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].