Descending Price Auctions with Bounded Number of Price levels and Batched Prophet Inequality

34 Pages Posted: 28 Mar 2022

See all articles by Saeed Alaei

Saeed Alaei

affiliation not provided to SSRN

Ali Makhdoumi

Fuqua School of Business; Massachusetts Institute of Technology (MIT)

Azarakhsh Malekian

University of Toronto - Rotman School of Management; Massachusetts Institute of Technology (MIT) - Electrical Engineering and Computer Science

Rad Niazadeh

University of Chicago - Booth School of Business

Date Written: March 2, 2022

Abstract

We consider descending price auctions for selling m units of a good to unit demand i.i.d. buyers where there is an exogenous bound of k on the number of price levels the auction clock can take. The auctioneer's problem is to choose price levels p_1 > p_2 >... > p_k for the auction clock such that auction expected revenue is maximized. The prices levels are announced prior to the auction. We reduce this problem to a new variant of prophet inequality, which we call "batched prophet inequality", where a decision-maker chooses k (decreasing) thresholds and then sequentially collects rewards (up to m) that are above the thresholds with ties broken uniformly at random. For the special case of m=1 (i.e., selling a single item), we show that the resulting descending auction with k price levels achieves 1- 1/e^k of the unrestricted (without the bound of k) optimal revenue. That means a descending auction with just 4 price levels can achieve more than 98% of the optimal revenue. We then extend our results for m>1 and provide a closed-form bound on the competitive ratio of our auction as a function of the number of units m and the number of price levels k.

Keywords: Clock auction, First price auction, Bayesian online selection, Optimal auction

Suggested Citation

Alaei, Saeed and Makhdoumi, Ali and Malekian, Azarakhsh and Niazadeh, Rad, Descending Price Auctions with Bounded Number of Price levels and Batched Prophet Inequality (March 2, 2022). Available at SSRN: https://ssrn.com/abstract=4048181 or http://dx.doi.org/10.2139/ssrn.4048181

Saeed Alaei

affiliation not provided to SSRN

Ali Makhdoumi

Fuqua School of Business ( email )

Box 90120
Durham, NC 27708-0120
United States

HOME PAGE: http://https://www.fuqua.duke.edu/faculty/ali-makhdoumi

Massachusetts Institute of Technology (MIT) ( email )

77 Massachusetts Avenue
50 Memorial Drive
Cambridge, MA 02139-4307
United States

Azarakhsh Malekian

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S 3E6 M5S1S4
Canada

Massachusetts Institute of Technology (MIT) - Electrical Engineering and Computer Science ( email )

77 Massachusetts Avenue
Cambridge, MA 02139-4307
United States

Rad Niazadeh (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S Woodlawn Ave
Chicago, IL 60637

HOME PAGE: http://radniazadeh.github.io/

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
72
Abstract Views
377
Rank
482,146
PlumX Metrics