Topological Price of Anarchy Bounds

for Clustering Games on Networks

Pieter Kleer

1

and Guido Sch¨afer

1,2(

B

)

1

Centrum Wiskunde & Informatica (CWI), Networks and Optimization Group,

Amsterdam, The Netherlands

{kleer,schaefer}@cwi.nl

2

Department of Econometrics and Operations Research,

Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

Abstract. We consider clustering games in which the players are

embedded in a network and want to coordinate (or anti-coordinate) their

choices with their neighbors. Recent studies show that even very basic

variants of these games exhibit a large Price of Anarchy. Our main goal is

to understand how structural properties of the network topology impact

the ineﬃciency of these games. We derive topological bounds on the Price

of Anarchy for diﬀerent classes of clustering games. These topological

bounds provide a more informative assessment of the ineﬃciency of these

games than the corresponding (worst-case) Price of Anarchy bounds. As

one of our main results, we derive (tight) bounds on the Price of Anarchy

for clustering games on Erd˝os-R´enyi random graphs, which, depending

on the graph density, stand in stark contrast to the known Price of Anar-

chy bounds.

Keywords: Clustering games

· Coordination games · Price of

Anarchy

· Random graphs · Nash equilibrium existence

1 Introduction

Motivation. Clustering games on networks constitute a class of strategic games

in which the players are embedded in a network and want to coordinate (or anti-

coordinate) their choices with their neighbors. These games capture several key

characteristics encountered in applications such as opinion formation, technology

adoption, information diﬀusion or virus spreading on various types of networks

(e.g., the Internet, social networks, biological networks, etc.).

Diﬀerent variants of clustering games have recently been studied intensively

in the algorithmic game theory literature, both with respect to the existence and

the ineﬃciency of equilibria (see, e.g., [3,4,11,15,16,18,20,21]). Unfortunately,

several of these studies reveal that the strategic choices of the players may lead to

equilibrium outcomes that are highly ineﬃcient. Arguably the most prominent

notion to assess the ineﬃciency of equilibria is the Price of Anarchy (PoA)

[19], which refers to the worst-case ratio of the optimal social welfare and the

c

Springer Nature Switzerland AG 2019

I. Caragiannis et al. (Eds.): WINE 2019, LNCS 11920, pp. 241–255, 2019.

https://doi.org/10.1007/978-3-030-35389-6

_18

242 P. Kleer and G. Sch¨afer

social welfare of a (pure) Nash equilibrium. It is known that even the most

basic clustering games exhibit a large (or even unbounded) Price of Anarchy

(see below for details). These negative results naturally trigger the following

questions: Is this high ineﬃciency inevitable in clustering games on networks?

Or, can we trace more precisely what causes a large ineﬃciency? These questions

constitute the starting point of our investigations: Our main goal in this paper is

to understand how structural properties of the network topology impact the Price

of Anarchy in clustering games.

In general, our idea is that a more ﬁne-grained analysis may reveal topological

parameters of the network which can be used to derive more accurate bounds on

the Price of Anarchy; we term such bounds topological Price of Anarchy bounds.

Given the many applications of clustering games on diﬀerent types of networks,

our hope is that such topological bounds will be more informative than the

corresponding worst-case bounds. Clearly, this hope is elusive for a number of

fundamental games on networks whose ineﬃciency is known to be independent

of the network topology, the most prominent example being the selﬁsh routing

games studied in the seminal work by Rougharden and Tardos [22]. But, in con-

trast to these games, clustering games exhibit a strong locality property induced

by the network structure, i.e., the utility of each player is aﬀected only by the

choices of her direct neighbors in the network. This observation also motivates

our choice of quantifying the ineﬃciency by means of topological parameters

(rather than other parameters of the game).

We derive topological bounds on the Price of Anarchy for diﬀerent classes of

clustering games. Our bounds reveal that the Price of Anarchy depends on dif-

ferent topological parameters in the case of symmetric and asymmetric strategy

sets of the players and, depending on these parameters, stand in stark con-

trast to the known worst case bounds. As one of our primary benchmarks, we

use Erd˝os-R´enyi random graphs [13] to obtain a precise understanding of how

these parameters aﬀect the Price of Anarchy. More speciﬁcally, we show that the

Price of Anarchy of clustering games on random graphs, depending on the graph

density, improves signiﬁcantly over the worst case bounds. To the best of our

knowledge, this is also the ﬁrst work that addresses the ineﬃciency of equilibria

on random graphs.

1

We note that the applicability of our topological Price of Anarchy bounds

is not limited to the class of Erd˝os-R´enyi random graphs. The main reason for

using these graphs is that their structural properties are well-understood. In

particular, our topological bounds can be applied to any graph class of interest

(as long as certain structural properties are well-understood).

Our Clustering Games. We study a generalization of the unifying model

of clustering games introduced by Feldman and Friedler [11]: We are given an

undirected graph G =(V,E)onn = |V | nodes whose edge set E = E

c

∪ E

a

is partitioned into a set of coordination edges E

c

and a set of anti-coordination

1

We note that Valiant and Roughgarden [23] study Braess’ paradox in large random

graphs (see Related Work).

Topological Price of Anarchy Bounds for Clustering Games on Networks 243

edges E

a

.

2

Further, we are given a set [c]={1,...,c} of c>1 colors and edge-

weights w : E → R

≥0

.

3

Each node i corresponds to a player who chooses a color

s

i

from her color set S

i

⊆ [c]. We say that the game is symmetric if S

i

=[c]

for all i ∈ V and asymmetric otherwise. An edge e = {i, j}∈E is satisﬁed if

it is a coordination edge and both i and j choose the same color, or if it is an

anti-coordination edge and i and j choose diﬀerent colors. The goal of player i

is to choose a color s

i

∈ S

i

such that the weight of all satisﬁed edges incident to

i is maximized.

We consider a generalization of these games by incorporating additionally:

(i) individual player preferences (as in [21]), and (ii) diﬀerent distribution rules

(as in [3]): We assume that each player i has a preference function q

i

: S

i

→ R

≥0

which encodes her preferences over the colors in S

i

. Further, player i has a split

parameter α

ij

≥ 0 for every incident edge e = {i, j} which determines the share

she obtains from e:ife is satisﬁed then i obtains a proportion of α

ij

/(α

ij

+ α

ji

)

of the weight w

e

of e. The utility u

i

(s)ofplayeri for choosing color s

i

∈ S

i

is

then the sum of the individual preference q

i

(s

i

) and the total share of all satisﬁed

edges incident to i. We consider the standard utilitarian social welfare objective

u(s)=

i

u

i

(s).

We use ¯α

e

to denote the disparity of an edge e = {i, j}, deﬁned as ¯α

e

=

max{α

ij

/α

ji

,α

ji

/α

ij

},andlet¯α = max

e∈E

¯α

e

refer to the maximum disparity

of all edges. We say that the game has the equal-split distribution rule if ¯α = 1

(equivalently, α

ij

= α

ji

for all {i, j}∈E).

Our clustering games generalize several other strategic games, which were

studied extensively in the literature before, such as max cut games and not-

all-equal satisﬁability games [15], max k-cut games [16], coordination games [4],

clustering games [11]andanti-coordination games [20].

Main Contributions. We derive results for symmetric and asymmetric clus-

tering games. Due to space restrictions, we elaborate on our main ﬁndings for

symmetric clustering games only below; our results for the asymmetric case are

discussed in Sect. 5. An overview of the bounds derived in this paper is given in

Table 1.

1. Topological Price of Anarchy Bound. We show that the Price of Anar-

chy for symmetric clustering games is bounded as a function of the maximum

subgraph density of G which is deﬁned as ρ(G) = max

S⊆V

{|E[S]|/|S|}, where

|E[S]| is the number of edges in the subgraph induced by S. More speciﬁcally,

we prove that PoA ≤ 1+(1+¯α)ρ(G) and that this bound is tight (even for

coordination games). Using this topological bound, we are able to show that

the Price of Anarchy is at most 4 + 3¯α for clustering games on planar graphs

and 1 + 2ρ(G) for coordination games with equal-split distribution rule. We

also derive a (qualitatively) reﬁned bound of PoA ≤ 5+2ρ(G[E

c

]) for cluster-

ing games with equal-split distribution rule which reveals that the maximum

2

The game is called a coordination game if all edges are coordination edges and an

anti-coordination game (or cut game) if all edges are anti-coordination edges.

3

In this paper, we use [k] to denote the set {1,...,k} for a given integer k ≥ 1.

244 P. Kleer and G. Sch¨afer

Table 1. Overview of our topological Price of Anarchy bounds for symmetric and

asymmetric clustering games. A “+” or “1” in the column “distr. α” indicates whether

the distribution rule α is positive or equal-split, respectively. ¯α is the maximum dis-

parity, and c is the number of colors. ρ(G)andΔ(G) refer to the maximum subgraph

density and the maximum degree of G, respectively. The stated bounds for random

graphs hold with high probability.

Symmetric clustering games

Graph topology Coord.

only

Indiv.

pref.

Distr.

α

Topological PoA (our bounds) PoA

(prev. work)

Arbitrary ✗✓+ 1+(1+¯α) ρ(G)(Theorem1) c [3,11]

Planar

✗✓+ ≤ 4+3¯α (Corrollary 1)

Arbitrary ✗✓1 1+2ρ(G) (Corrollary 2)

Arbitrary ✗✓1 ≤ 5+2ρ(G

c

)(Theorem2)

Sparse random ✓✓1 Θ(1) (Corrollary 3)

Dense random ✓✗ 1 Ω(c)(Theorem3)

Asymmetric clustering games

Graph topology Coord.

only

Indiv.

pref.

Distr.

α

(, k) -topological PoA (our bounds) (, k)-PoA

(prev. work)

Arbitrary ✓✗ 1 ≤ 2Δ(G)(Theorem5)

Arbitrary ✓✗ 1 ≥ (

Δ(G)

k−1

− 1) (Theorem 5) ≤ 2

n−1

k−1

Dense random ✓✗ 1 Ω(n) ≥ 2

n−k

k−1

+1

Sparse random

✓✗ 1 Θ

ln(n)

ln ln(n)

(Theorem 6)

[21]

+ common color

✓✗ 1 O(1) (Theorem 7)

subgraph density with respect to the graph G[E

c

] (or simply G

c

) induced by the

coordination edges E

c

only is the crucial topological parameter determining the

Price of Anarchy.

These bounds provide more reﬁned insights than the known (tight) bound of

PoA ≤ c (number of colors) on the Price of Anarchy for (i) symmetric coordina-

tion games with individual preferences and arbitrary distribution rule [3], and (ii)

clustering games without individual preferences and equal-split distribution rule

[11] (both being special cases of our model). An important point to notice here is

that this bound indicates that the Price of Anarchy is unbounded if the number

of colors c = c(n) grows as a function of n. In contrast, our topological bounds

are independent of c and are thus particularly useful when this number is large

(while the maximum subgraph density is small). Moreover, our reﬁned bound of

5+2ρ(G[E

c

]) mentioned above provides a nice bridge between the facts that for

max-cut (or anti-coordination) games the price of anarchy is known to be con-

stant, whereas for coordination games the price of anarchy might grow large.

2. Price of Anarchy for Random Coordination games. We derive the

ﬁrst price of anarchy bounds for coordination games on random graphs. We

focus on the Erd˝os-R´enyi random graph model [13] (also known as G(n, p)),

where each graph consists of n nodes and every edge is present (independently)

with probability p ∈ [0, 1]. More speciﬁcally, we show that the Price of Anarchy

is constant (with high probability) for coordination games on sparse random

Topological Price of Anarchy Bounds for Clustering Games on Networks 245

graphs (i.e., p = d/n for some constant d>0) with equal-split distribution

rule. In contrast, we show that the Price of Anarchy remains Ω(c) (with high

probability) for dense random graphs (i.e., p = d for some constant 0 <d≤ 1).

Note that our constant bound on the Price of Anarchy for sparse random

graphs stands in stark contrast to the deterministic bound of PoA = c [3,11]

(which could increase with the size of the network). On the other hand, our

bound for dense random graphs reveals that we cannot signiﬁcantly improve

upon this bound through randomization of the graph topology.

It is worth mentioning that all our results for random graphs hold against

an adaptive adversary who can ﬁx the input of the clustering game knowing the

realization of the random graph. To obtain these results, we need to exploit some

deep probabilistic results on the maximum subgraph density and the existence

of perfect matchings in random graphs.

3. Convergence of Best-Response Dynamics. In general, pure Nash equilib-

ria are not guaranteed to exist for clustering games with arbitrary distribution

rules α, even if the game is symmetric (see, e.g., [3]). While some suﬃcient condi-

tions for the existence of pure Nash equilibria, or, the convergence of best-response

dynamics (see also [3]) are known, a complete characterization is elusive so far.

In this work, we obtain a complete characterization of the class of distribu-

tion rules which guarantee the convergence of best-response dynamics in clus-

tering games on a ﬁxed network topology. Basically, we prove that best-response

dynamics converge if and only if α is a generalized weighted Shapley distribution

rule (Theorem 4). Our proof relies on the fact that there needs to be some form

of cyclic consistency similar to the one used in [14].

Prior to our work, the existence of pure Nash equilibria was known for certain

special cases of coordination games only, namely for symmetric coordination

games with individual preferences and c =2[3], and for symmetric coordination

games without individual preferences [11]. To the best of our knowledge, this is

the ﬁrst characterization of distribution rules in terms of best-response dynamics

(which, in particular, applies to the settings in which pure Nash equilibria are

guaranteed to exist for every distribution rule [3,11]).

4

Related Work. The literature on clustering and coordination games is vast; we

only include references relevant to our model here. The proposed model above

is a mixture of (special cases of) existing models in [3,4,11,21].

Anshelevich and Sekar [3] consider symmetric coordination games with indi-

vidual preferences and (general) distribution rules. They show existence of -

approximate k-strong equilibria,(, k)-equilibria for short, for various combina-

tions; in particular, (2,k)-equilibria always exist for any k. Moreover, they show

that the number of colors c is an upper bound on the PoA. Apt et al. [4] study

asymmetric coordination games with unit weights, zero individual preferences,

and equal-split distribution rules. They derive an almost complete picture of the

4

In the full version, we extend our ideas and provide a characterization of the existence

of pure Nash equilibria in symmetric coordination games, complementing a result

by Anshelevich and Sekar [3].