This paper studies firstly the existence and uniqueness of variational Wardrop equilibrium in nonatomic aggregative games with coupling aggregative constraints, where a continuum of players have heterogeneous compact convex pure-action sets and cost functions. It then examines the convergence of a sequence of variational Nash equilibrium in auxiliary finite-player games to the variational Wardrop equilibrium.
Background. Aggregative games form a large class of non-cooperative games. In such a game, a player’s payoff is determined by her own action and the aggregate of all the players’ actions . The setting of aggregative games is particularly relevant to the study of nonatomic games , games with a continuum of players. There, a player has an interaction with the other players only via an aggregate-level profile of their actions, for example, the distribution of certain actions, while she has no interest or no way to know the behavior of any particular player or the identity of the player making a certain choice.
Nonatomic games are readily adapted to many situations in industrial engineering or public sectors where a huge number of users, such as traffic commuters and electricity consumers, are involved. Those users have no direct interaction with each other except through the aggregate congestion or consumption to which they are contributing simultaneously. These situations can often be modeled as a congestion game, a special class of aggregative games, both in nonatomic version and finite-player version. The latter, called atomic congestion game, was formally formulated by Rosenthal in 1973 , while related research work in transportation and traffic analysis, mostly in the nonatomic version, appeared much earlier [51, 9]. The theory of congestion games has also found numerous applications in telecommunications , distributed computing , energy management , and so on.
The concept of equilibrium in nonatomic games is captured by the so called Wardrop equilibrium (WE) . A nonatomic player neglects the impact of her deviation on the aggregate profile of the whole population’s actions, in contrast to a finite player. For the computation of WE, existing results are limited to particular classes of nonatomic games, such as population games [37, 29, 44], where finite types of players are considered, each type sharing the same finite number of pure actions and the same payoff function. Convergence of some dynamical systems describing the evolution of pure-action distribution in the population has been established for some particular equilibria in some particular classes such as linear games , potential games [9, 43] and stable games [47, 28]. Algorithms corresponding to discretized versions of such dynamical systems for the computation of WE have been studied, in particular for congestion games [20, 52].
Motivation. This paper is mainly motivated by two gaps in the literature on nonatomic games.
Firstly, in engineering applications of nonatomic games such as the management of traffic flow or energy consumption, individual commuters or consumers often have specific choice sets due to individual constraints, and specific payoff functions due to personal preferences. Also, unlike for a transportation user who usually choose a single path, an electricity consumer faces to a resource allocation problem where she has to divide the consumption of a certain quantity of energy onto different time periods. Hence, her pure-action set is no longer a finite, discrete set as a commuter but a compact convex set in where is the total number of time periods. Fewer results exist for the computation of WE for the case where players have infinitely many different types (i.e. action sets and payoff functions) or where they have continuous action sets. For example, most of the works in line with Schmeidler [36, 41, 30, 10] use fixed-point theorems to prove the existence of WE, though in fairly general settings. Besides, most of the existing work assumes smooth cost functions of players which is somewhat strong.
Secondly, in the above-mentioned applications of aggregative games, coupling constraints, especially those at aggregative level, commonly exist . Examples are capacity constraints of the network or power-grid, and ramping constraints on the variation of total energy consumption between time periods. In this regard, there are even fewer studies in game-theoretical modeling of nonatomic game. However, the presence of coupling constraints adds non trivial difficulties in the analysis of WE and their computation. Indeed, an appropriate definition of equilibrium is already not obvious. An analog to the so-called generalized Nash equilibrium  for finite-player games does not exist for nonatomic games because a nonatomic player’s behavior has no impact on the aggregative profile. Moreover, dynamical systems and algorithms used to compute Wardrop equilibria in population games cannot be straightforwardly extended to this case. Indeed, in these dynamics and algorithms, players adapt their strategies unilaterally in their respective strategy spaces, which can well lead to a new strategy profile violating the coupling constraint.
In view of these two gaps, the main objective of this paper is to provide a model of nonatomic aggregative games with infinitely many player-specific, compact convex pure-action sets and infinitely many player-specific nonsmooth payoff functions, then introduce a general form of coupling aggregative constraints into these games, choose an appropriate equilibrium notion, study their properties such as existence and uniqueness and, finally, their computation.
Main results. After defining a pure-action profile in a nonatomic aggregative game where players have specific compact convex pure-action sets lying in , and specific cost functions, convex in their own action variable but nonsmooth, Theorem 3.1 characterizes a Wardrop equilibrium (WE) as a solution to an infinite-dimensional generalized variational inequality (IDGVI).
Theorem 3.3 proves the existence of WE and variational Wardrop equilibrium (VWE), equilibrium notion in the presence of coupling constraints defined by a similar IDGVI, in monotone nonatomic games by showing the existence of solutions to the characteristic IDGVI. Then, Theorem 3.4 shows the uniqueness of WE and VWE in case of strictly monotone or aggregatively strictly monotone games. The definition of monotone games is an extension of the stable games , also called dissipative games , in population games with a finite types of nonatomic players to the case with infinitely many types.
Theorem 4.1 is the main result of this paper. It shows that, for a sequence of finite-player aggregative games, if the players’ pure-action sets converge to those of a strongly monotone (resp. aggregatively strongly monotone) nonatomic aggregative game, and if the aggregative constraints in these finite-player games converge to the aggregative constraint in the nonatomic game, then a sequence of so-called variational Nash equilibria (VNE) in these finite-player games converge, in pure-action profile (resp. in aggregate-action profile), to the VWE. We provide an upper bound on the distance between the VNE and VWE, specified as a function of the parameters of the finite-player and nonatomic games.
This result allows the construction of an auxiliary sequence of finite-player games with finite-dimensional VNE so as to approximate the infinite-dimensional VWE in the special class of strongly or aggregatively strongly monotone nonatomic aggregative games, with or without aggregative constraints. Since there are much more results  on the resolution of finite-dimensional variational inequalities characterizing VNE, we can therefore obtain an approximation of the VWE with arbitrary precision.
Finally, we show how to construct an AAS for two general classes of nonatomic games.
Related work. Extensive research has been conducted on Wardrop equilibria in nonatomic congestion games via their formulation with variational inequalities , while the similar characterization of Nash equilibria in atomic splittable games, where players have continuous action sets in contrast to unsplittable games where players have finite action sets, has been less studied [27, 40]. In addition to their existence and uniqueness, the computational and dynamical aspects of equilibria as solutions to variational inequalities have also been studied [46, 54, 52, 11]. However, in most cases, the variational inequalities involved have finite dimensions, as opposed to the case of Wardrop equilibrium in this paper. Marcotte and Zhu  consider nonatomic players with continuous types (leading to a characterization of the Wardrop equilibrium as an infinite-dimensional variational inequality) and studied the equilibrium in an aggregative game with nonatomic players differentiated through a linear parameter in their cost function.
Some results have already been given to quantify the relationship between Nash and Wardrop equilibria. Haurie and Marcotte  show that in a sequence of atomic splittable games where atomic splittable players are replaced by smaller and smaller equal-size players with constant total weight, 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  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.
Gentile et al.  consider a specific class of finite-player aggregative games with linear coupling constraints. They use the variational inequality formulations for the unique generalized Nash equilibrium and the unique generalized Wardrop-type equilibrium (which consists in letting each finite player act as if she was nonatomic) of the same finite-player game to show that, when the number of players grows, the former can be approximated by the latter. There are several differences between our model and theirs. Firstly, we consider nonatomic games with players of infinitely-many different types instead of finite-player games only. Secondly, we consider variational Nash and Wardrop equilibria instead of generalized equilibria (which does not exist in nonatomic games). In contrast to generalized equilibria, a variational equilibrium is not characterized by a best reply condition for each of the finite or nonatomic players, which makes the study of its properties much more difficult, as shown in Section 4. Thirdly, we allow for nonsmooth cost functions and general form of coupling constraints while they consider differentiable cost functions and linear coupling constraints.
Milchtaich  studies finite and nonatomic crowding games (similar to aggregative games), where players have finitely many pure actions, and shows that, if each player in an -person game is replaced by identical replicas with constant total weight, pure Nash equilibria generically converge to the unique equilibrium of the limit nonatomic game as goes to infinity. His proof is not based on a variational inequality formulation.
Structure. The remaining of the paper is organized as follows. Section 2 recalls the definition of finite-player aggregative games with and without aggregative constraints, the notion of equilibrium in these cases and their properties. Section 3 is dedicated to nonatomic aggregative games with and without aggregative constraints. After defining Wardrop equilibrium and variational Wardrop equilibrium, we concentrate on the special class of monotone games and show the existence and uniqueness of equilibria there via generalized infinite dimensional variational inequalities. In Section 4, we give the definition of an approximating sequence of finite-player games for a nonatomic aggregative game with or without coupling constraints, and present the main theorem of the paper on the convergence of the sequence of (variational) Nash equilibria of the approximating finite-player games to the (variational) Wardrop equilibrium of the nonatomic game. The construction of such a sequence of approximating finite-player games is shown for two important classes of nonatomic games.
Notations.Vectors are denoted by a bold font (e.g. ) as opposed to scalars (e.g. ).
The transpose of vector is denoted by .
The closed unit ball in a metric space, centered at and of radius , is denoted by .
For a nonempty convex set in a Hilbert space (over ),
is the tangent cone of at ;
is the linear span of ;
is the affine hull of ;
is the relative interior of ;
is the relative boundary of in , i.e. the boundary of in .
The inner product of two points and in any Euclidean space is denoted by . The -norm of is denoted by .
We denote by the Hilbert space of measurable functions from (equipped with the Lebesgue measure ) to that are square integrable with respect to the Lebesgue measure . The inner product of two vector functions and is denoted by . The Hilbert space is endowed with -norm: .
The distance between a point and a set is denoted by , where is omitted or is equal to , depending on whether we consider an Euclidean space or .
Similarly, the Hausdorff distance between two sets and is denoted by , which is defined as . Later, we will define new metrics indexed by in Euclidean spaces. The point-set distances and Hausdorff distances are defined similarly and denoted with index .
The subdifferential, i.e. set of subgradients of a convex function at in its domain , which is a convex set in , is denoted by . Recall that if vector is a subgradient of at , then for all , .
For a function of two explicit variables, convex in , we denote by the (nonempty) subdifferential of function for any fixed .
2 Finite-player aggregative games
This section recalls the definition of finite-player aggregative games with and without coupling aggregative constraints, and some notions of equilibrium in these games as well as their characterization by generalized variational inequalities.
Definition 2.1 (Finite-player aggregative game).
A finite-player aggregative game is a non-cooperative game specified by:
(i) a finite set of players ,
(ii) a set of feasible pure actions for each player , where a constant, with a typical pure action ,
(iii) a cost function for each player , so that a player’s cost is determined by her own action and the aggregate action profile.
Denote . The pure-action profile of the players induces an aggregate-action profile load attributed to arc , which is denoted by where . Denote the set of feasible aggregate-action profiles by .
Denote the game by .
The following assumptions and notations are adopted in this paper.
Assumption 1 (Convex costs).
For each , the function , where , is continuous in and in , and is convex in for all .
Assumption 2 (Convex and compact strategy sets).
For each , the set is a nonempty, convex and compact subset of .
Recall the definition of Nash Equilibrium in finite-player non-cooperative games.
Definition 2.2 (Nash Equilibrium ()  ).
A (pure) Nash equilibrium of is a profile of pure actions such that for all and all .
Define a correspondence by
Since the cost functions are convex in players’ own strategies, NE can be characterized as solutions to generalized variational inequalities (GVI) .
Proposition 2.1 (GVI formulation of NE).
Remark 2.1 (Generalized VI and Generalized NE are different things).
The variational inequalities (VI) are of generalized type here because the subdifferentials of cost functions are not necessarily singled valued. In the case that cost functions are differentiable with respect to the players’ own actions, the GVI are reduced to a VI. Do not confuse with generalized NE in generalized games (cf. Definition 2.3 ) which are characterized by (generalized-)quasi-VI.
The existence of an NE is obtained by a classical result in game theory for finite-player continuous game, since the players have convex continuous cost functions and convex compact pure-action sets. No differentiability condition is needed.
Remark 2.2 (NE is a unilateral level stability condition).
The NE condition ensures stability not only in terms of a single player’s behavior but also in terms of their collective welfare. Indeed, on the one hand, condition (1a) is equivalent to for all for each , i.e. a unilaterally feasible deviation of player increases her cost; on the other hand, condition (1b) is equivalent to for all , i.e. a collectively feasible deviation of all the players increases their total costs. The two conditions are equivalent because the players have independent pure-action spaces so that , i.e. any collectively feasible deviation can be decomposed into unilaterally feasible deviations. This remark is important because it is no longer the case when one introduces a coupling constraint in the game.
The coupling aggregative constraint considered in this paper is of the following general form: There is a nonempty, convex and compact subset of , whose intersection with is not empty, such that the aggregate-action profile . An example is .
Definition 2.3 (Finite-player aggregative game with aggregative constraints).
Its only difference from the game defined in Definition 2.1 is that, for each player , given the profile of pure actions of the others players , her feasible pure-action set becomes , where is a subset of defined by
This game is denoted by or simply .
Finite-player non-cooperative games with coupling constraints are called generalized Nash games . The extension from games to generalized games is not trivial. In the case with no coupling constraint, the pure-action spaces of the players are independent so that any collectively feasible deviation can be decomposed into unilaterally feasible deviation. This property does not always hold with a coupling constraint. To see this, we recall the following notion of generalized equilibrium in generalized games.
Definition 2.4 (Generalized Nash Equilibrium (GNE), ).
A profile of pure actions is a generalized Nash equilibrium of if
Proposition 2.3 (GQVI formulation of GNE).
Remark 2.3 (Problematics of GNE).
The notion of GNE can be problematic. Given a pure-action profile , firstly simultaneous unilateral deviations can lead to a new profile out of . Secondly, if is a proper set of , profiles in are not feasible. Indeed, the unilateral stability condition (2a) is equivalent to collective stability only among those collective deviations composed by unilaterally feasible deviations, i.e. condition (2b), because . Collective deviations towards are not composed by unilaterally feasible deviations while they may effectively decrease the total cost or even each player’s cost (cf.  for an example). A GNE can thus lose its stability when such collective deviations are allowed. To answer to this issue, we consider the stronger notion of equilibrium defined below:
Definition 2.5 (Variational Nash Equilibrium (VNE), ).
A solution to the following GVI problem:
is called a variational Nash equilibrium of . In particular, if , a VNE is a NE.
A pure-action profile is a VNE if and only if a collective deviation to any profile in is not collectively beneficial. Indeed, VNE is also unilaterally stable, because it is a GNE [25, Theorem 3].
Remark 2.4 (VNE refines GNE).
Hence, VNE is adopted in this paper as the equilibrium notion in the presence of aggregative constraints. Moreover, in Section 3.3 it is argued that the notion of generalized equilibrium cannot even be established in nonatomic games.
From the convexity of each , we deduce that is a nonempty, convex, compact valued, upper hemicontinuous correspondence. Then [12, Corollary 3.1] shows that the GVI problem (3) admits a solution on the finite dimensional convex compact . (In the case that the is partially differentiable with respect to , then the GVI is reduced to a VI, and Lemma 3.1 in  suffices to show the existence of a solution.) ∎
3 A Continuum of Players: Nonatomic Framework
3.1 Nonatomic aggregative games
In nonatomic aggregative games considered here, players have compact pure-action sets, and heterogeneous pure-action sets as well as heterogeneous cost function. This model is in line with the Schmeidler’s seminal paper , but in contrast to most of the population games studied in game theory [29, 44] where nonatomic players are grouped into several populations, with players in the same population have the same finite pure-action set and the same cost function.
Definition 3.1 (Nonatomic aggregative game).
A nonatomic aggregative game is defined by:
(i) a continuum of players represented by points on the real interval endowed with Lebesgue measure,
(ii) a set of feasible pure actions for each player , with a constant, and
(iii) a cost function for each player , where and denotes the aggregate-action profile.
The set of feasible pure-action profiles is defined by:
Denote the game by .
The definition of a nonatomic game asks the pure-action profile to be a measurable and integrable function on instead of simply being a collection of for . In other words, a coupling constraint is inherent in the definition of nonatomic games and the notion of WE. This is in contrast to finite-player games.
The set of feasible aggregate actions is .
Further assumptions are necessary for to be nonempty and for the existence of equilibria to be discussed later.
Assumption 3 (Nonatomic pure-action sets).
The correspondence has nonempty, convex, compact values and measurable graph , i.e. is a Borel subset of . Moreover, for all , , with a constant.
Under Assumption 3, a sufficient condition for to be in is that is measurable.
Assumption 4 (Nonatomic convex cost functions).
For all , is defined on , where is a neighborhood of , and is bounded on , and for each aggregate profile ,
(i) function is measurable.
(ii) for each , function is continuous and convex on ;
(iii) There is such that for all subgradients for each , each , and each .
Assumption 4.(iii) implies that ’s are Lipschitz in the first variable with a uniform Lipschitz constant on for all . Besides, if is differentiable on , then contains one element , the gradient of at .
Wardrop equilibrium extends the notion of Nash equilibrium in the framework of nonatomic games, where a single player of measure zero has a negligible impact on the others.
Definition 3.2 (Wardrop Equilibrium (WE), ).
A pure-action profile is a Wardrop equilibrium of nonatomic game if
Before characterizing WE by infinite-dimensional GVI, let us introduce some notions and a technical assumption ensuring that the infinite-dimensional GVI is well-defined.
First, define a correspondence as follows:
In other words, is the collection of measurable (and integrable because of Assumption 4.(iii)) selections of a subgradient for each .
Next, define a best-reply correspondence from the set of aggregate-action profiles to the set of pure-action profiles :
Finally, fix and , define a correspondence from to as follows:
Clearly, this is a nonempty and closed-valued correspondence.
For all and all , is a measurable correspondence.
Theorem 3.1 (GVI formulation of WE).
We need the following lemma for the proof of Theorem 3.1.
(1) For all , is nonempty.
(2) For all , is nonempty.
(3) Under Assumption 5, for all and all , there exists a measurable mapping such that for each .
(1) For each , the subdifferential is nonempty and compact valued, so that a measurable selection exists according to the compact-valued selection theorem .
(2) Fix . A consequence of Assumption 4.(i-ii) is that the function is a Carathéodory function, that is, (i) is measurable on for each , and (ii) is continuous on for each . Thus, according to the measurable maximum theorem [1, Thm. 18.19] applied to , there exists a selection such that is a measurable function on .
Proof of Theorem 3.1.
For the converse, suppose that satisfies condition (6b) but not (6a). Then there must be a subset of with strictly positive measure such that for each , . In particular, for any , . By the same argument as in the proof of Lemma 3.1, one can select a for such that is measurable. By defining for , one has is measurable and hence belongs to . However, , contradicting (6b). ∎
Condition (6a) is equivalent to for all for each . The interpretation is the same as for atomic players at NE: no unilateral deviation is profitable. However, since each nonatomic player has measure zero, when considering a deviation in the profile of pure actions, one must let players in a set of strictly positive measure deviate: (6b) means that the collective deviation of players of any set of strictly positive measure increases their cost. Note that the GVI problem (6b) has infinite dimensions.
The existence of WE is obtained by an equilibrium existence theorem for nonatomic games.
Theorem 3.2 (Existence of a WE, ).
The conditions required in Remark 8 in Rath’s 1992 paper  on the existence of WE in aggregate games are satisfied. ∎
No convexity of ’s are needed.
3.2 Monotone nonatomic aggregative games
For the uniqueness of WE and the existence of equilibrium notion to be introduced in the next subsection for the case with coupling constraints, let us introduce the following notions of monotone nonatomic games.
The nonatomic aggregative game is monotone if
It is strictly monotone if the equality in (7) holds if and only if almost everywhere.
It is aggregatively strictly monotone if the equality in (7) holds if and only if .
It is strongly monotone with modulus if
It is aggregatively strongly monotone with modulus if
A recent paper of Hadikhanloo  generalizes the notion of stable games in population games  to monotone games in anonymous games, an extension of population games with players having heterogeneous compact pure-action sets but the same payoff function. He defines the notion of monotonicity directly on the distribution of pure-actions among the players instead of pure-action profile as we do. The two approaches are compatible.
Examples of aggregative games are given by cost functions of the form:
Here specifies the per-unit cost (or negative of per-unit utility) of each of the “public products”, which is a function of the aggregative contribution to each of the “public products”. Player ’s cost (resp. negative of utility) associated to these products is scaled by her own contribution . The function measures the private utility of player (resp. negative of private cost) for the contribution .
For instance, in a public goods game, is the common per-unit payoff for using public good , determined by the total contribution , while is player ’s private cost of supplying to the public goods; in a Cournot competition, is the common market price for product , determined by its total supply , while is player ’s private cost of producing unit of product for each product ; in a congestion game, is the common per-unit cost for using arc in a network, determined by the aggregate load on arc , while is player ’s private utility of her routing or energy consuming choice .
Under Assumptions 5, 4 and 3, in a nonatomic aggregative game with cost functions of form (10), assume that is monotone on and, for each , is a concave function on . Then:
(1) is a monotone game.
(2) If is strictly concave on for all , then is a strictly monotone game.
(3) If is strictly monotone on , then is an aggregatively strictly monotone game.
(4) If is strongly concave on with modulus for each and , then is a strongly monotone game with modulus .
(5) If is strongly monotone on with , then is an aggregatively strongly monotone game with modulus .
(1) Let and , . For each , . Then, given , with and for each , one has because is concave so that is a monotone correspondence on .
Then because is monotone. Hence is a monotone game.
The proof for (2)-(5) is omitted. ∎
In particular, if , then is monotone if ’s are all non-decreasing, and is strongly monotone if ’s are all strictly increasing.
3.3 Nonatomic aggregative games with aggregate constraints
Let us consider the aggregative constraint in nonatomic aggregative game : , where is a convex compact subset of such that . Let be a subset of defined by . Denote the nonatomic game with aggregative constraint .
In contrast to games with finitely many players, a generalized equilibrium in the style of Definition 2.4 is not well-defined in a nonatomic game. Indeed, since the impact of a nonatomic player’s choice on the aggregative profile is negligible, the feasible pure-action set of a nonatomic player facing the choices of the others in a game with coupling constraint is not a well-established notion: either then , or then . Departing from a pure-action profile in , simultaneous unilateral deviations by the players can lead to any profile in . If only profiles in are allowed to be attained, then one lands on a notion similar to VNE. Indeed, the most natural notion of equilibrium with the presence of aggregative constraint is the notion of variational Wardrop equilibrium, where feasible deviations are defined on a collective basis.
Definition 3.4 (Variational Wardrop Equilibrium (VWE)).
A solution to the following infinite dimensional GVI problem:
is called a variational Wardrop equilibrium of , where the correspondence is as defined by Eq. 4.
Remark 3.7 (Justification for VWE).
From a game theoretical point of view, the notion of VWE can be problematic as well. Each nonatomic player can deviate unilaterally without having any impact on the aggregate profile. The unilateral stability as for NE and VNE (cf. Remarks 2.4 and 2.2) is lost. However, our main theorem, Theorem 4.1, shows that VWE can be seen as the limit of a sequence of VNE which are unilaterally stable. Finally, in the literation of congestion games, the equilibrium notion characterized by VI of form (11) but in finite dimension and with smooth cost functions has long been studied. For example, see [32, 33, 14, 53] and references therein.
The following facts are needed for later use. Under Assumption 3:
is a nonempty, convex, closed and bounded subset of ;
is a nonempty, convex and closed subset of ;
and are nonempty, convex and compact subsets of .
Assumption 6 (Continuity of cost function in aggregate action).
For each and , is continuous on .
Theorem 3.3 (Existence of VWE).
Let us apply [16, Corollary 2.1] to show that Eq. 11 has a solution. This theorem states that if is bounded, closed and convex in , and if is a monotone correspondence which is upper hemicontinuous from the line segments in to the weak* topology of , then (11) admits a solution. We only need to show the upper hemicontinuity property. First notice that has closed values. Take and in , consider sequence with , and sequence such that and with . Let us show that .
Denote and . Then converges to in -norm.
By definition of , for each , for each , . Since is continuous in both variables, and . Besides, , and because , while because ’s are uniformly bounded by . Therefore,