Nawar, AbdelHameed and Sen, Debapriya (2018): kth price auctions and Catalan numbers. Forthcoming in: Economics Letters

PDF
MPRA_paper_88553.pdf Download (275kB)  Preview 
Abstract
This paper establishes an interesting link between kth price auctions and Catalan numbers by showing that for distributions that have linear density, the bid function at any symmetric, increasing equilibrium of a kth price auction (k is 3 or higher) can be represented as a finite series of k2 terms whose lth term involves the lth Catalan number. Using an integral representation of Catalan numbers, together with some classical combinatorial identities, we derive the closed form of the unique symmetric, increasing equilibrium of a kth price auction for a nonuniform distribution.
Item Type:  MPRA Paper 

Original Title:  kth price auctions and Catalan numbers 
Language:  English 
Keywords:  kth price auction; the revenue equivalence principle; Catalan numbers; Jensen's identity; HagenRothe's identity 
Subjects:  C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games 
Item ID:  88553 
Depositing User:  Debapriya Sen 
Date Deposited:  29 Aug 2018 06:23 
Last Modified:  29 Aug 2018 06:23 
References:  Chu, W. 2010. Elementary proofs for convolution identities of Abel and HagenRothe, Electronic Journal of Combinatorics, 17, #N24 Gardner, M. 1976. Catalan numbers: an integer sequence that materializes in unexpected places, Scientific American, 234(6):120125 Gould, H.W. 1956. Some generalizations of Vandermonde's convolution, American Mathematical Monthly, 63:8491 Gould, H.W., Quaintance, J. 2010. Combinatorial Identities: Table II: Advanced Techniques for Summing Finite Series. (from the seven unpublished manuscripts of H.W. Gould, edited and compiled by J. Quaintance) https://www.math.wvu.edu/~gould/Vol.5.PDF Guo, V.W.J. 2011. Jensen's and related combinatorial identities, Applicable Analysis and Discrete Mathematics, 5:201211 Kagel, J.H., Levin, D. 1993. Independent private value auctions: bidders behavior in first, second and thirdprice auction with varying numbers of bidders, Economic Journal, 103:868879 Krishna, V. 2002. Auction Theory. Academic Press Mathews, T., Schwartz, J.A. 2017. A note on kprice auctions with complete information when mixed strategies are allowed, Economics Letters, 153:68 Monderer, D., Tennenholtz, M. 2000. kprice auctions, Games and Economic Behavior, 31:220244 Myerson, R. 1981. Optimal auction design, Mathematics of Operations Research, 6:5873 Pak, I. 2015. History of Catalan numbers. In: Catalan Numbers by R.P. Stanley, Cambridge University Press, pp. 177190 (Appendix B) Penson, K., Sixdeniers J.M. 2001. Integral representation of Catalan and related numbers, Journal of Integer Sequences, 4:Article 01.2.5 Riley, J., Samuelson, W. 1981. Optimal auctions, American Economic Review, 71:381392 Sloane, N.J.A. The Online Encyclopedia of Integer Sequences https://oeis.org Stanley, R.P. 2015. Catalan Numbers. Cambridge University Press Tauman, Y. 2002. A note on kprice auctions with complete information,Games and Economic Behavior, 41: 161164 Vickrey, W. 1961. Counterspeculation, auctions and competitive sealed tenders, Journal of Finance, 16:837 Vickrey, W. 1962. Auctions and bidding games. In: Recent Advances in Game Theory. Princeton Conference Series, 29, Princeton University Press, pp. 1527 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/88553 