## Section: New Results

### Models of dynamics in Networks

Participant : Octave Boussaton.

We considered a model of interdomain routing proposed by a partner from SOGEA project that is based on the well known BGP protocol. We proved that the model has no pure Nash equilibria, even for 4 nodes. Proof of convergence of the fictious player dynamics for the corresponding network has been established for some specific cases.

We reviewed the different models of dynamism in literature in game theory, in particular models from evolutionary game theory. We presented some ways to use them to realize distributed computations in [56] . Considered models are particular continuous time models, and hence are also covered by the survey [55] . Octave Boussaton, who has now completed his PhD, is currently working on the theory of learning equilibria, in particular in Wardrop routing networks. The proof of the convergence of a specific learning strategy has been established for some networks. The result has been presented in [48] .

We analyzed the behavior of providers on a specific scenario, mainly by considering the simple but not simplistic case of one source and one destination. The analysis of the centralized transit price negotiation problem shows that the only one non cooperative equilibrium is when the lowest cost provider takes all the market. The perspective of the game being repeated makes cooperation possible while maintaining higher prices. Then, we considered the system under a distributed framework. We simulated the behavior of the distributed system under a simple price adjustment strategy and analyzed whether it matches the theoretical results or not. This work is published in [49] .

Moreover, we presented both a game theoretic and an algorithmic approach for solving the routing problem of choosing the best path in a path based protocol such as BGP. We proposed a distributed learning algorithm which is able to learn Nash equilibria in a Wardrop network. This work was published in Parallel Processing Letters [12] and a newer result on the time of convergence was published in [19] . The complexity of the method depends on the total number of paths, which can become unsustainable if the network is too large. We subsequently developed another approach that is able to narrow down the complexity of the method which is now based on the number of nodes in the graph that represents the network. This work has not been published yet, it appears in Octave Boussaton's PhD and will soon lead to a submission.