Topics arising from fictitious play dynamics

• Main Author: Ostrovski, Georg
• Published: University of Warwick 2013
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.589897

In this thesis, we present a few different topics arising in the study of the learning dynamics called fictitious play. We investigate the combinatorial properties of this dynamical system describing the strategy sequences of the players, and in particular deduce a combinatorial classification of zero-sum games with three strategies per player. We further obtain results about the limit sets and asymptotic payoff performance of fictitious play as a learning algorithm. In order to study coexistence of regular (periodic and quasi-periodic) and chaotic behaviour in fictitious play and a related continuous, piecewise affne flow on the threesphere, we look at its planar first return maps and investigate several model problems for such maps. We prove a non-recurrence result for non-self maps of regions in the plane, similar to Brouwer’s classical result for planar homeomorphisms. Finally, we consider a family of piecewise affne maps of the square, which is very similar to the first return maps of fictitious play, but simple enough for explicit calculations, and prove several results about its dynamics, particularly its invariant circles and regions.