## Section: New Results

### Multi-User Systems

Participants : Bruno Gaujal, Arnaud Legrand, Corinne Touati, Jean-Marc Vincent.

#### Strategies in Allocation Games

In [25] , we consider allocation games and we investigate the following question: under what conditions does the replicator dynamics select a pure strategy? By definition, an allocation game is a game such that the payoff of a player when she takes an action only depends on the set of players who also take the same action. Such a game can be seen as a set of users who share a set of resources, a choice being an allocation to a resource. A companion game (with modified utilities) is introduced. From the payoffs of an allocation game, we define the reper- cussion utilities: for each player, her repercussion utility is her payoff minus the decrease in marginal payoff that her presence causes to all other players. The corresponding allocation game with repercussion utilities is the game whose payoffs are the repercussion utilities. A simple characterization of those games is given. In such games, if the players select their strategy according to a stochastic approximation of the replicator dynamics, we show that it converges to a Nash equilibrium of the game that is a locally optimal for the initial game. The proof is based on the construction of a potential function for the game. Furthermore, a spectral study of the dynamics shows that no mixed equilibrium is stable, so that the strategies of all players converge to a set of Nash equilibria. Then, martingale argument prove the convergence of the stochastic approximation to a pure point. A discussion of the global/local optimality of the limit points is also included.

There are several approaches of sharing resources among users. There is a noncooperative approach wherein each user strives to maximize its own utility. The most common optimality notion is then the Nash equilibrium. Nash equilibria are generally Pareto inefficient. On the other hand, we consider a Nash equilibrium to be fair as it is defined in a context of fair competition without coalitions (such as cartels and syndicates). In [33] , we show a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium. We consider this Pareto optimum to be ”Nash equilibrium based fair”. We futher define a ”Nash proportion- ately fair” Pareto optimum. We then provide conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium. As examples that fit in the above framework, we consider noncooperative flow- control problems in communication networks, for which we show the conditions on the existence of Nash-proportionately fair Pareto optimal allocations.

#### A Fair User-Network Association Algorithm for Wireless Networks

Recent mobile equipment (as well as the norm IEEE 802.21) now offers the possibility for users to switch from one technology to another (vertical handover). This allows flexibility in resource assignments and, consequently, increases the potential throughput allocated to each user. In [24] , we design a fully distributed algorithm based on trial and error mechanisms that exploits the benefits of vertical handover by finding fair and efficient assignment schemes. On the one hand, mobiles gradually update the fraction of data packets they send to each network based on the rewards they receive from the stations. On the other hand, network stations send rewards to each mobile that represent the impact each mobile has on the cell throughput. This reward function is closely related to the concept of marginal cost in the pricing literature. Both the station and the mobile algorithms are simple enough to be implemented in current standard equipment. Based on tools from evolutionary games, potential games and replicator dynamics, we analytically show the convergence of the algorithm to solutions that are efficient and fair in terms of throughput. Moreover, we show that after convergence, each user is connected to a single network cell which avoids costly repeated vertical handovers. Several simple heuristics based on this algorithm are proposed to achieve fast convergence. Indeed, for implementation purposes, the number of iterations should remain in the order of a few tens. We also compare, for different loads, the quality of their solutions.