18 research outputs found
Payoff Performance of Fictitious Play
We investigate how well continuous-time fictitious play in two-player games
performs in terms of average payoff, particularly compared to Nash equilibrium
payoff. We show that in many games, fictitious play outperforms Nash
equilibrium on average or even at all times, and moreover that any game is
linearly equivalent to one in which this is the case. Conversely, we provide
conditions under which Nash equilibrium payoff dominates fictitious play
payoff. A key step in our analysis is to show that fictitious play dynamics
asymptotically converges the set of coarse correlated equilibria (a fact which
is implicit in the literature).Comment: 16 pages, 4 figure
Topics arising from fictitious play dynamics
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
Increasing the Action Gap: New Operators for Reinforcement Learning
This paper introduces new optimality-preserving operators on Q-functions. We
first describe an operator for tabular representations, the consistent Bellman
operator, which incorporates a notion of local policy consistency. We show that
this local consistency leads to an increase in the action gap at each state;
increasing this gap, we argue, mitigates the undesirable effects of
approximation and estimation errors on the induced greedy policies. This
operator can also be applied to discretized continuous space and time problems,
and we provide empirical results evidencing superior performance in this
context. Extending the idea of a locally consistent operator, we then derive
sufficient conditions for an operator to preserve optimality, leading to a
family of operators which includes our consistent Bellman operator. As
corollaries we provide a proof of optimality for Baird's advantage learning
algorithm and derive other gap-increasing operators with interesting
properties. We conclude with an empirical study on 60 Atari 2600 games
illustrating the strong potential of these new operators