Coalition Resilient Outcomes in Max k-Cut Games
Raffaello Carosi, Simone Fioravanti, Luciano Gualà, Gianpiero Monaco
CCoalition Resilient Outcomes in Max k -CutGames Raffaello Carosi , Simone Fioravanti , Luciano Gualà , and Gianpiero Monaco Gran Sasso Science Institute, Italy, [email protected] University of Rome “Tor Vergata”, Italy, [email protected]@mat.uniroma2.it DISIM - University of L’Aquila, Italy, [email protected]
Abstract.
We investigate strong Nash equilibria in the max k -cut game ,where we are given an undirected edge-weighted graph together with aset { , . . . , k } of k colors. Nodes represent players and edges capture theirmutual interests. The strategy set of each player v consists of the k colors.When players select a color they induce a k -coloring or simply a coloring.Given a coloring, the utility (or payoff ) of a player u is the sum of theweights of the edges { u, v } incident to u , such that the color chosen by u is different from the one chosen by v . Such games form some of the basicpayoff structures in game theory, model lots of real-world scenarios withselfish agents and extend or are related to several fundamental classes ofgames.Very little is known about the existence of strong equilibria in max k -cutgames. In this paper we make some steps forward in the comprehensionof it. We first show that improving deviations performed by minimalcoalitions can cycle, and thus answering negatively the open problemproposed in [13]. Next, we turn our attention to unweighted graphs. Wefirst show that any optimal coloring is a 5-SE in this case. Then, weintroduce x -local strong equilibria, namely colorings that are resilient todeviations by coalitions such that the maximum distance between everypair of nodes in the coalition is at most x . We prove that 1-local strongequilibria always exist. Finally, we show the existence of strong Nashequilibria in several interesting specific scenarios. We consider the max k -cut game . This is played on an undirected edge-weightedgraph where the n nodes correspond to the players and the edges capture theirmutual interests. The strategy space of each player is a set { , . . . , k } of k avail-able colors (we assume that the colors are the same for each player). Whenplayers select a color they induce a k -coloring or simply a coloring. Given a col-oring, the utility (or payoff ) of a player u is the sum of the weights of edges a r X i v : . [ c s . G T ] O c t u, v } incident to u , such that the color chosen by u is different from the onechosen by v . The objective of every player is to maximize its own utility.This class of games forms some of the basic payoff structures in game theory,and can model lots of real-life scenarios. Consider, for example, a set of companiesthat have to decide which product to produce in order to maximize their revenue.Each company has its own competitors (for example the ones that are in the sameregion), and it is reasonable to assume that each company wants to minimizethe number of competitors that produce the same product. Another possiblescenario is in a radio setting; radio towers are players and their goal is selectinga frequency such that neighboring radio-towers have a different one in order tominimize the interference.In such games on graphs it is beneficial for each player to anti-coordinateits choices with the ones of its neighbors (i.e., selecting a different color). As aconsequence, the players may attempt to increase their utility by coordinatingtheir choices in groups (also called coalitions). Therefore, in our studies we focuson equilibrium concepts that are resilient to deviations of groups. Along thisdirection, a very classic notion of equilibrium is the strong Nash equilibrium (SE)[2] that is a coloring in which no coalition, taking the actions of its complementsas given, can cooperatively deviate in a way that benefits all of its members,in the sense that every player of the coalition strictly improves its utility. Thenotion of SE is a very strong equilibrium concept. A weaker one is the notion of q -Strong Equilibrium ( q -SE), for some q ≤ n , where only coalitions of at most q players are allowed to cooperatively change their strategies. Notice that the1-SE is equivalent to the Nash equilibrium (NE), while the n -SE is equivalentto the SE.When it exists, an SE is a very robust state of the game and it is alsomore sustainable than an NE. However, while NE always exists in these games[8,15,18], little is known about the existence of strong equilibria in Max k -cutgames. Indeed, to the best of our knowledge, there are basically two papers ofthe literature dealing with such issue. In [12] the authors show that an optimalstrategy profile (or optimal coloring), i.e., a coloring that maximizes the sum ofthe players’ utilities or equivalently, a coloring that maximizes the k -cut, is an SEfor the max 2-cut game, and it is a 3-SE, for the max k -cut game, for any k ≥ k ≥
3. In [13] they show that, if the number of colors is at least thenumber of players minus two, then an optimal strategy profile is an SE. Finally,they show that the dynamics, where at each step a coalition can deviate so thatall of its members strictly improve their utility by changing strategy, can cycle.The main consequence of this latter fact is that no strong potential function can exist for the game, and hence the existence of an SE cannot be proved bysimply exhibiting it. It is worth noticing that strong potential functions are oneof the main tools used to prove the existence of an SE.All the above results suggest that it is hard to understand whether SE alwaysexist for max k -cut games. In this paper, although we do not prove or disprove See Section 2 for the definition of strong potential function. hat every instance of the max k -cut game possesses a strong equilibrium, wemake some step forward in the comprehension of it. Our results.
As pointed out in [13], sometimes the existence of an SE is provedby means of a potential function in which the set of deviating coalitions is re-stricted to minimal coalitions only, where a coalition is minimal if none of itsproper subsets can perform an improvement themselves (see for example [14]).Understanding whether this approach can be used in the max k -cut game ismentioned as an open problem in [13]. We answer this question negatively (seeProposition 2) by showing an instance in which there is a cycle of improvingdeviations performed by minimal coalitions only.We then focus on the unweighted case, where the utility of a player in acoloring is simply the number of neighbors with different color from its own, andwe provide some non-trivial existential results for it. In particular, in Section 4we show that 5-SE always exist for the max k -cut game. This is an improvementwith respect to the existence of 3-SE [12].Besides q -SE, we also consider another equilibrium concept that is weakerthan the notion of SE. Observe that in a q -SE two players can form a coalitioneven if they are far from each other in the graph. This is unrealistic in manypractical scenarios. In oder to encompass this aspect, in Section 5, we introducethe concept of x -Local SE ( x -LSE). A coloring is an x -LSE if it is resilient todeviations by coalitions such that the shortest path between every pair of nodesin the coalition is long at most x . Therefore, the notion of x -LSE also takesinto account that certain players may not have the possibility of communicatingto each other and thus to form a coalition. This seems an important point toconsider when modeling a situation of strategic interaction between agents. Herewe suppose that the input graph also represents knowledge between players, thatis two nodes know each other if they are connected by an edge. In this paper wefocus on the case x = 1, that is each player in the coalition must have a socialconnection (namely an edge) towards every deviating player. We show that, forany k , a 1-LSE always exists. Interestingly enough, our analysis also provides acharacterization of the set of local strong equilibria which relates 1-LSE to q -SE.Finally, in Section 6, we show that an SE always exists for some special classesof unweighted graphs. More precisely, in Corollary 1, we prove that in graphswith large girth, any optimal strategy profile is an SE, for any k ≥
2. Moreover,in Proposition 9, we prove that whenever the number of colors k is large enoughwith respect to the maximum degree of the graph, then any optimal strategyprofile is an SE.Some proofs have been moved to the appendix. Further related work.
The max k -cut game has been first investigated in[15,18], where the authors show that, when the graph is unweighted and undi-rected, it is possible to compute a Nash Equilibrium in polynomial time byexploiting the potential function method. When the graph is weighted undi-rected, even if the potential function ensures the existence of NE the problemof computing an equilibrium is PLS-complete even for k = 2 [22]. In fact, forsuch a value of k , it coincides with the classical max cut game. In [8] the authorshow the existence of NE in generalized max k -cut games where players alsohave an extra profit depending on the chosen color. When the graph is directed,the max k -cut game in general does not admit a potential function. Indeed, inthis case, even the problem of understanding whether they admit a Nash equi-librium is NP-complete for any fixed k ≥ v has to choose a color among k available onesand its payoff is equal to the number of nodes in the graph that have chosenits same color, unless some neighbor of v has chosen the same color, and in thiscase the payoff of v is 0. They prove that this is a potential game and that aNash equilibrium can be found in polynomial time.Max k -cut games are related to many other fundamental games considered inthe scientific literature. One example is given by the graphical games introducedin [17]. In these games the payoff of each agent depends only on the strategies ofits neighbors in a given social knowledge graph defined over the set of the agents,where an arc ( i, j ) means that j influences i ’s payoff. Max k -cut games can alsobe seen as a particular hedonic game (see [3] for a nice introduction to hedonicgames) with an upper bound (i.e., k ) to the number of coalitions. Specifically,given a k -coloring, the agents with the same color can be seen as members ofthe same coalition of the hedonic game. In order to get the equivalence amongthe two games, the hedonic utility of an agent v can be defined as the overallnumber of its neighbors minus the number of agents of its neighborhood that arein the same coalition. Nash equilibria issues in hedonic games have been largelyinvestigated under several different assumptions [5,6,20] (just to cite a few).Concerning local coalitions, a notion of equilibrium close in spirit to ourLSE has been studied in the context of network design games in [19]. Moreover,locality aspects have been also considered when restricting the strategy space insingle-player deviations (see for example [4,10]).Finally, it is worth mentioning the classical optimization max cut problem,a very famous problem in graph theory that was proven to be NP-Hard byKarp [16]. Preliminaries
Let G = ( V, E, w ) be an undirected weighted graph, where | V | = n , | E | = m ,and w : E → R + . Let δ v ( G ) = P u ∈ V : { v,u }∈ E w ( { v, u } ) denote the degree of v , that is the sum of the weights of all the edges incident to v . Let δ M ( G ) = max v ∈ V δ v ( G ) denotes the maximum degree in G . Given a set of nodes V ⊆ V ,let G ( V ) = ( V , E , w ) be the subgraph induced by V , where E = {{ v, u } ∈ E | v ∈ V ∧ u ∈ V } . For any pair of nodes v, u ∈ V , the distance dist G ( v, u )between v and u in G is equal to the length of the shortest path from v to u .Given G and a set of colors K = { , . . . k } , the max k-cut problem is topartition the vertices into k subsets V , . . . , V k such that the sum of the weightsof the edges having the endpoints in different sets is maximized. A strategicversion of the max k-cut problem is the max k-cut game , and it is defined asfollows. There are | V | players, and each node of G is controlled by exactly onerational player. Players have the same strategy set, and it is equal to the setof colors { , . . . k } . A strategy profile , or coloring σ : V → K , is a labeling ofnodes of G in which each player v is colored σ ( v ). Given a coloring σ , let E ( σ ) = {{ u, v } : σ ( u ) = σ ( v ) } be the edges that are proper with respect to σ , and let δ iu ( σ ) = P v ∈ V w ( { u, v } ) σ ( v )= i be the sum of the weights of the edges incident to u and towards nodes colored i in σ . The utility (or payoff) of player u is definedas µ u ( σ ) = P v ∈ V : { u,v }∈ E ∧ σ ( u ) = σ ( v ) w ( { u, v } ). The cut-value , or size of the cut ,of a coloring S ( σ ) is defined as follows: S ( σ ) = P { u,v }∈ E ∧ σ ( u ) = σ ( v ) w ( { u, v } ).The social welfare of a coloring σ is defined as the sum of players’ utilities, thatis SW ( σ ) = P v ∈ V µ v ( σ ) = 2 S ( σ ). Moreover, an optimal strategy profile (oroptimal coloring) is defined to be a strategy profile which maximizes the sum ofthe players’ utilities and thus the cut-value.Given a coalition C ⊆ V and a coloring σ , let C K ( σ ) = { i ∈ K | ∃ v ∈ C s.t. σ ( v ) = i } be the set of colors used by the coalition C in σ . Moreover, foreach color i , let C i ( σ ) = { v ∈ C | σ ( v ) = i } be the set of players in C that arecolored i in σ .Given a strategy profile σ , a player v and a coalition C , we denote by σ − v and σ − C the strategy profile σ besides the strategy played by v and by C ,respectively. Moreover, we denote by σ C the coloring σ restricted only to playersin C , and we use ( σ − v , σ ( v )) and ( σ − C , σ C ) to denote σ .A profile σ is a Nash Equilibrium (NE) if no player can improve its payoff bydeviating unilaterally from σ , that is, µ v ( σ − v , i ) ≤ µ v ( σ ) for each player v ∈ V and for each color i ∈ K . For each 1 ≤ q ≤ n , σ is a q -Strong Equilibrium ( q -SE) if there exists no coalition C with | C | ≤ q that can cooperatively deviatefrom σ C to σ C in such a way that every player in C strictly improves its utilityin ( σ − C , σ C ). The 1-strong equilibrium is equivalent to the Nash equilibrium,while for q = n an n -strong equilibrium is called strong equilibrium (SE) . Whena coalition C deviates so that all of its members strictly improve their utility,then we say it performs a strong improvement . A strong improvement is said Even if the graph is weighted, we consider here the hop-distance , where the lengthof a path is defined as the number of its edges. o be minimal if no proper subsets of the deviating coalition can perform animprovement themselves, and the coalition itself is said to be minimal. A strongimproving dynamics (shortly dynamics) is a sequence of strong improving moves.A game is said to be convergent if, given any initial state, any sequence ofimproving moves leads to a strong Nash equilibrium. Given a coloring σ , if acoalition C induces a new coloring σ after deviating, then we say that the setof edges E ( σ ) \ E ( σ ) enters the cut, and that the set of edges E ( σ ) \ E ( σ ) leaves the cut.A potential function Φ is a function mapping strategy profiles into real valuesin such a way that, for each coloring σ and each player v , whenever v canprofitably deviate from σ yielding a new coloring σ , it holds that Φ ( σ ) > Φ ( σ ).When this is true also for profitably deviations performed by coalitions, thefunction is called strong potential function .We conclude this section by stating some properties about minimal coalitionsthat will be useful later. The proofs of these properties can be found in theappendix. Proposition 1.
Let σ be a coloring, and let C be a minimal coalition that canperform a strong improvement from σ . Let σ be the resulting coloring. Then,the following properties hold: (i) C K ( σ ) = C K ( σ ) ; and (ii) if G ( C ) is acyclic,then changing from σ to σ strictly increases the size of the cut. Fig. 1: Instance for which the strong improvement dynamics cycles.In this section we focus on weighted graphs and, as discussed in the intro-duction, we close an open problem stated by Gourvès and Monnot in [13] byroviding an instance in which there is a cycle of improving deviations per-formed by minimal coalitions only. More specifically, the loop is composed bythe deviation of a clique, followed by four improvements performed by singleplayers.
Proposition 2.
No strong potential function exists for the max k-cut game,even if only minimal coalitions are allowed to deviate.Proof.
Consider the graph G and the coloring σ depicted in Figure 1, where, ifa node v is contained in the dashed ellipse labeled i , then v is colored i in σ , andwhere M and ε denote a very large and small positive value, respectively.Consider coalition C = { a, b, d, g } and consider the deviation σ where σ ( a ) =2, σ ( b ) = 3, σ ( d ) = 1, σ ( g ) = 1. It is easy to check that this deviation is prof-itable for players in C . In fact, they all improve their utility by ε .Player a has utility 3 + M in σ , and since it has an edge of weight M towardsnode i having color 3, a can only deviate to color 2. Hence, a strictly improvesits utility from 3 + M to 3 + M + ε only if node d changes color. Analogously,in σ , d has one edge of weight M towards node h having color 3. Thus, d canonly switch to color 1 and this is convenient for it only if both a and b leavecolor 1. If this happens, d ’s utility increases by at least ε . Similarly to a , player b deviates to color 3 only if player g switches to color 1, and this happens only ifboth a and b deviates too. To sum up, both d and g deviate if and only if a and b deviate too. Thus, C is minimal. Note that edge { d, g } becomes monochromatic,but d and g ’s new payoffs make the deviation worth it anyway, since they bothincrease their utility by ε .After the players in C jointly deviate from σ , player a , who is now colored 2,can go back to color 1, improving its utility from 3 + ε + M to 4 + M . Because of a ’s deviation, d ’s utility goes down to 1 / ε + M . Thus, it goes back to color2, achieving 1 + ε + M . Also g , whose utility is now 1 / ε + M , deviates to itsold color in σ , that is color 3, and it gets 1 / ε + M . In this configuration b ’utility is 5 / ε + M . Thus going back to color 1 its utility improves to 3 + M and we are now back to the initial configuration σ . From now on we will focus on unweighted graphs. In [12] it is shown that inthe weighted case, any optimal strategy profile is always a 3-SE, and there areweighted graphs in which every optimal coloring is not a 4-SE. In this sectionwe improve this result for unweighted graphs, by showing that a 5-SE alwaysexists. This also establishes a separation between the weighted and unweightedcase. In particular, we show that by performing minimal strong improvementswith coalitions of size at most five, the cut value increases. It implies that thecut value is a potential function and thus the dynamics converges to 5-SE. Westart by showing a simple lemma that is used in the rest of the section.
Lemma 1.
Let σ be an NE and let C be a minimal coalition which would profitby deviating from σ to σ . If there exists two players u, x ∈ C such that:i) σ ( u ) = σ ( x ) (ii) σ ( u ) = σ ( x ) (iii) { y ∈ C |{ u, y } ∈ E, σ ( y ) = σ ( x ) } = { x } then δ σ ( u ) u ( σ ) = δ σ ( u ) u ( σ ) . Moreover, if there exists a third player v ∈ C suchthat σ ( v ) = σ ( x ) and σ ( v ) = σ ( x ) , then { u, v } / ∈ E .Proof. Since σ is an NE we know that u cannot improve its utility by deviatingalone to σ ( u ) which implies: δ σ ( u ) u ( σ ) ≤ δ σ ( u ) u ( σ ) . By (iii) we know that, moving from σ to σ , the only neighbor of u whichleaves σ ( u ) is x which means that its new neighbors colored σ ( u ) are at least δ σ ( u ) u ( σ ) −
1, because, a priori, other players could move to the same strategyin σ , hence δ σ ( u ) u ( σ ) ≥ δ σ ( u ) u ( σ ) −
1. Moreover, player u strictly improves itsutility, which means δ σ ( u ) u ( σ ) < δ σ ( u ) u ( σ ). Using both inequalities we obtain: δ σ ( u ) u ( σ ) > δ σ ( u ) u ( σ ) − . As a consequence, we have δ σ ( u ) u ( σ ) ≥ δ σ ( u ) u ( σ ), and hence δ σ ( u ) u ( σ ) = δ σ ( u ) u ( σ ),which in turn implies in particular that δ σ ( u ) u ( σ ) = δ σ ( u ) u ( σ ) −
1, i.e. u ’s utilityimproves exactly by one. Thus, given a player v like in the hypothesis, if { u, v } ∈ E then u ’s utility would not increase after the deviation.Proposition 1 and Lemma 1 can be used to prove the following proposition,which shows that when the size of a deviating coalition C is related in a cer-tain way to the number of colors used by the players in C , then the improvingdeviation always increases the size of the cut. Proposition 3.
Let σ be an NE and let C be a minimal coalition which wouldprofit by deviating from σ to σ . If | C K ( σ ) | ∈ { , | C | − , | C |} , then the deviationstrictly improves the size of the cut. Gourvès and Monnot [12] show that in weighted graphs an optimal solutionis always a 3-strong equilibrium, that is, it is resilient to any joint deviation byat most three players. Proposition 3 already extends this result since it impliesthat unweighted graphs admit a potential function when minimal coalitions ofat most four players are allowed to deviate, implying that 4-SE always exists.We now prove that the cut value is a potential function even when the deviationis extended to coalitions of size at most five. This implies that a 5-SE alwaysexists in unweighted graphs.
Theorem 4.
Any optimal strategy profile is a 5-SE.
Local strong equilibria
In this section we introduce and discuss local strong equilibria. As our mainresult, we show that, for any k , such an equilibrium always exists. Interestinglyenough, our analysis also provides a characterization of the set of local strongequilibria which relates them to q -SE.Let C ⊆ V be a set of players. We say that C is an x -local coalition if thedistance in G between any two players in C is at most x . Moreover, we define an x -Local Strong Equilibrium ( x -LSE) to be a coloring in which no x -local coalitioncan profitably deviate. In this section, we will consider only the case x = 1, thatis, the coalition C induces a clique. We will use LSE in place of 1-LSE.Let us introduce some additional notation. Given a node u and a strategyprofile σ , we denote by c u ( σ ) the cost of u in σ , namely the number of neighbors of u that have the same color of u in σ , i.e. δ σ ( u ) u ( σ ). Notice that c u ( σ ) = δ u − µ u ( σ ).Given a coalition C , we also define c u,C ( σ ) = |{ ( u, v ) ∈ E | v ∈ C, σ ( v ) = σ ( u ) }| .We now prove a technical lemma which gives some necessary conditions fora clique to deviate profitably from an NE. Lemma 2.
Let σ be an NE. Suppose there exists a deviation σ such that allthe members of C can lower their cost changing from σ to σ . The followingconditions must hold :(i) | C i ( σ ) | = | C i ( σ ) | for all i = 1 , . . . , k ;(ii) c u ( σ ) − c u ( σ ) = 1 for all u ∈ C ;(iii) c u ( σ ) = c u ( σ − u , σ ( u )) for all u ∈ C . The following lemma underlines a very interesting property of deviatingcliques, which allows us to study only cliques formed by at most k players. Lemma 3.
Let C be a clique which profits by deviating from an NE σ to σ .Then there exists j ≤ | C K ( σ ) | and a subcoalition C = { u i , . . . , u j } ⊆ C whoseplayers can improve their payoffs by deviating alone to the strategy they use in σ . Moreover it holds :(i) σ ( u i ) = σ ( u i +1 ) for all i = 1 , . . . , j − ;(ii) σ ( u j ) = σ ( u ) .Proof. Consider a player v ∈ C and assume without loss of generality that σ ( v ) = 1 and σ ( v ) = 2. By Lemma 2, there is at least one player, say v , in C such that σ ( v ) = 2. If σ ( v ) = 1, then C = { v , v } . Otherwise, call withoutloss of generality σ ( v ) = 3. Then, once again by Lemma 2 there is a node in C ,say v , with σ ( v ) = 2. We can iterate this argument until we get a node v h with ‘ := σ ( v h ) ∈ { , , . . . , h − } . We set C = { v ‘ , v ‘ +1 , . . . , v h } and set j = | C | .Notice that C already satisfies the properties 1 and 2 of the statement of thelemma (observe also that it could be C = C ).Let σ ∗ be the strategy profile in which the players in C play as in σ whilethe others play as in σ . It remains to show that all players in C is improving itsutility by changing from σ to σ ∗ . Let ν = σ ∗ ( v i ). By definition of σ ∗ , we claim E 2-SE LSEk-SESE
Fig. 2: Equilibria in the unweighted max-k-cut gamethat c v i ( σ ∗ ) = c v i ( σ − v i , ν ) −
1. This is true because there is exactly one playerin C that leaves color ν and thus the number of v i ’s neighbors with such a colordecreases by 1. Hence, c v i ( σ ∗ ) = c v i ( σ − v i , ν ) − c v i ( σ − v i , σ ( v i )) − c v i ( σ ) − , where in the last equality we used property (iii) of Lemma 2 on σ .Lemma 3 allows us to prove the main result of this section, which is thefollowing: Theorem 5.
Any optimal strategy profile is an LSE.Proof.
Let σ be an optimal strategy profile and assume σ is not an LSE. Clearly, σ is an NE. Then there exists a coalition C = { u , . . . , u j } of j ≤ k players anda strategy profile σ which satisfy the conditions of Lemma 3. We will show thatthe size of the cut increases by exactly j from σ to σ , which is a contradiction.First of all, observe that all the edges between players of C are in the cutboth in σ and σ . Moreover, from property (ii) of Lemma 2, we have that theutility of each u i ∈ C increases exactly by one. Let E i = {{ u i , v }| v / ∈ C } . As aconsequence, we have that, for each u i ∈ C , the number of edges in E i crossingthe cut increases by exactly one. Since E i and E j are disjoint for i = j , the sizeof the cut increases exactly by j from σ to σ .We conclude this section by discussing some consequences of our analysisabout how LSE is related to q -SE: these results are depicted in Figure 2. Someinclusions are straightforward from the definition of q -SE. Here we show that anLSE is always a 2-SE and a k -SE is always an LSE. Concerning the former fact,note that a coalition of 2 players can profitably deviate from an NE σ if and onlyif there exists an edge between them. In fact, otherwise, they could profitablyeviate alone from σ . This means that such a coalition is a clique of two players,and hence a local coalition. As far as the latter relation is concerned, we provethe following: Proposition 6. A k -SE is always an LSE.Proof. Let σ be a k -SE and, by contradiction, let C be a clique which wouldprofit deviating to σ . By Lemma 3 there exists a minimal subcoalition C of atmost k players which can profit deviating alone, which is a contradiction.It is worth noticing that, as a consequence, when k = 2 the set of LSEcoincides exactly with the set of 2-SE. On the other hand, for k ≥ In this section we show that an SE always exists for some special classes ofunweighted graphs. More precisely, we prove that in graphs with large girth orlarge degree, any optimal strategy profile is an SE. It is worth noticing that,for general graphs, we have already proved that any optimal coloring is both a5-SE and an LSE. We conjecture that it is indeed always an SE, even if thisseems to be challenging to prove in general. A natural approach could be thatof using the size of the cut as a strong potential function, that is Φ S ( σ ) = S ( σ ),as it has already been done for proving that max k-cut games admit a Nashequilibrium [15] [18]. However, it can be argued that this approach cannot workin general, since a profitable coalition deviation could sometimes result in a cut-value decrease. This is stated in the following proposition whose proof can befound in the appendix. Proposition 7.
The size of the cut is not a strong potential function for themax k -cut game on unweighted graphs. Even though there exist strong improvements that can decrease Φ S , it doesnot mean that such function cannot be used in some interesting special setting.Indeed, there are cases in which Φ S ’s value always increases after a strong im-provement, that is, they admit a strong potential function. From now on weassume that only minimal coalitions can deviate. Bounded girth
Given a graph G , let ρ ( G ) be its girth , that is the size of theminimum cycle. We show that a graph with girth ρ ( G ) always admits a q-SE,for q ≤ ρ ( G ) −
3. This implies that when ρ ( G ) ≥ ( | V | + 3) / Proposition 8.
Given an unweighted graph G with girth ρ ( G ) and any numberof colors k , an optimal coloring is a (2 ρ ( G ) − -SE. Corollary 1. If ρ ( G ) ≥ ( | V | + 3) / , then an optimal coloring is always an SE. ounded degree Here we show that whenever the number of colors k is largeenough with respect to the maximum degree of the graph, then any optimalstrategy profile is an SE. More precisely, we prove the following: Proposition 9.
Any optimal strategy profile is an SE when k ≥ (cid:6)(cid:0) δ M + 1 (cid:1) / (cid:7) .Proof. Let σ ∗ be an optimal strategy profile and assume σ ∗ is not an SE. Thena coalition C and a strategy profile σ exist such that all players in C strictlyimproves their utility by deviating to σ . We will show that in this case the sizeof the cut will strictly increase in σ , which contradicts the optimality of σ ∗ .As we already pointed out, σ ∗ is NE. Moreover, consider any node u . Sinceits degree δ u is at most δ M ≤ k −
1, we have that in any coloring, by thepigeonhole principle, there must exist a color that appears at most once in u ’sneighborhood. As a consequence, since σ ∗ is an NE, it holds that µ u ( σ ∗ ) ≥ δ u − C must strictly improve their utility, wehave that, for every u ∈ C , µ u ( σ ∗ ) = δ u − µ u ( σ ) = δ u . This impliesthat the size of the cut must strictly increase. Indeed, consider the edge set F = {{ u, v }| u ∈ C or v ∈ C } . Clearly, only edges in F can enter or leave thecut when the strategy profile changes from σ ∗ to σ . Moreover, all edges in F belong to the cut E ( σ ) while there is at least an edge that is not in E ( σ ∗ ). We investigated coalition resilient equilibria in the max k -cut game. We solvedan open problem proposed in [13] on weighted graphs by showing that improvingdeviations performed by minimal coalitions can cycle. We then provided somepositive results on unweighted graphs. More precisely, we proved that any opti-mal coloring is both a 5-SE and a 1-LSE. We also showed that SE exist for somespecial cases, namely, when the graph has a large girth or the number of colorsis large enough with respect to the maximum degree.Even though we made a progress on the topic, the problem of understandingwhether any instance of the max k -cut game admits strong equilibria is stillopen on both weighted and unweighted graphs. We conjecture that an optimalstrategy profile is always an SE in the unweighted case. However, proving thatseems to be really challenging. Another possible way to prove the existence ofan SE would be that of providing a strong potential function. We proved inProposition 2 that such function cannot exist on weighted graphs even whenonly minimal coalitions can deviate but it is still unknown whether a strongpotential function exists or not on unweighted graphs. Along this direction, aninteresting intermediate step could be that of proving the existence of q -SE forpossibly non-constant values of q > x -local coalitions, our results are only about the case x = 1 onunweighted graphs. Some other research questions could be the study of theexistence of x -local strong equilibrium for x >
1, and how to extend our resultsto weighted graphs. For instance, it would be interesting to investigate whetherany instance of the max k -cut game on weighted graphs admits local strongequilibria. eferences
1. K. R. Apt, B. de Keijzer, M. Rahn, G. Schäfer, and S. Simon. Coordination gameson graphs.
Int. J. Game Theory , 46(3):851–877, 2017.2. R. J. Aumann. Acceptable points in games of perfect information.
Pacific Journalof Mathematics , 10:381 – 417, 1960.3. H. Aziz and R. Savani. Hedonic games. In
Handbook of Computational SocialChoice , chapter 15. Cambridge University Press, 2016.4. D. Bilò, L. Gualà, S. Leucci, and G. Proietti. Locality-based network creationgames. In , pages 277–286, 2014.5. V. Bilò, A. Fanelli, M. Flammini, G. Monaco, and L. Moscardelli. Nash stable out-comes in fractional hedonic games: Existence, efficiency and computation.
Journalof Artificial Intelligence , 62:315–371, 2018.6. A. Bogomolnaia and M. O. Jackson. The stability of hedonic coalition structures.
Games and Economic Behavior , 38:201–230, 2002.7. R. Carosi, M. Flammini, and G. Monaco. Computing approximate pure nashequilibria in digraph k-coloring games. In
Proceedings of the 16th Conference onAutonomous Agents and MultiAgent Systems, AAMAS , pages 911–919, 2017.8. R. Carosi and G. Monaco. Generalized graph k-coloring games. In
Proceedings ofthe 24th International Conference on Computing and Combinatorics, COCOON ,pages 268–279, 2018.9. G. Christodoulou, V. S. Mirrokni, and A. Sidiropoulos. Convergence and approx-imation in potential games. In
Proceedings of the 23rd Annual Symposium onTheoretical Aspects of Computer Science, STACS , pages 349–360, 2006.10. A. Cord-Landwehr and P. Lenzner. Network creation games: Think global - actlocal. In
Proceedings of the 40th International Symposium on Mathematical Foun-dations of Computer Science, MFCS , pages 248–260, 2015.11. M. Feldman and O. Friedler. A unified framework for strong price of anarchy inclustering games. In
Proceedings, Part II, of the 42nd International Colloquiumon Automata, Languages, and Programming, ICALP , pages 601–613, 2015.12. L. Gourvès and J. Monnot. On strong equilibria in the max cut game. In
Pro-ceedings of the 5th International Workshop on Internet and Network Economics,WINE , pages 608–615, 2009.13. L. Gourvès and J. Monnot. The max k -cut game and its strong equilibria. In Proceedings of the 7th Annual Conference on Theory and Applications of Modelsof Computation, TAMC , pages 234–246, 2010.14. T. Harks, M. Klimm, and R. H. Möhring. Strong nash equilibria in games with thelexicographical improvement property.
Int. J. Game Theory , 42(2):461–482, 2013.15. M. Hoefer.
Cost sharing and clustering under distributed competition . PhD thesis,University of Konstanz, 2007.16. R. M. Karp. Reducibility among combinatorial problems. In
Proceedings of asymposium on the Complexity of Computer Computations , pages 85–103, 1972.17. M. J. Kearns, M. L. Littman, and S. P. Singh. Graphical models for game theory. In
Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence, UAI ,pages 253–260, 2001.18. J. Kun, B. Powers, and L. Reyzin. Anti-coordination games and stable graphcolorings. In
Proceedings of the 6th International Symposium on Algorithmic GameTheory, SAGT , pages 122–133, 2013.9. S. Leonardi and P. Sankowski. Network formation games with local coalitions. In
Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Dis-tributed Computing, PODC , pages 299–305, 2007.20. G. Monaco, L. Moscardelli, and Y. Velaj. Stable outcomes in modified fractionalhedonic games. In
Proceedings of the 17th International Conference on AutonomousAgents and MultiAgent Systems, AAMAS , pages 937–945, 2018.21. P. N. Panagopoulou and P. G. Spirakis. A game theoretic approach for efficientgraph coloring. In
Proceedings of the 19th International Symposium on Algorithmsand Computation, ISAAC , pages 183–195, 2008.22. A. A. Schäffer and M. Yannakakis. Simple local search problems that are hard tosolve.
SIAM J. Comput. , 20(1):56–87, 1991. ppendix A: omitted proofs
Proof of Proposition 1
Let us start with the proof of property (i). In order to prove the equality weshow that C K ( σ ) ⊆ C K ( σ ) and C K ( σ ) ⊆ C K ( σ ). – if v ∈ C and σ ( v ) / ∈ C K ( σ ), then it is easy to see that v can deviate aloneand this is a contradiction with the fact that C is minimal; – if there is a color i ∈ C K ( σ ) \ C K ( σ ), then again it is easy to see that thecoalition C \ C i ( σ ) can perform a strong improvement and this is a contra-diction with the fact that C is minimal.In order to prove property (ii), we first show the following lemma that willsubsequently be used. Lemma 4.
In undirected weighted graphs any acyclic coalition C , with | C | > ,that can perform a minimal strong improvement must be placed in no more thantwo colors, that is | C K ( σ ) | ≤ .Proof. Given a minimal acyclic coalition C , if σ is the resulting coloring after C deviates, we know from property (i) that C K ( σ ) = C K ( σ ). We build a directedgraph G = ( C, E ) where an edge ( v, u ) is in E if and only if (i) { v, u } ∈ E ,(ii) σ ( v ) = σ ( u ), and (iii) if u does not change color, then v has no interestin deviating, that is, µ v ( σ ) − w ( { v, u } ) ≤ µ v ( σ ). In other words, the node v has interest in changing color only if also node u changes color. G representsall those moves that are mandatory in order to make the deviation profitableto every player in C . Since C performs a minimal strong improvement, everyplayer v ∈ C must have at least one ingoing edge and one outgoing edge, sinceotherwise v could be removed from C , i.e., there exists a sub coalition of C notcontaining v that can perform a strong improvement, or v could deviate alone,respectively. This implies the existence of at least one oriented cycle in G , andconsequently the existence of a cycle in G ( C ) too. Since we know that G ( C )is acyclic, the only type of cycle that can happen in G is the one limited toadjacent vertices, and this happens only when the number of colors used by C is at most 2.Lemma 4 says that a minimal coalition C must use at most two colors when G ( C ) is acyclic. If | C K ( σ ) | = 1 then | C | = 1 by the minimality condition,and we already know that the cut-value always strictly increases after a Nashimprovement [9]. If | C K ( σ ) | = 2 we know from [12] that each strong improvementperformed by a coalition that uses only two colors (i.e., the classical max cutproblem) always increases the size of the cut. roof of Proposition 3 By hypothesis we know that | C K ( σ ) | ∈ { , | C |− , | C |} . Let us consider the threecases in detail: – | C K ( σ ) | = 2. This result derives from [12], which shows that in the max cutgame (i.e., k = 2), any strong improvement always strictly increases the sizeof the cut. – | C K ( σ ) | = | C | . For each color i ∈ C K ( σ ) there is a unique player u ∈ C such that σ ( u ) = i . By Proposition 1, property (i), we know that nopair of players can share the same color in σ , since otherwise C would notbe minimal. Moreover, for each player u ∈ C it holds by Lemma 1 that δ σ ( u ) u ( σ ) = δ σ ( u ) u ( σ ), namely the number of edges that leave the cut is equalto the number of edges towards non-deviating players that enter the cut. Inaddition, since players’ utilities must increase there must be at least | C | − σ , that is the cut-value increases. – | C K ( σ ) | = | C | −
1. There must be exactly two vertices colored the same in σ , let them be u and v . Analogously, in σ there must be exactly two nodescolored the same too, let them be x and y . There are three possible cases: • u = x and v = y , that is u and v stay together in both σ and σ . If { u, v } / ∈ E , then by Lemma 1 | C K ( σ ) | = | C | , δ σ ( u ) u ( σ ) = δ σ ( u ) u ( σ ) and δ σ ( v ) v ( σ ) = δ σ ( v ) v ( σ ) and similarly to the case | C K ( σ ) | = | C | , u and v provide a new edge each to the cut which increases its size. Conversely,if { u, v } ∈ E then again by Lemma 1 at least one player among u and v does not improve its utility after the deviation, thus C is not a strongimprovement. • u = x and v = y , that is in σ two new nodes x, y ∈ C share the samecolor, while u and v are now separated. If { x, y } ∈ E , then this edgeleaves the cut in σ . If x and y switch to a color from which only one playerdeviates, then by Lemma 1 their utility do not strictly increase. Thus, itmust necessarily be that x and y benefit from the deviation of at leasttwo players, and the only chance is u and v . Therefore, σ ( x ) = σ ( y ) = σ ( u ) = σ ( v ), and { x, u } , { x, v } , { y, u } , { y, v } ∈ E . In order to strictlyincrease x and y ’s utility it must be that δ σ ( x ) x = δ σ ( u ) x and δ σ ( x ) y = δ σ ( u ) y ,because they earn two edges and they lose edge { x, y } . Moreover, weknow from Lemma 1 that δ σ ( u ) u = δ σ ( u ) u and δ σ ( u ) v = δ σ ( v ) v . So, four newedges enter the cut and edge { x, y } leaves the cut, that is the cut-valueincreases. Conversely, if { x, y } / ∈ E , then the potential increases in a waysimilarly to the one described in the previous case. • One node among u and v , say u , shares color σ ( u ) with another node x ∈ C . It must be that { u, x } / ∈ E otherwise, by Lemma 1, x would haveno desire to deviate. Thus, the cut-value must increase. roof of Theorem 4 We will argue that any improving deviation performed by a coalition of size atmost 5 strictly increases the size of the cut.Let C = { u, v, w, x, y } be a minimal coalition of size five that can deviatefrom a stable coloring σ inducing coloring σ . We already know from Proposition3 that when C is placed in either two, four or five colors in σ , then any of itsminimal improvement always increases the cut-value. Thus, the remaining caseis when | C K ( σ ) | = 3. Let the colors used be 1, 2, and 3. Let us consider how thenodes in C are located in σ .First, there could be three nodes u, v, w sharing the same color 1, while nodes x and y are colored 2 and 3, respectively. Since u, v and w have to choose whichcolor to deviate among 2 and 3, at least two of them have to be colored the samein σ too. Let us assume without loss of generality that u an v share the samecolor in σ . If { u, v } ∈ E , then by Lemma 1 they do not improve their utilitybecause they benefit from the deviation of only one player. This implies that thenodes in { u, v, w } that are colored the same in σ cannot have any edge betweenthem. Moreover, if one of the three nodes, say w , deviates to the same color asanother player of the coalition, say x , then { w, x } / ∈ E again by Lemma 1. Thus,the potential value increases for such configuration.The other case is when two nodes u, v are colored 1, two nodes w, x arecolored 2 and the fifth node y is colored 3. Let us analyze how the players in C can be located in σ after the deviation. – There are three nodes paired together and two nodes alone. The triplet mustnecessarily be composed by a pair of nodes that are paired together also in σ , let them be u, v , plus a third node. The number of edges in the subgraphinduced by such triplet is at most one. In fact, there is a player in suchsubgraph with degree 2 then, since it benefits from the deviation of at mosttwo players and σ is stable, its utility does not increase after the deviation.Moreover, there are two players in C that are alone in σ , that is they addat least one edge each to the cut. Therefore, at most one edge leaves the cutand at least two edges enters the cut, namely Φ increases. – There are two pairs of nodes together, and a node alone. If the node aloneis y in both σ and σ , then the two pairs of nodes in σ must be the samein σ , that is σ ( u ) = σ ( v ) and σ ( w ) = σ ( x ). One of the two pairs, say u, v deviates to σ ( y ), that is it benefits only from the deviation of y , andby Lemma 1 this implies that { u, v } / ∈ E . The other pair w, x deviates to σ ( u ), and if { w, x } ∈ E it implies a deviation similar to the one described inProposition 3 for | C K ( σ ) | = | C | −
1, case 2, and we know that such type ofdeviation increases the cut-value. If { w, x } / ∈ E , then the cut-value increasessimilarly to Proposition 3, case 1 of | C K ( σ ) | = | C | −
1. Conversely, supposethat y is now paired with another node, say u . If { y, u } ∈ E then in orderto make player y deviate profitably, it must have two edges { y, w } , { y, x } towards color σ ( w ), and it must be that σ ( y ) = σ ( w ). As described inprevious cases, this kind of deviation increases the cut-value. If { y, u } / ∈ E ,then by Lemma 1 y and u add at least two edges to the cut. The otherair is necessarily colored σ ( y ), that is the two players benefit only from y ’sdeviation. Lemma 1 implies that they cannot have an edge among them,therefore the edges in the cut increases. Proof of Lemma 2
Let u ∈ C be a node in the coalition and let ν = σ ( u ) be the new color of u .We claim that 0 < c u ( σ ) − c u ( σ ) ≤ c u,C ( σ − u , ν ) − c u,C ( σ ) (1)Indeed, first notice that u strictly decreases its cost deviating to σ , and hence c u ( σ ) < c u ( σ ). Moreover, since σ is an NE we also have that c u ( σ ) ≤ c u ( σ − u , ν ).As a direct consequence we get the following double inequality:0 < c u ( σ ) − c u ( σ ) ≤ c u ( σ − u , ν ) − c u ( σ ) . Since the players outside the coalition do not change their colors, it holds that( c u ( σ ) − c u,C ( σ )) = ( c u ( σ − u , ν ) − c u,C ( σ − u , ν )). And hence we have c u ( σ ) = c u ( σ − u , ν ) − c u,C ( σ − u , ν ) + c u,C ( σ ), which implies: c u ( σ − u , ν ) − c u ( σ ) == c u ( σ − u , ν ) − ( c u ( σ − u , ν ) − c u,C ( σ − u , ν ) + c u,C ( σ )) == c u,C ( σ − u , ν ) − c u,C ( σ ) . Replacing this expression in the double inequality we have derived before, weget Equation (1).Since C is a clique and σ ( u ) = ν we know also that c u,C ( σ ) = | C ν ( σ ) | − c u,C ( σ − u , ν ) = | C ν ( σ ) | . Replacing these terms in Equation (1) we obtain:0 < c u ( σ ) − c u ( σ ) ≤ | C ν ( σ ) | − | C ν ( σ ) | + 1 (2)which implies that | C ν ( σ ) | ≤ | C ν ( σ ) | . The arguments used so far hold for anyplayer u ∈ C , and thus the above inequality also holds that for all colors i ∈ C K ( σ ), i.e. | C i ( σ ) | ≤ | C i ( σ ) | for each i ∈ C K ( σ ). Observe that this impliesalso that C K ( σ ) ⊆ C K ( σ ).By summing up over all colors in C K ( σ ), we have that: X i ∈ C K ( σ ) | C i ( σ ) | = X i ∈ C K ( σ ) | C i ( σ ) | = | C | and as a consequence we get | C i ( σ ) | = | C i ( σ ) | for all i ∈ C K ( σ ) that is exactly(i). Thus we can rewrite Equation (2) as:0 < c u ( σ ) − c u ( σ ) ≤ c u ( σ ) = |{ ( u, v ) | v / ∈ C, σ ( v ) = ν }| + |{ ( u, v ) | v ∈ C \ { u } , σ ( v ) = ν }| = |{ ( u, v ) | v / ∈ C, σ ( v ) = ν }| + | C ν ( σ ) | − |{ ( u, v ) | v / ∈ C, σ ( v ) = ν }| + | C ν ( σ ) | − c u ( σ − u , ν ) − c u ( σ ) = c u ( σ ) + 1, we can conclude (iii). Proof of Proposition 7
Fig. 3: Instance G and coloring σ for which the minimal strong improvementstrictly decreases the cut-value.Consider the graph G and a coloring σ . Both are depicted in Figure 3. K denotes a clique composed by nodes a, b, . . . , g . Due to space shortage, only theneeded edges are explicitly reported. Regarding the omitted edges, we assumethat (i) none of the non-labelled nodes vertices desire to change color, (ii) node g does not want to deviate to any other color, (iii) nodes h, i, l, m, n, and o havesufficient edges towards all the other colors, except color 1, to prevent them fromdeviating to such colors, and (iiii) a, b, c, d, e , and f have sufficient edges towardsll the other colors, except colors 2 , , , ,
6, and 7 respectively, to prevent themfrom deviating to such colors. Given these assumptions, the only minimal coali-tion that can perform an improving move is C = { a, b, c, d, e, f, h, i, l, m, n, o } .In fact, a has six edges towards both color 1 and color 2, thus it is willing todeviate to color 2 only one of its neighbors leaves that color. Node h has oneedge towards color 2 and two edges towards nodes a and b , that are colored 1.Thus, h moves to color 1 only if players a and b deviate. Similarly to node a , b deviates to color 3 only if i changes color, and so on and so forth. Finally, player o deviates to color 1 only if a and f change color. To sum up, the coloring σ obtained after C ’s deviation is such that σ ( a ) = 2 , σ ( b ) = 3 , σ ( c ) = 4 , σ ( d ) =5 , σ ( e ) = 6 , σ ( f ) = 7 , σ ( h ) = . . . = σ ( o ) = 1, while the other nodes’ strategiesremain unchanged. It is not difficult to check that Φ ( σ ) − Φ ( σ ) = − Φ ’s value strictly decreases even after a strong minimal improvement. Proof of Proposition 8
Fig. 4: A coalition C whose size is upper-bounded as defined in Proposition 8has at most two cycles.Let σ ∗ be an optimal coloring and let C be a minimal coalition that deviatesfrom σ ∗ to σ , where | C | ≤ (2 ρ ( G ) − G ( C ) according to | C | . – Trivially, if | C | < ρ ( G ) then G ( C ) must not contain any cycle; – If | C | ≥ ρ ( G ) then G ( C ) can contain at least one cycle; – Given a cycle, since it has length at least ρ ( G ) there always exist a pair ofnodes u, v , such that dist G ( C ) ( u, v ) ≥ b ρ ( G ) / c , that is the shortest pathbetween u and v is made of at least b ρ ( G ) / c + 1 nodes, u and v included.Thus, by having at least d ρ ( G ) / e − G ( C ) is allowed tohave a second cycle that makes use of the shortest path between u and v ; – Since | C | ≤ (2 ρ ( G ) − C cannot contain more than twocycles in G ( C ), otherwise there would be a cycle of length strictly less than ρ ( G ).Let us assume that G ( C ) contains two cycles c , c with some nodes in common.Let u and v be the first and last node in the intersection of the cycles, respectively.oreover, since there cannot be any more cycles, each node of the cycles can bethe root of a (possible empty) tree of deviating players (if we do not considerthe other nodes in the cycles). A sketch of G ( C ) is depicted in Figure 4.First, we know from Proposition 1 that the deviation performed by a player w that belongs to C but not to c and c always increases the size of the cut.If it was not so then it would mean that w is not increasing its utility because,since it does not belong to any of the cycles, its adjacent players who deviatealways select a color different from its own in σ , otherwise some node could beremoved from C , contradicting the minimality assumption.Let w ∈ c ∪ c be a node that belongs to one of the two cycles. The only w ’sdeviating neighbors that can be in the same color as w in σ are the ones thatbelong to c or c . In fact, if there is a node x that is adjacent to w , is not in c or c and σ ( x ) = σ ( w ), then by removing from C the subtree with root x westill have a deviating coalition. For the same reason, there cannot be a deviatingnode x that is adjacent to w , is not in c and c and such that σ ∗ ( w ) = σ ∗ ( x ),because w does not need x to deviate. Therefore, we can assume in the followingthat C is composed only of c and c .Node w must have at least one adjacent node x colored differently from itin σ ∗ , that is σ ∗ ( w ) = σ ∗ ( x ), otherwise w would have no reason to deviatejointly because it could perform a Nash deviation itself. Moreover, since (i)2 ≤ δ w ( G ( C )) ≤ ∀ w ∈ C , (ii) each deviating player has at least one properedge in G ( C ) with respect to σ , (iii) by Lemma 1, players with degree 2 cannotshare the same color with some other deviating nodes in σ , otherwise theycould have deviated alone, and (iiii) there are only two not-neighboring players u, v with degree 3 in G ( C ), it must necessarily be that no edge in G ( C ) ismonochromatic with respect to σ . Moreover, the difference between the numberof properly colored edges incident to w ∈ C , where δ w ( G ( C )) = 2, and towardsnodes outside C in σ and σ ∗ , respectively, is greater or equal than 0. This istrue otherwise w ’s utility has not strictly improved after the deviation, since w can improve its utility by at most 1 from the deviation of its neighbors because σ ∗ is Nash stable by definition. Thus, the deviating players with degree in C equal to 2 contributes to the cut a total of | C | − σ ∗ is a 5-SE, we canassume | C | >
5, that is more than three edges enters the cut. On the other hand,the only two nodes u, v with degree δ u ( G ( C )) = δ v ( G ( C )) = 3 can decrease thenumber of the edges in the cut and towards nodes not in the coalition by at most1 each. This is true otherwise if one of them decreases the cut-value by at least2 then it does not strictly improve its utility. Thus, the cut-value increases, butthis contradicts the fact that σ ∗ is optimal. Appendix B
In this section we show that the inclusions depicted in Figure 2 are all proper.ig. 5: An example of an NE which is not a 2-SE in the Max-Cut game.First we give an example of NE which is not a 2-SE in the Max-Cut game.The example is shown in Figure 5. The strategy profile σ is an NE, but thecouple {c, d} can improve their utility by deviating to σ . a b cdef (a) Initial strategy profile. a b cdef (b) Strategy profile afterthe deviation of b, d and e. Fig. 6: Example of a 2-SE which is not an LSE in the Max-3-Cut game.In Figure 6 we have an example of a 2-SE (Figure 6a) which is not an LSE.Figure 6b shows a profitable deviation of players b, d and e, which form a cliqueof three nodes. Thus the initial strategy profile is not a LSE.Now we show an example of a LSE being not a k -SE. In the following example k = 3. The example is given in Figure 7. It is easy to check that the strategyprofile shown in Figure 7a is an NE. Moreover, the nodes a, d, g, i, m have thehighest possible utility, thus no deviating coalition can contain them, includedthe two cliques of 3 nodes. All possible couples either contain one of the playersmentioned above, or does not satisfy the conditions given in 2, thus they do notprofit deviating. Thus, the strategy profile is an LSE. On the other hand, if e and f change to blue and c to yellow they all improve by 1 their payoff. Thus{ c, e, f } is a non-local coalition which breaks the equilibrium. bcd e f gh i lm (a) Initial strategy profile. abcd e f gh i lm (b) Strategy profile after thedeviation of c, e and f. Fig. 7: Example of a LSE which is not a 3-SE in the Max-3-Cut game.There is only one case left. In Figure 8 is given an instance of the Max-Cutgame along with a 2-SE which is not an SE (if A, B and C deviate in the onlypossible way they all improve their utility by 1).