WAN Cheng

Aggregative games have many industrial applications, and computing an equilibrium in those games is challenging when the number of players is large. In the framework of atomic aggregative games with coupling constraints, we show that variational Nash equilibria of a large aggregative game can be approximated by a Wardrop equilibrium of an auxiliary population game of smaller dimension. Each population of this auxiliary game corresponds to a group of atomic players of the initial large game. This approach enables an efficient computation of an approximated equilibrium, as the variational inequality characterizing the Wardrop equilibrium is of smaller dimension than the initial one. This is illustrated on an example in the smart grid context.

We consider a network of prosumers involved in peer-to-peer energy exchanges, with differentiation price preferences on the trades with their neighbors, and we analyze two market designs: (i) a centralized market, used as a benchmark, where a global market operator optimizes the flows (trades) between the nodes, local demand and exibility activation to maximize the system overall social welfare. (ii) a distributed peer-to-peer market design where prosumers in local energy communities optimize selfishly their trades, demand, and exibility activation. We first characterize the solution of the peer-to-peer market as a Variational Equilibrium and prove that the set of Variational Equilibria coincides with the set of social welfare optimal solutions of market design (i). We give several results that help understanding the structure of the trades at an equilibrium or at the optimum. We characterize the impact of preferences on the network line congestion and renewable energy surplus under both designs. We provide a reduced example for which we give the set of all possible generalized equilibria, which enables to give an approximation of the price of anarchy. We provide a more realistic example which relies on the IEEE 14-bus network, for which we can simulate the trades under dierent preference prices. Our analysis shows in particular that the preferences have a large impact on the structure of the trades, but that one equilibrium (variational) is optimal. Finally, the learning mechanism needed to reach an equilibrium state in the peer-to-peer market design is discussed together with privacy issues.

This paper shows the existence of O(1/n^γ)-Nash equilibria in n-player noncooperative aggregative games where the players' cost functions depend only on their own action and the average of all the players' actions, and is lower semicontinuous in the former while γ-Hölder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes congestion games with γ being 1, a proximal best-reply algorithm is used to construct an O(1/n)-Nash equilibria with at most O(n^3) iterations. These results are applied in a numerical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when n tends to infinity.

We define and analyze the notion of variational Wardrop equilibrium for nonatomic aggregative games with an infinity of players types. These equilibria are characterized through an infinite-dimensional varia-tional inequality. We show, under monotonicity conditions, a convergence theorem enables to approximate such an equilibrium with arbitrary precision. To this end, we introduce a sequence of nonatomic games with a finite number of players types, which approximates the initial game. We show the existence of a symmetric Wardrop equilibrium in each of these games. We prove that those symmetric equilibria converge to an equilibrium of the infinite game, and that they can be computed as solutions of finite-dimensional variational inequalities. The model is illustrated through an example from smart grids: the description of a large population of electricity consumers by a parametric distribution gives a nonatomic game with an infinity of different players types, with actions subject to coupling constraints.

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Computing an equilibrium in congestion games can be challenging when the number of players is large. Yet, it is a problem to be addressed in practice, for instance to forecast the state of the system and be able to control it. In this work, we analyze the case of generalized atomic congestion games, with coupling constraints, and with players that are heterogeneous through their action sets and their utility functions. We obtain an approximation of the variational Nash equilibria-a notion generalizing Nash equilibria in the presence of coupling constraints-of a large atomic congestion game by an equilibrium of an auxiliary population game, where each population corresponds to a group of atomic players of the initial game. Because the variational inequalities characterizing the equilibrium of the auxiliary game have smaller dimension than the original problem, this approach enables the fast computation of an estimation of equilibria in a large congestion game with thousands of heterogeneous players.

After defining a pure-action profile in a nonatomic aggregative game, where players have specific compact convex pure-action sets and nonsmooth convex cost functions, as a square-integrable function, we characterize a Wardrop equilibrium as a solution to an infinite-dimensional generalized variational inequality. We show the existence of Wardrop equilibrium and variational Wardrop equilibrium, a concept of equilibrium adapted to the presence of coupling constraints, in monotone nonatomic aggregative games. The uniqueness of (variational) Wardrop equilibrium is proved for strictly or aggregatively strictly monotone nonatomic aggregative games. We then show that, for a sequence of finite-player aggregative games with aggregative constraints, if the players' pure-action sets converge to those of a strongly (resp. aggregatively strongly) monotone nonatomic aggregative game, and the aggregative constraints in the finite-player games converge to the aggregative constraint of the nonatomic game, then a sequence of so-called variational Nash equilibria in these finite-player games converge to the variational Wardrop equilibrium in pure-action profile (resp. aggregate-action profile). In particular, it allows the construction of an auxiliary sequence of games with finite-dimensional equilibria to approximate the infinite-dimensional equilibrium in such a nonatomic game. Finally, we show how to construct auxiliary finite-player games for two general classes of nonatomic games.

We consider an instance of a nonatomic routing game. We assume that the network is parallel, that is, constituted of only two nodes, an origin and a destination. We consider infinitesimal players that have a symmetric network cost, but are heterogeneous through their set of feasible strategies and their individual utilities. We show that if an atomic routing game instance is correctly defined to approximate the nonatomic instance, then an atomic Nash Equilibrium will approximate the nonatomic Wardrop Equilibrium. We give explicit bounds on the distance between the equilibria according to the parameters of the atomic instance. This approximation gives a method to compute the Wardrop equilibrium at an arbitrary precision.

We consider an instance of a nonatomic routing game. We assume that the network is parallel, that is, constituted of only two nodes, an origin and a destination. We consider infinitesimal players that have a symmetric network cost, but are heterogeneous through their set of feasible strategies and their individual utilities. We show that if an atomic routing game instance is correctly defined to approximate the nonatomic instance, then an atomic Nash Equilibrium will approximate the nonatomic Wardrop Equilibrium. We give explicit bounds on the distance between the equilibria according to the parameters of the atomic instance. This approximation gives a method to compute the Wardrop equilibrium at an arbitrary precision.

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We study games with finitely many participants, each having finitely many choices. We consider the following categories of participants: (I) populations: sets of nonatomic agents, (II) atomic splittable players, (III) atomic non splittable players. We recall and compare the basic properties, expressed through variational inequalities, concerning equilibria, potential games and dissipative games, as well as evolutionary dynamics. Then we consider composite games where the three categories of participants are present, a typical example being congestion games, and extend the previous properties of equilibria and dynamics. Finally we describe an instance of composite potential game.

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An important scenario for smart grids which encompass distributed electrical networks is given by the simultaneous presence of aggregators and individual consumers. In this work, an aggregator is seen as an entity (a coalition) which is able to manage jointly the energy demand of a large group of consumers or users. More precisely, the demand consists in charging an electrical vehicle (EV) battery. The way the EVs user charge their batteries matters since it strongly impacts the network, especially the distribution network costs (e.g., in terms of Joule losses or transformer ageing). Since the charging policy is chosen by the users or the aggregators, the charging problem is naturally distributed. It turns out that one of the tools suited to tackle this heterogenous scenario has been introduced only recently namely, through the notion of composite games. This paper exploits for the first time in the literature of smart grids the notion of composite game and equilibrium. By assuming a rectangular charging profile for an EV, a composite equilibrium analysis is conducted, followed by a detailed analysis of a case study which assumes three possible charging periods or time-slots. Both the provided analytical and numerical results allow one to better understand the relationship between the size (which is a measure) of the coalition and the network sum-cost. In particular, a social dilemma, a situation where everybody prefers unilaterally defecting to cooperating, while the consequence is the worst for all, is exhibited.

In a network composite congestion game, two types of players (nonatomic player of weight zero and atomic splittable player with positive weight) have different strategic behaviors. Statistic properties of equilibrium have been studied recently. This work considers the dynamical aspects, i.e. the evolution of the players' behavior at disequilibium states. Suppose that players adjust their strategies in a selfish and myopic way. The evolution of the strategy profile can be approximated by a continuous-time dynamical system. One shows that several dynamics well-known in the framework of population games (thus with only nonatomic players) such as replicator, BNN, Smith, and two projection dynamics can well be adapted to this more general framework with heterogeneous players. Their asymptotic properties are analysed. Explicitly, one investigates under which condition these dynamics converge to the equilibria.

The main approach of the work is to characterize an equilibrium as a solution to a variational inequality problem (VIP). Then some known "gap functions", whose minimum points coincide with the solutions of the VIP, are natural candidates to serve as Lyapunov function in the corresponding dynamics.

This work is the first to study dynamics in composite congestion games or more generally in games with similar heterogeneous players (which are not rare in economics). Besides, some of the dynamics can be adapted (via a quasi-variational inequality formulation of the equilibrium) to generalized equilibrium problems where the strategy space of a player depends on his opponents' strategies. This setting is more realistic when one takes the real world constraints such as road capacity or production capacity into account.

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This work studies a new strategic game called delegation game. A delegation game is associated to a basic game with a finite number of players where each player has a finite integer weight and her strategy consists in dividing it into several integer parts and assigning each part to one subset of finitely many facilities. In the associated delegation game, a player divides her weight into several integer parts, commits each part to an independent delegate and collects the sum of their payoffs in the basic game played by these delegates. Delegation equilibrium payoffs, consistent delegation equilibrium payoffs and consistent chains inducing these ones in a delegation game are defined. Several examples are provided.

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No summary available.

This thesis is contributed to evolutionary games and congestion games.After a survey of the studies on network congestion games in Chapter 1, Chapters 2 and 3 consider the relation between the composition of the players (nonatomic, atomic, composite) and the equilibrium cost. In particular, the impact of the formation of coalitions is examined.Chapters 4 and 5 introduce the behavior of delegation in composite games and integer-splittable games. Several delegation games and a delegation process are defined and studied in different contexts.Finally, dynamic aspects in games are considered. Chapter 6 focuses on a two-level dynamics which models the phenomenon of multilevel selection. The thesis is concluded by a survey of the studies on the dynamics of replicator type in Chapter 7.

This thesis is contributed to evolutionary games and congestion games. After a survey of the studies on network congestion games in Chapter 1, Chapters 2 and 3 consider the relation between the composition of the players (nonatomic, atomic, composite) and the equilibrium cost. In particular, the impact of the formation of coalitions is examined. Chapters 4 and 5 introduce the behavior of delegation in composite games and integer-splittable games. Several delegation games and a delegation process are defined and studied in different contexts. Finally, dynamic aspects in games are considered. Chapter 6 focuses on a two-level dynamics which models the phenomenon of multilevel selection. The thesis is concluded by a survey of the studies on the dynamics of replicator type in Chapter 7.