[D2] - Stochastic perturbations in game theory and applications to networks

P. Mertikopoulos. PhD thesis, University of Athens, 2010.

Abstract

The main issue addressed in this dissertation is what happens if, in addition to the interactions between the players of a game (e.g., the users of a network), the situation is exacerbated by the interference of an assortment of exogenous and unpredictable factors, commonly referred to as “nature”. We find that this random interference differentiates crucially between evolutionary and learning approaches, leading to different (stochastic) versions of the replicator dynamics (perhaps the most widely studied model of game dynamics). Rather surprisingly, in the case of learning, we find that many aspects of rationality remain unperturbed by the effects of noise: regardless of the fluctuations’ magnitude, players are still able to identify suboptimal actions, something which is not always possible in the evolutionary setting. Even more to the point, the strict (Nash) equilibria of the game (an important class of steady states) turns out to be stochastically stable and attracting, again irrespective of the noise level. From the viewpoint of network theory (where stochastic perturbations are ubiquitous), the importance of these results is that they guarantee the robustness of the replicator dynamics against the advent of noise. In this way, if the users of a stochastically fluctuating network adhere to a replicator learning scheme and are patient enough, we show that the flow of traffic in the network converges to an invariant (stationary) distribution which is sharply concentrated in a neighborhood of the network’s equilibrium point.

A version in greek can be found here.

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