A Counter-Example to Karlin's Strong Conjecture for Fictitious Play

Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by [Brown 1949], and shown to converge by [Robinson 1951]. Samuel Karlin conjectured in 1959 that fictitious play converges at rate $O(1/\sqrt{t})$ with the number of steps $t$. We disprove this conjecture showing that, when the payoff matrix of the row player is the $n \times n$ identity matrix, fictitious play may converge with rate as slow as $\Omega(t^{-1/n})$.Comment: 55th IEEE Symposium on Foundations of Computer Scienc

Thyroid dysfunction and risk of cutaneous malignant melanoma: a bidirectional two-sample Mendelian randomization study

BackgroundEpidemiologic and observational data have found a risk association between thyroid dysfunction and cutaneous malignant melanoma (CMM), however, the cause and direction of these effects are yet unknown. By using a bidirectional two-sample Mendelian randomization (MR) methodology, we hoped to further investigate the causal link between thyroid dysfunction and CMM in this work.MethodsA genome-wide association study (GWAS) of 9,851,867 single nucleotide polymorphisms (SNPs) in a European population was used to develop genetic tools for thyroid dysfunction. Hypothyroidism was linked to 22,687 cases and 440,246 controls. For hyperthyroidism, there were 3545 cases and 459,388 controls. A total of 3751 cases and 372016 controls were included in the genetic data for CMM from UK Biobank (http://www.nealelab.is/uk-biobank) (the Dataset: ieu - b - 4969). Among them, inverse variance weighting (IVW) is the main MR Analysis method for causality assessment. MR-Egger method, MR Pleiotropic residual and outlier test (MR-PRESSO), and simple and weighted median (VM) were used to supplement the IVW method. Sensitivity analyses, mainly Cochran’s Q test, leave-one-out analysis, and MR Egger intercept test were performed to assess the robustness of the outcomes.ResultsThe two-sample MR Analysis results revealed a negative correlation between genetically predicted hypothyroidism and the probability of CMM (OR=0.987, 95%CI =0.075-0.999, p=0.041). The supplemental MR Analysis did not reveal any statistically significant differences, although the direction of the effect sizes for the other approaches was consistent with the IVW effect sizes. The results of the causal analysis were relatively robust, according to a sensitivity analysis. The risk of CMM was unaffected by hyperthyroidism (p>0.05). No correlation between CMM and thyroid dysfunction was seen in the reverse MR analysis.ConclusionAlthough the magnitude of the causal association is weak and further investigation of the mechanism of this putative causal relationship is required, our findings imply that hypothyroidism may be a protective factor for CMM

Mapping our Universe in 3D with MITEoR

Mapping our universe in 3D by imaging the redshifted 21 cm line from neutral hydrogen has the potential to overtake the cosmic microwave background as our most powerful cosmological probe, because it can map a much larger volume of our Universe, shedding new light on the epoch of reionization, inflation, dark matter, dark energy, and neutrino masses. We report on MITEoR, a pathfinder low-frequency radio interferometer whose goal is to test technologies that greatly reduce the cost of such 3D mapping for a given sensitivity. MITEoR accomplishes this by using massive baseline redundancy both to enable automated precision calibration and to cut the correlator cost scaling from N^2 to NlogN, where N is the number of antennas. The success of MITEoR with its 64 dual-polarization elements bodes well for the more ambitious HERA project, which would incorporate many identical or similar technologies using an order of magnitude more antennas, each with dramatically larger collecting area.Comment: To be published in proceedings of 2013 IEEE International Symposium on Phased Array Systems & Technolog

Mapping our universe in 3D with MITEoR

Mapping our universe in 3D by imaging the redshifted 21 cm line from neutral hydrogen has the potential to overtake the cosmic microwave background as our most powerful cosmological probe, because it can map a much larger volume of our Universe, shedding new light on the epoch of reionization, inflation, dark matter, dark energy, and neutrino masses. We report on MITEoR, a pathfinder low-frequency radio interferometer whose goal is to test technologies that greatly reduce the cost of such 3D mapping for a given sensitivity. MITEoR accomplishes this by using massive baseline redundancy both to enable automated precision calibration and to cut the correlator cost scaling from N[superscript 2] to N log N, where N is the number of antennas. The success of MITEoR with its 64 dual-polarization elements bodes well for the more ambitious HERA project, which incorporates many identical or similar technologies using an order of magnitude more antennas, each with dramatically larger collecting area.National Science Foundation (U.S.) (Grant AST-0908848)National Science Foundation (U.S.) (Grant AST-1105835)MIT Kavli Instrumentation FundMassachusetts Institute of Technology. Undergraduate Research Opportunities Progra

A counter-example to Karlin's strong conjecture for fictitious play

Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 23-26).Fictitious play is a natural dynamic for equilibrium play in zero-sum games, proposed by Brown , and shown to converge by Robinson . Samuel Karlin conjectured in 1959 that fictitious play converges at rate O(t- 1/ 2) with respect to the number of steps t. We disprove this conjecture by showing that, when the payoff matrix of the row player is the n x n identity matrix, fictitious play may converge (for some tie-breaking) at rate as slow as [Omega](t- 1/n).by Qinxuan Pan.M. Eng

MITEoR: a scalable interferometer for precision 21 cm cosmology

We report on the MIT Epoch of Reionization (MITEoR) experiment, a pathfinder low-frequency radio interferometer whose goal is to test technologies that improve the calibration precision and reduce the cost of the high-sensitivity 3D mapping required for 21 cm cosmology. MITEoR accomplishes this by using massive baseline redundancy, which enables both automated precision calibration and correlator cost reduction. We demonstrate and quantify the power and robustness of redundancy for scalability and precision. We find that the calibration parameters precisely describe the effect of the instrument upon our measurements, allowing us to form a model that is consistent with χ[superscript 2] per degree of freedom <1.2 for as much as 80 per cent of the observations. We use these results to develop an optimal estimator of calibration parameters using Wiener filtering, and explore the question of how often and how finely in frequency visibilities must be reliably measured to solve for calibration coefficients. The success of MITEoR with its 64 dual-polarization elements bodes well for the more ambitious Hydrogen Epoch of Reionization Array project and other next-generation instruments, which would incorporate many identical or similar technologies