Finding All Nash Equilibria of a Finite Game Using Polynomial Algebra

The set of Nash equilibria of a finite game is the set of nonnegative solutions to a system of polynomial equations. In this survey article we describe how to construct certain special games and explain how to find all the complex roots of the corresponding polynomial systems, including all the Nash equilibria. We then explain how to find all the complex roots of the polynomial systems for arbitrary generic games, by polyhedral homotopy continuation starting from the solutions to the specially constructed games. We describe the use of Groebner bases to solve these polynomial systems and to learn geometric information about how the solution set varies with the payoff functions. Finally, we review the use of the Gambit software package to find all Nash equilibria of a finite game.Comment: Invited contribution to Journal of Economic Theory; includes color figure

Simulation and physical model based gamma-ray burst afterglow analysis

Advances in our numerical and theoretical understanding of gamma-ray burst afterglow processes allow us to construct models capable of dealing with complex relativistic jet dynamics and non-thermal emission, that can be compared directly to data from instruments such as Swift. Because afterglow blast waves and power law spectra are intrinsically scale-invariant under changes of explosion energy and medium density, templates can be generated from large-scale hydrodynamics simulations. This allows for iterative template-based model fitting using the physical model parameters (quantifying the properties of the burster, emission and observer) directly as fit variables. Here I review how such an approach to afterglow analysis works in practice, paying special attention to the underlying model assumptions, possibilities, caveats and limitations of this type of analysis. Because some model parameters can be degenerate in certain regions of parameter space, or unconstrained if data in a limited number of a bands is available, a Bayesian approach is a natural fit. The main features of the standard afterglow model are reviewed in detail.Comment: Invited contribution to Journal of High Energy Astrophysics special issue "Swift: 10 years of discovery". Replaced with expanded version matching JHEAP publicatio

Lie group analysis for multi-scale plasma dynamics

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov-Bogoliubov-Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory demonstrate the potentialities of the suggested approach. Several examples of invariant solutions for the system of the Vlasov-Maxwell equations for the two-component (electron-ion) plasma are presented.Comment: Latex, 15 pages, 7 figure. This is an enlarged contribution to Journal of Nonlinear Mathematical Physics v.18, Suppl. 1 (2011) p.163-175 with modest stylistic corrections introduced mainly in the third Sectio

Substitution and Complementarity in the Creation and Communication of Australian University Research

The generation of research is one of the major functions of the University sector. In most disciplines, journal articles continue to be the main outlet for the communication of research findings. However, in Australia, government induced distortions have rewarded refereed conference papers an equal status to refereed journal papers. The aim of this paper is to explore the association between research published in journals and research published in conference proceedings. We use a panel dataset of the research output of 36 Australian universities, for the period 1995-2004. Cobb-Douglas research production functions are estimated, as well as a system of research production functions that allows for simultaneity. The results indicate that journals and conferences are contemporaneous substitutes – an expansion in conference publications displaces journal publications. There is also a “DEST effect”. On average, conference papers are not converted into subsequent journal papers. The DEST effect is found also through analysis of the publication histories of 152 business and law academics. Post-graduate enrollments are shown to contribute only to conferences and have no effect on journal publications. Research income has a positive effect on both conferences and journal publications.Journals, conferences, DEST effect, research production functions, Australian

spatiotemporal filtering and motion illusions

We are perplexed by Clarke et al.'s (2013) criticisms on our recent contribution to Journal of Vision (Pooresmaeili, Cicchini, Morrone, & Burr, 2012). Our group has long championed the idea that perceptual processing of information can be anchored in a dynamic coordinate system that need not correspond to the instantaneous retinal representation. Our recent evidence shows that temporal duration (Burr, Tozzi, & Morrone, 2007; Morrone, Cicchini, & Burr, 2010), orientation (Zimmermann, Morrone, Fink, & Burr, 2013), motion (Melcher & Morrone, 2003; Turi & Burr, 2012) and saccadic error-correction (Zimmermann, Burr, & Morrone, 2011) are all processed to some extent in spatiotopic coordinates. Imaging studies reinforce these studies (d'Avossa et al., 2007; Crespi et al., 2011). Much earlier, we showed that the processing of smoothly moving objects was not anchored in instantaneous, retinotopic coordinates, but in the reference frame given by the trajectory of motion. There is an effective interpolation along the trajectory, so temporal offsets in spatially collinear stimuli causes them to appear spatially offset, corresponding to the physical reality of stimuli moving over large regions of space, behind occluders (Burr, 1979; Burr & Ross, 1979). Our explanation for this surprising effect was that it could be a direct consequence of the spatiotemporal orientation of the impulsive response of motion detectors, providing the spatiotemporal reference frame needed to account for the interactions between time and space (Burr & Ross, 1986; Burr, Ross, & Morrone, 1986; Burr & Ross, 2004; Nishida, 2004). Recently, we have applied the concept of spatiotemporal oriented receptive fields to account for ''predictive remapping,'' the ''nonretinotopic'' effects that occur on each saccadic eye-movement (Burr & Morrone, 2010; Burr & Morrone, 2012; Cicchini, Binda, Burr, & Morrone, 2012). We were most impressed by the compelling demonstrations of Herzog's group, clearly showing that the reference frame of processing is not the instantaneous retinal position, but is flexible, depending not only on real physical motion, but on an illusory apparent motion where the stimuli do not actually move (Boi, Ogmen, Krummenacher, Otto, & Herzog, 2009). This seemed to us important, worthy of quantitative measurement and modeling, particularly to see whether these new effects may fall within the framework that so successfully explained previous demonstrations, such as spatiotemporal interpolation. It is reassuring that Clarke et al. (2013) confirm our results, albeit with some variability between subjects. But more importantly add a very nice result in showing that our simplified version of the ''litmus test'' can be enhanced by attending to the motion. This is an excellent point that we overlooked. The strength of this type of motion is well known to depend on attention (Cavanagh, 1992), and it is indeed interesting that the strength of motion-induced effects depends not only on the physical conditions, but on internal states such as attention. Perhaps attention may also provide the flexibility in choosing the most appropriate scale for analysis, which in this case would be lower, given that attention is diverted to the periphery. This would add strength to our model, and an idea worth following up