## 1 Introduction

*Congestion games* constitute one of the most important and well-studied class
of games in the field of

*algorithmic game
theory*

*weighted*congestion games, each player has her own (positive) weight and the cost of a resource is a nondecreasing function of the total weight of players using it. An important special case is that of unweighted games, where all players have the same weight. The cost of a resource then depends only on the number of players using it.

Players are selfish and each one chooses a set of resources that
minimizes her own cost. On the other hand, a central authority would aim at minimizing *social cost*, that is, the sum of players’ costs. It is well known that
these two objectives do not, in general, align: due to the selfish behavior of
players, the game may reach a stable state (i.e., a Nash
equilibrium [25, 30]) that is suboptimal in terms of social
cost. This gap is formally captured by the fundamental notion of *Price of
Anarchy* (PoA) [27], defined as the ratio between the social cost of the worst equilibrium
and that of an optimal solution enforced by a centralized authority.

From the seminal work of Rosenthal [32], we know that unweighted congestion
games always have *(pure Nash) equilibria*^{1}^{1}1In this paper we focus
exclusively on *pure* Nash equilibria; this is standard in the congestion games
literature.. This is a direct consequence of the fact that they are *potential
games* [29]. However, finding such a stable state is, in general,
computationally hard [1, 17]. An important question
then is whether one can efficiently compute *approximate*
equilibria [6, 36]. These are states of the game where
no player can unilaterally deviate and improve her cost by more than a factor of
; (exact) equilibria correspond to the special case where .

The situation becomes even more challenging in the general setting of weighted congestion games [7], where exact equilibria may not even exist [22]. Consequently, weighted congestion games do not generally admit a potential function. Thus, in this setting one needs more sophisticated approaches and approximation tools to establish computability of approximate equilibria. This is precisely the problem we study in the present paper: the efficient computation of -approximate equilibria in weighted congestion games, with as small as possible. We focus on resource cost functions that are polynomials (with nonnegative coefficients), parametrized by their degree ; this is a common assumption in the literature of congestion games (see, e.g., [35, 2, 10, 14, 22]).

### 1.1 Related Work

The potential function approach has long become a central tool for obtaining results about existence and computability of approximate equilibria in weighted congestion games. The concept of a potential function for unweighted congestion games was proposed by Rosenthal [32], who used it as a tool for proving existence of pure Nash equilibria in such games. Later, Monderer and Shapley [29] formally introduced and studied extensively the class of potential games. As it turns out, weighted congestion games do not admit a potential function in general, even for“well-behaved” instances [28, 22, 20, 24]. Exceptions include games with linear and exponential resource cost functions [24]. However, recently, Christodoulou et al. [14] showed that polynomial weighted congestion games have an approximate analogue of a potential function, which they called Faulhaber’s potential. An exact potential function decreases whenever a player improves her cost, and even by the same amount. In contrast, the approximate potential of [14] is only guaranteed to decrease when a player deviates and improves her cost by a factor of at least . Factor is greater than , but at most , where is the degree of the game. We will use this approximate potential function in the analysis of our algorithm. Other approximate potential functions have been successfully used before [14] to establish the existence of approximate equilibria in congestion games; see [8, 12, 23].

On the algorithmic side, there have been many negative results concerning exact
equilibria in various classes of congestion games. Fabrikant et al. [17] showed
that even in the unweighted case of a network congestion game, computing equilibria
is PLS-complete. Dunkel and Schulz [16] showed that it is strongly NP-complete to
determine whether an equilibrium exists in a given weighted congestion game. As a
further negative result, Ackermann et al. [1] proved that it is PLS-complete to
compute equilibria even in the linear unweighted case. These hardness results
motivated the search for efficient methods to compute *approximate* equilibria.
In general, this is again a computationally hard problem; Skopalik and Vöcking [36]
showed that for any polynomially computable , finding a -approximate
equilibrium in a congestion game is a PLS-complete problem. The focus of research
thus shifted towards searching for positive algorithmic results for
-approximate equilibria of various special classes of weighted congestion
games. The first such result was obtained by Chien and Sinclair [9], who showed
convergence of the best-response dynamics, in symmetric unweighted congestion games
with “well-behaved” cost functions, to -approximate equilibria in polynomially many steps with respect to
and the number of players.

The next significant positive result of this kind was obtained for approximate
equilibria in polynomial unweighted congestion games by Caragiannis et al. [6].
They showed how to efficiently compute -approximate equilibria in such
games. Subsequently, Caragiannis et al. [7] extended this result to the weighted
case, achieving an approximation factor of . To the best of our
knowledge, this has remained state-of-the-art ever since.^{2}^{2}2As far as
deterministic solutions are concerned. See below for a discussion of the recent
randomized algorithm by Feldotto et al. [19]. In the present paper, we reduce this
factor to . The algorithm in [7] first transforms
the original game into a so-called approximating game defined for the same
players and states. Specifically, it runs on the so-called -game, which is an
exact potential game where players’ costs are within a factor of from their
costs in the original polynomial game. Then it finds and returns a state which is a
-approximate equilibrium in the new -game. In the original game,
the approximation factor of this state (which is the approximation guarantee of the
algorithm) is worse by the mentioned factor of and thus becomes .

In the past few years, several new positive results have been obtained based on the algorithm in [6, 7]. Feldotto et al. [18] explored PoA-like bounds on the potential function in unweighted games. This enabled them to bound from above the approximation factor that the algorithmic framework of [6] yields when applied to unweighted games with general cost functions. Most recently, Feldotto et al. [19] designed a algorithm similar to [7] which computes -approximate equilibria in weighted congestion games. However, it is important to note that their algorithm is randomized and finds -approximate equilibria

*only with high
probability*

rule to share the total cost of a resource among the players that use it. Computing players’ costs using this scheme is in fact computationally hard, so instead those costs are estimated using a sampling approach.

The study of the Price of Anarchy () was initiated by Koutsoupias and Papadimitriou [27]. One of the
first significant results concerning tight bounds on the of atomic congestion
games was obtained by Christodoulou and Koutsoupias [11]. They proved the tight bound of on the
exact of linear unweighted congestion games. In the next few years, several
results were obtained. For example, Gairing and Schoppmann [21] provided various
upper and lower bounds for the exact of
singleton^{3}^{3}3See Footnote 8 for a formal definition.
unweighted congestion games. Subsequently, Aland et al. [2] introduced a systematic
approach to upper-bounding the exact of polynomial weighted congestion games,
which was later extended to general classes of cost functions and named
*smoothness framework* in [3, 34].
Aland et al. [2] gave the tight bound of on the of exact
pure NE in polynomial weighted games, where is the unique root of the
equation .
Based on the same technique, Christodoulou et al. [12] provided a
tight bound on the PoA of -approximate equilibria in unweighted congestion
games. It turned out to be equal to , where
is the maximum integer that satisfies . In our
notation, this is equivalent to .

Since the development of the smoothness method, other approaches to finding tight bounds on the have been investigated. Recently, Bilò [4] was able to rederive, through the use of a primal-dual framework, the upper bound on the of linear unweighted games from [11]. He also provided a simplified lower bound instance. Furthermore, he was able to show the upper bound of on the of -approximate equilibria for the special case of linear weighted games. It turns out to be equal to , the special case for of the general tight bound that we present below. Moreover, he provided matching instances with equal to , for in a certain subset of .

### 1.2 Our Results and Techniques

We study approximate (pure Nash) equilibria in polynomial weighted congestion games of degree . Our main result is a polynomial-time algorithm for computing -approximate pure Nash equilibria in such games. Improving the bound on the approximation guarantee of efficiently computable approximate equilibria in polynomial weighted games was posed as an open problem by Caragiannis et al. [7]. As a matter of fact, their paper provided the best known bound on this approximation factor, namely , before our paper.

Our algorithm, as well as the outline of its analysis, is based on the ideas in
[7]. However, there is a significant difference between our
approach and previous ones. Our algorithm builds a polynomially-long sequence of
best-response moves in the *actual game* itself that, from any given state, leads to a
-equilibrium. We then utilize, in the absence of an exact potential
function for the game, an approximate potential introduced by
Christodoulou et al. [14] in order to analyze the runtime and approximation
guarantee. In contrast, the original algorithm of [7] first replaces the input game with
a modified, exact-potential approximating game (called -game), then computes a
equilibrium there, and then projects it back into the original game, at
the expense of increasing the approximation guarantee by a factor of , which
results in the approximation guarantee .

Our approach has a number of advantages, at the cost of our proofs being more
involved due to the use of an approximate, instead of an exact, potential. First,
unlike in our algorithm, the sequence of moves found by the algorithm in
[7] need not be a best-response sequence when projected back
into the original game.^{4}^{4}4In an attempt to address this issue,
Caragiannis et al. [7] themselves present also a modification of their main
algorithm that actually runs in the original game, but unfortunately this leads to
significantly worse approximation guarantees of in their case.
As a matter of fact, it may contain moves that increase individual costs.
Secondly, once an approximate equilibrium is found, we do not need to project it back
into the input game. This saves us a factor of compared to the original
algorithm. This is the main reason why our approximation factor goes down to
.
Third, it turns out from our analysis that essentially our approximation guarantee
corresponds to the value of the PoA for -approximate equilibria in the input game; so, we can deploy known frameworks
for studying the of polynomial weighted games (see next paragraph) to
derive the necessary bound on the PoA, and thus to the approximation factor
itself.

As a necessary tool for proving the approximation guarantee of the algorithm and as
a result of independent interest, we obtain a tight bound on the PoA of
-approximate equilibria, denoted by , of polynomial weighted
congestion games of degree , for any any and degree . It
turns out to be equal to , where is the
unique positive solution of the equation . This bound
generalizes the following results: the tight bound of on the
(of exact equilibria) of weighted congestion games [2]; the tight bound
on the -approximate of unweighted congestion games
[12]; and the upper bound on the -approximate of
linear weighted congestion games [4].
Our matching lower bound proof extends an example from [21] that
bounds the of singleton unweighted congestion games. As such, our lower bound
is easily verified to hold for singleton and network weighted congestion
games.^{5}^{5}5The fact that the worst-case PoA can be realized at such simple
singleton games should come as no surprise, due to the work
of Bilò and Vinci [5, Theorem 1]. Our contribution here lies in determining the
actual value of the PoA.
To prove the upper bound, we essentially utilize the *smoothness method*
developed in [2, 3, 34]. The smoothness
approach automatically extends the validity of our tight bound from pure Nash
to mixed Nash and correlated equilibria as well
[3, 34].

One further contribution is an analytic upper bound , involving the Lambert-W function. This bound adds to the understanding of the different asymptotic behavior of with respect to each of the two parameters, and , and plays an important role in deriving the desired approximation factor of in the analysis of the main algorithm of our paper. It is interesting to note here that this bound also generalizes, in a smooth way with respect to , a similar result presented in [14] for the special case of exact equilibria (i.e., ).

All proofs omitted from the main text can be found in the Appendix.

## 2 Model and Notation

We denote by and the set of real and nonnegative real numbers, respectively, and by the class of polynomials of degree at most with
nonnegative coefficients^{6}^{6}6Formally,
..
A well-established notation in the literature of congestion games is that of , for
a positive integer, as the unique positive root of the equation . Notice
how, the special case of corresponds to the *golden ratio* constant
. In this paper, we introduce a further generalization by
defining, for all , to be the unique positive root of
the equation .
Also, we shall make use of (the principal real branch
of) the classical function known as the *Lambert-W
function* [15]: for , is defined to be
the unique solution to the equation .

A *polynomial ^{7}^{7}7We shall usually omit the word “polynomial” and refer
to these games as weighted congestion games of degree , or simply as congestion
games, when this causes no confusion. (weighted) congestion game of degree *,
with a positive integer, is a tuple . Here, a (finite) set of players and is a
finite set of resources. Each resource has a
polynomial cost function .
Every player has a set of strategies

and each vector

will be called a state (or strategy profile) of the game . Following standard game-theoretic notation, for any and , we denote by the profile of strategies of all players if we remove the strategy of player ; in this way, we have . Finally, each player has a real positive weight . However, we may henceforth assume that for all , as we can without loss of generality appropriately scale player weights and cost functions, without affecting our results in this paper.Given , we let denote the total weight of players using resource in state . Generalizing this definition to any group of players , we let . The cost of a player at state is defined as

Players are selfish and rational, and thus choose strategies as to minimize their
own cost. Let be a *best-response* strategy of player
to the strategies of the other players, that is, (in case of ties, we make an arbitrary selection).
A state of the game is a (pure Nash) equilibrium, if all players are already playing best-responses, that is, no player can unilaterally improve her costs; formally, for all and .

For a real parameter , a unilateral deviation of player to strategy
from state is called a -move if .
Extending the notion of an equilibrium in two directions, we call a state
a *-approximate equilibrium* (or simply a -equilibrium) for a given
group of players , if none of the players in has a -move;
formally, for all and
. If this holds for , then we simply refer to as a
-equilibrium of our game. We use to denote the set
of all -equilibria of game .

Ideally, our objective is to find states that induce low total cost in our game; we capture this notion by defining the social cost of a state to be the sum of the players’ costs, i.e., . Extending this to any subset of players , we also denote

Clearly, .

The standard way to quantify the inefficiency due to selfish behaviour, is to study
the worst-case ratio between any equilibrium and the optimal solution, quantified by
the notion of *the Price of Anarchy (PoA)*. Formally, given a game and
a parameter , the PoA of -equilibria (or simply the -PoA) of
is , where . Finally, taking the worst case over all polynomial congestion games of
degree , we can define the -PoA of degree as

## 3 The Price of Anarchy

In this section we present our tight bound on the PoA of -approximate equilibria for (weighted) congestion games. We first extend the smoothness method of Aland et al. [2] to obtain the upper bound on the (Theorem 1), and then explicitly construct an example that extends a result of Gairing and Schoppmann [21] and provides the matching lower bound on the (Theorem 2). We note here that there is a specific reason that this section precedes Section 4, where our algorithm for computing approximate pure Nash equilibria is presented. The estimation of the approximation guarantee of the algorithm requires the use of the closed-form bound on the we provide in Theorem 1 and, furthermore, the “Key Property” of our algorithm (Theorem 3) rests critically on an application of Lemma 1 below.

### 3.1 Upper Bound

We formulate our upper bound on the PoA as the following theorem:

###### Theorem 1.

The Price of Anarchy of -approximate equilibria in (weighted) polynomial congestion games of degree , is at most , where is the unique positive root of the equation . In particular,

where denotes the Lambert-W function.

Theorem 1 is a direct consequence of the following Lemma 1, applied with , and Lemma 3. The reason we are proving a more general version of Lemma 1 than what’s needed for just establishing our PoA upper bound of Theorem 1, is that we will actually need it for the analysis of our main algorithm in Section 4.3.

###### Lemma 1.

For any group of players , let and be states such that is a -equilibrium for group and every player in uses the same strategy in both and . Then, the social cost ratio of the two states is bounded by .

The proof of Lemma 1 will essentially follow the smoothness technique [34] (see, e.g., [35, Theorem 14.6]). However, special care still needs to be taken related to the fact that only a subset of players is deviating between the two states and . In particular, the key step in the smoothness derivation is captured by the following lemma (proved in Appendix A) that quantifies the PoA bound:

###### Lemma 2.

For any constant and positive integer ,

The constraint that Lemma 2 imposes on parameters and is
slightly more general than the analogous lemma in the smoothness derivation
of Aland et al. [2]; namely, our condition contains an extra variable . This is
a consequence of exactly the aforementioned fact that Lemma 2 is tailored
to upper bounding a generalization of the PoA for *groups* of players.

###### Proof of Lemma 1.

Assume that and are parameters such that, for any polynomial of degree with nonnegative coefficients and for any , it is

Applying this for the cost function of any resource , and replacing , , and (the last equality holding due to the fact that every player in uses the same strategy in and we have that

(1) |

Here we also used that , , and . Summing (1) over all resources , we obtain the following inequality:

(2) |

Next, using the fact that is a -equilibrium we can upper-bound the social cost of the players in by

(3) |

Now, observe that for any player and any resource that player uses in profile , it is

The first inequality holds because player is the only one deviating between states and , while the second one because definitely uses resource in profile . Using the above, due to the monotonicity of the cost functions , the bound in (3) can be further developed to give us

and thus, deploying the bound from (2), we finally arrive at

which is equivalent to

Taking the infimum of the right-hand side, over the set of all feasible parameters and , Lemma 2 gives us desired upper bound of . ∎

We conclude this section by presenting the following useful bound on the generalized golden ratio , which is used in Theorem 1 to get the corresponding analytic expression for our bound. As discussed in the introduction of the current section, we will use it in the proof of Theorem 6 in Section 4, for deriving the improved approximation guarantee of our algorithm.

###### Lemma 3.

For any and any positive integer ,

where is the Lambert-W function.

###### Proof.

Recall that is the solution of equation , which can be rewritten as

By the proof of Lemma 11 (see function ), this equation has a unique positive root and the left side is monotonically decreasing as a function of ; thus, to conclude the proof of our lemma, it suffices to prove that

(4) |

Indeed, substituting for convenience , we have that

For the inequality we used the fact that for all . The last equality is a direct consequence of the definition of the Lambert-W function. ∎

### 3.2 Lower Bound

To prove a matching lower bound to the upper bound of the previous Section 3.1, we provide a simple instance involving players and resources. Each player has just strategies, and each strategy consists of a single resource; letting , we obtain the desired lower bound of . This bound extends smoothly the lower bound of for the PoA of exact () equilibria by Gairing and Schoppmann [21, Theorem 4]. We also want to mention here that the construction below can be extended to apply to network congestion games (see, e.g., [13, Proposition 1]).

###### Theorem 2.

Let be a positive integer and . For every , there
exists a (singleton^{8}^{8}8In *singleton*
congestion games the strategies of all players consist of a single resource.
Formally , for any and all .) weighted
polynomial congestion game of degree , whose -approximate is at least
, where is the unique positive
root of the equation .

###### Proof.

Consider the following congestion game, with players and resources . Each player has a weight of , where . Resources have cost functions

Each player has only two available strategies, denoted by and : either use only resource , or only resource . Formally, , where and . The social cost of profile , where every player uses the -th resource, is

while that of , where player uses the -th resource is

We now claim that profile is a -equilibrium. Indeed, for player ,

and for all players ,

the last equality coming for the definition of the generalized golden ratio ; thus, no player can unilaterally deviate from and gain (strictly) more than a factor of .

Since is a -equilibrium, the -PoA of our game is at least

as grows arbitrarily large. This concludes our proof. ∎

## 4 The Algorithm

In this section we describe and study our algorithm for computing -approximate equilibria in weighted congestion games of degree . The algorithm, as well as the general outline of its analysis, are inspired by the work of Caragiannis et al. [7]. However, as discussed in Section 1.2, here we are using the approximate potential function of Christodoulou et al. [14]. This will be crucial in proving that the algorithm is indeed poly-time and, more importantly, has the improved approximation guarantee of .

In Section 4.1, we introduce the aforementioned potential function from [14], along with some natural extensions that will be useful for our analysis — partial and subgame potentials. This is complemented by a set of technical lemmas through which our use of this potential function will be instantiated in the rest of the paper. In Section 4.2, we describe our approximation algorithm and next, in Section 4.3, we show that it indeed runs in polynomial time; the critical step in achieving this is proving the “Key Property” (Theorem 3) of our algorithm, that appropriately bounds the potential of certain groups of players throughout it execution. Finally, in Section 4.4 we establish the desired approximation guarantee of the algorithm.

### 4.1 The Potential Function Technique

Our subsequent proofs regarding both the runtime and the approximation guarantee of
our algorithm, will rely heavily on the use of an approximate potential function. In
particular, we will use a straightforward variant of the *Faulhaber potential*
function, introduced by Christodoulou et al. [14]. Consider a polynomial weighted
congestion game of degree . For every resource with cost function
, we set

(5) |

The potential of any state of the game is then defined as

The potential function introduced above satisfies a crucial property given in the
following lemma, and which we will extensively use in the rest of our paper. To
state it, we will first define^{9}^{9}9The reason for introducing extra notation
here, and not making the arguably simpler choice to directly use the actual value of
instead of variable , for the rest of the paper, is that we want to
assist readability: by using , we make clear where exactly this factor from our
approximate potential comes into play. an auxiliary constant

(6) |

###### Proof.

From the proof^{10}^{10}10In particular, see Eq. (19) in the proof of
[13, Claim 2] and the related [13, Lemma 4],
both applied for the special case of here. of Theorem 3
of Christodoulou et al. [13] we know that, if for resource with cost function
we define

(7) |

then

We now multiply each -th term of the sum for in (7) by , defining

which coincides with our definition of the potential in (5). It is not difficult to see (by following the same proof of [13, Thereom 3]) that the above translates into essentially scaling the entire potential by , resulting in the desired bound in the statement of Lemma 4. ∎

The notions of partial and subgame potential were introduced in [7], and we now adapt them for the setting of our new approximate potential. The subgame potential with respect to a group of players of a state is defined as:

The partial potential with respect to a group of players is then defined as:

(8) |

In the original work by Christodoulou et al. [14], the variable in Lemma 4 was interpreted as a single player’s weight, and as the total remaining weight on resource . In our analysis, however, and will each play the role of the total weight of players, from some group, using resource . Lemma 4 then becomes a very powerful algebraic tool that relates social cost of groups of players and partial potentials in a variety of settings, and plays an important role in many of our proofs. In many cases, it will be present in such proofs implicitly, through the following corollary (Lemma 5), which says that the chosen potential function is cost-revealing in a strong sense. That is, for any group of players, the ratio of the partial potential and the social cost of that group is bounded from both above and below.

###### Lemma 5.

For any group of players ,

###### Proof.

The following lemma provides a relation between the change in potential due to a single player’s deviation and a linear combination of that player’s old and new costs. For an exact potential function, the latter would simply be the difference in the cost experienced by that player. However, in our case, the player’s cost in one of the two states is weighted by an additional factor of compared to her cost in the other state. Thus, Lemma 6 only implies a decrease in the potential function if the deviating player has improved her cost by at least a factor .

###### Lemma 6.

Let be a player, be an arbitrary subset
of players with , and and be two states that differ
*only* in the strategy of player . Then,

###### Proof.

First observe that, since each player in plays the same strategy in both profiles and , it must be that . Using this, we can derive that

and thus

where the first equality is due to the definition of partial potentials (8).

### 4.2 Description of the Algorithm

We shall now describe our algorithm for finding -approximate equilibria
in weighted congestion games of degree (see Algorithm 1). We remark, once again, that it is inspired by a similar algorithm by Caragiannis et al. [7]. However, a critical difference is that our algorithm runs directly in the *actual* game (using the original cost functions, and thus, players’ deviations that are best-responses with respect to the actual game); as a result, we also need to appropriately calibrate the original parameters from Caragiannis et al. [7].
First, we fix the following constant that essentially captures our *target approximation factor*:

(9) |

The following lemma captures a critical property of the above parameter, which is the one that will essentially give rise to the specific approximation factor of our algorithm (see Theorem 6).

###### Lemma 7.

###### Proof.

First, recall from (6) that . Next, we note that the Lambert-W function is increasing on the positive reals (see Corless et al. [15]) and so, for any ,

Furthermore, the -approximate Price of Anarchy is also nondecreasing with respect to the approximation parameter (since the set of allowable approximate equilibria gets larger; see Section 2). Thus, from Theorem 1 we can see that

From this, we can finally get that indeed

The equality in the statement of our lemma is just a consequence of the fact that the -PoA is exactly equal to (from Section 3; see Theorem 1 and Theorem 2). ∎

The input to the algorithm consists of the description of a weighted polynomial congestion game of fixed degree and an (arbitrary) initial state of . We now let and . Here, 0 denotes the “empty state”, that is, a fictitious state in which all players’ strategies are empty sets. Recall also (see Section 2) that returns a best-response move of player at a given state. Observe that is the maximum cost of a player in state , and can be used as a lower bound on the cost of any player in any state of our game. We also define parameter . Notice that is polynomial on the input (that is, the description of our game ). Next, we introduce a factor (see creftype 4 of Algorithm 1), that depends (polynomially) on both the aforementioned parameter of our game and the approximation factor . Using this, we set “boundaries” so that for the player costs (see discussion below).

The algorithm runs in phases, indexed by

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