Section: New Results

Advances in Methodological Tools

Participants : Eitan Altman, Konstantin Avrachenkov, Ilaria Brunetti.

In [19] K. Avrachenkov in collaboration with O. Habachi (UAPV) and A. Piunovskiy and Y. Zhang (both from Univ. of Liverpool, UK) investigate infinite horizon deterministic optimal control problems with both gradual and impulsive controls, where any finitely many impulses are allowed simultaneously. Both discounted and long run time average criteria are considered. They establish very general and at the same time natural conditions, under which the dynamic programming approach results in an optimal feedback policy. The established theoretical results are applied to the Internet congestion control, and by solving analytically and nontrivially the underlying optimal control problems, they obtain a simple threshold-based active queue management scheme, which takes into account the main parameters of the transmission control protocols, and improves the fairness among the connections in a given network.

Standard Evolutionary Game framework is a useful tool to study large interacting systems and to understand the strategic behavior of individuals in such complex systems. Adding an individual state to model a local feature of each player in this context, allows one to study a wider range of problems in various application areas as networking, biology, etc. In [47] , Ilaria Brunetti and Eitan Altman in collaboration with Yezekael Hayel (UAPV) introduce such an extension of evolutionary game framework and particularly, focus on the dynamical aspects of this system. Precisely, the authors study the coupled dynamics of the strategies and the individual states inside a population of interacting individuals. They consider here a two-strategies evolutionary game. They first obtain a system of combined dynamics and they show that the rest-points of this system are equilibria of the evolutionary game with individual state. Second, by assuming two different time scales between states and strategy dynamics, one can compute explicitly the equilibria. Then, by transforming the evolutionary game with individual states into a standard evolutionary game, the authors obtains an equilibrium which is equivalent, in terms of occupation measure, to the previous one. Finally, they show a generalization of the model. All the results are illustrated with numerical results.

Motivated by uncertainty in the value of the interest rate, in [62] K. Avrachenkov in collaboration with A. Varava (KTH, Sweden) study discounted zero-sum stochastic games with and arbitrary discount factor. Their general goal is to obtain a power series expansion of the value of the game with respect to the discount factor around its nominal value. They consider a specific but important class of stochastic games – completely mixed stochastic games. As an illustrative example they take a tax evasion model.