Load Balancing Congestion Games and their Asymptotic Behavior

A central question in routing games has been to establish conditions for the uniqueness of the equilibrium, either in terms of network topology or in terms of costs. This question is well understood in two classes of routing games. The first is the non-atomic routing introduced by Wardrop on 1952 in the context of road traffic in which each player (car) is infinitesimally small; a single car has a negligible impact on the congestion. Each car wishes to minimize its expected delay. Under arbitrary topology, such games are known to have a convex potential and thus a unique equilibrium. The second framework is splitable atomic games: there are finitely many players, each controlling the route of a population of individuals (let them be cars in road traffic or packets in the communication networks). In this paper, we study two other frameworks of routing games in which each of several players has an integer number of connections (which are population of packets) to route and where there is a constraint that a connection cannot be split. Through a particular game with a simple three link topology, we identify various novel and surprising properties of games within these frameworks. We show in particular that equilibria are non unique even in the potential game setting of Rosenthal with strictly convex link costs. We further show that non-symmetric equilibria arise in symmetric networks. I. INTRODUCTION A central question in routing games has been to establish conditions for the uniqueness of the equilibria, either in terms of the network topology or in terms of the costs. A survey on these issues is given in [1]. The question of uniqueness of equilibria has been studied in two different frameworks. The first, which we call F1, is the non-atomic routing introduced by Wardrop on 1952 in the context of road traffic in which each player (car) is infinitesimally small; a single car has a negligible impact on the congestion. Each car wishes to minimize its expected delay. Under arbitrary topology, such games are known to have a convex potential and thus have a unique equilibrium [2]. The second framework, denoted by F2, is splitable atomic games. There are finitely many players, each controlling the route of a population of individuals. This type of games have already been studied in the context of road traffic by Haurie and Marcotte [3] but have become central in the telecom community to model routing decisions of Internet Service Providers that can decide how to split the traffic of their subscribers among various routes so as to minimize network congestion [4]. In this paper we study properties of equilibria in two other frameworks of routing games which exhibit surprisin

Non-Cooperative Scheduling of Multiple Bag-of-Task Applications

Multiple applications that execute concurrently on heterogeneous platforms compete for CPU and network resources. In this paper we analyze the behavior of $K$ non-cooperative schedulers using the optimal strategy that maximize their efficiency while fairness is ensured at a system level ignoring applications characteristics. We limit our study to simple single-level master-worker platforms and to the case where each scheduler is in charge of a single application consisting of a large number of independent tasks. The tasks of a given application all have the same computation and communication requirements, but these requirements can vary from one application to another. In this context, we assume that each scheduler aims at maximizing its throughput. We give closed-form formula of the equilibrium reached by such a system and study its performance. We characterize the situations where this Nash equilibrium is optimal (in the Pareto sense) and show that even though no catastrophic situation (Braess-like paradox) can occur, such an equilibrium can be arbitrarily bad for any classical performance measure

Le jeu des Kirlis et des Gourlus

National audienceCet article présente un jeu éducatif a destination des lycéens ayant pour but de présenter quelques exemples contre-intuitifs de théorie des jeux (paradoxe de l'information). La résolution des scénarios leur permet de se familiariser avec l'utilisation d'arbres de décisions et de tableaux et de faire des calculs simples de probabilités

A Dynamic Game Analysis and Design of Infrastructure Network Protection and Recovery

Infrastructure networks are vulnerable to both cyber and physical attacks. Building a secure and resilient networked system is essential for providing reliable and dependable services. To this end, we establish a two-player three-stage game framework to capture the dynamics in the infrastructure protection and recovery phases. Specifically, the goal of the infrastructure network designer is to keep the network connected before and after the attack, while the adversary aims to disconnect the network by compromising a set of links. With costs for creating and removing links, the two players aim to maximize their utilities while minimizing the costs. In this paper, we use the concept of subgame perfect equilibrium (SPE) to characterize the optimal strategies of the network defender and attacker. We derive the SPE explicitly in terms of system parameters. Finally, we use a case study of UAV-enabled communication networks for disaster recovery to corroborate the obtained analytical results.Comment: 6 page

Adaptive Power Allocation and Control in Time-Varying Multi-Carrier MIMO Networks

In this paper, we examine the fundamental trade-off between radiated power and achieved throughput in wireless multi-carrier, multiple-input and multiple-output (MIMO) systems that vary with time in an unpredictable fashion (e.g. due to changes in the wireless medium or the users' QoS requirements). Contrary to the static/stationary channel regime, there is no optimal power allocation profile to target (either static or in the mean), so the system's users must adapt to changes in the environment "on the fly", without being able to predict the system's evolution ahead of time. In this dynamic context, we formulate the users' power/throughput trade-off as an online optimization problem and we provide a matrix exponential learning algorithm that leads to no regret - i.e. the proposed transmit policy is asymptotically optimal in hindsight, irrespective of how the system evolves over time. Furthermore, we also examine the robustness of the proposed algorithm under imperfect channel state information (CSI) and we show that it retains its regret minimization properties under very mild conditions on the measurement noise statistics. As a result, users are able to track the evolution of their individually optimum transmit profiles remarkably well, even under rapidly changing network conditions and high uncertainty. Our theoretical analysis is validated by extensive numerical simulations corresponding to a realistic network deployment and providing further insights in the practical implementation aspects of the proposed algorithm.Comment: 25 pages, 4 figure

Coalition Formation Game for Cooperative Cognitive Radio Using Gibbs Sampling

This paper considers a cognitive radio network in which each secondary user selects a primary user to assist in order to get a chance of accessing the primary user channel. Thus, each group of secondary users assisting the same primary user forms a coaltion. Within each coalition, sequential relaying is employed, and a relay ordering algorithm is used to make use of the relays in an efficient manner. It is required then to find the optimal sets of secondary users assisting each primary user such that the sum of their rates is maximized. The problem is formulated as a coalition formation game, and a Gibbs Sampling based algorithm is used to find the optimal coalition structure.Comment: 7 pages, 2 figure

The Social Medium Selection Game

We consider in this paper competition of content creators in routing their content through various media. The routing decisions may correspond to the selection of a social network (e.g. twitter versus facebook or linkedin) or of a group within a given social network. The utility for a player to send its content to some medium is given as the difference between the dissemination utility at this medium and some transmission cost. We model this game as a congestion game and compute the pure potential of the game. In contrast to the continuous case, we show that there may be various equilibria. We show that the potential is M-concave which allows us to characterize the equilibria and to propose an algorithm for computing it. We then give a learning mechanism which allow us to give an efficient algorithm to determine an equilibrium. We finally determine the asymptotic form of the equilibrium and discuss the implications on the social medium selection problem

How to measure efficiency?

In the context of applied game theory in networking environments, a number of concepts have been proposed to measure both efficiency and optimality of resource allocations, the most famous certainly being the price of anarchy and the Jain index. Yet, very few have tried to question these measures and compare them one to another, in a general framework, which is the aim of the present article

Comment le mode de scrutin peut influencer le résultat d'unéuné election

National audienceLes années 2016 et 2017 ontétéontété riches enmatì ere d'´ electionsàelectionsà travers le monde. Nous avons tous entendu parler du référendum au Royaume-Uni sur le maintien ou non dans l'union européenne, l'´ election présidentielle américaine, française, lesélectionslesélections législatives françaises. Certains des résultats de cesélectionscesélections ont surpris les observateurs, et beaucoup se sont demandés si le choix des r` egles des scrutins ou la di↵usion de sondages avaient eu un impact dans les résultats. Dans cet atelier, nous allons donc jouer un petit jeu de rôle pour uné election fictive afin de voir si, pour une population donnée, le choix du scrutin peut meneràmenerà des résultats di↵érents et s'il existe un mode de scrutin optimal. 1 Le jeu Dans ce jeu chaqué electeur (joueur) reçoit un carton lui indiquant quelles sont ces opinions pour chacun des 4 candidats d'uné election fictive. Chaqué electeur peut choisiràchoisirà sa guise de communiquer son avisàavisà ses voisins (pasàpasà la population totale) ou bien de le garder secret. On informe ensuite aux joueurs le type de scrutin qui va se dérouler et chaque joueur est alors libre de voter comme bon lui semble, mais bien sûr de façon rationnelle, c'est-` a-dire de sorte a maximiser les chances que son candidat favori soitélusoitélu, ou si cela n'est pas possible, que le candidat qui será elu soit aussi apprécié que possible de la part de l'´ electeur. Les bulletins des joueurs sont fournis en fin de document (découpables). Pour chacun desélecteursdesélecteurs, le carton peutêtrepeutêtre compris comme suit : ï¿¿ï¿¿ C'est le candidat idéal ï¿¿ Un candidat apprécié même s'il n'est pas favori ï¿¿ Un candidat peu apprécié, sansêtresansêtre détesté ï¿¿ï¿¿ C'est le candidat que l'on souhaitè a tout prixéviterprixéviter Pour que les résultats suivants soient valables, il convient d'utiliser les cartes dans l'ordre de leur numé-rotation. Ainsi pour un jeu avec 18 joueurs, on utilisera les cartes 1 ` a 18 par exemple. La description qui suit a ´ eté faite pour un exemple avec 16 joueurs, mais les résultats restent valables pour un nombre supérieur de joueurs. (Pour créer d'avantage de cartons, il sut de répéter les cartons 21à21à 24 dans le même ordre, autant que fois que nécessaire)