When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,...,N}, chosen randomly according to a binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the random subset is almost surely difference dominated, as N --> oo, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums and differences in the random subset are almost surely distinct, and in particular the difference set is almost surely about twice as large as the sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_{f,g} N^{-1/2}, for some computable constant c_{f,g}, such that one form almost surely dominates below the threshold, and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p.Comment: Version 2.1. 24 pages. Fixed a few typos, updated reference

Evolving dynamic multiple-objective optimization problems with objective replacement

This paper studies the strategies for multi-objective optimization in a dynamic environment. In particular, we focus on problems with objective replacement, where some objectives may be replaced with new objectives during evolution. It is shown that the Pareto-optimal sets before and after the objective replacement share some common members. Based on this observation, we suggest the inheritance strategy. When objective replacement occurs, this strategy selects good chromosomes according to the new objective set from the solutions found before objective replacement, and then continues to optimize them via evolution for the new objective set. The experiment results showed that this strategy can help MOGAs achieve better performance than MOGAs without using the inheritance strategy, where the evolution is restarted when objective replacement occurs. More solutions with better quality are found during the same time span

No-regret Dynamics and Fictitious Play

Potential based no-regret dynamics are shown to be related to fictitious play. Roughly, these are epsilon-best reply dynamics where epsilon is the maximal regret, which vanishes with time. This allows for alternative and sometimes much shorter proofs of known results on convergence of no-regret dynamics to the set of Nash equilibria

Evolving Non-Dominated Parameter Sets for Computational Models from Multiple Experiments

© Peter C. R. Lane, Fernand Gobet. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY-NC 3.0)Creating robust, reproducible and optimal computational models is a key challenge for theorists in many sciences. Psychology and cognitive science face particular challenges as large amounts of data are collected and many models are not amenable to analytical techniques for calculating parameter sets. Particular problems are to locate the full range of acceptable model parameters for a given dataset, and to confirm the consistency of model parameters across different datasets. Resolving these problems will provide a better understanding of the behaviour of computational models, and so support the development of general and robust models. In this article, we address these problems using evolutionary algorithms to develop parameters for computational models against multiple sets of experimental data; in particular, we propose the ‘speciated non-dominated sorting genetic algorithm’ for evolving models in several theories. We discuss the problem of developing a model of categorisation using twenty-nine sets of data and models drawn from four different theories. We find that the evolutionary algorithms generate high quality models, adapted to provide a good fit to all available data.Peer reviewedFinal Published versio

There's more to volatility than volume

It is widely believed that fluctuations in transaction volume, as reflected in the number of transactions and to a lesser extent their size, are the main cause of clustered volatility. Under this view bursts of rapid or slow price diffusion reflect bursts of frequent or less frequent trading, which cause both clustered volatility and heavy tails in price returns. We investigate this hypothesis using tick by tick data from the New York and London Stock Exchanges and show that only a small fraction of volatility fluctuations are explained in this manner. Clustered volatility is still very strong even if price changes are recorded on intervals in which the total transaction volume or number of transactions is held constant. In addition the distribution of price returns conditioned on volume or transaction frequency being held constant is similar to that in real time, making it clear that neither of these are the principal cause of heavy tails in price returns. We analyze recent results of Ane and Geman (2000) and Gabaix et al. (2003), and discuss the reasons why their conclusions differ from ours. Based on a cross-sectional analysis we show that the long-memory of volatility is dominated by factors other than transaction frequency or total trading volume.Comment: 25 pages, 9 figure