Analytical solution of k th price auction
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Analytical solution of kth price auction Martin Mihelich1,2 · Yan Shu3 Accepted: 20 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We provide an exact analytical solution of the Nash equilibrium for the kth price auction by using inverse of distribution functions. As applications, we identify the unique symmetric equilibrium where the valuations have polynomial distribution, fat tail distribution and exponential distributions. Keywords Vickrey auctions · kth Price auctions · Inverse distribution functions
1 Introduction In a kth price auction with k or more bidders, the highest bidder wins the bid and pays the kth highest bid as price. The kth price auction has been studied by many researchers in recent years. Readers can refer to Klemperer and Klemperer (1999), Krishna (2010), Maskin (2004), Wilson (1992) for related literature. The Revenue Equivalence Theorem (RET) (see Myerson 1981; Riley and Samuelson 1981) can be used to characterize equilibrium strategies of kth price auction. Monderer and Tennenholtz (2000) proved the uniqueness of the equilibrium strategies in kth price auctions for k = 3 . Under some regularity assumptions, they also provided a characterization equation of the equilibrium bid function (see theorem 2.1 below). Wolfstetter (2001) solved the equilibrium kth price auctions for a uniform distribution. Recently, Nawar and Sen (2018) represented the solution of Monderer and Tennenholtz’s characterization equation as a finite series involving Catalan numbers. With
* Martin Mihelich [email protected] Yan Shu [email protected] 1
Open Pricer, Paris, France
2
École normale supérieure, Paris, France
3
Walnut Algorithms, Paris, France
13
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M. Mihelich, Y. Shu
their representation, they provided a closed form solution of the unique symmetric, increasing equilibrium of a kth price auction for a second degree polynomial distribution. In this paper, we analysis Monderer and Tennenholtz’s characterization equation by using a method involving inverse of distribution functions. We provide a new representation of the equilibrium bid function of kth price auction with this representation. For applications, we extend Nawar and Sen’s results and provide a closed form solution of a kth price auction for polynomial distribution, fat tail distribution and exponential distributions. After recalling the framework of the problem in Sect. 2, we prove our main result in Sect. 3. Then, in Sect. 4, we compare our result with those of Nawar and Sen. Finally in Sect. 5 we provide a closed-form solution of the equilibrium bid function for polynomial distribution, exponential distribution and a class of fat tail distributions.
2 Notations and assumptions In this section we present our assumptions and recall the result on the uniqueness of the equilibrium strategies provided by Monderer and Tennenholtz. Consider a kth price auction with n bidders, where the highest bidder wins, and pays only the kth highest bid. Let k ⩾ 2 and n > k . We make the fo
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