Elsevier

Operations Research Letters

Volume 33, Issue 5, September 2005, Pages 531-534
Operations Research Letters

A note on expected rent in auction theory

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Abstract

The expected rent in a reverse (buyer's) auction is shown to be monotonically nonincreasing in the number of bidders if the distribution of valuations is DRHR (IFR). Conditions under which an increase in the spread of bids results in an increase in the expected rent are established as well.

Introduction

This note deals with the second-price auction model. In a buyer's auction, the bidder with the highest bid is awarded the good at a price that is some function of the submitted bids. The valuations of the bidders are i.i.d. random variables and prices are bid in an ascending sequence by individual bidders until only one bidder remains, the winner. The winner pays the last price reached in the sequence, and the average rent of the winner is the expectation of the difference between the largest price reached and the second largest price reached from bidders. On the other hand, in a reverse auction, the lowest bidder is awarded the good at a price that is also some function of the bids submitted. In a reverse English auction, the sellers bid in a descending sequence until only one single bidder remains. Now each bidder possesses a valuation of the good and the rent of the winner is the difference between his valuation and the price. For more details of the auction model, see [2], [7].

This note extends recently published results in [7] on the effect of the variability of valuation on the expected rent in a second-price auction model. The expected rent of the winner in a reverse (buyer's) auction is shown to be monotonically nonincreasing in the number of bidders if the common distribution of the valuations is DRHR (IFR). We also show that an increase in the spread of bids in the sense of the excess wealth order results in an increase in average of winner's rent in the case of a reverse auction.

Section snippets

Main results

For two nonnegative random variables X with distribution F and Y with distribution G, denote F¯=1-F and G¯=1-G their reliability functions, F-1(p) and G-1(p) give the p-quartiles of X and Y, respectively. Let X1,X2,,Xn and Y1,Y2,,Yn be n independent copies of X and of Y, respectively, denote their corresponding order statistics by X1:n,X2:n,,Xn:n and Y1:n,Y2:n,,Yn:n. According to [7], for a reverse auction, the expected rent with n bidders is given by the mean of the first sample spacing,

Acknowledgements

I would like to thank the editor and a referee of Applied Stochastic Models in Business and Industry for their encouraging comments on the earlier version. I am also grateful to the referee of Operation Research Letters for a number of valuable comments which have greatly enhanced the presentation of this note.

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The research is supported by National Natural Science Foundation of China under Grant no.10201010.

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