ths contribting to ( a ) fetches an tility of 1 + 3α in eery soltion. Howeer contribting to edge () (or edge (w)) can fetch an tility of at most 1 (or 1 + α). Ths in any candidate soltion for PE, will contribte 1 to edges ( a ), ( b )and( c ). Analogos conclsions hold for the other ertices of 21

22 the central pentagon. Hence to examine the possibility of an integral PE, we only need to look at the last remaining contribtion that these ertices make. For the remaining contribtion, each ertex of the central pentagon has to choose between two edges of the central pentagon that it is part of. Recall that it cannot contribte on the dotted edges as the dotted edges are in G F bt not in G C. Gien these choices for the remaining contribtions for the ertices of the central pentagon, now we will show that any of the reslting soltions cannot be an integral PE. To see this, let s make a simple obseration that after each ertex of the central pentagon chooses one of the adjoining edges of the pentagon to make its last contribtion, there is at least one ertex sch that if it contribtes to edge e of the central pentagon then the other endpoint does not contribte to it. Withot loss of generality, say this ertex is and it contribtes to (w) bt w does not contribte to (w). Ths does not get any direct or two-hop benefit by contribting to (w) by sing Eqation (47). Now we hae two cases: contribtingto() or(z). We will show that both cases do not lead to an integral PE. contribtes to (): Here can choose to contribte to () instead of (w), ths making = min(, ) = 1 to gain a direct benefit of 1 whereas its preios contribtion to (w) fetched no tility. Ths this case does not reslt in an integral PE. contribtes to (z): Here we will show that both and can instead choose to contribte to () and increase their tility, thereby proing that this case does not lead to an integral PE. By contribting to (z)node cannot get any two-hop benefit as there is no other neighbor of z in G C from which there is an edge to in G F (Refer Eqation 47 to see when a node can obtain two-hop benefit). Howeer, it can get a direct benefit of 1 if z also happens to be contribting to (z), reslting in x z = min(x z,x z z )=1. Ths its contribtion (z) canfetchnode an tility of at most 1. Now sppose and choose to contribte to (), making = min(, )=1.Thsboth and get a direct benefit of 1 by their contribtion to (). This leads to an increase in the tility of as its earlier contribtion to edge (w) did not fetch any tility to as arged before. In addition to the direct benefit of 1, node also gets a two-hop benefit of α becase now the term x a becomes 1 and we hae ( a ) G F.Thsinthis process, node gains a total tility of 1 + α which more than compensates for the tility of 1 that it cold hae been getting by contribting to (z). Ths both and improe their tility in this process and hence this case does not lead to an integral PE. Hence we hae shown that an integral PE does not exist for the instance shown in Figre 6 ths proing that an integral PE may not exist for M2H games. Theorem 10 For the M2H game: 1. If G C G F then 2-PoA 2+2αk. 2. For the special case of G F =G C = K n, a PE always exists and 2-PoA tends to 1+αk 1+α(k 1) as n. Proof (Thm 10, Proof of case G C G F ). First let s introdce some notation. Let s define the qantities P (M) ands(m) for a soltion M as: P (M) = S(M) = :() G C G F (w) G F (),(w) G C (48) α x w (49) The qantity P (M) is the combined direct benefit in M of all the nodes, and S(M) is the combined twohop benefit in M of all the nodes. Note that U(M) =P (M) +S(M). We will proe the 2-PoA bond for G C G F in two steps. In the first step, we will proe that for any optimm soltion M and a PE M, we hae P (M ) 2 U(M). Then in the second step, we will proe that S(M ) 2αk U(M). Combining these two steps with the obseration U(M )=P(M )+S(M ) proes the desired 2-PoA bond. Now let s proe P (M ) 2 U(M). Let y s (or r s) denote the contribtions made by in M (or in M). Similarly, let y = min(y,y )andr = min(r,r ). To begin with, we claim that for an edge () G F G C s.t. y >r, at least one of its endpoints, say obtains a direct benefit of exactly 22

23 min(k, d )inm (note that no node can obtain a direct benefit of more than min(k, d )). In other words, for eery edge () B, at least one of its endpoints belongs to set A wherewedefineseta and B as follows: A = { : r =min{k, d }} () B = {() G C G F : y >r } Sppose this claim is not tre and that for some edge () B node (and node ) obtains a direct benefit less than min(k, d )(lessthanmin(k, d )) in the PE M. SincewehaeG C G F,thisimpliesthat (w) rw < min(k, d )and (y) ry < min(k, d ). Now combining (w) rw < min(k, d ) with the constraint (w) rw = min(k, d ) we get that there mst exist an edge in G C adjoining, say(z) s.t. r z >r z. Similarly, for, there mst exist an adjoining edge in G C,say(p) s.t. r p >r p. Now let s break the analysis intwo two cases: Let (z) =() bt (p) (): Since (z) =() wehaemin(r,r ) <r, i.e. r <r.we already hae r p >rp p.now,let decrease rp by an infinitesimal qantity ɛ and increase r p by ɛ. Notice that since r p > min(r p,rp p ) decreasing rp by a tiny constant does not change the tility of. Howeer since r <r, increasing r by ɛ increases r (i.e., min(r,r ). Gien that () is also in G F, in this process the direct benefit of increases, increasing the tility of in trn. This contradicts or assmption of M being a PE. Let (z) () and(p) (): The analysis of this case is similar to the preios case, except that both and simltaneosly will be able to increase their contribtions to () by a tiny constant (this is permitted as for () B we hae r < 1) and both will be able to increase their tility. This contradicts or assmption of M being a PE. Hence we hae proed that for eery edge () B, at least one of its endpoints belongs to A. Becase of this, the following ineqality mst hold: () B y A () () B Now consider the edges () G F G C which are not in B. y + y () B = () G F G C () B () G F G C y A () () B A () y (50) y + y + () G F G C () B A () () B Let s analyze the terms that appear on the right hand side of Eqation (51): y y (51) First let s consider the terms () y for A. We know that this is in total at most min{k, d }, since this is the largest amont can contribte to all the incident edges. Howeer, by definition of A, we also know that min{k, d } = () r. Ths, this term is at most () r.thisisatmost the direct benefit obtained by in M since G C G F and hence is at most U (r, r ). Now consider the terms () () B y for A. Since we are only smming oer edges not in B, andby definition of B this means that y r, then the aboe term is at most () r. As arged in the preios case, this pper bond is at most U (r, r )forg C G F. Using the aboe, we get () G F G C y A U (r, r )+ / A U (r, r ) = P (M )/2 U(M) 23

24 Hence we hae proed that P (M ) 2 U(M) for M2H games wheneer G C G F. Now let s proe that S(M ) 2αk U(M) for M2H games wheneer G C G F. Consider the two-hop benefit obtained in M by a particlar node. It can be bonded as αy y w αy y w y αk (52) (),(w) (w) G F,w () (w) () Becase G C G F,wehaethat () y is at most the direct benefit obtained by in M.Ths smming the aboe bond oer all the nodes, we get S(M ) αkp (M ). We hae already proed that P (M ) 2 U(M). Ths we hae S(M ) 2αkU(M). Ths we hae proed that wheneer G C G F,wehaeP (M ) 2 U(M) ands(m ) 2αkU(M). Combining these obserations with U(M )=P (M )+S(M ), we get the desired 2-PoA bond of 2 + 2αk. Proof (Thm 10, Proof of case G C = G F = K n ). Now consider the case of G F = G C = K n. For the existence of a PE, it can be erified that eery node making a contribtion of k/(n 1) to eery adjacent link is a PE. Now we gie a brief otline of how to proe the bond on 2-PoA. We will assme that n is large enogh, in particlar at least 2k. Ths eery node has degree of at least 2k 1. First we proe that when G F = G C = K n, for a pairwise eqilibrim, if two nodes and satisfy (w) xw <kand (w) xw <krespectiely then = 1. We se this claim to bond the cardinality of the set T defined by T = { : (w) xw <k}. Then sing this bond on cardinality, we bond the tility U(M) obtained in a PE M from which the 2-PoA bond will follow. Now we will proceed to the proof. We claim that when G F = G C = K n, for a pairwise eqilibrim, if two nodes and satisfy (w) xw < k and (w) xw <krespectiely then = 1. To see this, notice that wheneer (w) xw <k,there exists an edge (z) sch that x z >xz z and ths can decrease its contribtion x z withot affecting its tility. Sch an edge exists becase otherwise if x z xz z holds for all the adjoining edges of then (w) min(xw,x w w ) will become eqal to min(k, d ) sing the fact that (w) xw = k. Ths wheneer (w) xw <k, there exists an edge adjoining sch that can can decrease its contribtion x z withot affecting its tility. Sppose for also similar ineqality holds, i.e. (w) xw <k.inschacase,ifthe mtal consent is less than 1 then and can increase their contribtions to () withot decreasing the tility obtained becase of their contribtions from other edges. This wold be a contradiction to or assmption of pairwise eqilibrim. Ths we hae proed the claim that for G F = G C = K n, in any PE, if two nodes and satisfy (w) xw <kand (w) xw <krespectiely then = 1. We will call an edge () with mtal consent = 1 as fll edge. We hae jst proed that all the nodes belonging to the set T = { : (w) xw <k} form a cliqe in the sense that all the edges between sch nodes are fll. Each node in this set can hae at most k 1 fll edges adjoining it becase if it were k for any node in set T then we will hae (w) xw = k, contradicting the criterion for a node to be inclded in T.Ths, T k. Let s denote the cardinality of T by q, whichisatmostk as arged aboe. For large n, the cardinality of T (complement of T ) will also be at least k. For each node in T, the direct benefit component of U (x, x )isexactlyk. Ths the cmlatie direct-benefit component of the nodes in T is gien by k T. Now the two-hop benefit component of a node in T wold be α x w + α x w α (k 1) + α (q 2) () T (w),w () T (w),w () T () T The aboe ineqality holds becase for a node T we hae (w) xw = k and for a node T we hae (w) xw q 1. Expressing q 2as(k 1) (k q + 1), a lower bond on the two-hop benefit component of a node in T can be frther expressed as α (k 1) α (k q +1) = αk(k 1) α (k q + 1) (53) () () T 24 () T

25 Adding it oer all the nodes in T, the cmlatie two-hop benefit obtained by nodes in T is lower bonded by αk(k 1) T α (k q +1) αk(k 1) T αq(k q +1) 2 (54) () T, T The aboe ineqality holds becase for any node in T, it is contribting q 1toedgestoothernodesin T, and ths can only contribte at most k q + 1 to edges to nodes in T. Ths, the total contribtions on edges between T and T are at most q(k q +1). Adding it to the cmlatie direct benefit component k T of all the nodes in T, the total tility of all the nodes in T, and ths the tility U(M) ofapem is lower bonded by U(M) k T + αk(k 1) T αq(k q +1) 2 (55) It is easy to see that an pper bond on the tility U(M ) of an optimm soltion M is gien by From Eqations (55) and (56), we obtain U(M ) n(k + αk 2 ) (56) U(M) U(M ) T 1+α(k 1) αq(k q +1)2 n 1+αk n(k + αk 2 ) (57) As n grows, T /n 1 and the second term in the aboe bond decreases, ths we get 2-PoA 1+αk 1+α(k 1) as n (58) 5 Empirical Findings We begin by presenting the models and the settings sed for the simlations we carried ot for or experiments. Then we will present the findings that we obtained, relating them to the theoretical reslts for Weighted S2H and Weighted M2H games. Let s start by describing the base model that we se for the simlations. In the base model, we consider networks of 100 nodes placed niformly at random inside a nit sqare. Ths each node corresponds to a point (x, y) inside a nit sqare. Each node has an edge joining to eery other node in the network. These edges hae weights f which depend on the distance d(, ) between the nodes. We will specifically consider the following three kinds of weight fnctions: 1. Inerse: For the inerse weight fnction, we set f =1/d(, ). 2. Exponential: For the exponential weight fnction, we set f = e d(,) e 2. The weight fnction has 1 e 2 been normalized to take ale 1 when d(, ) =0and0whend(, ) = 2, with 2 being the largest distance between any two nodes located in a nit sqare. 3. Linear: For the linear weight fnction, we set f =1 d(, )/ 2. Again, the weight fnction has been normalized to take ale 1 when d(, ) =0and0whend(, ) = 2. The attenation with respect to distance becomes steeper as the weight fnctions change from linear to exponential to inerse. In the model that we consider for simlations, we allow the ariables to take ales only from {0, 1}, i.e. we allow the system to conerge only to integral eqilibria. We set k =3.The nodes in this base model engage in either S2H or M2H games. Now we describe the settings for the simlations. For each new instance of simlations, we start by placing 100 nodes niformly at random inside a nit sqare. Then in the second step, the ales of its contribtions are set for each node. Here we explore two possibilities: for a node, either we choose any other three 25

26 nodes randomly and make contribte (with a contribtion of 1) to the edges joining to these nodes or we make s contribtions as 1 on the edges joining to the three closest nodes. Note that as k = 3, a node can contribte 1 corresponding to only three adjoining edges. After this initial assignment of the ariables, a random permtation of nodes is generated in the third step. In each iteration of the instance the nodes are examined in the order gien by this permtation. When a node gets examined, we execte a better response for node (if a better response exists for node ) whereanode increases its contribtion to1onsomeedge() by remoing its contribtion on some other edge. Note that this better response is exected for M2H games only if it is a better response also for node, since for M2H games we are interested in pairwise deiations. If none of the nodes hae any better response, then it means we hae fond a NE (PE) for Weighted S2H (Weighted M2H) games. We rn the simlations for a large nmber of iterations, typically We fond that wheneer an instance conerged, it took a mch less nmber of iterations, typically of the order of 10 iterations. Hence we consider that an instance is not likely to conerge if it fails to conerge within 500 iterations. We will ealate the simlations based on two criteria: Existence of an (integral) eqilibrim: If a simlation instance conerges within 1000 iterations, it implies we hae fond a NE if it was an instance of Weighted S2H games (or a PE for Weighted M2H games). Howeer, if a simlation instance does not conerge, then it may or may not hae an eqilibrim. Based on how many instances conerged for a model nder consideration, we can ealate the likelihood of an instance for that model to hae an eqilibrim. Price of Anarchy: If we are considering an instance of Weighted S2H games that conerges (to a NE M) then the tility U(M) of the resltant NE is gien by adding the tilities obtained by all the nodes. Comptationally it is difficlt to compte the worst NE as well as optimm soltion in or settings. HoweerifwetaketheratioofU(M) to an pper bond U(M ) on the ale of an optimm soltion then we hae a bond on the qality on the resltant NE. We will slightly abse the notation to call this ratio as the price of anarchy in this section. Ths in this section, price of anarchy means the qality of the soltion to which the simlation conerged. An pper bond on U(M ) for Weighted S2H games is gien by 2(1 + αk) (w 1 + w 2 + w 3 )wherew 1,w 2,w 3 are the maximm 3 ales of weight fnctions among the weights of the edges adjoining a node. To see why this is an pper bond, notice that the maximm tility that a node canobtainisgienby2(1+αk) (w 1 + w 2 + w 3 )asper Eqation (40). We will see that the aerage ale of the eqilibria wheneer the instances conerged is close to the ale of an optimm soltion. For the instances of Weighted M2H games, we ealate the pper bond on U(M ) in the similar way except that the pper bond on U(M )isgienby (1 + αk) (w 1 + w 2 + w 3 ). We will examine how the aerage ale of the eqilibria obtained aries with α. Now we present the reslts of the simlations in Section 5.1 and Section Existence of an (Integral) Eqilibrim Althogh in Section 3 we gae an instance of S2H games where an integral NE does not exist, we fond that in simlations better-response dynamics conerge to an integral NE almost all of the time, at least for the types of graph strctres that are considered in the simlation settings. To gie specific nmbers, we fond that for linear, exponential, and inerse weight fnctions, we hae conergence for 99%, 97% and 73% of the simlation instances for Weighted S2H games. Moreoer, the conergence, when it occrs, is extremely fast: all the instances either conerged within 10 ronds, or did not conerge within 500 iterations. The same is not tre for Weighted M2H games: oer 65 percent of or instances did not conerge to a PE een after 5000 ronds. Ths, in the settings that we examine, we fond that we are mch more likely to hae an (integral) NE in S2H games (with extremely fast conergence to a stable soltion) compared to Weighted M2H games. See Figre 7 and Figre 8 for some typical otcomes. We explore these otcomes in details later. 5.2 Price of Anarchy The qality of NEs that or simlations conerged to was extremely close to the qality of the an optimal soltion, sally within a few percent of the centralized optimal soltion, indicating that or theoretical 26

27 bonds are trly only for the worst case, not aerage case. Table 1 shows the aerage ales of the eqilibria obtained for or simlations when the weight fnctions are linear. The ales for exponential and inerse weight fnctions also showed similar trends. We can see that on aerage, the ale of the eqilibria obtained is ery close to the ale of an optimm soltion. The ales are also consistent with worst-case PoA being 1+αk, which decreases as α decreases. As α decreases, the NEs and optimm soltions both conerge to nodes following the strategy of contribting to edges leading to their closest neighbors. Ths there is less tendency to deiate from the strategy in OPT, reslting in better eqilibria. α Aerage PoA (with standard deiation) Aerage PoA (with standard deiation) Weighted S2H games Weighted M2H games 1/ (0.0021) (0.0063) 1/ (0.0018) (0.0049) 1/ (0.0016) (0.0040) 1/ (0.0025) (0.0043) Table 1: Aerage ale of eqilibria obtained with α for n = 100, k = 3 for linear weight fnctions. PoA here refers to the ratio between the compted NE and the optimm soltion. Fig. 7: Typical pairwise eqilibria compted for the Weighted M2H games with linear weight fnction and (Left) α =1/2 (Middle) α =1/8 (Right)α =1/16. As α decreases or as the weight fnctions become steeper from linear to exponential to inerse, we also obsered that nodes increasingly form a backbone -type network by connecting to the closest nodes. We explain this as follows: As α decreases, the contribtion of two-hop benefit to node tility starts losing its significance. When α is small, nodes simply attempt to connect to their closest k nodes and little clstering takes place, instead forming a backbone-type of network. This effect is especially prononced in M2H games (see Figre 7). In Figre 7, for α = 1/2, two-hop benefit plays a significant part in the tility obtained by a node. Becase of the bidirectional natre of contribtions, the strctres that can increase two-hop benefit for a node in M2H games are cliqes as it is easy to see in a cliqe, a node is a part of maximm possible k(k 1) two-hop paths. Howeer, for α =1/8 andα =1/16 in Figre 7, two-hop benefit plays increasingly insignificant role in node tility, ths the otcome is a backbone-type network as predicted. It wold be interesting to inestigate frther if a phase transition occrs as α decreases, where the clstering effect sddenly disappears, or whether it disappears gradally. We plot the effect of increasing the steepness of weight fnctions in Figre 8, where as the weight fnctions become steeper the nodes increasingly form a backbone -type of network by connecting to the k closest nodes. Althogh the rationale is similar to the effect of decreasing α with M2H games, there are slight differences between Figres 7 and 8 as they correspond to eqilibria of S2H and M2H games respectiely. 27

28 Fig. 8: Typical NE compted for the Weighted S2H games with linear weight fnction (Left) and exponential weight fnction (Middle) and inerse weight fnction (Right) with α=1/6. For steeper weight fnctions with S2H games, two-hop benefit plays a significant part in node tility bt it manifests in a slightly different way than M2H games. In S2H games, the natre of contribtions is nidirectional, ths a node has no control oer incoming (one-hop or two-hop) paths. Ths if a node were to contribte on an edge () where the other endpoint has already contribted, then can be a part of at most (k 1) otgoing two-hop paths throgh node. Howeer, if a node were to contribte on an edge () sch that does not contribte to (), then can be a part of k otgoing paths throgh node. Ths in S2H games, in order to obtain a higher two-hop benefit as encoraged by a less steep (linear) weight fnction, nodes tend to contribte to edges where the other endpoint does not contribte, ths redcing the tendency of nodes to form cliqes and also making the edges that get contribted close to 2kn. This leads to a dense network with higher connectiity in a typical otcome, as shown in Figre 8 for linear weight fnctions. Note that sch a typical otcome also has higher nmber of nidirectional (orange-ble) edges as predicted. When the weight fnctions become steeper, two-hop benefit contribtion to node tility becomes increasingly insignificant, ths nodes tend to contribte to the edges leading to closest k neighbors. If is among k closest neighbors of then is also highly likely to be among k closest neighbors of becase of the natre of node distribtion (random in a nit sqare) ths it is highly likely that an edge that gets contribted to, gets contribted from both its endpoints as two-hop benefit becomes insignificant. This can be obsered in increasing percentage of ble-colored edges (denoting contribtion from both endpoints) in Figre 8 as weight fnctions change from linear to exponential to inerse (increasing their steepness). Most of the contribted edges being bidirectional with nodes connecting to k closest neighbors also means a typical otcome in S2H games with steep weight fnction (e.g., inerse weight fnction) resembles to a typical otcome with small α in M2H games as obsered from the rightmost sbfigres of Figre 7 and Figre 8. Let s smmarize the simlation reslts. We inestigated Weighted S2H and Weighted M2H games with nodes scattered niformly at random inside a nit sqare. We sed three different classes of weight fnctions linear, exponential and inerse with the attenation of weight fnction becoming steeper with distance as they change from linear to exponential to inerse. We fond that for Weighted S2H games, we are mch more likely to hae an (integral) NE compared to Weighted M2H games. The qality of NEs that or simlations conerged to was extremely close sally within a few percent of the centralized optimal soltion. We also saw that as α decreases or as the weight fnctions go from linear to exponential to inerse, the nodes stop forming clsters and instead form a backbone -type network reslting from their connecting to the closest possible neighbors. 6 Conclsion Most game theoretic ersions of network formation games consider agents deriing benefit either from their immediate neighborhood or from the entire network. In many settings, howeer, the agents are neither so 28