Section: New Results

Network Engineering Games

Participants : Eitan Altman, Konstantin Avrachenkov, Ilaria Brunetti, Julien Gaillard, Majed Haddad, Manjesh Kumar Hanawal, Alexandre Reiffers.

Association problem

In [32] , A. Silva, in collaboration with H. Tembine, E. Altman and M. Debbah, study a non-cooperative association game where mobiles associate to Base Stations. The authors solve the problem using the theory of optimal transportation after incorporating in it the effect of network congestion. They are able to find a closed form expression for its solution. The authors also solve a global optimization problem for minimizing the total power needed by the mobile terminals over the whole network.

Cognitive radio

In [52] O. Habachi considers a non-cooperative Opportunistic Spectrum Access (OSA) where Secondary Users (SUs) access opportunistically the spectrum licensed for Primary Users (PUs) in TV white spaces (TVWS). As sensing licensed channels is time and energy consuming, the author considers a hierarchical Cognitive Radio (CR) architecture, where CR base stations sense a subset of the spectrum in order to locate some free frequencies. Thereafter, a SU that needs to communicate through TVWS sends a request to a CR base station for a free channel. The author models the problem using a Partially Observable Stochastic Game (POSG), and he takes into consideration the energy consumption of CR base stations and the Quality of Service of SUs. Since solving POSG optimally may require a significant amount of time and computational complexity, the author then models the OSA problem using a game theoretical approach, and proposes a symmetric Nash equilibrium solution concept. Finally, the simulations that validate the theoretical findings are provided.

In [24] , J. Elias (Univ. Paris Descartes), F. Martignon (Univ. Paris Sud), L. Chen and E. Altman address the joint pricing and network selection problem in cognitive radio networks. The problem is formulated as a Stackelberg game where first the Primary and Secondary operators set the network subscription price to maximize their revenue. Then, users perform the network selection process, deciding whether to pay more for a guaranteed service, or use a cheaper, best-effort secondary network, where congestion and low throughput may be experienced. They use the Nash equilibrium concept to characterize the equilibria for the price setting game. On the other hand, a Wardrop equilibrium is used in the network selection game.

Cooperative games in wireless networks

We have pursued this year our new activity on cooperative games in wireless communications. We have pursued our work on coalition games and started working on the area of matching games. In [56] , E. Altman, C. Hasan and J.-M. Gorce (both from Inria project-team Socrate ) have addressed the problem of association of mobiles to base stations which can be viewed as a coalition game. They formulated the game using a stochastic geometric approach (one Poisson point process representing the base stations and another one representing the mobiles) and studied the impact of switching off base stations (for energy efficient operation).

An important class of games within cooperative games is the matching games. They have been used in stable marriage games (in which a bi-partite graph called matching is to be proposed between a group of men and women based on mutual ranking between this group). A second well-known application of matching games is the college admission problem in which students are assigned to colleges based on their preferences as well as on the preferences of the colleges. We introduced and solved two matching games in wireless communication using the theory of matching games. In [55] the same authors study a game similar to the above ones to match pairs of mobiles where one mobile serves as a relay for the other in the absence of a good direct channel to the base station. The utilities studied here are the outage probabilities. In [65] , R. Vaca-Ramirez, E. Altman, J. S. Thompson and V. Ramos-Ramos propose a distributed algorithm for energy efficient virtual Multiple-input/Multiple-output coalition formation. They model cooperation as a game derived from the concept of stable marriage with incomplete lists. Single antenna devices such as mobile and relay stations cooperate in order to improve the user's and system's energy efficiency. In both problems above, the performance of the equilibrium is shown to be close to the social optimum and yet the complexity for achieving the equilibrium is only polynomial (whereas that of computing a global optimal matching is NP hard).

In [40] K. Avrachenkov, L. Cottatellucci (EURECOM) and L. Maggi (CREATE-NET, Italy) study multiple access channels whose channel coefficients follow a quasi-static Markov process on a finite set of states. The authors address the issue of allocating transmission rates to users in each time interval, such that optimality and fairness of an allocation are preserved throughout a communication, and moreover all the users are consistently satisfied with it. First, it is shown how to allocate the rates in a global optimal fashion. The authors provide a sufficient condition for the optimal rates to fulfill some fairness criteria in a time-consistent way. Then the authors utilize the game-theoretical concepts of time consistent Core and Cooperation Maintenance. It is demonstrated that in the model the sets of rates fulfilling these properties coincide and they also coincide with the set of global optimal rate allocations. The relevance of the presented dynamic rate allocation to LTE systems is also shown.

Bayesian games in networking

K. Veeraruna, E. Altman, R. El-Azouzi and S. Rajesh have studied in [29] a power control problem in which a base station allocates power according to the channel state as reported by the mobiles. The paper addresses the question of how to allocate the power, given that the channel reported by some non-cooperative mobile may be unreliable. They obtain the equilibrium allocation after formulating the problem as a Bayesian game.

In [38] , E. Altman and T. Jiménez consider both a cooperative as well as non-cooperative admission into an M/M/1 queue. The only information available is a signal that says whether the queue size is smaller than some value L or not. They first compute the globally optimal and the Nash equilibrium stationary policy as a function of L. They compare the performance to that of full information and of no information on the queue size. They identify the value of L that optimizes the equilibrium performance.

In [58] , K. Ibrahimi, E. Altman and M. Haddad introduce a signaling game approach to power control. They consider two players named player I and player II. They assume that player I only knows his channel state without any information about the channel state of player II and vice-versa. Player I moves first and sends a signal to player II which can be accurate or distorted. Player II chooses his power control strategy based on this information and his belief about the nature of the informed player's information. In order to analyze such a model, the proposed scheme game is transformed into an equivalent 4x4 matrix game. The authors establish the existence of Nash equilibria and then derive it numerically and study its properties.

In [53] , M. Haddad and E. Altman, in collaboration with P. Wiecek and H. Sidi, present a Bayesian game theoretic framework for determining the decision to which cell a given mobile user should associate in LTE two-tier Heterogeneous Networks. Users are assumed to compete to maximize their throughput by picking the best locally serving cell with respect to their own measurement, their demand and a partial statistical channel state information of other users. In particular, the authors investigate the properties of a hierarchical game, in which the macro-cell BS is a player on its own. They derive analytically the utilities related to the channel quality perceived by users to obtain the equilibria. They show in the Stackelberg formulation, how the operator, by dynamically choosing the offset about the state of the channel, can optimize its global utility while end-users maximize their individual utilities.

Network neutrality and collusion

Representatives of several Internet access providers have expressed their wish to see a substantial change in the pricing policies of the Internet. In particular, they would like to see content providers pay for use of the network, given the large amount of resources they use. This would be in clear violation of the “network neutrality” principle that had characterized the development of the wireline Internet. We proposed and studied possible ways of implementing such payments and of regulating their amount. M. K. Hanawal and E. Altman have pursued in [54] working on network neutrality studying various ways of collusion between an ISP and a content provider and in particular, another form of non-neutrality in which a content provider signals to an ISP information on the popularity of its content and hides this information from other ISPs. They define and compute the price of collusion and study the impact of such signalling on the ISP that is in collusion as well as on the other ones.

In the situation just described, the demand is modelled to be elastic. In contrast, in [62] , A. Reiffers and E. Altman study in collaboration with Y. Hayel pricing issues in non-neutral network with non-elastic traffic. A Stackelberg equilibrium is derived and the price of collusion is computed.

Our research on network neutrality started already on 2010 with a research report [83] that has now been published in [14] . We already reported on this publication in 2011 when it became available electronically.

Competition over popularity in social networks

In [39] E. Altman, P. Kumar, S. Venkatramanan and A. Kumar consider a situation where several content producers send their content to some subscriber of a social network. These posts appear on the subscriber's timeline which is assumed to have finite capacity. Whenever a new post arrives to the timeline, an older post leaves it. Therefore to be visible, a source has to keep sending contents from time to time. Each source is modelled as a player in a non-cooperative game in which one trades between the utility for being visible on the timeline and the cost (or effort) for keeping sending content. This game is solved in a Markovian setting the performance measures of interest are computed.

In [37] , E. Altman in cooperation with F. De Pellegrini (CREATE-NET), D. Miorandi, T. Jiménez and R. El-Azouzi study situations in which subscribers of a social network take the decision whether to access or not some content, based on the number of views that the content has. Their analysis aims at understanding the way in which information about the quality of a given content can be deduced from view counts when only part of the viewers that access the content are informed about its quality. In this paper they present a game formulation for the behavior of individuals using a mean-field model: the number of individuals is approximated by a continuum of atomless players and for which the Wardrop equilibrium is the solution concept. They derive conditions on the problem's parameters that result in the emergence of threshold equilibria policies. But they also identify some parameters in which other structures are obtained for the equilibrium behavior of individuals.

Evolutionary games

Evolutionary game theory is a relatively young mathematical theory that aims at formalizing in mathematical terms evolution models in biology. In recent years this paradigm has penetrated more and more into other areas such as the linguistics, economics and engineering. The current theory of evolutionary game makes an implicit assumption that the evolution is driven by selfishness of individuals who interact with each other. In mathematical terms this can be stated as “an individual equals a player in a non-cooperative game model”. This assumption turns out to be quite restrictive in modeling evolution in biology. It is now more and more accepted among biologist that the evolution is driven by the selfish interests of large groups of individuals; a group may correspond for example to a whole beehive or to an ants' nest. In [43] and [71] , I. Brunetti and E. Altman propose an alternative paradigm for modeling evolution where a player does not necessarily represent an interacting individual but a whole class of such individuals. In [71] in particular, they use Markov Decision Evolutionary Games (MDEG) to allow a parent and a child represent the same individual at different states. This is yet another enhancement in what we understand as a player. An important contribution is in the study of the Hawk and Dove game in these new frameworks.

In [27] , M. Haddad, J. Gaillard, E. Altman and D. Fiems (Ghent Univ.) study an evolutionary game in the MDEG framework of power control. Aging is taken into account by assuming that as the battery of the mobile becomes empty, high power is not available anymore. The goal of a mobile is to use power that maximizes the amount of traffic it can transmit during its lifetime. We restrict in this work to policies that are state independent and compute the equilibrium.