## Section: New Results

### Network Engineering Games

Participants : Eitan Altman, Konstantin Avrachenkov, Giovanni Neglia, Nessrine Trabelsi.

#### Network formation games

Network formation games have been proposed as a tool to explain the topological characteristics of existing networks. They assume that each node is an autonomous decision-maker, ignoring that in many cases different nodes are under the control of the same authority (e.g. an Autonomous System) and then they operate as a team. In [11] K. Avrachenkov and G. Neglia in collaboration with V.V. Singh (LRI, Univ. Paris-Sud, France) introduce the concept of network formation games for teams of nodes and show how very different network structures can arise also for some simple games studied in the literature. Beside extending the usual definition of pairwise stable networks to this new setting, we define a more general concept of stability toward deviations from a specific set C of teams’ coalitions (C-stability). We study then a trembling-hand dynamics, where at each time a coalition of teams can create or sever links in order to reduce its cost, but it can also take wrong decisions with some small probability. We show that this stochastic dynamics selects C-stable networks or networks from closed cycles in the long run as the error probability vanishes.

#### Routing Games

A central question in routing games has been to establish conditions for uniqueness of the equilibrium, in terms of network topology or of costs. This question is well understood in two classes of routing games. In [27], E. Altman and C. 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, 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.

#### Game theory applied to the Internet and social networks

In [25] E. Altman, A. Jain (UAPV) and C. Touati (Inria Grenoble - Rhône-Alpes) in collaboration with N. Shimkin (Technion), present an overview of the use of dynamic games for analyzing competition in the Internet and in on-line social networks. A special emphasis is put on identifying phenomena and tools that are novel with respet to game theory applied to other types of networks.

#### Resilience of Routing in Parallel Link Networks

E. Altman, C. Touati and A. Singhal (Inria Grenoble - Rhône-Alpes), in collaboration with J. Li (Tsukuba Univ. Japan), use a game approach in [26] to study the resilience problem of routing traffic in a parallel link network with a malicious player. They consider two players: the first 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. We 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. We identify best responses strategies of each player and compute both the max-min and the min-max values of the game. We provide stable algorithms for computing both max-min and min-max strategies as well as for best responses.

#### A game theoretic solution for Resource Allocation in LTE Cellular Networks

Due to Orthogonal Frequency Division Multiple Access (OFDMA) mechanism adopted in LTE cellular networks, intra-cell interference is nearly absent. Yet, as these networks are designed for a frequency reuse factor of 1 to maximize the utilization of the licensed bandwidth, inter-cell interference coordination remains an important challenge. In both homogeneous and heterogeneous cellular networks, there is a need for scheduling coordination techniques to efficiently distribute the resources and mitigate inter-cell interference. In [54], N. Trabelsi and E. Altman in collaboration with C. S. Chen, L. Roullet from Nokia Bell Labs and with R. El-Azouzi from UAPV propose a dynamic solution of inter-cell interference coordination performing an optimization of frequency sub-band reuse and transmission power in order to maximize the overall network utility. The proposed framework, based on game theory, permits to dynamically define frequency and transmission power patterns for each cell in the coordinated cluster.