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⋆ Coordination Games on Graphs KrzysztofR.Apt1,BartdeKeijzer1,MonaRahn1,GuidoScha¨fer1,2,andSunil Simon3 6 1 CWI,Amsterdam,TheNetherlands 1 2 VUUniversityAmsterdam,TheNetherlands 0 3 IITKanpur,Kanpur,India 2 t c O Abstract. We introduce natural strategic games on graphs, which capture the 8 ideaofcoordinationinalocalsetting.Westudytheexistenceofequilibriathat 1 areresilienttocoalitionaldeviationsofunboundedandboundedsize(i.e.,strong equilibriaandk-equilibriarespectively).WeshowthatpureNashequilibriaand ] 2-equilibriaexist,andgiveanexampleinwhichno3-equilibriumexists.More- T over,weprovethatstrongequilibriaexistforvariousspecialcases. G Wealsostudythepriceofanarchy (PoA)andpriceofstability(PoS)forthese . solutionconcepts. WeshowthatthePoSforstrongequilibriais1inalmostall s c ofthespecialcasesforwhichwehaveprovenstrongequilibriatoexist.ThePoA [ forpureNashequilbriaturnsouttobeunbounded, evenwhenwefixthegraph onwhichthecoordinationgameistobeplayed.ForthePoAfork-equilibria,we 3 showthatthepriceofanarchyisbetween2(n−1)/(k−1)−1and2(n−1)/(k−1). v 8 Thelatterupperboundistightfork=n(i.e.,strongequilibria). 8 Finally, weconsider the problems of computing strong equilibria and of deter- 3 miningwhetherajointstrategyisak-equilibriumorstrongequilibrium.Weprove 7 that,givenacoordinationgame,ajointstrategy s,andanumberkasinput,itis 0 co-NPcompletetodeterminewhether sisak-equilibrium.Onthepositiveside, . wegivepolynomialtimealgorithmstocomputestrongequilibriaforvariousspe- 1 0 cialcases. 5 1 : 1 Introduction v i X Ingametheory, coordination gamesareusedtomodelsituations inwhichplay- r a ersarerewardedforagreeingonacommonstrategy,e.g.,bydecidingonacom- mon technological or societal standard. In this paper we introduce and study a verysimple classofcoordination games,whichwecallcoordination gameson graphs: Wearegivenafinite(undirected) graph,ofwhichthenodescorrespond to the players of the game. Each player chooses a color from a set of colorsavailabletoher.Thepayoffofaplayeristhenumberofneighbors whochoose thesamecolor. ⋆Anextendedabstractofthispaperappearedin[3].Partofthisresearchhasbeencarriedout whilethesecondauthorwasapost-doctoralresearcheratSapienzaUniversityofRome,Italy. Our main motivation for studying these games is that they constitute a natural classofstrategicgamesthatcapturethefollowingthreekeycharacteristics: 1. Join the crowd property [34]: the payoff of each player weakly increases whenmoreplayers chooseherstrategy. 2. Asymmetricstrategy sets:playersmayhavedifferentstrategysets. 3. Local dependency: the payoff of each player depends only on the choices madebycertain groupsofplayers(i.e.,neighbors inthegivengraph). The above characteristics are inherent to many applications. As a concrete example,considerasituationinwhichseveralclientshavetochoosebetweenmul- tiple competing providers offering the same service (or product), such as peer- to-peer networks, social networks, photo sharing platforms, and mobile phone providers. Here the benefit of a client for subscribing to a specific provider increases with the number of clients who opt for this provider. Also, each client typicallycaresonlyaboutthesubscriptionsofcertainotherclients(e.g.,friends, relatives, etc.). In coordination games on graphs it is beneficial for each player to align her choices with the ones ofher neighbors. Asaconsequence, the players may attempt to increase their payoffs by coordinating their choices in groups (also calledcoalitions).Inourstudieswethereforefocusonequilibriumconceptsthat areresilienttodeviationsofgroups;morespecificallywestudystrongequilibria [6] and k-equilibria (also known as k-strong equilibria) of coordination games on graphs. Recall that in a strong equilibrium no coalition of players can prof- itablydeviateinthesensethateveryplayerofthecoalitionstrictlyimprovesher payoff. Similarly, in ak-equilibrium withk ∈ {1,...,n}, where n isthe number ofplayers, nocoalition ofplayers ofsizeatmostkcanprofitablydeviate. Ourcontributions. Thefocusofthispaperisontheexistence, inefficiency and computability of strong equilibria and k-equilibria of coordination games on graphs. Ourmaincontributions areasfollows: 1.Existence.WeshowthatNashequilibriaand2-equilibriaalwaysexist.Onthe other hand, k-equilibria for k ≥ 3 do not need to exist. We therefore derive a complete characterization of the values of k for which k-equilibria exist in our games. We also show that strong equilibria exist if only two colors are available. Further, we identify several graph structural properties that guarantee the existence of strong equilibria: in particular they exist if the underlying graph is a pseudoforest4,andwheneverypairofcyclesinthegraphisedge-disjoint. Also, 4Recallthatinapseudoforesteachconnectedcomponenthasatmostonecycle. 2 they exist if the graph is color complete, i.e., if for each available color x the components ofthesubgraph induced bythenodeshaving color xarecomplete. Moreover, existence of strong equilibria is guaranteed in case the coordination game is played on a color forest, i.e., for every color, the subgraph induced by theplayerswhocanchoosethatcolorisaforest. Wealso address the following question. Given acoordination game denote itstransitionvalueasthevalueofkforwhichak-equilibriumexistsbuta(k+1)- equilibrium does not. Thequestion thenistodetermine forwhichvalues ofk a gamewithtransition valuekexists.Weexhibitagamewithtransition value4. Inallourproofstheexistenceofstrongequilibria isestablished byshowing a stronger result, namely that the game has the coalitional finite improvement property,i.e.,everysequenceofprofitablejointdeviationsisfinite(seeSection2 foraformaldefinition). 2.Inefficiency.Wealsostudytheinefficiencyofequilibria.Inourcontext,theso- cialwelfareofajointstrategyisdefinedasthesumofthepayoffsofallplayers. Thek-priceofanarchy[1](resp.k-priceofstability) referstotheratiobetween the social welfare of an optimal outcome and the minimum (resp. maximum) socialwelfareofak-equilibrium5. We show that the price of anarchy is unbounded, independently of the un- derlyinggraphstructure,andthestrongpriceofanarchyis2.Ingeneral,forthe k-priceofanarchywithk ∈ {2,...,n−1}wederivealmostmatching lowerand upper bounds of 2n−1 − 1 and 2n−1, respectively (given a coordination game k−1 k−1 thathasak-equilibrium). Wealsoprovethatthestrongpriceofstabilityis1for the cases that there are only two colors, or the graph is a pseudoforest or color forest. Our results thus show that as the coalition size k increases, the worst-case inefficiency of k-equilibria decreases from ∞ to 2. In particular, we obtain a constant k-priceofanarchyfork = Ω(n). 3.Complexity. Wealsoaddressseveralcomputational complexity issues.Given a coordination game, a joint strategy s, and a number k as input, it is co-NP completetodeterminewhether sisak-equilibrium.However,weshowthatthis problemcanbesolvedinpolynomialtimeincasethegraphisacolorforest.We alsogivepolynomial timealgorithmstocomputestrongequilibria forthecases ofcolorforests, colorcompletegraphs, andpseudoforests. Related work. Our coordination games on graphs are related to various well- studiedtypesofgames.Weoutline someconnections below. 5Thek-priceofanarchyisalsocommonlyknownasthek-strongpriceofanarchy. 3 First, coordination games on graphs are polymatrix games. Recall that a polymatrix game(see[24,26])isafinitestrategic gameinwhichthe payoff for each player is the sum of the payoffs obtained from the individual games the player plays with each other player separately. Cai and Daskalakis [13] considered a special class of polymatrix games which they call coordination-only polymatrix games. These games are identical to coordination games on graphs withedgeweights.TheyshowedthatpureNashequilibriaexistandthatfinding one is PLS-complete. The proof of the latter result crucially exploits that the edge weights can be negative. Note that negative edge weights can be used to enforcethatplayersanti-coordinate. Ourcoordination gamesdonotexhibitthis characteristic andaretherefore differentfromtheirs. Second,ourcoordination gamesarerelatedtoadditively separable hedonic games (ASHG)[10,12], which were originally proposed in acooperative game theory setting. Here the players are the nodes of an edge weighted graph and formcoalitions.Thepayoffofanodeisdefinedasthetotalweightofalledgesto neighbors thatareinthesamecoalition. Iftheedge weightsaresymmetric, the corresponding ASHG is said to be symmetric. Recently, a lot of work focused on computational issues of these games (see, e.g., [8,9,18]). Aziz and Brandt [7]studied theexistence ofstrong equilibria inthesegames.ThePLS-hardness resultestablishedin[18]doesnotcarryovertoourcoordination gamesbecause it makes use of negative edge weights, which we do not allow in our model. NotealsothatinASHGseveryplayercanchoosetoentereverycoalitionwhich is not necessarily the case in our coordination games. Such restrictions can be imposedbytheuseofnegativeedgeweights(seealso[18])andourcoordination games therefore constitute a special case of symmetric ASHGs with arbitrary edgeweights. Third, our coordination games on graphs are related to congestion games [32]. In particular, they are isomorphic to a special case of congestion games withweaklydecreasingcostfunctions(assumingthateachplayerwantstomin- imize her cost). Rozenfeld and Tennenholtz [33] derived a structural characterization of strategy sets that ensure the existence of strong equilibria in such games. By applying their characterization to our (transformed) games one ob- tainsthatstrongequilibriaexistiftheunderlyinggraphofthecoordinationgame is a matching or complete (both results also follow trivially from our studies). Bilo` et al. [11] studied congestion games where the players are embedded in a (possibly directed) influence graph (describing how the players delay each other). They analyzed the existence and inefficiency of pure Nash equilibria in these games. However, because the delay functions are assumed to be linearly increasing in the number of players, these games do not cover the games we studyhere. 4 Further, coordination games on graphs are special cases of the social net- workgamesintroducedandanalyzedin[4](ifoneusesinthemthresholdsequal to0).Thesearegamesassociatedwithathresholdmodelofasocialnetworkin- troduced in[2]whichisbasedonweightedgraphswiththresholds. Coordination gamesarealsorelatedtotheproblem ofclustering, wherethe taskistopartition thenodes ofagraph inameaningful manner. Ifweviewthe strategies as possible cluster names, then a Nash equilibrium of our coordination gameon agraph corresponds to a “satisfactory” clustering of the underly- inggraph. Hoefer[22]studied clustering gamesthatarealsopolymatrix games based on graphs. Each player plays one of two possible base games depend- ing on whether the opponent is a neighbor in the given graph or not. Another morerecentapproachtoclustering through gametheoryisbyFeldman,Lewin- Eytan and Naor [17]. In this paper both a fixed clustering of points lying in a metric space and a correlation clustering (in which the distance is in [0,1] and eachpointhasaweightdenotingits‘influence’)isviewedasastrategichedonic game.However,inbothreferenceseachplayerhasthesamesetofstrategies,so theresulting gamesarenotcomparable withours. Strategic games that involve coloring of the vertices of a graph have also been studied in the context of the vertex coloring problem. These games are motivatedbythequestionoffindingthechromaticnumberofagraph.Asinour games,theplayersarenodesinagraphthatchoosecolors. However,thepayoff function differsfrom theoneweconsider here:itis0ifaneighbor chooses the same color and it is the number of nodes that chose the same color otherwise. Panagopoulou andSpirakis[30]showedthatanefficientlocalsearchalgorithm can be used to compute a good vertex coloring. Escoffier, Gourve`s and Mon- not[15]extended thisworkbyanalyzing socially optimaloutcomes andstrong equilibria. Chatzigiannakis et al. [14] studied the vertex coloring problem in a distributed setting and showed that under certain restrictions a good coloring canbereached inpolynomial time. Strong and k-equilibria in strategic games on graphs were also studied in Gourve`s and Monnot [19,20]. These games are related to, respectively, the MAX-CUT and MAX-k-CUT problems. However, they do not satisfy the join the crowdproperty, so,again,theresultsarenotcomparable withours. To summarize, in spite of these close connections, our coordination games ongraphsaredifferentfromallclassesofgamesmentionedabove.Notably,this isduetothefactthatourgamescombinethethreeproperties mentioned above, i.e., join the crowd, asymmetric strategy sets and local dependencies modeled bymeansofanundirected graph. Research reported here was recently followed in two different directions. In [5] and [35] coordination games on directed graphs were considered, while 5 in[31]coordination gamesonweightedundirected graphswereanalyzed. Both setupsleadtosubstantiallydifferentresultsthatarediscussedinthefinalsection. Finally, [16] studied the strong price of anarchy for a general class of strategic games that, in particular, include as special cases our games and the MAX-CUT gamesmentionedabove. Asafinalremark,letusmentionthatthecoordination gamesongraphs are examplesofgamesonnetworks, avastresearch areasurveyedin[25]. Our techniques. Most of our existence results are derived through the applica- tionofonetechnicalkeylemma.Thislemmarelatesthechangeinsocialwelfare caused by a profitable deviation of a coalition to the size of a minimum feedback edge set of the subgraph induced by the coalition6. This lemma holds for arbitrary graphs and provides a tight bound on the maximum decrease in social welfare caused by profitable deviations. Using it, we prove our existence resultsbymeansofageneralizedordinalpotentialfunctionargument.Inpartic- ular, this enables us to show that every sequence of profitable joint deviations isfinite.Further,weusethegeneralized ordinal potential function toprovethat thestrong priceofanarchy is1andthatstrong equilibria canbecomputed effi- cientlyforcertaingraphclasses. Thenon-existenceproofof3-equilibriaisbasedonaninstancewhosegraph essentiallycorrespondstotheskeletonofanoctahedronandwhosestrategysets are set up in such a way that at most one facet of the octahedron can be unicolored. We then use the symmetry of this instance to prove our non-existence result. Theupperbound onthek-priceofanarchy isderivedthrough acombinato- rial argument. We first fix an arbitrary coalition of size k and relate the social welfareofak-equilibrium tothesocialwelfareofanoptimumwithinthiscoali- tion. We then extrapolate this bound by summing over all coalitions of size at most k. We believe that this approach might also prove useful to analyze the k-priceofanarchyinothercontexts. 2 Preliminaries A strategic game G := (N,(S ) ,(p) ) consists of a set N := {1,...,n} i i∈N i i∈N of n > 1 players, a non-empty set S of strategies, and a payoff function p : i i S ×···×S → R for each player i ∈ N. We denote S ×···×S by S, call 1 n 1 n each element s ∈ S a joint strategy, and abbreviate the sequence (s ) to s . j j,i −i Occasionally wewrite(s,s )insteadof s. i −i 6Recallthatafeedbackedgesetisasetofedgeswhoseremovalmakesthegraphacyclic. 6 We call a non-empty subset K := {k ,...,k } of N a coalition. Given a 1 m jointstrategy sweabbreviate thesequence (s ,...,s )ofstrategies to s and k1 km K S ×···×S toS .Wealsowrite(s ,s )instead of s.Ifthereisastrategy k1 km K K −K xsuchthat s = xforallplayersi∈ K,wealsowrite(x ,s )for s. i K −K Given two joint strategies s′ and s and a coalition K, we say that s′ is a deviation oftheplayers in K from sif K = {i ∈ N | s , s′}.Wedenote this by i i s→K s′.Ifinaddition p(s′)> p(s)holdsforalli∈ K,wesaythatthedeviation s′ i i from sisprofitable. Further, wesaythattheplayers in K canprofitably deviate from sifthereexistsaprofitabledeviation oftheseplayers from s. Next, we call a joint strategy s a k-equilibrium, where k ∈ {1,...,n}, if no coalition of at most k players can profitably deviate from s. Using this defini- tion, a Nash equilibrium is a 1-equilibrium and a strong equilibrium [6] is an n-equilibrium. Given a joint strategy s, we call the sum SW(s) = p(s) the social i∈N i welfare of s. When the social welfare of s is maximal, wePcall s a social optimum.Givenafinitegamethathasak-equilibrium, itsk-priceofanarchy (resp. stability) is the ratio SW(s)/SW(s′), where s is a social optimum and s′ is a k- equilibriumwiththelowest(resp.highest)socialwelfare7.The(strong)priceof anarchy refers to the k-price of anarchy with k = 1 (k = n). The (strong) price ofstability isdefinedanalogously. A coalitional improvement path, in short a c-improvement path, is a maximal sequence (s1,s2,...) of joint strategies such that for every k > 1 there is a coalition K such that sk is a profitable deviation of the players in K from sk−1.Clearly, ifac-improvement path isfinite, itslast elementisastrong equilibrium. We say that G has the finite c-improvement property (c-FIP) if every c-improvement path is finite. So if G has the c-FIP, then it has a strong equilibrium. Further, we say that the function P : S → A (where A is any set) is a generalizedordinalc-potentialforGifthereexistsastrictpartialordering≻on the set A such that if s→K s′ is a profitable deviation, then P(s′) ≻ P(s). A gen- eralizedordinalpotentialisalsocalledageneralizedstrongpotential[21,23].It is easy tosee that if a finite gameadmits ageneralized ordinal c-potential then thegamehasthec-FIP.Theconversealsoholds:afinitegamethathasthec-FIP admits a generalized ordinal c-potential. The latter fact is folklore; we give a self-contained proofinAppendixA. Notethatinthedefinitionofaprofitabledeviationofacoalition,weinsisted that all members of the coalition change their strategies. This requirement is irrelevant for the definitions of the k-equilibrium and the c-FIP, but it makes someargumentsslightly simpler. 7Inthecaseofdivisionbyzero,wedefinetheoutcomeas∞. 7 {a,b} {b,c} {c,a} 2 4 6 {a,c} {b,a} 1 8 3 5 7 {a,b} {b,c} {c,a} Fig.1.Agraphwithacolorsetassignment.Theboldedgesindicatepairsofplayerschoosingthe samecolor. 3 Coordination gamesongraphs We now introduce the games we are interested in. Throughout the paper, we fix a finite set of colors M of size m, an undirected graph G = (V,E) without self-loops,andacolorassignment A.Thelatterisafunctionthatassignstoeach nodeianon-empty set A ⊆ M.Anode j ∈ V isaneighbor ofthenodei ∈ V if i {i, j} ∈ E.Let N denote thesetofallneighbors ofnode i.Wedefineastrategic i gameG(G,A)asfollows: – theplayersareidentifiedwiththenodes, i.e.,N = V, – thesetofstrategies ofplayeriis A, i – thepayofffunction ofplayeriis p(s) := |{j ∈ N | s = s }|. i i i j So each node simultaneously chooses a color from the set available to her and thepayofftothenodeisthenumberofneighborswhochosethesamecolor.We callthesegamescoordination gamesongraphs, fromnowonjustcoordination games. Example1. Consider the graph and the color assignment depicted in Figure 1. Takethejointstrategythatconsistsoftheunderlinedstrategies.Thenthepayoffs areasfollows: – 1forthenodes1,6,7, – 2forthenodes2,3, – 3forthenodes4,5,8. ItiseasytoseethattheabovejointstrategyisaNashequilibrium.However, it is not a strong equilibrium because the coalition K = {1,4,5,6,7} can prof- itablydeviatebychoosing colorc. ⊔⊓ 8 We now recall some notation and introduce some terminology. Let G = (V,E) be a graph. Given a set of nodes K, we denote by G[K] the subgraph of G induced by K and by E[K] the set of edges in E that have both endpoints in K. So G[K] = (K,E[K]). Further, δ(K) denotes the set of edges that have one node in K and the other node outside of K. Also, given a subgraph C ofG we use V(C) and E(C) to refer to the set of nodes and the set of edges of C, respectively. Furthermore, we define SW (s) := p(s). Given a joint strategy s we K i∈K i denote by E+s the set of edges {i, j} ∈ EPsuch that si = sj. We call these edges unicoloredins.(InFigure1,thesearetheboldedges.)NotethatSW(s) = 2|E+|. s Finally, we call a subgraph unicolored in s if all its nodes have the same color in s. 4 Existence ofstrong equilibria We begin by studying the existence of strong equilibria and k-equilibria of coordination games. We first prove our key lemma and then show how it can be appliedtoderiveseveralexistenceresults. 4.1 Keylemma Recall that an edge set F ⊆ E is afeedback edge set of the graphG = (V,E)if thegraph(V,E\F)isacyclic. Lemma1 (Key lemma). Suppose s→K s′ is a profitable deviation. Let F be a feedback edge set ofG[K]. Denote SW(s′)−SW(s)by ∆SW and for acoalition LdenoteSW (s)−SW (s)by∆SW .Then L L L ∆SW = 2(∆SW −|E+ ∩E[K]|+|E+∩E[K]|) (1) K s′ s and ∆SW > 2(|F ∩E+|−|F ∩E+|). (2) s s′ Proof. Let N denote the set of neighbors of nodes in K that are not in K. K Abbreviate SW(s′) − SW(s) to ∆SW, and analogously, for a coalition L, let ∆SW = SW (s′) − SW (s). The change in the social welfare can be written L L L as ∆SW = ∆SW +∆SW +∆SW . K NK V\(K∪NK) Wehave SW (s)= 2|E+∩E[K]|+|E+∩δ(K)| K s s 9 andanalogously for s′.Thus ∆SW = 2(|E+ ∩E[K]|−|E+∩E[K]|)+|E+ ∩δ(K)|−|E+∩δ(K)|. K s′ s s′ s Itfollowsthat ∆SW = |E+ ∩δ(K)|−|E+∩δ(K)| NK s′ s = ∆SW −2(|E+ ∩E[K]|−|E+∩E[K]|). K s′ s Furthermore, the payoff of the players that are neither in K nor in N does not K changeandhence∆SW = 0.Puttingtheseequalitiestogether,weobtain V\(K∪NK) (1). LetFc = E[K]\F.Then |E+∩E[K]|−|E+ ∩E[K]| = |E+∩F|−|E+ ∩F|+|E+∩Fc|−|E+ ∩Fc|. s s′ s s′ s s′ Weknowthat(K,Fc)isaforestbecause F isafeedbackedgeset.So|Fc| < |K|. Hence |E+∩Fc|−|E+ ∩Fc| ≥ −|Fc|> −|K|. s s′ Furthermore, each player in K improves his payoff when switching to s′ and hence∆SW ≥ |K|.So,plugging intheseinequalities in(1)weget K ∆SW > 2(|K|+|E+∩F|−|E+ ∩F|−|K|) = 2(|E+∩F|−|E+ ∩F|), s s′ s s′ whichproves(2). ⊔⊓ Letτ(K)bethesizeofaminimalfeedback edgesetofG[K],i.e., τ(K) = min{|F||G[K]\F isacyclic}. (3) Equation(2)thenyieldsthatSW(s′)−SW(s)> −2τ(K).Thefollowingexample showsthatthisboundistight. Example2. We define a graph G = (V,E) and a color assignment as follows. Consider a clique on l nodes and let K be the set of nodes. Every i ∈ K can choose between two colors {c,x}, where c , c for every j , i. Further, every i i j node i ∈ K is adjacent to(l−2) additional nodes of degree one, each of which hasthecolorset{c}.Notethatwhendefiningajointstrategy s,itissufficientto i specify s for every i ∈ K because the remaining nodes have only one color to i choosefrom. 10