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Section: New Results

Game theory applied to networking

Participants : Eitan Altman, Konstantin Avrachenkov, Veeraruna Kavitha, Giovanni Neglia.

Power save mechanisms

Nowadays energy saving and reduction of electromagnetic pollution become important issues. One approach to these problems is the introduction of taxes on the energy dissipation. In [33] , E. Altman and K. Avrachenkov, in collaboration with A. Garnaev (St.Petersburg State Univ.), investigate a taxation game between a user or a provider or a group of users and the taxation authority. This is a Stackelberg game where the taxation authority acts as a leader and users or service providers act as followers. They focus on the problem of finding taxation strategy in closed form and investigate how incomplete information of authorities about users impacts the equilibrium strategy.

Pricing mechanisms

Typically the cost of a product has many components. Various components correspond to the production chain steps through which the product goes before meeting a customer. This also takes place in the price formation in wireless networks. For instance, before transmitting customer data, a network operator has to buy some frequency range and also to establish contracts with electricity providers. In [50] , E. Altman and K. Avrachenkov, in cooperation with Y. Hayel (LIA/Univ. of Avignon) and A. Garnaev (St. Petersburg State Univ.), establish the tarif formation scheme in wireless networks. Specifically, a hierarchical Stackelberg game with three levels, the user, the provider and the authority is analyzed.

Mobile association based on partial channel information

In [48] , E. Altman, in collaboration with S-E. Elayoubi, M. Haddad and Z. Altman (all from Orange Labs), addresses the question of what decisions should be left to the user and what decision should be taken by the base station. In case the mobile takes the decision of association, it has to do so based on the channel information which is available but not in details: the base station only transmits information on whether the state is good, bad or in between. The problem is solved using a game theoretic formulation. These authors further study in [51] the question of how should the base station choose optimally which channel states should be declared as good, bad or in between, respectively.

Fair scheduling in presence of non-cooperation

In a cellular network, using an $ \alpha$ -fair scheduler, the base station (BS) has to assign the slot to one of the mobiles based on truthful information from mobiles about their time-varying channel gains. A non-cooperative mobile may misrepresent its signal to the BS so as to maximize its throughput. In [55] , V. Kavitha, E. Altman, R. Elazouzi (LIA/Univ. Avignon) and R. Sundaresan (IISc, Bangalore), have shown that the presence of non-cooperative users results in an $ \alpha$ -fair bias in the channel assignment for small values of $ \alpha$ while the existing schedulers are robust at high values of $ \alpha$ . When the BS is aware of the non-cooperative mobiles and when the BS has additional knowledge of the statistics of the signals of the mobiles, new robust policies are proposed, which elicit the truthful signals from mobiles and achieve a Truth Revealing Equilibrium. The popular, iterative fair scheduling algorithms, proposed by H. J. Kushner and P. A. Whiting, are shown to fail under non-cooperation and are robustified against non-cooperation.

Jamming the signaling channel

In collaboration with S. Sarkar and P. Vaidyanathan (Univ. Pennsylvania), E. Altman investigates in [28] a game in which n channels are available to a mobile. The authors consider some adversarial node that can prevent the mobile from obtaining information on the state of k out of the n channels. This is the extended journal version that corresponds to a conference paper presented in the 2009 Maestro activity report.

The problem of jamming plays an important role in ensuring the quality and security of wireless communications, especially 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 [19] , E. Altman and K. Avrachenkov, in cooperation with A. Garnaev (St. Petersburg State Univ.), consider jamming in wireless networks in the framework of zero-sum games with a-fairness utility functions. The base station has to distribute the power fairly among the users in the presence of a jammer. The jammer in turn tries to distribute its power among the channels to produce as much harm as possible. The Shannon capacity and the SNIR optimization are particular cases of the proposed more general a-fairness SNIR based utility functions.

Cognitive radio networks

Spectrum sharing in cognitive radio enables an efficient use of the scarce frequency spectrum by allowing the coexistence of licensed and unlicensed users in the same spectrum. In [72] , K. Avrachenkov, in cooperation with X. Lei, L. Cottatellucci (Institut Eurecom), and A. Garnaev (St. Petersburg State Univ.), consider a slow fading multiuser environment with primary and secondary users. The secondary users have only partial knowledge of the channel and are subjected to transmitted power constraints by the primary users. Their communications are intrinsically affected by outage events. The authors propose and analyze two algorithms for joint rate and power allocation. In one algorithm, the secondary transmitters cooperate to maximize a common utility function accounting for the total throughput of the network. In a second approach based on a game framework, the secondary users aim at maximizing selfishly their own utilities. The latter approach shows better fairness properties at the expense of some global performance loss compared to the optimum cooperative approach.

Hierarchical routing games

Many structural results are known in routing games in which the link cost density (i.e. cost per unit of flow) is the same for all users and is given as a function of the total flow in that link. In [54] , V. Kamble (Univ. Berkeley), E. Altman, R. El-Azouzi (LIA/Univ. Avignon) and V. Sharma (IISc, Bangalore), relax this structure and explore hierarchical routing games in which some flows (called primary flows) have strict priority over other flows (called secondary). The link cost density for primary flows is a function of the total amount of high priority flow in the link, whereas for secondary users. the density link cost depends on the total flow on the link from all classes. Uniqueness of the equilibrium is established under general conditions.

Control of epidemics with applications to propagation in computer networks

Back in 2008, E. Altman, in collaboration with T. Başar (Univ. Illinois) and F. De Pellegrini (Create-Net, Italy), started investigating and developing the theory of control of epidemic models having as goal to use these tools in computer networks. Their main achievements are listed below.

Epidemic with monotone structure

The first results obtained were in developing a theory adapted to epidemics with some monotone structure in the dynamics and in the cost. These results have now appeared in an extended journal version in [23] . On the other hand, they have extended these results in [45] from a scalar state space to a vector valued one. The authors identify optimal policies with a switching structure where one action is used until some time threshold and then another one is used thereafter. The context of these works as well as a number of generalizations which we are specified below, has been DTNs (Delay Tolerant Networks) and the object that is propagated is assumed to be some content (such as music or data files). In [34] , A. Azad joins the three coauthors and studies the problem in which not only do we control the power of mobiles but also their activation time.

Adversarial problems

In [38] , E. Altman, T. Başar (Univ. Illinois) and V. Kavitha consider a multi-criteria control problem that arises in a delay tolerant network with two adversarial controllers: the source and the jammer. Open loop as well as closed loop optimal policies have similar structures. When the jammer has a tighter constraint on its energy resources than the source, both the policies have two switching times. Before the first switching time, the source and jammer policies are inner (are not pure) and are given by equalizer policies. After the first switching time, the jammer switches off and the source continues transmitting at maximum probability. After the second switching time, both the source and jammer are off. When the source has a tighter constraint on its energy resources than the jammer, there exists only one switching time before which the source and the jammer use inner equalizer policies and after which both are switched off. Dynamic programming techniques are used to obtain the above results.

A particular important adversarial situation arises in the propagation of e-viruses or of malware within a network. As these are often designed to damage the network as much as possible, the aim is to come up with methods for combatting those efficiently. E. Altman, in collaboration with S. Sarkar and M.H.R. Khouzani (Univ. Pennsylvania), has used the maximum principle in order to obtain the structure of optimal network policies against viruses [59] , [58] [89] .

Adversarial modeling is also used in robust control: when some parameters evolve in an unpredictable way, or when some noise that cannot be modelled well affects the dynamics or observations, one resorts to a worst case design, and attempts to obtain a strategy that can guarantee the best performance under any behavior of the noise or of the unknown parameter. In [38] , E. Altman, in collaboration with A. Aram (Univ. Pennsylvania), T. Başar (Univ. Illinois), C. Touati (Mescal , Inria ) and S. Sarkar (Univ. Pennsylvania), carries on this approach in a problem where both the dynamic and the cost are linear in the state and control. The author uses robust control techniques and manage to obtain an explicit solution for the optimal control as a function of the available information.

Applying risk sensitive control to delay tolerant networks

Our most important theoretical contribution to the theory of control of epidemics in computer network has been to identify tools from risk sensitive control which enable to obtain much more precise solutions as were available before. Indeed, consider a message that propagates within a set of mobile nodes, and assume that one is interested in maximizing probability that it would reach some destination (thanks to relaying) by some time T . This quantity can be expressed as the expectation of the exponential function of some integral cost. In the past, to minimize this cost, one maximized directy the integral. We have noticed that the original cost is the same as the risk sensitive cost which we often find in financial mathematics, and for which many algorithms are available. In [83] , E. Altman and K. Kavitha, in collaboration with F. De Pellegrini (Create-Net, Italy) V. Kamble (Univ. Berkeley) and V. Borkar (TATA Institute, Mumbia), use risk sensitive theory in order to solve power control problems in DTNs.

Network design with socially-aware users

In many scenarios network design is not enforced by a central authority, but arises from the interactions of several self-interested agents. This is the case of the Internet itself. K. Avrachenkov and G. Neglia, together with J. Elias (Politecnico di Milano) and F. Martignon (Univ. Bergamo), have proposed two novel socially-aware network design games [49]   [88] . In the first game they have incorporated a socially-aware component in the users' utility functions, while in the second game they have adopted a Stackelberg approach, where a leader (e.g., the network administrator) architects the desired network buying an appropriate subset of network's links, driving in this way the users to overall efficient Nash equilibria. In [75] the same researchers, together with Leon Petrosyan (St Petersburg State Univ.), have studied the advantages to use Nash Bargaining Solution, rather than Shapley Value, to solve Cooperative Network Formation games.

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. G. Neglia, together with I. Tinnirello and L. Giarré (Univ. Palermo) have been studying the interaction of selfish nodes in the last two years (the research activity is described in Maestro 2009 activity report). In [68] , they have proposed some new mechanisms for channel access, that incite s elfish nodes to operate at Pareto optimal equilibria. The issue of how to implement these mechanisms has been addressed as well.


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