Section: New Results

Game theory and networks

We studied the traffic routing problem in networks whose users try to minimize their latencies by employing a distributed learning rule inspired by the replicator dynamics of evolutionary game theory. The stable states of these dynamics coincide with the network's (Wardrop) equilibrium points. Despite this abundance of stable states, we find that (almost) every solution trajectory converges to an equilibrium point at an exponential rate. When network latencies fluctuate unpredictably we show that the time-average of the traffic flows of sufficiently patient users is still concentrated in a neighborhood of evolutionarily stable equilibria and we estimate the corresponding stationary distribution and convergence times [42] .

We also analyzed the distributed power allocation problem in parallel multiple access channels (MAC) by studying an associated non-cooperative game which admits an exact potential function. We show that the parallel MAC game admits a unique equilibrium almost surely. Furthermore, if the network's users employ a distributed learning scheme based on the replicator dynamics, we show that they converge to equilibrium from almost any initial condition, even though users only have local information at their disposal [41] .

Using a large deviations approach we calculate the probability distribution of the mutual information of MIMO channels in the limit of large antenna numbers. We calculate the full distribution, including its tails which strongly deviate from the Gaussian behavior near the mean. This calculation provides us with a tool to obtain outage probabilities analytically at any point in the parameter space, as long as the number of antennas is not too small [20] .