Annotation of the results obtained in 2023
1. Published an article : T.V. Bogachev, A.V. Kolesnikov, “On the monopolist problem and its dual”, Matem. Notes, 114:2 (2023),181–194; Math. Notes, 114:2 (2023), 147–158. https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=13933&option_lang=rus . The main result of the work is the proof of the duality theorem in the monopolist problem under broad assumptions about the function φ (convex, lower semicontinuous, finite on the unit cube and equal to infinity outside). The proof is relatively simple and uses a suitable minimax principle, namely the Rockafellar-Fenchel duality theorem. 2. A survey article Kolesnikov A.V., Auctions and mass transportation, https://arxiv.org/pdf/2312.08077.pdf Talks: 1. Third conference of Mathematical Centers in Russia, Maikop, October 2023 г., "Auction theory and transportation problem", https://mc-conf.adygnet.ru/upload/ter_ver.pdf 2. Vienna Seminar in Mathematical Finance and Probability, Vienna, November 2023, "Transportation problem and auction theory", https://fam.tuwien.ac.at/events/vs-mfp/

 

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Annotation of the results obtained in 2022
All the results declared in the project were obtained and published in the following preprint: Alexander V. Kolesnikov, Fedor Sandomirskiy, Aleh Tsyvinski, Alexander P. Zimin, Beckmann's approach to multi-item multi-bidder auctions. https://arxiv.org/abs/2203.06837 1. The reduction of the problem with n bidders to the problem with 1 bidder based on the application of Border’s theorem The reduction of the n-bidder problem to the 1-bidder problem with additional constraints is obtained in Appendix 1. The novel fact here is that we get an equivalent formulation in terms of maximization of a linear functional depending on utility function, where the constraints depend on the number of bidders. The constraints are given in terms of distribution of the first order partial derivatives. These distributions should satisfy a condition of second order stochastic dominance. On the proof we use a result of S. Hart, P. J. Reny (an earlier result is known as Border’s theorem). 2. Construction of numerical examples and simulations, visualization. Computation of asymptotic with number of bidders tending to infinity. In the paper we made numerical computation of solution of the problem with 2 items, 2 bidders and uniform distribution of types on the unit square. A numerical algorithm is described and the convergence of approximations is theoretically proved. There results are new. The algorithm is based of the approach from the work I. Ekeland and S. Moreno-Bromberg. An algorithm for computing solutions of variational problems with global convexity constraints. Numerische Mathematik, 115(1):45–69, 2010/ It was adapted and extended for our purposes. In particular, the approximation of the additional second order stochastic domination constraints in the multi-bidder problem was based a generalization of the classical Strassen theorem about existence of a martingale with given marginals. The solution of the dual problem is computed. Visualization: see Figures 1-2, asymptotic: Figure 3. 3 First-order approach testing (the negative result is expected). The “first-order approach” was tested, but gives no results, that is why this was not described in the publication. 4. Duality theorem for the auction problem in terms of naturally appearing dual objects (vector fields, couples of conjugated convex functions), economical interpretation of the result. We prove a duality theorem, which is a central theoretical result of our work. Duality results include theorem 1 and theorem 2. Theorem 1 demonstrates the connection of the auction problem with the classical Beckmann’s continuous transportation problem (M. Beckmann. A continuous model of transportation. Econometrica: Journal of the Econometric Society, pages 643–660, 1952). The Beckmann’s problem is formulated in terms of «commodity flow», generated by a vector filed c. Thus, the dual problem can be formulated in terms of such a field. An economical interpretation is given (see page 41). Theorem 2 improves theorem 1 establishing existence of a minimizer. The proof is based on the minimax principle and technically involved, because the classical minimax principle works under assumption of compactness of at least one of two space, but this is not our case. The conjugated one-dimensional convex functions appearing in the dual problem turn out to be related to an auxiliary monopolist problem (see, for instance, J.-C. Rochet and P. Chone. Ironing, sweeping, and multidimensional screening. Econometrica, pages 783–826, 1998), where they can be interpreted as production costs. 5. New pointwise and integral a priori estimates to solution of the auction problem With the help of duality theorem some explicit estimates for the auctioneer’s revenue are obtained (see D4). A new important a priori estimate with clear economical interpretation is obtained in proposition 3. 6. Interpretation of the results from the transportation viewpoint (connections to Daskslakis-Deckelbaum-Tzamos work and the weak transportation problem). It our work we investigate the relation to the optimal transportation problem in the case of 1 bidder. The new result here is the explicit solution to the dual problem for the 2-item problem with uniform distribution of types on the unit square (this problem is equivalent to a weak transportation problem according to Daskslakis-Deckelbaum-Tzamos). See also section D4, where one-dimensional transportation problem is used for the estimation of the auctioneer’s revenue.

 

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