On the Convergence of Learning Algorithms in Bayesian Auction Games

Equilibrium problems in Bayesian auction games can be described as systems of differential equations. Depending on the model assumptions, these equations might be such that we do not have a rigorous mathematical solution theory. The lack of analytical or numerical techniques with guaranteed convergence for the equilibrium problem has plagued the field and limited equilibrium analysis to rather simple auction models such as single-object auctions. Recent advances in equilibrium learning led to algorithms that find equilibrium under a wide variety of model assumptions. We analyze first- and second-price auctions where simple learning algorithms converge to an equilibrium. The equilibrium problem in auctions is equivalent to solving an infinite-dimensional variational inequality (VI). Monotonicity and the Minty condition are the central sufficient conditions for learning algorithms to converge to an equilibrium in such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity holds for the respective VIs. The second-price auction's equilibrium is a Minty-type solution, but the first-price auction is not. However, the Bayes--Nash equilibrium is the unique solution to the VI within the class of uniformly increasing bid functions, which ensures that gradient-based algorithms attain the {equilibrium} in case of convergence, as also observed in numerical experiments

Retinal nerve fibre layer thickness profile in normal eyes using third-generation optical coherence tomography

Aims To establish four normal retinal nerve fibre layer (RNFL) thickness radial profiles based on third-generation optical coherence tomography (OCT) and to compare them with previously reported histologic measurements.Methods A total of 20 normal eyes were studied. A circular scan was adjusted to the size of the optic disc and three scans were performed with this radius and every 200 mu m thereafter, up to a distance of 1400 mu m. Four different radial sections (superotemporal, superonasal, inferonasal, and inferotemporal) were studied to establish RNFL thickness OCT profiles. Additionally, two radial scans orientated at 45 and 1351 crossing the optic disc centre were performed in six of 20 eyes, and RNFL thickness was measured at disc margin.Results Quadrant location and distance from disc margin interaction in RNFL thickness was statistically significant (P < 0.001). the RNFL thickness decreased (P < 0.001) as the distance from the disc margin increased for all sections. the measurements automatically generated by the OCT built-in software were thinner (P < 0.001) than histologic ones close to the disc margin.Conclusions Four normal OCT RNFL profiles were established and compared with histological data obtained from the same area. RNFL measurements assessed by OCT 3 were significantly thinner close to the optic disc margin.Hosp Olhos Araraquara, Glaucoma Sect, BR-14802530 Araraquara, SP, BrazilHosp Olhos Araraquara, Retina Diagnost & Treatment Div, BR-14802530 Araraquara, SP, BrazilUniversidade Federal de São Paulo, São Paulo, BrazilUSP, Inst Fis Sao Carlos, Sao Carlos, SP, BrazilUniv So Calif, Doheny Eye Inst, Dept Ophthalmol, Los Angeles, CA USAUniversidade Federal de São Paulo, São Paulo, BrazilWeb of Scienc