Price of Anarchy for Graph Coloring Games with Concave Payoff

We study the price of anarchy in a class of graph coloring games (a subclass of polymatrix common-payoff games). In those games, players are vertices of an undirected, simple graph, and the strategy space of each player is the set of colors from $1$ to $k$. A tight bound on the price of anarchy of $\frac{k}{k-1}$ is known (Hoefer 2007, Kun et al. 2013), for the case that each player's payoff is the number of her neighbors with different color than herself. The study of more complex payoff functions was left as an open problem. We generalize by computing payoff for a player by determining the distance of her color to each of her neighbors, applying a non-negative, real-valued, concave function $f$ to each of those distances, and then summing up the resulting values. Denote $f^*$ the maximum value that $f$ attains on the possible distances $0,\dots,k-1$. We prove an upper bound of $4$ on the price of anarchy for general $f$, a bound of $3 \, (1-\frac{1}{k})$ if $f(\frac{k}{4}) \geq f^*$, a tight bound of $2$ for non-decreasing $f$, and a tight bound of $2$ if $f$ attains $f^*$ in $\lfloor\frac{k}{2}\rfloor$ and in $\lceil\frac{k}{2}\rceil$ (tight if $k$ is even). The latter includes what is also known as cyclic payoff. Graph coloring, especially in distributed and game-theoretic settings, is often used to model spectrum sharing scenarios, such as the selection of WLAN frequencies. For such an application, our framework would allow to express a dependence of the degree of radio interference on the distance between two frequencies.

rks " {1, . . . , k}. Vertices of the graph represent players, with the set of colors rks being the strategy space for each player. We sometimes call the set rks the spectrum. Clearly, k-colorings are exactly the strategy profiles of this game. 1 Given a coloring c, define the payoff for player v as µ v pcq :" where f is a non-negative, real-valued function defined on r0, ks (choosing this domain instead of {0, . . . , k´1} is technically easier). Given such f , we denote f˚:" max iP{0,...,k´1} f piq the maximum value that f attains on the possible distances between two colors, and X˚pf q :" {i P N ; 0 ď i ď k´1^f piq " f˚} the set of distances where f˚is attained. We call f |cpvq´cpwq| the contribution of edge {v, w}. So f˚is an upper bound on the contribution of any edge, and it is attained if the two players v and w manage to put a distance between each other which is in the set X˚pf q.
Let c be a k-coloring. For a player v P V and a color t P rks we write the coloring where v changes to color t as crv Ð ts, so In other words, stable coloring are exactly the pure Nash equilibria of the game. The social welfare or welfare of c is W pcq :" vPV µ v pcq. We denote the optimum welfare as W OPT :" max c W pcq, where c runs over all k-colorings; optimum colorings exist since there are only finitely many colorings. If we are dealing with multiple functions taking the role of f , we indicate by putting them in superscript, like W f OPT or W f pcq. Finally, the price of anarchy is: max This notion was introduced by Koutsoupias and Papadimitriou [24]. It measures the worst-case performance-loss due to non-cooperative behavior.
It is easy to see that stable colorings always exist, by a potential function argument. This has been observed before in the broader class of polymatrix common-payoff games [7], 2 and we repeat that simple argument here for completeness. Note that we can write social welfare as follows: 1 The term "graph coloring game" in the literature also names a class of maker-breaker style games, which we are not referring to here.
If a player v makes an improvement step resulting in coloring c 1 (i. e., there is t P rks such that c 1 " crv Ð ts and µ v pcq ă µ v pc 1 q) then {v,w}PE f |cpvq´cpwq| ă {v,w}PE f |c 1 pvq´c 1 pwq| , that is, the part of the sum in (3) with edges incident in v strictly increases. All other terms in the sum are maintained. Hence W pcq ă W pc 1 q. So in particular any optimal coloring is stable, and existence of optimal colorings is guaranteed. Note however that there can also be suboptimal stable colorings, which motivates the study of the price of anarchy.
We use (3) also to compute welfare in many places. In particular, if each edge gives the same contribution, say p, then welfare is 2mp, where m is the number of edges in the graph.

Previous, Related Work and Our Contribution
Previous and Related Work. Graph coloring has been a theme in Combinatorics and Combinatorial Optimization for many decades. A coloring c for a graph G " pV, Eq is called proper or legal if cpvq ‰ cpwq for all {v, w} P E, and given v P V we call a neighbor w P N pvq properly colored if cpvq ‰ cpwq. Determining the minimal k such that a given graph admits a proper k-coloring, i. e., the chromatic number χpGq, is NP-hard [22]. One of the classical highlights is Brooks' theorem [6,27], stating that for each graph G which is neither a complete graph nor a cycle of odd length, a proper ∆-coloring can be constructed by a combinatorial algorithm in polynomial time, where ∆ is the maximum vertex degree in G. Another important algorithmic technique for finding proper colorings with provable worst-case performance is semidefinite programming with randomized hyperplane rounding [21]. In addition, for more than three decades, distributed algorithms for proper colorings have been studied, see [4] for a recent survey. In the distributed model, each vertex is a processor and can communicate with its neighbors, where communication is done in rounds. For example, in one round, each vertex could communicate its color to all of its neighbors. In 2006, a study with human subjects was conducted by Kearns et al. [23], where each subject controlled the color of one vertex, to be selected from a set of χpGq colors, and each subject could see the colors of her neighbors. The goal was to construct a proper coloring. The study used graphs with n " 38 vertices and focused on certain classes of graphs, like cycles, small world graphs, and random graphs from the preferential attachment model. It is reported how the time required to reach a proper coloring is influenced by the structure of the graph.
A game-theoretic view on proper colorings was given in 2008 by Panagopoulou and Spirakis [28]. In their model, payoff for a player v is 0 whenever there exists a non-properly colored neighbor of v, otherwise payoff is the total number of players with color cpvq in the graph. The idea is to incentivize players to create a proper coloring with few different colors. Indeed, two of the main results in [28] are that Nash equilibria can be constructed by improvement steps in polynomial time, and that they are proper colorings using a total number of different colors that is upperbounded by several known upper bounds on χpGq. So [28] yields a constructive proof for all of those bounds. For the same model, improved bounds and an extension to coalitions were later given by Escoffier et al. [12]; and Chatzigiannakis et al. [9] gave algorithmic improvements including experimental studies. Also in 2008, Chaudhuri et al. [10] considered the simpler payoff function that is 1 for player v if all of v's neighbors are properly colored and 0 otherwise. They showed that if the total number of available colors is ∆`1 and if players do improvement steps in a randomized manner, then an equilibrium (which also is a proper coloring) is reached in Oplogpnqq steps with high probability.
A generalization of proper colorings are distance-constrained labelings, which are connected to the frequency assignment problem, where colors correspond to frequencies [20]. Given integers p 1 , . . . , p l , a coloring c is called an Lpp 1 , . . . , p l q-labeling if for all i P rls and all v, w P V we have |cpvq´cpwq| ě p i whenever dist G pv, wq " i, where dist G pv, wq is the graph-theoretical distance between v and w, i. e., the length of a shortest v-w path in G. The notion of Lp1qlabeling coincides with that of proper coloring. The case of Lpp, qq-labelings, that is, where l " 2, has received special attention, in particular in connection with frequency assignment, see, e. g., [8,18,37]. As a variation of this, not the distance |cpvq´cpwq| between colors is considered, but instead it is required that min |cpvq´cpwq|, k´|cpvq´cpwq| ě p i , whenever dist G pv, wq " i, as considered in [35]. Another variation are T -colorings [16,31]: given a set of integers T , a T -coloring is one where |cpvq´cpwq| R T whenever {v, w} P E. This problem also arises in frequency assignment where T is the set of forbidden separations between neighboring senders, which are known to cause interference.
In our model, we allow colorings being evaluated by the payoff function and not only ask whether they have a certain property (e. g., being proper, a T -coloring, etc.) or not. For example, using as the function f , payoff for each player v is the number of her properly colored neighbors. We call this basic payoff. Note that this is a concave function. With this payoff, the game is also known as a max-k-cut game (in the unweighted version), since we partition the vertices in k clusters and payoff for v is the total number of cut edges incident to v, i. e., edges incident to v and running between clusters. These games were studied by Hoefer [19] in 2007, and a tight bound of k k´1 on the price of anarchy was proved, where tightness is already attained on bipartite graphs. The upper bound works by a mean-value argument: for each coloring and for each player v, there is a color t that occurs on at most degpvq k neighbors of v. In a stable coloring, v chooses this t, or even better if possible, hence obtaining a payoff of at least k´1 k degpvq. Since optimum welfare is upper-bounded by 2m, the theorem follows from this (using the handshaking lemma for the sum of degrees). In Section A we provide indication that already for f pxq " x, a straightforward generalization of this technique cannot yield a constant upper bound.
A weighted version of max-k-cut games is obtained by assigning a weight to each edge and defining payoff for v as the sum of the weights of the incident cut edges. This version under the aspect of coalitions among players was studied by Gourvès and Monnot [15] in 2009.
In 2013, Kun, Powers, and Reyzin [25] considered complexity issues for basic payoff combined with variations of the game. It is clear that since welfare for basic payoff can only change in integer steps and can never be more than 2m, we will reach a stable coloring after at most Opmq improvement steps (m is the number of edges in the graph). On the other hand, Kun et al. show that for basic payoff, it is NP-hard to decide whether a graph admits a strictly stable coloring, where the latter notion is defined by replacing ď for ă in (2). They also show that for basic payoff, it is NP-hard to decide whether a directed graph has a stable coloring, where for directed graphs, payoff is defined by having the sum in (1) only range over the out-neighbors of v.
Recently, in 2014, Apt et al. [3] used the function which counts the neighbors of the same color, together with the extension that each player has a set of colors to choose from (whereas we in our model always allow all colors for all players). They study the resulting game under different aspects, including coalitions, price of anarchy, and complexity. As for the price of anarchy without coalitions, it is easy to see that it can be unbounded in those games. The lower-bound construction depends on the fact that we can forbid certain colors for certain players: for example, for each player v let there be a distinct "private" color s v , and in addition there is a "common" color t. Player v can choose her color from the set {s v , t}. The social optimum, namely 2m, is obtained when each player chooses t, whereas a worst case stable coloring, with welfare 0, is obtained when each player v chooses s v . Without the ability to restrict players to certain colors, i. e., if we use our framework for the function f in (4), a tight bound on the price of anarchy of k is easy to see (Section B).
The graph coloring games studied in our work belong to the class of polymatrix games [7,36].
In such a game, we have a graph G " pV, Eq, for each player a set of strategies, and for each edge {v, w} P E a two-player matrix game Γ {v,w} . Each player v chooses one strategy and has to play this same strategy in all the games {Γ {v,w} } wPN pvq corresponding to incident edges. The payoff for v is the sum of the payoffs over all those two-player games. A special case is that of polymatrix common-payoff games, which means that each Γ {v,w} is a common-payoff game, i. e., it always yields the same payoff for v as for w. 3 Thus our graph coloring games belong to this class, since each edge {v, w} contributes the same value f |cpvq´cpwq| to the payoffs of v and of w. Very recently, in 2015, Rahn and Schäfer [30] studied polymatrix common-payoff games with coalitions. They consider pα, lq-equilibria, that are α-approximate equilibria under coalitions of size l. For the corresponding price of anarchy, they give a lower bound of 2αpn´1q{pl´1q`1´2α and an upper bound of 2αpn´1q{pl´1q. Note that in our work, we have l " 1 since we do not consider coalitions, and for this case their bounds are 8. This follows already from the example given in [3], which is based on restricting certain players to certain colors (the "private" and "public" colors).
Our Direction. In 2013, Kun et al. [25] named as open problems the study of the price of anarchy for payoff as per (1) and induced by f being the identity, i. e., the contribution of edge {v, w} is the distance |cpvq´cpwq|. We call this distance payoff. A variation, which Kun et al. also refer to, is the distance notion studied by van den Heuvel et al. [35] in the context of frequency assignment, namely the contribution of {v, w} is min {|cpvq´cpwq|, k´|cpvq´cpwq|}, which in our notation means f pxq " min {x, k´x}. We call this cyclic payoff. It rewards players for keeping a "medium" distance from others. This has an additional interpretation in connection with an example often given in the context of proper colorings, namely where colors correspond to skills and people inside of an organization try to develop skills that are complementary to nearby colleagues (see, e. g., [10,23]). Cyclic payoff refines that idea: subjects are incentivized to develop different but still related (that is, not too far away) skills. Distance and cyclic payoff both result from concave functions f .
Spectrum sharing and frequency (or channel) assignment problems, that is, when a multitude of participants compete for using the same or similar frequencies, has received much attention lately, see, e. g., [11,13,17,29]. Many works in that field use some form of graph coloring (colors corresponding to frequencies) in a distributed or game-theoretic setting. It is common in the frequency assignment literature to consider not only feasible versus infeasible colorings but instead to quantify the "degree of interference". To this end, the spectral distance |cpvq´cpwq| between players v and w is often used as a measure or as a substantial ingredient to a measure. In addition to the references we give above for distance-constrained labeling and for T -coloring, this is documented for example in [ A first game-theoretic study of the distance and cyclic payoff in our framework was conducted by Schink [33] in 2014. He observed that a stable k-coloring for distance payoff can be constructed from a stable 2-coloring for basic payoff by replacing color 2 with k. A similar approach works for cyclic payoff if k is even, replacing color 2 with k 2`1 . Since a stable coloring for basic payoff can be constructed in Opmq improvement steps, this gives a runtime guarantee independent of k. Since welfare can reach up to Ωpmkq for distance and cyclic payoff, such a k-independent guarantee is not easily possible by a direct argument using improvement steps. Interestingly, for cyclic payoff and odd k, we have no k-independent runtime guarantee for the construction of stable colorings at this time. For price of anarchy, Schink proved an upper bound of ∆pGq, the maximum vertex degree in G, for cyclic payoff. Apart from that, we are not aware of any bounds on the price of anarchy for graph coloring games with concave payoff, not even conjectures. The work by Rahn and Schäfer [30], for general polymatrix commonpayoff games, can be considered orthogonal to ours since they do not consider the effects of restricting the two-player games Γ {v,w} to certain classes, whereas we restrict to such games arising from applying a concave function to the color distance |cpvq´cpwq|, resulting in small constant bounds on the price of anarchy. Moreover, [30] allows restricting players to certain sets of colors (making finite bounds on the price of anarchy impossible without coalitions), whereas in our model, each player has the same set of colors to choose from.
Our Contribution and Techniques. We prove constant upper bounds on the price of anarchy for several classes of concave functions f . We give a tight bound of 2 for distance as well as cyclic payoff (tightness requires k being even for cyclic). The upper bound directly generalizes to all concave functions f that are non-decreasing, or fulfill the following condition: This condition (5) essentially means that the maximum value f˚is attained in the middle of the spectrum. Our proofs make extensive use of the concavity of f . However, for distance and cyclic payoff, standard concavity arguments appear to be insufficient. Instead, technical calculations using the particular shape of those functions are used. All our proofs work by local arguments. That is, if we are to prove an upper bound of ρ ě 1 on the price of anarchy, we show the following: for each player v, given the colors {cpwq} wPN pvq of her neighbors, there is a color t P rks such that wPN pvq f |cpwq´t| ě degpvq¨fρ .
Clearly, in a stable coloring, each player chooses such a color t, or better. Hence W pcq ě 2mfρ for a stable c. Since W OPT ď 2mf˚, the bound ρ follows. (For affine f pxq " ax`b, Theorem 1, the expression is slightly different, but in principle follows the same idea. A rigorous formal exposition is given in Section E, covering all the cases.) Our type of arguments reduce the study of the price of anarchy essentially to the study of concave functions under certain aspects. This fits into a long line of results initiated over a decade ago by Roughgarden [32] in the context of selfish routing. In selfish routing, tight bounds on the price of anarchy can be given by just looking at the latency functions on the edges of the network used for routing. Tightness is in the sense that a matching lower bound can be constructed by a network using latency functions of the same class, for example polynomials up to a certain degree. Moreover, the lower-bound examples are very simple in terms of network topology. This has led to the famous statement: "The price of anarchy is independent of the network topology." In this work, we exhibit a similar phenomenon -although structurally simpler -for the graph coloring game. The function f , which converts color distances into payoff, takes the role that the edge latency functions have in selfish routing. The graphs in our lowerbound examples are complete bipartite or cycles, hence of simple structure.

Open Problems.
Computer experiments suggest that upper bound 2 is possible for concave function which attain their maximum left of k 2 , and for the other cases 4 an upper bound of 2`1 5 . As a future challenge, we state: prove a bound on the price of anarchy that matches the bounds observed in experiments. Moreover, computational issues should be addressed, in particular the construction of stable colorings for cyclic payoff and odd k in a number of steps being independent of k. Another type of question is: what modifications to the model will yield a dependence of the price of anarchy on the graph topology? One possibility to explore would be to not only consider colors of neighbors (that is, at graph-theoretical distance 1) but also at larger graph-theoretical distances, for example also at graph-theoretical distance 2, as in the Lpp, qq-labeling problem.

Non-Decreasing Concave f
We start with affine f , which includes distance payoff, and then generalize. Theorem 1. Let a P R ą0 and b P R ě0 and f pxq " ax`b. Then the price of anarchy is upper-bounded by ρpa, b, kq :" 2 a pk´1q`b a pk´1q`2b ď 2. Proof. Let c be stable and v P V . First case: cpvq " 1. By stability, changing to color k cannot improve payoff, that is, µ v pcq ě µ v pcrv Ð ksq. We compute: It may be surprising at first that the situation is not symmetric. But in fact this is to be expected since for example, a player can always force all the distances to her neighbors to be on or below ⌊ k 2 ⌋ by choosing her own color as ⌊ k 2 ⌋`1, but it is not always possible to force all distances beyond 1. So there is an asymmetry between short and long distances. ðñ µ v pcq ě degpvq 1 2 a pk´1q`b An analogous computation can be done if cpvq " k, then we compare to payoff if she would change to color 1.
Second case: 2 ď cpvq ď k´1. We start by proving that v has only neighbors colored 1 or k, and moreover the same number of each. Denote N´pvq :" {w P N pvq ; cpwq ă cpvq} and N`pvq :" {w P N pvq ; cpvq ă cpwq} and N " pvq :" {w P N pvq ; cpwq " cpvq}, i. e., the neighbors of v with smaller, greater, or the same color. For brevity, we write n´:" |N´pvq| and n`:" |N`pvq| and n " :" |N " pvq|. If v changes to color 1, she increases distance by cpvq´1 to each neighbor in N`pvq Y N " pvq and decreases distance by at most cpvq´1 to each neighbor in N´pvq. Since the increase in payoff when changing to color 1 cannot be positive, we have 0 ě (n``n "´n´) pcpvq´1q ě (n`´n´) pcpvq´1q , (6) so n`ď n´since cpvq´1 ą 0. (Note that the shift by b cancels out when we look at differences, and that we can also omit a since a ą 0.) By considering a change to color k, we prove n`ď ní n the same way. It follows n`" n´. Now, assume there is a neighbor w P N pvq with color 2 ď cpwq ď cpvq. If v changes to color 1, then she looses strictly less than cpvq´1 in distance to w, precisely we have for the increase in payoff: This is a contradiction, hence there exists no such neighbor. By considering a change to color k, we exclude the existence of a neighbor w with cpvq ď cpwq ď k´1 in the same way.
We conclude that v has n´neighbors colored 1 and n`neighbors colored k, and no other neighbors. Recall that n´" n`, so each number is degpvq 2 . It follows: pcpvq´1q`pk´cpvqq `degpvq b " degpvq 1 2 a pk´1q`b So we get the same lower bound on µ v pcq as in the case of cpvq P {1, k}. In total, using a trivial upper bound on W OPT , we have: Remark 2. Let c be stable for f pxq " ax`b. For each t P rks denote V t :" {v P V ; cpvq " t} and V 1 :" ⋃ k´1 t"2 V t . Then the proof of the previous theorem shows that V 1 is an independent set in the graph and each v P V 1 has as many neighbors in V 1 as in V k , so in particular degpvq is an even number. Proposition 3. The bound in Theorem 1 is the best possible, and the worst case is assumed already on bipartite graphs.
Proof. Consider the complete bipartite graph K 2,2 and denote {u 1 , u 2 } the vertices of one partition and {w 1 , w 2 } those of the other (so edges are all {u i , w j } with i, j P {1, 2}). Define coloring c by: cpu 1 q :" 1 cpu 2 q :" k cpw 1 q :" ⌊ k`1 2 ⌋ cpw 2 q :" ⌈ k`1 2 ⌉ It is easy to see that c is stable: players w 1 and w 2 have payoff a pk´1q`2b each, no matter which color they choose. Players u 1 and u 2 also have payoff a pk´1q`2b each, but only for colors 1 and k; for all other colors they get less. An optimal coloring is obtained by cpu i q :" 1 and cpw i q :" k for i P {1, 2}, with each edge giving contribution a pk´1q`b. We have, using the number m " 4 of edges, Proof. Denote gpxq :" ax`b. By concavity of f , we have gpxq ď f pxq for all x P r0, k´1s. In the proof of Theorem 1, we have worked locally in that we have considered the behavior of only the single player v at a time. The colors of her neighbors could have been anything, not necessarily from a stable coloring. So what the proof in fact shows is that for each player v and for any choice {cpwq} wPN pvq of her neighbors' colors, v can -by choice of her own colorenforce a payoff (with respect to g) for herself of at least degpvq p 1 2 apk´1q`bq. So v can choose her color t so that, where the second inequality is since g ď f in the interval r0, k´1s. In a coloring c which is stable with respect to f , she will choose such t or better. Hence W pcq ě 2mp 1 2 apk´1q`bq. By monotonicity, f˚" g˚" apk´1q`b, so W OPT ď 2mpapk´1q`bq, and the claim follows.

General Concave f
The proofs in this section work by an interval technique. Let 1 ď a ď b ď k, and define the interval of colors I :" {a, . . . , b} Ď rks. Let v be a player. Clearly, at least half of v's neighbors have their color in I, or at least half of v's neighbors have their color outside I. By an appropriate choice of I, we obtain a sufficient concentration of colors in a certain range of the spectrum. This allows v to choose her color so that the distance to at least half of her neighbors lies in a range where f assumes large values. Of course, the interval I must not be too large or too small for this.
Theorem 5. Let f be concave. Then the price of anarchy is upper-bounded by 4.
Proof. Let k˚P X˚pf q, i. e., k˚P {0, . . . , k´1} with f pk˚q " f˚. Let v P V be a player and denote d :" degpvq the number of her neighbors. We are done if we can prove that for any k-coloring c, there is a color t P rks such that wPN pvq f |cpwq´t| ě df4 , since then player v chooses such t, or better, as her color in a stable coloring.
By concavity, f assumes a value of at least f2 between k˚and the half-way point to 0 as well as to the half-way point to k, i. e., for all x P H :" r k2 , k˚`k´k2 s " r k2 , k`k2 s we have f pxq ě f2 . One of the following two cases is given: cpwq ď ⌊ k 2 ⌋ for at least d 2 of the neighbors w P N pvq, or cpwq ě ⌊ k 2 ⌋`1 for at least d 2 of the neighbors w P N pvq. Assume the first case. We choose t :" ⌈ k`k2 ⌉ ď ⌈ 2k 2 ⌉ " k. Let w be a neighbor with cpwq ď ⌊ k 2 ⌋ (of which we have at least d 2 ones in this case). Then |cpwq´t| " t´cpwq ď t´1 " k`k2 ´1 ď k`k2 , and t´cpwq ě k`k2 ´⌊ k 2 ⌋ ě k`k2´k 2 " k2 . Hence |cpwq´t| P H, which means that w contributes at least f2 to v's payoff when v chooses color t.
Theorem 6. Let f be concave with f p k 4 q ě f˚and k ě 12. Then the price of anarchy is upper-bounded by 3 k k´1 . The intuition for the proof is as follows. If at least half of the neighbors of a player v have their color in the interval r k 6 , 3 6 ks, that is the third in the middle of the spectrum, then we choose a color for v at distance k 3 from k 2 , the center of this interval. This brings the distance to at least half the neighbors in the range where f is at least 2 3 f˚. In the other case, half of all neighbors are in r1, k 6 s Y r 3 6 , ks. Then choosing color k 2 for v creates a distance of k 3 to the centers of each of the two intervals, which has the same effect. The proof is rather technical mainly because of floor/ceiling issues.
Proof. By concavity, we know f pxq ě 2 3 f˚for all (An illustration for this is given in Section C.) Right of k 4 , the function f drops no faster than at the rate of 4f3 k . Hence for all x P r k 6 , k 2`1 2 s we have f pxq ě 2 3 f˚´1 2¨4 f3 k " 2 3 f˚¨p1´1 k q. Denote the colors t 1 :" ⌈ k 2 ⌉ and t 2 :" t 1`⌊ t 1´k 6 ⌋`1 ď 5 6 k`2 ď k, the latter since k ě 12. Moreover, denote a :" ⌊t 1´k 6 ⌋ and b :" t 1´1`⌈ k 6 ⌉. Let v P V be a player and d :" degpvq. One of the following two cases holds: (i) At least d 2 of the neighbors w P N pvq have: cpwq ď a or b`1 ď cpwq. (ii) At least d 2 of the neighbors w P N pvq have: a`1 ď cpwq ď b. We show that in each of the two cases, v can choose her color so that at least half of her neighbors are at a distance in H to v, hence each contributes at least 2 3 f˚¨p1´1 k q. Since payoff for v cannot be larger than df˚, the theorem follows. The colors to choose are t 1 or t 2 , respectively, as shown in the following.

Cyclic Payoff
Theorem 7. Let f pxq " min {x, k´x}. Then the price of anarchy with respect to f is upperbounded by 2.
Proof. Let c be a stable coloring and v P V . We show 1 2 ⌊ k 2 ⌋ degpvq ď µ v pcq. Like in our previous proofs, this is sufficient since the maximum contribution of an edge is ⌊ k 2 ⌋ here. Denote numbers of neighbors at certain distances: Payoff can be written as: Case cpvq ď ⌊ k 2 ⌋: then n 1 i " 0 for ⌊ k 2 ⌋ ď i ď k´1, since the maximum possible distance to neighbors of smaller color is ⌊ k 2 ⌋´1. Payoff simplifies to: The degree is degpvq " k´1 i"0 n i` i . Define t :" cpvq`⌊ k 2 ⌋ P rks. We write µ v pcrv Ð tsq using n i and n 1 i , by determining at which distance the respective neighbors are located with respect to t. For example, for each 0 ď i ď ⌊ k 2 ⌋, the n i neighbors at distance i above cpvq have distance ⌊ k 2 ⌋´i from t. Note that for distance x ě ⌊ k 2 ⌋`1, we have to use k´x to compute payoff. We have: By stability, µ v pcrv Ð tsq ď µ v pcq. We compute: The case cpvq ě ⌊ k 2 ⌋`1 can be treated likewise, we give the full computation in Section D.
Proposition 8. The bound in Theorem 7 is the best possible for even k, and the worst case is assumed already on a cycle of even length. For odd k, we have a lower bound of 3 2 p1´1 k q. Proof. For each i P N 0 define k i :" k 2 `i .
Then half of the edges have contribution 0, namely between players of the same color, and the other half has contribution k 0 each, so the welfare is nk 0 . We prove that this coloring is stable.
Let v be a player with cpvq " 1. Her payoff is k 0 . If she changes to a color 2 ď t ď k 1 , her new payoff will be pt´1q`pk 1´t q " k 1´1 " k 0 , so no improvement. If she changes to a color k 1`1 ď t ď k, her new payoff will be pt´k 1 q`k´pt´1q " k´k 1`1 " k 0´1`1 " k 0 , so also no improvement. The case cpvq " k 1 is treated likewise. An optimal coloring uses 1 and k 1 alternately and yields welfare 2nk 0 . This proves the claim. 5 For odd k, we take a cycle of length 6n for some n P N ě1 and color like so: 1, k 1 , k 2 , 1, k 1 , k 2 , . . .. The pattern 1, k 1 , k 2 can be repeated an integral number of times since the number of vertices is a multiple of 3. This yields welfare 2n pk 0`1`k0 q " 4nk 0`2 n, so in comparison with the optimum (still attained by using 1 and k 1 alternately, since number of vertices is even) we have We prove that this coloring is stable. Let v be a player with cpvq " 1. Her payoff is 2k 0 . If she changes to color t with 2 ď t ď k 1 , her new payoff will be pk 1´t q`pk 2´t q " 2k 0`3´2 t ď 2k 0´1 , so no improvement. If she changes to color t with k 2 ď t ď k " 2k 0`1 , her new payoff will be pt´k 1 q`pt´k 2 q " 2t´2k 0´3 ď 2p2k 0`1 q´2k 0´3 " 2k 0´1 , so also no improvement. Now let cpvq " k 1 . Her payoff is k 0`1 " k 1 . If she changes to color t with 2 ď t ď k 1´1 " k 0 , her new payoff will be pt´1q`pk 2´t q " k 2´1 " k 1 , so it is no improvement. If she changes color to 1, then her new payoff will be k´pk 2´1 q " k´k 1 " k´k 0´1 " k 1´1 " k 0 , so no improvement; note that k´k 0 " k 1 . If she changes to color t with k 1`1 ď t ď k, her new payoff will be k´pt´1q`pt´k 2 q " k`1´k 0´2 " k 1´1 , so also no improvement. The case cpvq " k 2 can be treated likewise and is omitted here.
Corollary 9. Let f be concave with condition (5). Then the price of anarchy is upper-bounded by 2.
The proof works by similar arguments as Corollary 4. First note that by definition of the price of anarchy, the bound of 2 from Theorem 7 also holds if we use a¨min {x, k´x} with a constant a P R ą0 . Now the crucial point is that any concave function f with (5) is -for integer arguments -on or above the function gpxq :" a¨min {x, k´x} with a :" f˚{⌊ k 2 ⌋, but at the same time has f˚" g˚, so we have the bound of 2mg˚on the optimal welfare for f . For completeness, we give a formal proof in Section E.
choosing this color will yield at least that much payoff for v. Hence W pcq ě vPV D Proof of Theorem 7