Section: New Results

Game Theory and Applications

As far as results in pure game theory are concerned, we studied in [12] a general framework of systems wherein there exists a Pareto optimal allocation that is Pareto superior to an inefficient Nash equilibrium and defined a ‘Nash proportionately fair' Pareto optima. In this context, we provided conditions for the existence of a Pareto-optimal allocation that is, truly or most closely, proportional to a Nash equilibrium – an approach with applications in non-cooperative flow-control problems in communication networks.

In a learning context, we also explored what happens beyond the standard first-order framework of continuous time game dynamics and introduced in [42] a class of higher order game dynamics, extending all first order imitative dynamics, and, in particular, the replicator dynamics to higher orders. In stark contrast to the first order case, we showed that weakly dominated strategies become eliminated in all $n$-th order payoff-monotonic dynamics for all $n>1$ and strictly dominated strategies become extinct in $n$-th order dynamics $n$ orders as fast as in first order. Finally, we also established a higher order analogue of the folk theorem of evolutionary game theory which shows that higher order accelerate the rate of convergence to equilibria in games.

In terms of applications, we also examined the distribution of traffic in networks whose users try to minimise their delays by adhering to a simple learning scheme inspired by the replicator dynamics of evolutionary game theory. A major challenge occurs in this context when the users' delays fluctuate unpredictably due to random external factors, but we showed that if users are not too greedy in their learning scheme, then the long-term averages of the users' traffic flows converge to the vicinity of an equilibrium [43] .