Section: New Results

Network Engineering Games

Participants : Eitan Altman, Konstantin Avrachenkov, Giovanni Neglia.

Matching games and the association problem

In [33] , Mikaël Touati, Jean-Marc Kélif (Orange Labs), Rachid El-Azouzi (UAPV), Marceau Coupechoux (Telecom ParisTech) and Eitan Altman propose two new algorithms for finding stable structures in ordinal coalition potential games. The first one is enumerative and it performs on a graph. The second one is a modified Deferred Acceptance Algorithm using counter-proposals. It finds a many-to-one matching. The authors illustrate with the example of video caching from a content creator's servers to a service provider's servers.

This is applied to the association of mobiles to IEEE 802.11-based WLANs in populated areas where many mobile terminals are covered by several Access Points (APs) [32] . These mobiles have the possibility to associate to the AP with the strongest signal (best-RSSI association scheme). This can lead to poor performances and overloaded APs. Moreover, the well-known anomaly in the protocol at the MAC layer may also lead to very unpredictable performances and affect the system throughput due to the presence of heterogeneous data rate nodes and the shared nature of the 802.11 medium. In [61] , the same authors solve the joint resource allocation and mobile user association after modeling it as a matching game with complementarities, peer effects and selfish players.

Normalized Nash Equilibria for power control with correlated constraints

When correlated constraints are introduced to a game (i.e. the set of actions of a player depends on the policies of other players) there may exist infinitely many Nash equilibria. Assume one wishes to select a particular one $u$. According to the Karush Kuhn Tucker theorem, there exist Lagrange mutipliers such that the best response when all players use their equilibrium policy is the same as that obtained by optimizing the corresponding Lagrangian of that player. The Lagrange multipliers can be interpreted as marginal costs such that if they are imposed on the player as some tax to pay then this induces the player to use Nash equilibrium. The following question arises: does there exist an equilibrium $u$ for which the corresponding Lagrange multipliers are player independent. If the answer is positive then this would make in many cases the billing scalable and simple to implement. An equilibrium $u$ for which the corresponding Lagrange multipliers are player independent is called a normalized Nash equilibrium (NNE). In [39] , [50] and [24] , Arnob Ghosh (Univ. of Pennsylvania), Laura Cottatellucci (Eurecom) and Eitan Altman provide new conditions for existence and uniqueness of NNE and apply this for power control games arising in cognitive radio [24] and in heterogeneous networks [39] , [50] .

Admission control to an infinite server queue

In [36] , Eitan Altman studies in collaboration with Piotr Wiecek (Wrocław Univ. of Technology) and Arnob Ghosh (Univ. of Pennsylvania) a mean field approximation of the M/M/$\infty$ queueing system. The problem they consider is quite different from standard games of congestion as they consider the case in which higher congestion results in smaller costs per user. This is motivated by a situation in which some TV show is broadcast so that the same cost is needed no matter how many users follow the show. Using a mean-field approximation, they show that this results in multiple equilibria of threshold type which is explicitly computed. The authors further derive the social optimal policy and compute the price of anarchy, and show that the mean-field approximation becomes tight as the workload increases, thus the results obtained for the mean-field model well approximate the discrete one.

Posting Time of Content over a Temporally-Ordered Shared Medium

In [17] , Eitan Altman in collaboration with Nahum Shimkin (Technion) consider a game of timing between a random number of content creators, who compete for position and exposure time over a shared medium such as an on-line classified list. Contents (such as ads, messages, multimedia items or comments) are ordered according to their submission times, with more recent submissions displayed at the top (and better) positions. The instantaneous effectiveness of each ad depends on its current display position, as well as on a time-dependent exposure function common to all. Each content creator may choose the submission time of her content within a finite time interval, with the goal of maximizing the total exposure of this content. The authors formulate the problem as a non-cooperative game, analyze its symmetric equilibrium, characterize it in terms of a differential boundary value problem and devise a numerical scheme for its computation.

Routing Games

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 [64] , Eitan Altman and Corinne Touati (Inria Grenoble-Rhône-Alpes) 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, they identify various novel and surprising properties of games within these frameworks. The authors show in particular that equilibria are non unique even in the potential game setting of Rosenthal with strictly convex link costs. They further show that non-symmetric equilibria arise in symmetric networks.

Resilience of Routing in Parallel Link Networks

Aniruddha Singhal, Corinne Touati (both from Inria Grenoble-Rhône-Alpes) in collaboration with Eitan Altman and Jie Li (Univ. of Tsukuba) revisit in [63] , the resilience problem of routing traffic in a parallel link network model with a malicious player using a game theoretic framework. Consider that there are two players in the network: the first player wishes to split its traffic so as to minimize its average delay, which the second player, i.e., the malicious player, tries to maximize. The first player has a demand constraint on the total traffic it routes. The second player controls the link capacities: it can decrease by some amount the capacity of each link under a constraint on the sum of capacity degradation. The authors first show that the average delay function is convex both in traffic and in capacity degradation over the parallel links and thus does not have a saddle point. They identify best responses strategies of each player and compute both the max-min and the min-max values of the game. One is especially interested in the min-max strategy as it guarantees the best performance under worst possible link capacity degradation. It thus allows to obtain routing strategies that are resilient and robust. The authors compare the results of the min-max to those obtained under the max-min strategies. They provide stable algorithms for computing both max-min and min-max strategies as well as for best responses.

The Social Medium Selection Game

In [72] , Fabrice Lebeau (ENS Lyon) Corinne Touati and Nof Abuzainab (Inria Grenoble-Rhône-Alpes) in collaboration with Eitan Altman, consider 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. The authors model this game as a congestion game and compute the pure potential of the game. In contrast to the continuous case, they show that there may be various equilibria. The authors show that the potential is M-concave which allows them to characterize the equilibria and to propose an algorithm for computing it. They then give a learning mechanism which leads to an efficient algorithm to determine an equilibrium. The authors finally determine the asymptotic form of the equilibrium and discuss the implications on the social medium selection problem.

Activation Games in Online Dating Platforms

In [41] , Eitan Altman in collaboration with Francesco De Pellegrini (CREATE-NET, Trento) and Huijuan Wang (Delft Univ. of Technology) describe a model for the activation level of users in online dating platforms (ODPs). Such popular systems are conceived in order to match individuals from two groups of potential mates. The business of such platforms pivots around the customers' expectancy to get in contact with their future dates: upon the payment of a fee to the platform owner, ODPs provide specific tools to improve reach and visibility. However, ODPs require a critical number of active users in order to sustain their operations (and their business). Customers of the platform trade off on the price for being more visible and attract mates' contacts. A user becomes inactive if he or she is not contacted by others for some time: being contacted by potential mates acts as an activation signal. The aim of the analysis is to propose a game theoretical framework to capture such a complex activation problem in strategic form. The authors unveil the structure of Nash equilibria and further derive a Stackelberg formulation. The latter is a hierarchical game where the platform owner aims at maximizing profits while preserving the ODP activity level above a critical epidemic threshold.

Epidemics in Networks

Stojan Trajanovski, Huijuan Wang, Piet Van Mieghem (all from Delft Univ. of Technology), in collaboration with Yezekael Hayel (UAPV) and Eitan Altman have pursued their work in the Congas European project concerning malware attacks modeled as SIS (for Susceptible-Infected-Susceptible) epidemics in networks. In [34] , the authors consider decentralized optimal protection strategies when a virus is propagating over a network. they assume that each node in the network can fully protect itself from infection at a constant cost, or the node can use recovery software, once it is infected. They model the system using a game theoretic framework and find pure, mixed equilibria, and the Price of Anarchy (PoA) in several network topologies. Further, they propose both a decentralized algorithm and an iterative procedure to compute a pure equilibrium in the general case of a multiple communities network. Finally, the authors evaluate the algorithms and give numerical illustrations of all results.

They then considered the game-formation problem while balancing multiple, possibly conflicting objectives like cost, high performance, security and resiliency to viruses. In [60] , Stojan Trajanovski, Fernando Antonio Kuiper and Piet Van Mieghem (all from Delft Univ. of Technology) in collaboration with Yezekael Hayel (UAPV) and Eitan Altman use a game-formation approach to network design where each player (node), aims to collectively minimize the cost of installing links, of protecting against viruses, and of assuring connectivity. In the game, minimizing virus risk as well as connectivity costs results in sparse graphs. They show that the Nash Equilibria are trees that, according to the Price of Anarchy (PoA), are close to the global optimum, while the worst-case Nash Equilibrium and the global optimum may significantly differ for small infection rate and link installation cost. Moreover, the types of trees, in both the Nash Equilibria and the optimal solution, depend on the virus infection rate, which provides new insights into how viruses spread: for a high infection rate, the path graph is the worst- and the star graph is the best-case Nash Equilibrium. However, for small and intermediate infection rates, trees different from the path and star graphs may be optimal.

Retrial games

In [46] K. Avrachenkov in collaboration with E. Morozov and R. Nekrasova (both from Petrozavodsk State Univ., Russia) consider a single-server retrial system with one and several classes of customers. In the case of several classes, each class has its own orbit for retrying customers. The retrials from the orbits are generated with constant retrial rates. In the single class case, the objective is finding an optimal retrial rate. Whereas in the multi-class case, a game theoretic framework is used and equilibrium retrial rates are found. The performance criteria balance the number of retrials per retrying customer with the number of unhappy customers.

Cooperative Network Design

The Network Design problem has received increasing attention in recent years. Previous works have addressed this problem considering almost exclusively networks designed by selfish users, which can be consistently suboptimal. In [18] K. Avrachenkov, J. Elias (Univ. Paris Descartes, France), F. Martignon (Univ. Paris Sud, France), G. Neglia and L. Petrosyan (St. Petersburg State Univ.) address the network design issue using cooperative game theory, which permits to study ways to enforce and sustain cooperation among users. Both the Nash bargaining solution and the Shapley value are widely applicable concepts for solving these games. However, the Shapley value presents several drawbacks in this context. For this reason, they solve the cooperative network design game using the Nash bargaining solution (NBS) concept. More specifically, they extend the NBS approach to the case of multiple players and give an explicit expression for users' cost allocations. They further provide a distributed algorithm for computing the Nash bargaining solution. Then, they compare the NBS to the Shapley value and the Nash equilibrium solution in several network scenarios, including real ISP topologies, showing its advantages and appealing properties in terms of cost allocation to users and computation time to obtain the solution.

Numerical results demonstrate that the proposed Nash bargaining solution approach permits to allocate costs fairly to users in a reasonable computation time, thus representing a very effective framework for the design of efficient and stable networks.