Section: New Results
Game theory applied to networking
Participants : Eitan Altman, Konstantin Avrachenkov, Veeraruna Kavitha, Giovanni Neglia, Sreenath Ramanath.
Wireline networks
Participants : Eitan Altman, Giovanni Neglia.
Fairness based on Nash equilibria
In [77] , E. Altman, jointly with H. Kameda (University of Tsukuba , Japan), C. Touati and A. Legrand (Inria project-team Mescal ), has proposed a new fairness concept for resource sharing among users. The starting point is that a Nash equilibrium is often considered to be fair since it is by definition an outcome of a competition with no collusions. On the other hand, a Nash equilibrium can be non-efficient since it can be dominated by other policies. The authors establish the existence of efficient policies that are on the one hand Pareto optimal and on the other hand proportional to a Nash equilibrium. These properties are used to define a new fairness criterion.
Size-based differentiation
In [92] , E. Altman, S. Soudan, D. Divakaran and P. Primet (Inria project-team Reso ) study competitive situations that occur as a result of the differentiation between short and long flows. A long session can be presented as a number of short sessions in order to avoid being recognized and treated as a long session. The authors characterize the equilibrium.
File sharing systems
In [84] G. Neglia, together with D. Menasche, D. Towsley and S. Zilberstein (University of Massachusetts at Amherst , USA) study the problem of how to combine files into bundles in swarming systems. In particular, they analyze the case of a monopoly where a single publisher decides how to aggregate its files so as to satisfy user demands while mitigating its serving costs. They establish conditions for the existence and uniqueness of an equilibrium and have shown how the publisher's bundling strategy affects its profit. Moreover, they also consider the competitive case where bundling decisions of one publisher affect the outcome of other publishers. Using normal form games they analyze the impact of different system parameters on the Nash equilibrium.
Wireless networks
Participants : Eitan Altman, Konstantin Avrachenkov, Giovanni Neglia.
Jamming in wireless networks
The problem of jamming plays an important role in ensuring the quality and security of wireless communications,
especially at nowadays when wireless networks are quickly becoming ubiquitous. Jamming is a form of a denial
of service attack in which an adversary can degrade the quality of the reception by creating interference.
One can study jamming both in the purpose of protecting a wireless network against such attack or, on the contrary,
in order to efficiently disrupt the communications of some adversary. In both cases jamming is part of a conflict
for which game theory is an appropriate tool. In a series of papers [41] , [40] , [39] E. Altman, K. Avrachenkov
and A. Garnaev (St. Petersburg State University , Russia) analyze the jamming in wireless networks with partial information available.
In particular, in [41]
they study the scenario where users do not know how jamming efforts are distributed among sub-carriers and also
do not know the fading channels gains with certainty. In [40] the authors consider the scenario
when a user either does not have a complete information about the total energy available to the jammer or the
user is not sure if the jammer is present in the environment. In [39] the same scenarios as in [40]
are considered but with a more general utility function, the 
-fairness utility function.
Finally, in [42] the case of several jammers is investigated.
Jamming the signaling channel
In collaboration with S. Sarkar (University of Pennsylvania , USA), R. El-Azouzi and Y. Hayel (University of Avignon ), E. Altman investigates in [34] a game in which n channels are available to a mobile. The authors consider some adversarial node that can prevent the mobile to obtain the information on the state of k out of the n channels. Using a zero-sum Bayesian game model, they answer the questions of which state-dependent jamming policy is the most harmful and what is the best way for the other mobile to react.
WiFi networks
In WiFi networks, mobile nodes compete for accessing the shared channel by means of a random access protocol called Distributed Coordination Function (DCF), which is long term fair. Selfish nodes could benefit from violating the protocol and increasing their transmission probability. In [74] , [73] , [96] , G. Neglia, I. Tinnirello and L. Giarre (University of Palermo , Italy) study the interaction of selfish nodes in two different scenarios. In the first, one mobile stations are simply interested to maximize their upload rate to a single Access Point (AP); in the second one, they are also interested in their downlink rate. The work highlights the poor performance in presence of selfish behaviors and study the role of the AP in WiFi networks in infrastructure mode. Simple changes to AP functionalities can be introduced to design the game in order to achieve optimal global performance at Nash equilibrium. This work has also been presented to a larger community of researchers through the signal processing magazine [25] .
Survey papers
In [28] S. Lasaulce, M. Debbah (Supelec ) and E. Altman present an overview of foundations and tools of game theory that are relevant to wireless communications. Both finite as well as infinite populations of players are considered and various solution concepts (equilibria) are overviewed. Another survey paper by E. Altman in collaboration with R. El-Azouzi, Y. Hayel and H. Tembine (University of Avignon ) focuses on evolutionary games and their applications to wireless networks (see [30] ).
Competition in cellular networks
Participants : Eitan Altman, Veeraruna Kavitha, Sreenath Ramanath.
Opportunistic scheduling in the presence of noncooperative mobiles
In collaboration with R. El-Azouzi (University of Avignon ) and R. Sundaresan (IISc , Bangalore, India), V. Kavitha and E. Altman study in [79] a centralized dynamic scheduling decision that has to be made by a base station to fair share the resources, based on the current channel gains signaled by the mobiles. Mobiles can be non-cooperative in the sense that they may erroneously signal to improve their own utilities. The authors formulate this non cooperative scheduling problem by using a signaling game, where (1) mobiles (lead players) signal the base station of their channel value, (2) base station (follower) makes a scheduling decision using the signals, (3) scheduling decision determines the payoff for all mobiles, base station. It is shown that this signaling game admits only babbling equilibria: mobile's strategy is to signal regardless of its channel. The base station's strategy is to ignore the signals from mobiles. Various approaches are then proposed so as to enforce truthful signaling of the radio channel conditions including a distributed stochastic approximation approach that combines estimation and control.
Location games applied to base station placement
In [49] , E. Altman with A. Kumar, C. Singh and R. Sundaresan (IISc , Bangalore, India) consider the question of determining locations of base stations (BSs) that may belong to the same or to competing service providers, taking into account the impact of these decisions on the behavior of intelligent mobile terminals who can connect to the base station that offers the best utility. They first study the Signal to Interference and Noise Ratio (SINR) association-game: the authors determine the cells corresponding to each base station, namely, the locations at which mobile terminals prefer to connect to a given base station than to others. SINR is used as the quantity that determines the association. They make some surprising observations: (i) displacing a base station a little in one direction may result in a displacement of the boundary of the corresponding cell to the opposite direction, and (ii) a cell corresponding to a BS may be the union of disconnected sub-cells. They further study the Stackelberg equilibrium in the combined BS location and mobile association problem: they determine where to locate the BSs so as to maximize the revenues obtained at the induced SINR mobile association game. They consider the cases of single frequency band and two frequency bands of operation. Finally, they study Stackelberg equilibria in two frequency systems with successive interference cancellations.
A hierarchical base station location game with differentiated services
In [78] E. Altman, jointly with G. Kasbekar and S. Sarkar (University of Pennsylvania , USA), considers a scenario in which a regulator owns the spectrum in a certain region. A number of service providers lease spectrum from the regulator. Each service provider sets up a base station to serve mobile subscribers in the region. This gives rise to a hierarchical game with players at three levels: the mobile subscribers (level 1), the service providers (level 2) and the regulator (level 3). In the game at level 3, the regulator chooses the price at which to lease the spectrum. In the game at level 2, each service provider chooses the quantity of spectrum to lease from the regulator and the rate at which to charge the mobile subscribers. The authors consider the situation in which each service provider has two kinds of mobile subscribers: primary users, who receive high-priority service and secondary users who receive low-priority service. In the game at level 1, each mobile subscriber chooses a service provider to join and whether to become a primary or secondary user. Using game theoretic tools the authors compute the equilibrium strategies at all three levels.
Uplink competitive resource allocation problem
In [48] , E. Altman, A. Kumar (IISc , Bangalore, India) and Y. Hayel (University of Avignon ) consider a resource allocation problem in a multichannel wireless access system being shared by several users for uplink transfer of elastic traffic. Each user can allocate its resources (e.g., radios, antennas or power) to one or more of the carriers. The authors consider a problem of noncooperative allocation of resources by the users, each user's objective being to maximize its own utility. The theory of potential games is used to solve this problem by transforming it into an equivalent global optimization one. Structural properties of the equilibrium policies are obtained using tools from Schur concave stochastic order. Finally, a distributed algorithm is introduced and is shown to converge to a Nash equilibrium of the system.
Distributed resource allocation algorithms in a slotted aloha system
In collaboration with E. Sabir, R. El-Azouzi, Y. Hayel (University Avignon ) and E.-H. Bouyakhf (University of Mohammed V , Morocco), V. Kavitha considers in [90] finite number of users, with infinite buffer storage, sharing a single channel using the Aloha medium access protocol. They investigate the uplink case of a cellular system where each user will select a desired throughput. The users then participate in a non cooperative game wherein they adjust their transmit rate to attain their desired throughput. In contrast to the saturated case, the authors show that the game either has no Nash equilibrium or has infinitely many Nash Equilibria (NE). They further show that the region of NE coincides with an appropriate “stability region”. The authors discuss the efficiency of the equilibria in term of energy consumption and congestion rate. Next, they propose two learning algorithms using a stochastic iterative procedure that converges to the best Nash equilibrium. They approximate the control iterations by an equivalent ordinary differential equation in order to prove that the proposed stochastic learning algorithm converges to a Nash equilibrium even in the absence of any coordination or extra information. Extensive numerical examples and simulations are provided to validate the results.
Evolutionary games
Participant : Eitan Altman.
In [16] E. Altman studies in cooperation with H. Tembine, Y. Hayel and R. El-Azouzi (University of Avignon ) the evolution of competing flow control protocols. In this work models of population dynamics (such as the replicator dynamics) are considered to understand under what conditions can one expect different variants of TCP protocols to coexist, and what fraction of users should be using each type. A shorter version of this work has been described in the last year Maestro activity report.
