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Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. Theoretical approaches to the problem have hit some limits in the past decades and analytical solutions are known for only a few simple settings. Computational approaches to the problem through the use of LPs have their own set of limitations. Building on the success of deep learning, a new approach was recently proposed by Duetting et al. (2019) in which the auction is modeled by a feed-forward neural network and the design problem is framed as a learning problem. The neural architectures used in that work are general purpose and do not take advantage of any of the symmetries the problem could present, such as permutation equivariance. In this work, we consider auction design problems that have permutation-equivariant symmetry and construct a neural architecture that is capable of perfectly recovering the permutation- equivariant optimal mechanism, which we show is not possible with the previous architecture. We demonstrate that permutation-equivariant architectures are not only capable of recovering previous results, they also have better generalization properties. 

A bstract Jet production in lead-lead (PbPb) and proton-proton (pp) collisions at a nucleon-nucleon center-of-mass energy of 5.02 TeV is studied with the CMS detector at the LHC, using PbPb and pp data samples corresponding to integrated luminosities of 404 μ b − 1 and 27.4 pb − 1 , respectively. Jets with different areas are reconstructed using the anti- k T algorithm by varying the distance parameter R . The measurements are performed using jets with transverse momenta ( p T ) greater than 200 GeV and in a pseudorapidity range of |η| < 2. To reveal the medium modification of the jet spectra in PbPb collisions, the properly normalized ratio of spectra from PbPb and pp data is used to extract jet nuclear modification factors as functions of the PbPb collision centrality, p T and, for the first time, as a function of R up to 1.0. For the most central collisions, a strong suppression is observed for high- p T jets reconstructed with all distance parameters, implying that a significant amount of jet energy is scattered to large angles. The dependence of jet suppression on R is expected to be sensitive to both the jet energy loss mechanism and the medium response, and so the data are compared to several modern event generators and analytic calculations. The models considered do not fully reproduce the data. 

A bstract We present the first study of charged-hadron production associated with jets originating from b quarks in proton-proton collisions at a center-of-mass energy of 5.02 TeV. The data sample used in this study was collected with the CMS detector at the CERN LHC and corresponds to an integrated luminosity of 27.4 pb − 1 . To characterize the jet substructure, the differential jet shapes, defined as the normalized transverse momentum distribution of charged hadrons as a function of angular distance from the jet axis, are measured for b jets. In addition to the jet shapes, the per-jet yields of charged particles associated with b jets are also quantified, again as a function of the angular distance with respect to the jet axis. Extracted jet shape and particle yield distributions for b jets are compared with results for inclusive jets, as well as with the predictions from the pythia and herwig++ event generators.