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

Density and molar volumes of liquid alloys of iron with C, B, Si, P, Cu, Al, Ni, Cr and Mn

The paper discusses density and molar volume of liquid alloys of iron with C, B, Si, P, Cu, Al, Ni, Cr and Mn. Liquid alloys of iron with other metals are considered as substitutional solutions using the hard-sphere model. Deviations of molar volumes of Fe-Cu and Fe-Al alloys from the ideal solutions were analysed using relationship between excess volume and entropy. Alloys of iron with Ni, Cr and Mn are close to the ideal solutions. Calculated volumes are in agreement with experimental data. Alloys of Fe with C, B, Si and P are considered as interstitial solutions using P. Gaskell’s model developed for amorphous alloys of transition metals with metalloids. Calculated partial molar volumes of C, B, Si and P in dilute binary solutions at 1823 K are (cm3/mol): 1.59, 3.37, 7.06 and 6.26 respectively. The model correctly describes the liquid alloys of iron with metalloids as interstitial solutions

CINEMA NO PROCESSO EDUCATIVO

O presente artigo visa relatar a vivencia pedaggica, utilizando as Tecnologias da Informao e Comunicao TICs, em sala de aula, com alunos dos cursos tcnicos integrados em Sade e Segurana no Trabalho e Tcnico em Qumica da Universidade Tecnolgica Federal do Paran UTFPR, campus Medianeira, na disciplina de Geografia II. Assim, apresenta-se a experincia da utilizao de filmes para o auxlio na aprendizagem do contedo escolar. Estar-se- apresentando uma fundamentao com base em bibliografias, que abaliza a utilizao do filme no processo educativo e demonstrao da metodologia utilizada na realizao da atividade

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