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 of hyperbolicity for classes of real transcendental entire functions and circle maps

We prove density of hyperbolicity in spaces of (i) real transcendental entire functions, bounded on the real line, whose singular set is finite and real and (ii) transcendental self-maps of the punctured plane which preserve the circle and whose singular set (apart from zero and infinity) is contained in the circle. In particular, we prove density of hyperbolicity in the famous Arnol'd family of circle maps and its generalizations, and solve a number of other open problems for these functions, including three conjectures by de Melo, Salom\~ao and Vargas. We also prove density of (real) hyperbolicity for certain families as in (i) but without the boundedness condition. Our results apply, in particular, when the functions in question have only finitely many critical points and asymptotic singularities, or when there are no asymptotic values and the degree of critical points is uniformly bounded.Comment: 46 pages, 3 figures. V5: Final peer-reviewed accepted manuscript, to appear in Duke Mathematical Journal. Only minor changes from the previous (significantly revised) version V

Decay of correlations in one-dimensional dynamics

We consider multimodal C^3 interval maps f satisfying a summability condition on the derivatives D_n along the critical orbits which implies the existence of an absolutely continuous f -invariant probability measure mu. If f is non-renormalizable, mu is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence D_n as n tends to infinity . We also give sufficient conditions for mu to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.Comment: To appear in Annales de l'Ecole Normale Superieure, 200

Participation and the City

The City of Ann Arbor is working on reducing its energy consumption and greenhouse gas emissions. Historically, it has focused on municipal operations but is now working on community-wide outreach. Currently the City’s focus is on increasing home weatherization rates. In this practicum I focused on creating an outreach and engagement model for the City. I chose “participation” as a core intervention strategy. Literature and case studies suggest that participation can be an effective intervention method, and one that could be used to address the City’s constraints (e.g. financial, staff limitations). I suggested that the City could use participation to identify barriers preventing weatherization in the community, build relationships with the community, communicate the City’s energy and greenhouse gas emission reduction targets, and engage and involve the community in finding local and applicable solutions. I argue that a participation strategy can be used to improve weatherization rates. However, I also suggested that an engagement strategy making use of participatory methods would benefit from a broad focus on residential energy use, rather than a sole focus on weatherization, and would help achieve the City’s goals of net greenhouse gas reductions.Master of ScienceNatural Resources and EnvironmentUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/90865/1/Van_Strien_practicum_2012.pd