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COS 444 COS 444 Internet Auctions:Internet Auctions:Theory and PracticeTheory and Practice

Spring 2008

Ken Steiglitz ken@cs.princeton.edu

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181 Landmark paper: constructs the Landmark paper: constructs the

benchmark theory for benchmark theory for optimaloptimal IPV IPV auctionsauctions with with reservesreserves

Elegant and deceptively simple-Elegant and deceptively simple-looking in retrospectlooking in retrospect

Paradoxically, gets more powerful Paradoxically, gets more powerful results more easily by generalizingresults more easily by generalizing

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181 The Class of auctions:The Class of auctions:

One seller, one indivisible objectOne seller, one indivisible object ReserveReserve b b0 0 (open reserve)(open reserve) N bidders, with valuations N bidders, with valuations vvi i i=1,…,n i=1,…,n Values Values iidiid according to according to cdf Fcdf F, which is, which is strictly increasing, differentiable, with support [0,1]strictly increasing, differentiable, with support [0,1] There is a symmetric equilibrium bidding function There is a symmetric equilibrium bidding function

b(v) b(v) (which we know must be increasing )(which we know must be increasing ) Highest acceptable bid winsHighest acceptable bid wins Rules are anonymousRules are anonymous

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

bid as if value = bid as if value = z z ::

Revenue = Revenue = vv11 F(z) F(z) n-1n-1 P(z)P(z)

General… for the entire class of auctions.General… for the entire class of auctions.

Then, differentiate wrt Then, differentiate wrt zz, set to zero at , set to zero at z=vz=v, impose boundary coundition at v, impose boundary coundition at v* * ……

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

… … leads to revenue at equilibriumleads to revenue at equilibrium

= = “marginal revenue”“marginal revenue” = = “virtual “virtual valuation”valuation”

1

*

)()(v

nrs vdFvMRR

)(

)(1)(

vf

vFvvMR

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

Result: All auctions in Riley & Result: All auctions in Riley & Samuelson’s class are revenue Samuelson’s class are revenue equivalent.equivalent.

Furthermore, expected revenue Furthermore, expected revenue depends only on the entry value depends only on the entry value vv** , which determines the set of , which determines the set of active biddersactive bidders

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Reserves: Test of benchmark Reserves: Test of benchmark theorytheory

““Field Experiments on the Effects of Field Experiments on the Effects of Reserve Prices in Auctions: More Reserve Prices in Auctions: More Magic on the Internet” Lucking-Magic on the Internet” Lucking-Reiley, 2000Reiley, 2000

Pre-eBay, first-price sealed-bid Pre-eBay, first-price sealed-bid auctionsauctions

with control over open reservewith control over open reserve

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Reserves: Test of benchmark Reserves: Test of benchmark theorytheory

Theory (more to come) predicts that Theory (more to come) predicts that higher reserves: higher reserves:

reduces # of bidders reduces # of bidders OKOK decreases prob. of sale decreases prob. of sale OKOK increases price conditional on sale increases price conditional on sale OKOK increases total revenue increases total revenue NO!NO! Bidders respond strategically to Bidders respond strategically to

increased reserve increased reserve OKOK

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Theory: Riley & Samuelson Theory: Riley & Samuelson 8181

111 )()()()()(*

nv

v

nn vFvbdxxFvvFvP

quick equilibrium:

which gives us immediately:

1

1

)(

)()( *

n

v

v

n

vF

dxxFvvb

Example…