1 Introduction
Motivation. Network routing games were first considered by Rosenthal [18] in their “atomic unsplittable” version, where a finite set of players share a network subject to congestion. Routing games found later on many practical applications not only in transport [20, 11], but also in communications [15], distributed computing [1] or energy [2]. The different models studied are of three main categories: nonatomic games (where there is a continuum of infinitesimal players), atomic unsplittable games (with a finite number of players, each one choosing a path to her destination), and atomic splittable games (where there is a finite number of players, each one choosing how to split her weight on the set of available paths).
The concept of equilibrium is central in game theory, for it corresponds to a “stable” situation, where no player has interest to deviate. With a finite number of players—an
atomic unsplittable game—it is captured by the concept of Nash Equilibrium [13]. With an infinite number of infinitesimal players—the nonatomic case—the problem is different: deviations from a finite number of players have no impact, which led Wardrop to its definition of equilibria for nonatomic games [20]. A typical illustration of the fundamental difference between the nonatomic and atomic splittable routing games is the existence of an exact potential function in the former case, as opposed to the latter [14]. However, when one considers the limit game of an atomic splittable game where players become infinitely many, one obtains a nonatomic instance with infinitesimal players, and expects a relationship between the atomic splittable Nash equilibria and the Wardrop equilibrium of the limit nonatomic game. This is the question we address in this paper.Main results. We propose a quantitative analysis of the link between a nonatomic routing game and a family of related atomic splittable routing games, in which the number of players grows. A novelty from the existing literature is that, for nonatomic instances, we consider a very general setting where players in the continuum have specific convex strategysets, the profile of which being given as a mapping from to . In addition to the conventional network (congestion) cost, we consider individual utility function which is also heterogeneous among the continuum of players. For a nonatomic game of this form, we formulate the notion of an atomic splittable approximating sequence, composed of instances of atomic splittable games closer and closer to the nonatomic instance. Our main results state the convergence of Nash equilibria (NE) associated to an approximating sequence to the Wardrop equilibrium of the nonatomic instance. In particular, Thm. 11 gives the convergence of aggregate NE flows to the aggregate WE flow in in the case of convex and strictly increasing price (or congestion cost) functions without individual utility; Thm. 14 states the convergence of NE to the Wardrop equilibrium in in the case of playerspecific strongly concave utility functions. For each result we provide an upper bound on the convergence rate, given from the atomic splittable instances parameters. An implication of these new results concerns the computation of an equilibrium of a nonatomic instance. Although computing an NE is a hard problem in general [10], there exists several algorithms to compute an NE through its formulation with finitedimensional variational inequalities [6]. For a Wardrop Equilibrium, a similar formulation with infinitedimensional variational inequalities can be written, but finding a solution is much harder.
Related work. Some results have already been given to quantify the relation between Nash and Wardrop equilibria. Haurie and Marcotte [8] show that in a sequence of atomic splittable games where atomic splittable players replace themselves smaller and smaller equalsize players with constant total weight, the Nash equilibria converge to the Wardrop equilibrium of a nonatomic game. Their proof is based on the convergence of variational inequalities corresponding to the sequence of Nash equilibria, a technique similar to the one used in this paper. Wan [19] generalizes this result to composite games where nonatomic players and atomic splittable players coexist, by allowing the atomic players to replace themselves by players with heterogeneous sizes.
In [7], the authors consider an aggregative game with linear coupling constraints (generalized Nash Equilibria) and show that the Nash Variational equilibrium can be approximated with the Wardrop Variational equilibrium. However, they consider a Wardroptype equilibrium for a finite number of players: an atomic player considers that her action has no impact on the aggregated profile. They do not study the relation between atomic and nonatomic equilibria, as done in this paper. Finally, Milchtaich [12] studies atomic unsplittable and nonatomic crowding games, where players are of equal weight and each player’s payoff depends on her own action and on the number of players choosing the same action. He shows that, if each atomic unsplittable player in an person finite game is replaced by identical replicas with constant total weight, the equilibria generically converge to the unique equilibrium of the corresponding nonatomic game as goes to infinity. Last, Marcotte and Zhu [11] consider nonatomic players with continuous types (leading to a characterization of the Wardrop equilibrium as a infinitedimensional variational inequality) and studied the equilibrium in an aggregative game with an infinity of nonatomic players, differentiated through a linear parameter in their cost function and their feasibility sets assumed to be convex polyhedra.
Structure. The remaining of the paper is organized as follows: in Sec. 2, we give the definitions of atomic splittable and nonatomic routing games. We recall the associated concepts of Nash and Wardrop equilibria, their characterization via variational inequalities, and sufficient conditions of existence. Then, in Sec. 3, we give the definition of an approximating sequence of a nonatomic game, and we give our two main theorems on the convergence of the sequence of Nash equilibria to a Wardrop equilibrium of the nonatomic game. Last, in Sec. 4 we provide a numerical example of an approximation of a particular nonatomic routing game.
2 Splittable Routing: Atomic and Nonatomic
2.1 Atomic Splittable Routing Game
An atomic splittable routing game on parallel arcs is defined with a network constituted of a finite number of parallel links (cf Fig. 1) on which players can load some weight. Each “link” can be thought as a road, a communication channel or a time slot on which each user can put a load or a task. Associated to each link is a cost or “latency” function that depends only of the total load put on this link.
Definition 1.
Atomic Splittable Routing Game
An instance of an atomic splittable routing game is defined by:

[leftmargin=*,wide,labelindent=1pt]

a finite set of players ,

a finite set of arcs ,

for each , a feasibility set ,

for each , a utility function ,

for each , a cost or latency function .
Each atomic player chooses a profile in her feasible set and minimizes her cost function:
(1) 
composed of the network cost and her utility, where . The instance can be written as the tuple:
(2) 
where and .
In the remaining of this paper, the notation will be used for an instance of an atomic game (Def. 1).
Owing to the network cost structure (1), the aggregated load plays a central role. We denote it by on each arc , and denote the associated feasibility set by:
(3) 
As seen in (1), atomic splittable routing games are particular cases of aggregative games: each player’s cost function depends on the actions of the others only through the aggregated profile .
For technical simplification, we make the following assumptions:
Assumption 1.
Convex costs Each cost function is differentiable, convex and increasing.
Assumption 2.
Compact strategy sets For each , the set is assumed to be nonempty, convex and compact.
Assumption 3.
Concave utilities Each utility function is differentiable and concave.
An example that has drawn a particular attention is the class of atomic splittable routing games considered in [15]. We add playerspecific constraints on individual loads on each link, so that the model becomes the following.
Example 1.
Each player has a weight to split over . In this case, is given as the simplex:
can be the mass of data to be sent over different canals, or an energy to be consumed over a set of time periods [9]. In the energy applications, more complex models include for instance “ramping” constraints .
Example 2.
An important example of utility function is the distance to a preferred profile , that is:
(4) 
where is the value of player ’s preference. Another type of utility function which has found many applications is :
(5) 
which increases with the weight player can load on .
Below we recall the central notion of Nash Equilibrium in atomic noncooperative games.
Definition 2.
Nash Equilibrium (NE)
An of the atomic game is a profile such that for each player :
Proposition 1.
Proof.
Since is convex, (6) is the necessary and sufficient first order condition for to be a minimum of . ∎
Def. 1 defines a convex minimization game so that the existence of an NE is a corollary of Rosen’s results [17]:
Theorem 2 (Cor. of Rosen, 1965).
Rosen [17] gave a uniqueness theorem applying to any convex compact strategy sets, relying on a strong monotonicity condition of the operator . For atomic splittable routing games [15], an NE is not unique in general [4]. To our knowledge, for atomic parallel routing games (Def. 1) under Asms. 3, 1 and 2, neither the uniqueness of NE nor a counter example of its uniqueness has been found. However, there are some particular cases where uniqueness has been shown, e.g. [9] for the case of Ex. 1.
However, as we will see in the convergence theorems of Sec. 3, uniqueness of NE is not necessary to ensure the convergence of of a sequence of atomic unsplittable games, as any sequence of NE will converge to the unique Wardrop Equilibrium of the nonatomic game considered.
2.2 Infinity of Players: the Nonatomic Framework
If there is an infinity of players, the structure of the game changes: the action of a single player has a negligible impact on the aggregated load on each link. To measure the impact of infinitesimal players, we equip real coordinate spaces with the usual Lebesgue measure .
The set of players is now represented by a continuum . Each player is of Lebesgue measure 0.
Definition 3.
Nonatomic Routing Game
An instance of a nonatomic routing game is defined by:

[leftmargin=*,wide,labelindent=1pt]

a continuum of players ,

a finite set of arcs ,

a pointtoset mapping of feasibility sets ,

for each , a utility function ,

for each , a cost or latency function .
Each nonatomic player chooses a profile in her feasible set and minimizes her cost function:
(7) 
where denotes the aggregated load. The nonatomic instance can be written as the tuple:
(8) 
For the nonatomic case, we need assumptions stronger than Asms. 3 and 2 for the mappings and , given below:
Assumption 4.
Nonatomic strategy sets There exists such that, for any , is convex, compact and , where is the ball of radius centered at the origin. Moreover, the mapping has a measurable graph .
Assumption 5.
Nonatomic utilities There exists s.t. for each , is differentiable, concave and . The function is measurable.
Def. 3 and Asms. 5 and 4 give a very general framework. In many models of nonatomic games that have been considered, players are considered homogeneous or with a finite number of classes [14, Chapter 18]. Here, players can be heterogeneous through and . Games with heterogeneous players can find many applications, an example being the nonatomic equivalent of Ex. 1:
Example 3.
Let be a density function which designates the total demand for each player . Consider the nonatomic splittable routing game with feasibility sets
As in Ex. 1, one can consider some upper bound and lower bound for each and each , and add the bounding constraints in the definition of .
Heterogeneity of utility functions can also appear in many practical cases: if we consider the case of preferred profiles given in Ex. 2, members of a population can attribute different values to their cost and their preferences.
Since each player is infinitesimal, her action has a negligible impact on the other players’ costs. Wardrop [20] extended the notion of equilibrium to the nonatomic case.
Definition 4.
Wardrop Equilibrium (WE)
is a Wardrop equilibrium of the game if it is a measurable function from to and for almost all ,
where .
Proposition 3.
Proof.
Given , (9) is the necessary and sufficient first order condition for to be a minimum point of the convex function . ∎
According to (9), the monotonicity of is sufficient to have the VI characterization of the equilibrium in the nonatomic case, as opposed to the atomic case in (6) where monotonicity and convexity of are needed.
Theorem 4 (Cor. of Rath, 1992 [16]).
Proof.
The conditions required in [16] are satisfied. Note that we only need and to be continuous functions.∎
The variational formulation of a WE given in Prop. 3 can be written in the closed form:
Proof.
From the characterization of the WE in Thm. 5 and Cor. 6, we derive Thms. 8 and 7 that state simple conditions ensuring the uniqueness of WE in .
Proof.
Suppose that and are both WE of the game. Let and . Then, according to Theorem 5,
(12)  
(13) 
By adding (12) and (13), one has
Since for each , is strictly concave, is thus strictly monotone. Therefore, for each , and equality holds if and only if . Besides, is monotone, hence . Consequently, , and equality holds if and only if for almost all , . (In this case, .) ∎
Theorem 8.
Proof.
Remark 1.
If for each , is (strictly) increasing, then is a (strictly) monotone operator from .
One expects that, when the number of players grows very large in an atomic splittable game, the game gets close to a nonatomic game in some sense. We confirm this intuition by showing that, considering a sequence of equilibria of approximating atomic games of a nonatomic instance, the sequence will converge to an equilibrium of the nonatomic instance.
3 Approximating Nonatomic Games
To approximate the nonatomic game , the idea consists in finding a sequence of atomic games with an increasing number of players, each player representing a “class” of nonatomic players, similar in their parameters.
As the players are differentiated through and , we need to formulate the convergence of feasibility sets and utilities of atomic instances to the nonatomic parameters.
3.1 Approximating the nonatomic instance
Definition 5.
Atomic Approximating Sequence (AAS)
A sequence of atomic games is an approximating sequence (AAS) for the nonatomic instance if for each , there exists a partition of cardinal of set , denoted by , such that:

[leftmargin=*,wide,labelindent=1pt]

,

where is the Lebesgue measure of subset ,

where is the Hausdorff distance (denoted by ) between nonatomic feasibility sets and the scaled atomic feasibility set:
(16) 
where is the distance (in ) between the gradient of nonatomic utility functions and the scaled atomic utility functions:
(17)
From Def. 5 it is not trivial to build an AAS of a given nonatomic game , one can even be unsure that such a sequence exists. However, we will give practical examples in Secs. 3.4.1 and 3.4.2.
A direct result from the assumptions in Def. 5 is that the players become infinitesimal, as stated in Lemma 9.
Lemma 9.
If is an AAS of a nonatomic instance , then considering the maximal diameter of , we have:
(18) 
Proof.
Let . Let and denote by the projection on . By definition of , we get:
(19)  
(20) 
∎
Lemma 10.
If is an AAS of a nonatomic instance , then the Hausdorff distance between the aggregated sets and is bounded by:
(21) 
Proof.
Let be a nonatomic profile. Let denote the Euclidean projection on for and consider . From (16) we have:
(22)  
(23)  
(24)  
(25) 
which shows that for all . On the other hand, if , then let us denote by the Euclidean projection on for , and for . Then we have for all , and we get:
(26)  
(27)  
(28) 
which shows that for all and concludes the proof. ∎
To ensure the convergence of an AAS, we make the following additional assumptions on costs functions :
Assumption 6.
Lipschitz continuous costs For each , is a Lipschitz continuous function on . There exists such that for each , .
Assumption 7.
Strong monotonicity There exists such that, for each , on
In the following sections, we differentiate the cases with and without utilities, because we found different convergence results in the two cases.
3.2 Players without Utility Functions: Convergence of the Aggregated Equilibrium Profiles
In this section, we assume that for each .
We give a first result on the approximation of WE by a sequence of NE in Thm. 11.
Theorem 11.
Proof.
Let denote the Euclidean projection onto and the projection onto . We omit the index for simplicity. From (11), we get:
(29) 
On the other hand, with , we get from (1):
(30)  
(31) 
with . From the CauchySchwartz inequality and Lemma 9, we get:
(32)  
(33)  
(34) 
Besides, with the strong monotonicity of and from (29) and (30):
which concludes the proof. ∎
3.3 Players with Utility Functions: Convergence of the Individual Equilibrium Profiles
In order to establish a convergence theorem in the presence of utility functions, we make an additional assumption of strong monotonicity on the utility functions stated in Asm. 8. Note that this assumption holds for the utility functions given in Ex. 2.
Assumption 8.
Strongly concave utilities For all , is strongly concave on , uniformly in : there exists such that for all and any :
Remark 2.
If is strongly concave, then the negative of its gradient is a strongly monotone operator:
(35) 
We start by showing that, under the additional Asm. 8 on the utility functions, the WE profiles of two nonatomic users within the same subset are roughly the same.
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