week 3 1

COS 444 Internet Auctions:

Theory and Practice

Spring 2010

Ken Steiglitz ken@cs.princeton.edu

week 3 2

Theory

• Order statistics

pdf of k-th highest

cdf of k-th highest

• Application to revenue calculation

week 3 3

pdf of k-th largest

1))(1( kxF

knxF )(

n choices for which drawn

1

1

k

n

1

1

k

nn f(x) dx 1))(1( kxF knxF )(

ways

week 3 4

knkk xFxFxf

k

nnxg

)())(1)((1

1)( 1

11 )()()( nxFxfnxg

nxFxG )()(1

22 )())(1)(()1()( nxFxFxfnnxg

nn xFnxnFxG )()1()()( 12

week 3 5

German tank problem

Nk

kdxxkxNdxxxgNEM k

1)([max]

1

0

11

0 1

Suppose tank numbers are sequential real numbers uniformon [0,N]. We capture k tanks. The expected max of k samples is

So a reasonable estimate is

Mk

kN

1 (when values are

integers, subtract 1)

week 3 6

Theory

• Order statistics

pdf of k-th highest

cdf of k-th highest

• Simple equilibrium calculation

week 3 7

bbbv

bvn 1

1

1

)/()(

}wins1{prob)(]surplus[E

First price, n biddersSuppose everyone else bids θvi

Differentiate wrt b, set to 0 b = (1-1/n)v1

(checks )b

so θ = (1-1/n) works as equilibrium

Quick and dirty equilibrium calculation for FP with iid uniform values

week 3 8

Theory

• Order statistics

pdf of k-th highest

cdf of k-th highest

• Simple equilibrium calculation

• Revenue equivalence w. uniform v’s

week 3 9

Revenue equivalence of FP & SP, uniform

• SP:

E[price paid] = E[Y2] = (n-1)/(n+1)

E[value of winner] = E[Y1] = n/(n+1)E[surplus of winner] = difference = 1/(n+1)

• FP:Equil. Bid is v*(n-1)/n

E[surplus of winner] = E[vmax/n] = 1/(n+1)

week 3 10

New directions: Sociology

M. Shohat and J. Musch “Online auctions as a research tool: A field experiment on ethnic discrimination” Swiss Journal of Psychology 62 (2), 2003, 139-145

week 3 11

Shohat & Musch ’03 typical field experiment

methodology

• Parallel items sold (30 DVDs), once with each experimental variable (mehmet.orgum vs. michael.ottersbach); and all other variables controlled as equal or randomized.

Eg., Mehmet auctioned one randomly chosen half (A) the first week, and the other half (B) the second week; Michael auctioned B the first week and A the second.

week 3 12

Shohat & Musch ’03 typical field experiment

methodology

• Other variables controlled as carefully as possible (for example, DVDs not shipped first week, so no feedback appeared for either “seller”)

• Statistical tests of results (usually probability of observing results assuming null hypothesis > 95%)

week 3 13

Shohat & Musch ’03 typical field experiment

methodology

Discussion, interpretation• Alternative explanations (not high enough

financial risk; not social setting; “Mehmet” not recognized as Turk; presentation might have seemed “German” (!); buyers may turn to Turkish sellers only after not finding it from a German seller, etc.)

• Other work

week 3 14

Shohat & Musch ’03 typical field experiment methodology

Privacy, ethics: Buyers not informed, no consent!

• Data hidden after experiment (“make profile private”, anonymity guaranteed)

• Experiments kept within rules (no surveys, eg)

• Deals kept scrupulously honest

week 3 15

Field experiments: Testing theory

• LR 99: tests FP-Dutch and SP-English revenue equivalence online (pre-eBay)

• Katkar-LR 00: tests public vs. secret reserve on eBay

• LR 00: tests effects of reserves in FP online (pre-eBay)

week 3 16

Lucking-Reiley 99:Magic on the internet(his PhD dissertation)

• Pre-eBay

• Tests revenue equivalence:

FP=Dutch and

SP=English

week 3 17

LR 99 (con’t)

• Newsgroup: rec.games.deckmaster, now an impossible experiment.

• 1995: 6000 messages/week, highest volume on the internet. 90% Magic cards.

• Variety of auctions observed, all but Vickrey• Sold matched pairs, first with FP first, then

Dutch; then with Dutch first, then FP. Similarly for English-SP.

• Experiments were profitable…$1600 purchase sold for $2000.

week 3 18

LR 99 (con’t)

Results:• Revenue in Dutch > FPDutch > FP by 30% (opposite to lab experiments) Explain! (while strategically equilvalent)• Revenue in English English ≈ SP≈ SP (while weakly strategically equivalent)• Revenue equivalence between pairs left open