Competitive Auctions and Digital Goods

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mnopqrst uvs wtnxysz n{ psyyqo| } o~zxst n{ qrsouq}y
quszp un nop~zstp vn s}v }ou } pqo|ys qusz }or uvs
quszp }ts }}qy}xys qo p~wwy ~oyqzqusr
p~wwy s zs}o uv}u squvst uvs psyyst v}p }u ys}pu
}p z}o quszp }p uvsts }ts nop~zstp nt uv}u uvs
psyyst }o tswtnr~s quszp no rsz}or }u os|yq|qxys
z}t|qo}y npu { w}tuq~y}t qoustspu }ts rq|qu}y quszp
p~v }p rnoyn}r}xys }~rqn ysp }or w}wstqs
znqsp quv ~oyqzqusr p~wwy nop~zst uvs
z}qz~z }zn~oup uv}u nop~zstp }ts qyyqo| un w}
{nt uvs qusz }ts uvs pnys {}unt rsustzqoqo| p}ys wtqsp
vs psyystp |n}y qp un z}qzqs uvsqt unu}y tsso~s
os } un psu wtqsp {nt quszp qo ~oyqzqusr p~wwy
qp un spuqz}us nop~zst ~uqyqu q} z}tsu }o}ypqp }or
uvso psu } sr wtqs s ts{st un uvqp zsuvnr }p
}wstqs znqsp }ts }o s}zwys n{ sr
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{nt }o ~oyqzqusr p~wwy z}tsu quv wst{su
onysr|s n{ nop~zst ~uqyquqsp
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wtqs }u vqv un psyy quszp ¡qsr wtqqo| |sost}yy
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}oonu }vqss uvqp qrs}y r~s un uvs qovstsou qo}~t}
n{ z}tsu }o}ypqp Í{ uvs wtqs qp psu unn vq|v onu
son~|v quszp z} xs pnyrÎ q{ uvs wtqs qp psu unn yn
qop~Ïqsou tsso~s z} xs nyysusr wst qusz
Ð~uqnop }~unz}uq}yy }rÑ~pu wtqsp un z}tsu
norquqnop s pu~r pqo|ys tn~or ps}ysr xqr }~uqnop
Ò~v }~uqnop v}s xsso pu~rqsr {nt quszp }}qy}xys qo
Ó vsts z}qzqqo| uvs tsso~s tsÔ~qtsp
uv}u }yy }}qy}xys
quszp xs pnyr vs }ts spwsq}yy
wt}uq}y vso uvs o~zxst n{ nop~zstp qp y}t|s Ío
w}tuq~y}t Õqts Ö×ØÙ qoutnr~sr } z~yuqqusz }~uqno
uv}u qp ÚÛ Ð ut~uv{~y }~uqno son~t}|sp xqrrstp
un xqr uvsqt ~uqyqu }y~s Ío }o ~out~uv{~y }~uqno
xqrrstp z} xqr pq|oq}ouy xsyn uvsqt ~uqyqu }y~sp
tsr~qo| }~uqno tsso~s
Ío } ut~uv{~y }~uqno t}uqno}y xqrrstp xqr uvsqt
~uqyquqsp Ío }rrquqno s n~yr yqs p~v }o }~uqno un
xs ÜÝ qu z~pu qsyr tsso~s quvqo } nopu}ou
{}unt n{ nwuqz}y sr wtqqo| n xs nzwsuquqs }
ut~uv{~y }~uqno z~pu }t vn z}o quszp }ts pnyr
rswsorqo| no uvs xqr }y~sp ¡nt s}zwys }p s pvn
qo Òsuqno Þ uvs z~yuqqusz Õqts }~uqno qp onu
nzwsuquqs q{ uvs psyyst vnnpsp uvs o~zxst n{ quszp
un psyy xs{nts onqo| uvs xqr }y~sp ß}or onu ut~uv{~y
q{ uvs psyyst vnnpsp uvs o~zxst n{ quszp }{ust onqo|
uvs xqr }y~spà Ðp quv sr wtqqo| psyyqo| unn {s
nt unn z}o quszp z} onu z}qzqs tsso~s v~p
uvs zsuvnr {nt vnnpqo| vn z}o xqrp un p}uqp{ qp }o
qous|t}y w}tu n{ } ut~uv{~y nzwsuquqs }~uqno
n n~t onysr|s }~uqnop v}s osst xsso pu~rqsr
qo } nzwsuquqs {t}zsnt ánt }ts }o sqpuqo|
}~uqnop nzwsuquqs qo uvs psops uv}u s qoutnr~s
vsts Ðp s swynts qo Òsuqnop ×× }or ×â uvqp
nzwsuquqs {t}zsnt qp ~ps{~y qo pu~rqo| }o qor
n{ }~uqno vsts qrsouq}y |nnrp }ts xsqo| pnyr onu Ñ~pu
}~uqnop {nt ~oyqzqusr p~wwy
Ð~uqnop }ts xsnzqo| } wnw~y}t wtqqo| zsv}
oqpz qo sysutnoq nzzsts xnuv {nt v~z}o ~pstp
}or {nt ut}rqo| }|soup ßxnupà Ío z}o }psp uvs
~ps n{ }~uqnop qp nzwyq}usr x uvs {}u uv}u } |nnr
xqrrqo| put}us| {nt } x~st tsÔ~qtsp }o ~orstpu}orqo|
n{ put}us|qsp }or ~uqyquqsp n{ nuvst x~stp t~uv{~y
}~uqno zsv}oqpzp z} xs }uut}uqs qo uvqp nousu
xs}~ps uvs }nqr uvqp nzwyq}uqno
Ío uvqp w}wst s pu~r } y}pp n{ ut~uv{~y }~uqnop
{nt ~oyqzqusr p~wwy s pu~r xnuv }~
uqnop vsts sst qooqo| xqrrst w}p uvs p}zs wtqs
}or }~uqnop vsts uvs wtqsp z} rq st Ío
}rrquqno un }~uqnop s pu~r
}~uqnop uv}u ~ps t}ornzq}uqno un rsqrs vqv xqrp
un yy }or }u v}u wtqs s rssynw usvoqÔ~sp {nt
rspq|o }or }o}ypqp n{ }~uqnop {nt ~oyqzqusr p~wwy
~t }wwtn}v qp tszqoqpsou n{ nzwsuquqs }o}ypqp n{
noyqos }y|ntquvzp Ö× × Ù vsts wst{ntz}os n{ }o no
yqos }y|ntquvz qp |}~|sr qo ustzp n{ wst{ntz}os n{
}o nwuqz}y n yqos }y|ntquvz sts uvs nwuqz}y n
yqos }y|ntquvz qp }o}yn|n~p un uvs nwuqz}y sr wtqqo|
zsv}oqpz
Ðyuvn~|v s rssynw n~t tsp~yup {nt ~oyqzqusr p~w
wy pnzs n{ uvs tsp~yup susor un Ó vsts
uvs o~zxst n{ quszp {nt p}ys qp xn~orsr x~u z}qzqqo|
tsso~s zq|vu onu tsp~yu qo }yy quszp xsqo| pnyr s
rqp~pp xn~orsr p~wwy qo Òsuqno ××
s qs }~uqnop }p }y|ntquvzp {nt rsqrqo| vqv
qow~u xqrp un yy }or }u v}u wtqs Ðp quv }o
}y|ntquvz nos ossrp un }rrtspp uvs qpp~sp n{ nttsuospp
sÏqso }or wst{ntz}os Ío uvs nousu n{ uvqp
w}wst }o }~uqno qp nttsu q{ uvs }~uqno qp ut~uv{~y
}or yyp s}v qooqo| xqr }u nt xsyn uvs xqr }y~s
Ïqso n{ }o }~uqno ts{stp un uvs uqzs ossrsr un
wtnspp xqrp vs }~uqnop qoutnr~sr qo uvqp w}wst }ts
st {}puÎ pntuqo| n{ uvs qow~u xqrp qp uvs znpu swsopqs
nzw~u}uqno}y nwst}uqno s wst{ntz Ðp rqp~ppsr
}xns s zs}p~ts }~uqno wst{ntz}os x qup tsso~s
tsy}uqs un uvs sr wtqqo| tsso~s vqp nzw~ust
pqsos }wwtn}v }yynp ~p onu noy un rspq|o os
ut~uv{~y }~uqnop x~u }ypn un |qs uvsntsuq}y |~}t}oussp
{nt uvsqt wst{ntz}os Ò~v wtn}xys wst{ntz}os
|~}t}oussp }ts os un uvs }ts} n{ }~uqno zsv}oqpz
rspq|o
n pu}us n~t tsp~yup znts {ntz}yy s qoutnr~s
uvs {nyynqo| onu}uqno su rsonus uvs o~zxst n{
xqrrstp quvn~u ynpp n{ |sost}yqu s }pp~zs uv}u uvs
ynspu xqr qp nos }or rsonus uvs vq|vspu xqr x su
uvs unu}y ~uqyqu xs uvs p~z n{ }yy xqrrstp ~uqyquqsp
qp }o nxqn~p ~wwst xn~or no uvs tsso~s uv}u }o
xs nxu}qosr {tnz uvqp psu n{ xqrrstp su xs uvs
nwuqz}y sr wtqqo| tsso~s mys}ty
qp
}o ~wwst xn~or no uvs tsso~s uv}u }o xs nxu}qosr
x }o sr wtqsr p}ys nt }o pqo|yswtqs }~uqno s
}ou tsso~sp n{ n~t ut~uv{~y }~uqnop un xs nzwsuquqs
quv s }pp~zs uv}u qp pz}yy nzw}tsr un
vqp }pp~zwuqno wtssoup } utqq}y ~wwst xn~or no uvs
tsso~sÎ pss Òsuqno â
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s pu}us pnzs n{ n~t tsp~yup qo ustzp n{ uvs unu}y
~uqyqu }or nuvstp qo ustzp n{ uvs nwuqz}y sr
wtqqo| tsso~s s pvn uv}u nzw}tsp {}nt}xy
un Î pwsq}yy
ß yn| à }or }ypn
ß yn| à vqp tsp~yu pvnp uv}u uvs nwuqz}y sr
wtqqo| tsso~s qp quvqo } zqoßyn| yn| à {}unt n{ uvs
tsso~s n{ }o wtqqo| pvszs s ~ps uvqp tsp~yu un
tsy}us }tqn~p xn~orp swtsppsr qo ustzp n{ un uvnps
swtsppsr qo ustzp n{
s qoutnr~s } y}pp n{ ut~uv{~y pqo|yswtqs }~
uqnop }or } y}pp n{ ut~uv{~y z~yuqwtqs }~uqnop Ð
t}ornzqsr }~uqno {tnz uvs tpu y}pp qp nzwsuquqsÝ qu
v}p }o swsusr tsso~s n{ ß à ß yn| à Ð r~}y
wtqs }tq}ou n{ uvqp }~uqno v}p tsso~s uv}u qp ynps un
q{
qp y}t|s son~|v Ð t}ornzqsr z~yuqwtqs
}~uqno {tnz uvs psnor y}pp v}p }o swsusr tsso~s
n{ ß yn| à v~p xnuv n{ uvsps }~uqnop v}s uvs
p}zs ntpu}ps xn~or qo ustzp n{ nsst s pvn
uv}u uvs y}uust }~uqno qp onu nzwsuquqs Íup swsusr
tsso~s qp ß yn| à }or uvqp xn~or qp uq|vuÝ no
stu}qo qow~up uvs swsusr tsso~s qp ß yn| à
vqp wtnqrsp p~wwntu {nt ~pqo| t}uvst uv}o
un rsos nzwsuquqs }~uqnop Ðonuvst tsp~yu wtnqrsp
{~tuvst p~wwntu s pvn uv}u {nt }o ut~uv{~y }~uqno
sso } z~yuqwtqs nos uvs swsusr tsso~s rnsp onu
sssr vqp tsp~yu qp pnzsv}u p~twtqpqo|Ý ~pqo|
pqo|yswtqs }~uqnop rnsp onu v~tu tsso~s x znts uv}u
} nopu}ou {}unt
Ð o}u~t}y Ô~spuqno un }p qp q{ uvsts qp } ut~uv{~y
rsustzqoqpuq }~uqno quv }o ß à tsso~s s pvn
uv}u uvsts qp onosÝ {nt }o p~v }~uqno uvsts qp } psu n{
xqrp uv}u ys}rp un }o ß à tsso~s qs tsso~s uv}u
qp } pz}yy {t}uqno n{ q{ qp y}t|s v~p {nt ntpu}ps
wst{ntz}os t}ornzqsr }~uqnop qsyr xsuust tsso~s
uv}o uvs rsustzqoqpuq nosp
~t uvsntsuq}y }o}ypsp }ts yqzqusr qo uv}u uvsqt
wst{ntz}os zsutqp }ts }~t}us noy ~w un } nopu}ou
{}unt Ðp } tsp~yu uvs }o}ypsp rn onu tss}y vsuvst
nos n{ n~t }~uqnop rnzqo}usp uvs nuvstp nt vqv
}~uqno qp xsuust {nt } o}u~t}y rqputqx~uqno Ðp }
p~wwyszsou un n~t uvsntsuq}y tsp~yup s wst{ntzsr }
o~zxst n{ pqz~y}uqnop un nzw}ts n~t }~uqnop quv
s}v nuvst }or un sr wtqqo| no } }tqsu n{ qow~u
{}zqyqsp ~t pqz~y}uqnop p~||spu uv}u no o}u~t}y
qow~up pnzs n{ n~t }~uqnop }uu}qo tsso~s st ynps un
q{ uvs o~zxst n{ xqrp qp y}t|s son~|v ¡~tuvstznts
n~t }~uqnop }o n~uwst{ntz sr wtqqo| quv z}tsu
}o}ypqp ~oyspp uv}u }o}ypqp qp {}qty }~t}us s }ypn
pvn } rsustzqoqpuq }~uqno uv}u rspwqus uvs ntpu
}ps tsp~yu rnsp st syy no o}u~t}y qow~up
s rssynw } {t}zsnt {nt } uvsntsuq}y }or
swstqzsou}y }o}ypqp n{ tsso~sz}qzqqo| ut~uv{~y
}~uqnop }or qoutnr~s }~uqnop uv}u wst{ntz syy qo
uvqp {t}zsnt s pvn vn }y|ntquvz }o}ypqp
usvoqÔ~sp }o xs ~psr quvqo uvqp {t}zsnt un nxu}qo
tsp~yup uv}u }ts qoustspuqo| }or qo pnzs }psp p~twtqpqo|
~s un uvs pw}s yqzqu s nzqu pnzs z}ustq}y {tnz
uvqp stpqno n{ uvs w}wst vs {~yy w}wst }wws}tp }u

h AkAk >gC @k@l
s nopqrst }~uqnop quv xqrrstp s}v xqrrst v}qo| } ~uqyqu }y~s }or xqrrqo| ! s }ypn
}pp~zs uv}u uvs xqrp }ts ntrstsr pn uv}u ! ! " Ío
}~uqnop vsts uqsp ossr un xs xtnso s }o }pp~zs
}o }txqut}t unu}y ntrst no uvs xqr }y~sp uv}u tspwsup
uvs w}tuq}y ntrst v}u qp s }o }pp~zs uv}u uvs ntrst
|qso x uvs qorqsp qp putqu s }pp~zs uv}u uvsts qp
on nyy~pqno }zno| uvs xqrrstp
# qso } psu n{ xqrp uvs n~unzs n{ }o }~uqno qp
uvs p~xpsu n{ xqrp uv}u }ts p}uqpsr }or } nttspwnorqo|
psu n{ p}ys wtqsp p~v uv}u {nt s}v qooqo| xqr !
uvs }ppnq}usr p}ys wtqs qp }u znpu ! Ð
Ú z}wp psup n{ xqrp un }~uqno n~u
nzsp Ð Ú z}wp psup n{
xqrp un wtnx}xqyqu rqputqx~uqnop no }~uqno n~unzsp
s ~ps $ un rsonus uvs }~uqno Ü {nt } w}tuq~y}t
}~uqno zsv}oqpz }or psu n{ xqrp $ qp uvs p~z n{ }yy
p}ys wtqsp ¡nt t}ornzqsr }~uqnop $ qp } t}ornz
}tq}xys
s p} uv}u }o }~uqno qp ÚÛ q{ xqrrqo| qp
} rnzqo}ou put}us| {nt xqrrst % nts pwsq}yy ysu
} & xs uvs rq stsos xsusso uvs xqrrstp
~uqyqu }y~s
}or uvs wtqs uvs xqrrst w}p q{ uvs qo uvs
}~uqno nt stn q{ uvs ynps vso }o }~uqno qp ut~uv{~y
q{ } xqrrstp wtnu ßnt swsusr wtnu {nt t}ornzqsr
}~uqnopà }p } {~ouqno n{ uvs xqrrstp xqr qp z}qzqsr
}u uvs xqrrstp ~uqyqu }y~s {nt }o sr }y~sp n{
uvs nuvst xqrrstp xqrp t~uv{~y }~uqnop son~t}|s
xqrrqo| }u ~uqyqu }y~s q{ uvs xqr }y~s uv}u z}qzqsp
uvs ßswsusrà wtnu qp ~oqÔ~s vso nopqrstqo|
ut~uv{~y }~uqnop s }pp~zs uv}u !
n so}xys }o}ypqp n{ }~uqno tsso~s s rsos
psst}y wtnwstuqsp n{ }o qow~u psu n{ xqrp Ðp pu}usr
qo uvs qoutnr~uqno qp uvs p~z n{ }yy n{ uvs xqrp
Ðo sÔ~q}ysou rsoquqno n{ qp uvs tsso~s r~s un
uvs nwuqz}y z~yuqwtqs ~out~uv{~y }~uqno uvs nos uv}u
p}uqpsp }yy xqrp }u uvsqt }y~s vs tsso~s {nt nwuqz}y
sr wtqqo| qp ánus uv}u }o }ypn xs qoustwtsusr
}p uvs tsso~s r~s un uvs nwuqz}y pqo|yswtqs ~out~uv{~y
}~uqno % nts rqp~ppqno n{ }or qup tsy}uqno un
uvs nwuqz}y pqo|yswtqs ~out~uv{~y }~uqno qp |qso qo
Òsuqno ' vs nuvst wtnwstu n{ uvs xqrp uv}u s ~ps
qp uvs vq|vspu xqr }y~s s }pp~zs uv}u xqrp }ts
p}ysr p~v uv}u uvs ynspu xqr }y~s qp ×
Ðo}yn|n~p un noyqos }y|ntquvz uvsnt s swtspp
}~uqno wst{ntz}os tsy}uqs un uv}u {nt nwuqz}y ~o
ut~uv{~y }~uqnop }p t}uqnp $ nt $ nsst s
pnys } z}qzq}uqno wtnxysz vqys noyqos }y|ntquvzp
pnys zqoqzq}uqno wtnxyszp v~p wnpquqs tsp~yup
vqv }ts ynst xn~orp no $ nt $ }ts swtsppsr
~pqo| ( ) Ízwnppqxqyqu tsp~yup vqv }ts ~wwst
xn~orp no $ nt $ {nt }o }~uqno qo } stu}qo
y}pp }ts swtsppsr ~pqo| ( )
}or }ts ~psr noy {nt }o}ypqp ~t }~uqnop
nt quvn~u onqo| uvsqt }y~sp qo }r}os
Í{ s rn onu qzwnps }o tsputquqnop no s |su uvs
~wwst xn~or n{ $
ß× à n pss uvqp qz}|qos
* × xqrp }u }y~s × }or nos xqr !+ }u Ðo }~uqno
uv}u qpvsp un rn xsuust uv}o ß× à z~pu x}ps uvs
psyyqo| wtqs no xqrrst p xqr nsst uvqp n~yr
son~t}|s xqrrst un xqr xsyn +
n wtssou uvqp ~wwst xn~or no }~uqno tsso~s s
}o z}s uvs }pp~zwuqno uv}u uvs nwuqz}y tsso~s
qp pq|oq}ouy y}t|st uv}o uvs vq|vspu xqr v}u qp
{nt pnzs nopu}ou , s }pp~zs uv}u ,
quv
uvqp }pp~zwuqno nwuqz}y sr wtqqo| psyyp }u ys}pu ,
quszp Ío pnzs }psp , qp } sr nopu}ou Ío nuvst
}psp p~spp wtnx}xqyqu }wwtn}vsp × }p , - .
s p} uv}u }o }~uqno qp Ü ~orst stu}qo
}pp~zwuqnop q{ vso uvs }pp~zwuqnop
vnyr uvs }~
uqnop tsso~s qp ß à nt sÔ~q}ysouy $
ß×à
/ 0 BkhB 1 @j A@l
~t tsp~yup }ts tsy}usr un uvs syr n{ zsv}oqpz rspq|o
uv}u nzxqosp zqtnsnonzq znuq}uqno quv |}zs
uvsntsuq unnyp }or qoy~rsp }~uqno zsv}oqpzp ¡nt
qoutnr~uqno un uvs }ts} pss {nt s}zwys Ö×2 ×ÞÙ Ío
w}tuq~y}t }~uqnop {nt p}ts p~wwy z}tsup v}s xsso
susopqsy pu~rqsr Òss Ö×' Ù {nt } p~ts Ònzs nt qo
uvs mnzw~ust Òqsos nzz~oqu nzxqosp snonzq nt
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rsustzqoqpuq }~uqnop
vs Ú Ú qp uvs
xqrqorswsorsou
}~uqno quv : nwu uvs nwuqz}y
uvtspvnyr {~ouqno rsosr qo Òsuqno ' vs noy
rq stsos xsusso uvs rsustzqoqpuq nwuqz}y uvtspvnyr
}~uqno }or uvs nwuqz}y sr wtqqo| zsv}oqpz qp uv}u
uvs {ntzst ~psp uvtspvnyr nwuß à {nt xqrrst }or uvs
y}uust ~psp nwuß à s}yy uv}u }or noy rq st qo
uv}u ! qp onu qo ¡tnz uvqp s zq|vu }ypn swsu
uv}u {nt y}t|s quv p~qu}xys noput}qoup no uvs
rsustzqoqpuq nwuqz}y uvtspvnyr }~uqno n~yr wst{ntz
un quvqo } nopu}ou {t}uqno n{ Ðp s qyy pss
pvntuy uvqp qp onu uvs }ps Ðo qoustspuqo| tsp~yu uv}u
s qyy onu pvn qo uvqp w}wst qp uv}u uvs rsustzqoqpuq
nwuqz}y uvtspvnyr }~uqno qp pqo|yswtqs
le B " hjgi hB A B kgk@AkD " ki #
fgi gi gA >jDAkhg@l Ío uvqp psuqno s wtns uvs
{nyynqo| ~wwst xn~orÝ
Ó ÚÛ
Ó , Ú
Û ÚÚ $
ß× à ,
Û su : xs uvs ßrsustzqoqpuqà {~ouqno uv}u rsosp
uvs }~uqno mnopqrst } xqwny}t qow~u quv xqrp
}u }y~s }or xqrp }u }y~s × sputqusr un p~v
qow~up : qp } {~ouqno n{ }or ánus uv}u s }o
}pp~zs quvn~u ynpp n{ |sost}yqu uv}u : u}sp no noy
un }y~sp × }or uvst }y~sp n{ : ys}r un pz}yyst
tsso~sp
s qpv un or }o qow~u {}zqy p~v uv}u uvs
xqrqorswsorsou }~uqno quv {~ouqno : v}p tsso~s
ß à n nxu}qo uvqp {}zqy s vnps }or qo
p~v } } uv}u : ß * × à × : ß * ×à
}or , ßqzwyqo| ,
à ¡nt p~v }o qow~u
}or < pn $
$
× ~t |n}y on qp
|qso } rsustzqoqpuq : un or }y~sp n{ }or uv}u
v}s uvs }xns wtnwstuqsp
mnopqrst uvs wy}os ¡nt } sr ' ynn }u
uvs yqos 5 }or ' * 5 }or nopqrst uvs yqos
ps|zsou noosuqo| ß2 ' à }or ß' 2 à s ossr un or
} }y~s n{ 5 quv 5 , vsts vso 5 qots}psp x nos
: v}o|sp {tnz × un
Òsu ' , Ðpp~zs uv}u : ß, ' * , à × Ðp
s qots}ps 5 {tnz 5 , uvs }y~s n{ : ß5 ' * 5 à
z~pu v}o|s {tnz × un xs}~ps {nt 5 ' s v}s
: ß5 ' * 5 à : ß' 2 à
v~p qu z~pu xs }u pnzs 5 uv}u : ß5 ' * 5 à }or : ß5 * × ' * 5 6 ×à × Í{
s on vnnps 5 }or ' * 5 6 × s p}uqp{
2
pn s v}s
n~t tqustq} uv}u , }or uv}u : ß * × à × }or wtspsou swstqzsou}y tsp~yup {nt un wtnxysz {}zqyqsp
Òss uvs nzwysus w}wst {nt znts swstqzsou}y tsp~yup
: ß * ×à
v~p $
ß× à
Ò~wwnps on uv}u n~t }pp~zwuqno uv}u : ß, ' *
, à × qp {}yps }or qopus}r qu qp vso {nt 5 ,
}or ' * 5 6 × s v}s ' 6 × , 6 × }or
$
, pn
Zipf Size Test
1
" 0.9
v~p {nt }o xqrqorswsorsou }~uqno quv rsust
zqoqpuq {~ouqno : uvsts sqpup } rqputqx~uqno p~v uv}u
$
ß à
BjA j A B kgk@AkD >jDAkhg@ CB " ki #
l
fgi gi gAl vs {nyynqo| yszz} }yynp ~p un susor
uvs tsp~yu n{ vsntsz 3 × un }txqut}t rsustzqoqpuq
}~uqnop
Ó ÚÛ
Òss uvs {~yy stpqno n{ uvs w}wst {nt uvs wtnn{
pqo| szz} 3 × s |sost}yqs vsntsz 3 × }p {nyynp
Ó ÚÛ
Ó , Ú Û ÚÚ
$
ß× à ,
Ío ustzp n{ }pzwunuq ntpu}ps wst{ntz}os
rsustzqoqpuq }~uqnop }ts pq|oq}ouy ntps uv}o t}o
rnzqsr }~uqnop vqp qp onu un p} uv}u rsustzqoqpuq
}~uqnop }ts x}r un ~ps {nt }yy qow~u {}zqyqsp Ío {}u
n~t swstqzsou}y tsp~yup tss}y uv}u {nt z}o {}zqyqsp
uvs rsustzqoqpuq nwuqz}y uvtspvnyr }~uqno ntp st
syy ÐrsÔ~}us onysr|s n{ uvs xqrrqo| rqputqx~uqno
z} z}s qu wnppqxys un ~ps } rsustzqoqpuq }~uqno
e Bk gAC 1 @j A@l
~t uvsntsuq}y }o}ypqp n{ }~uqnop v}p yqzqu}uqnop
ntpu}ps }o}ypqp }|}qopu nt yn| ys}sp } yn|
|}w {nt qow~up vqv nzs {tnz (uwq}y) ß}p nwwnpsr un
u}qyntsr un xs v}trà rqputqx~uqnop Ío }rrquqno nopu}ou
{}untp s nxu}qo qo n~t }o}ypqp }ts n{uso unn wsp
pqzqpuq vsntsuq}y }o}ypqp {nt pwsq rqputqx~uqnop
psszp Ô~qus rqÏ~yu sso {nt pqzwys rqputqx~uqnop
Ío wt}uqs nopu}ou {}untp n{ uvs }~uqno tsso~s
}ts qzwntu}ou s qoutnr~sr psst}y }~uqno zsv}
oqpzp uv}u wtn}xy wst{ntz quvqo } nopu}ou {}unt
n{ s}v nuvst qo ntpu}ps nsst s rn onu on
vqv nos qp xsuust s n~yr }ypn yqs un on vn
uvsps }~uqnop nzw}ts un sr wtqqo| quv qzwst{su
z}tsu }o}ypqp
s u~to un swstqzsoup un }opst uvsps Ô~spuqnop
Ío n~t swstqzsoup s pqz~y}us }tqn~p }~uqnop no ps
st}y qow~u {}zqyqsp }or pss vn uvs nzw}ts syn s
0.8
R/F
$ 0.7
0.6
DSO
SSO
WP
DOT
FPFP+
0.5
0.4
10
100
1000
Number of bids
10000
¡q|~ts ×Ý qw{ßuvsu} 2 vq|v oà quv Ö×2 ×225 Ù
Ío uvs tpu wtnxysz {}zqy s}v xqr qp vnpso
qorswsorsouy {tnz uvs qw{ rqputqx~uqno Ö×4Ù quv
× â vqp qp } |sost}yq}uqno n{ uvs rqputqx~uqno
quv 42 n{ uvs unu}y xqr }y~s nzqo| {ntz â 2
n{ uvs xqrrstp ¡nt qo uvs qoust}y Ö× ÚÚÙ s
rsos 0 BÖ Ù
quv vnpso pn uv}u uvs
wtnx}xqyquqsp qous|t}us un nos ¡q|~ts
× pvnp uvs tsp~yup
n{ pqz~y}uqno {nt uvs qw{ {}zqy }p uvs o~zxst n{
xqrrstp }tqsp xsusso ×2 }or ×22 222 vqp {}zqy
v}p uvs wtnwstu uv}u }o ~oq{ntzy vnpso t}ornz
p~xpsu n{ uvs xqrp v}p uvs p}zs rqputqx~uqno }p uvs
ntq|qo}y s}~ps n{ uvqp wtnwstu uvs t}ornz p}zwyqo|
}~uqnop wst{ntz st syy {nt uvsps {}zqyqsp Ío uvs wynu
t}uqnp $ }ts tswntusr {nt uvs {nyynqo| zsv}oqpzpÝ
Ò r~}ywtqs p}zwyqo| nwuqz}y uvtspvnyr
ÒÒ pqo|yswtqs p}zwyqo| nwuqz}y uvtspvnyr
'

sq|vusr w}qtqo|
rsustzqoqpuq nwuqz}y uvtspvnyr
¡ * sr wtqqo| quv nwuqz}y wtqs yspp â
¡ 6 sr wtqqo| quv nwuqz}y wtqs wy~p â
¡nt y}t|s Ò }or }ts uvs xspu }~
uqnop Ðp qots}psp uvs t}uqn n{ uvsqt tsso~s un
}wwtn}vsp × vsts}p uvs t}uqnp {nt ¡ * }or ¡ 6
}wwtn}v } nopu}ou yspp uv}o × Ðp } tsp~yu sso
{nt znrst}usy y}t|s n~t xspu }~uqnop wst{ntz xsuust
uv}o sr wtqqo| quv } â wtqs sttnt o }st}|s
}ps rqputqx~uqnop qp uvs ntpu }~uqnoÎ qup t}uqn
}pzwunuq}yy }wwtn}vsp â Þ
Ío uvs xqwny}t {}zqy xqrp v}s noy un }y~sp ×
}or }or uvs t}uqn n{ vq|v xqrp un uvs unu}y o~zxst n{
xqrp }tqsp vs tsp~yup }wws}t qo ¡q|~ts â
vsts }ts psst}y s uvqo|p un onus }xn~u uvs
xqwny}t {}zqy ¡qtpu qu rsznoput}usp uvs wtnxysz quv
Bipolar Ratio Test (n=100)
1
0.9
0.8
0.7
R/F
0.6
0.5
p~wwy qp }onuvst sutszs Ío uvs p}ts p~wwy }ps uvs
nwuqz}y sr wtqqo| tsso~s qp z}qzqsr x psyyqo| }yy
uvs quszp vso uvs o~zxst n{ }}qy}xys quszp qp }tn~or
5 tsqn~p nt no }~uqnop nosout}usr no uvs p}ts
p~wwy }ps vs z~yuqqusz Õqts }~uqno qp uvs xspu
pqo|yswtqs }~uqno {nt uvs p}ts p~wwy }ps ß}or qu qp
nzwsuquqsà ~t tsp~yup susorsr un xn~orsr p~wwy
}ts nzwsuquqs qo uvs {~yy t}o|s n{ uvs p~wwyrsz}or
pwsut~z quv noy uvs }pp~zwuqno uv}u , ; 8
s on pvn vn un susor n~t nwuqz}y uvtspvnyr
p}zwyqo| }~uqnop un uvs 5 qusz xn~orsr p~wwy }ps
su nwu 8 xs uvs {~ouqno uv}u |qso } psu n{ xqrp
tsu~top uvs nwuqz}y uvtspvnyr uv}u psyyp 5 quszp nt
yspp vqp {~ouqno no uvs souqts psu n{ xqrp |qsp
uvs uvtspvnyr un ~ps {nt 8 vs pqo|yswtqs p}zwyqo|
nwuqz}y uvtspvnyr zsv}oqpzp }o on xs znrqsr un
~ps uvtspvnyr {~ouqno nwu 8 +7 no p}zwys n{ pqs
' Í{ uvqp tsp~yup qo unn z}o xqrp xsqo| p}uqpsr
}txqut}tqy ßs| }u t}ornzà tsÑsu xqrp ~ouqy uvsts }ts
noy 5 ys{u os }o pvn uv}u quv vq|v wtnx}xqyqu
uvs o~zxst n{ xqrp tsÑsusr qyy xs pz}yy }or uv}u uvqp
}~uqno qp nzwsuquqs
Ío uvs r~}ywtqs }~uqno quv p}zwys pqs ' â
~ps nwu 8 pn uv}u }xn~u 5 â xqrp }ts psysusr {tnz s}v
n{ uvs p}zwys }or uvs onop}zwys os }o pvn uv}u
uvs tsp~yuqo| }~uqno qp }ypn ut~uv{~y }or nzwsuquqs
s v}s pvno uv}u qo } tsy}uqsy put}q|vu{nt}tr
} n~t p}zwyqo| }~uqnop susor {tnz uvs ~oyqzqusr
un uvs xn~orsr p~wwy }ps os }o susor uvs
rsustzqoqpuq nwuqz}y uvtspvnyr }~uqno un xn~orsr
p~wwy }p syy Ðypn }yy n{ n~t ~wwst xn~orp }wwy un
xn~orsr p~wwy xs}~ps qu |sost}yqsp ~oyqzqusr p~wwy
n |sost}yqs vsntsz ' × {nt uvs 5 qusz }ps ts}yy
uvs }yusto}uqs rsoquqno n{ }p uvs tsso~s r~s un uvs
nwuqz}y ~out~uv{~y z~yuqwtqs }~uqno sos 8 un xs
uvs tsso~s n{ uvs nwuqz}y z~yuqwtqs }~uqno tsputqusr
un noy p}uqp{qo| 5 xqrrstp ßqs uvs p~z n{ uvs vq|vspu
5 xqrpà }or 8 }p }xns vso uvs |sost}yqsr tsp~yu qp
8 < 8 ßâ yn| à
os tsp~yu s rn onu on vn un susor un
xn~orsr p~wwy qp uv}u {nt uvs sq|vusr w}qtqo| }~uqno
e hgD jikg % 1 CB @l
s v}s rsznoput}usr uv}u uvsts sqpu ut~uv{~y }~
uqnop {nt ~oyqzqusr p~wwy z}tsup s v}s pvno
t}ornzqsr }~uqnop uv}u }ts nzwsuquqs qo uv}u uvs
qsyr tsso~s uv}u qp quvqo } nopu}ou {}unt n{ nwuqz}y
sr wtqqo| s v}s pvno uv}u uvqp tsp~yu qp uq|vu
~w un } nopu}ou {}unt sso {nt z~yuqwtqs }~uqnop
s v}s }ypn pvno uv}u on rsustzqoqpuq }~uqno qp
nzwsuquqs qo uvs ntpu}ps ¡qo}yy q} pqz~y}uqnop
s v}s }t|~sr uv}u n~t }~uqnop nzw}ts {}nt}xy un
sr wtqqo| quv z}tsu }o}ypqp
0.4
0.3
DSO
SSO
WP
DOT
0.2
0.1
0
0.2
0.4
0.6
Ratio: # highs / # total
0.8
1
¡q|~ts âÝ qwny}tßyn ×vq|v ×2 t}uqn Ö2 ×Ùà
uvs }~uqno vsts qp } pv}tw rqw qo uvs tsso~s
n{ wtsqpsy vso uvs o~zxst n{ vq|v xqrp qp ×2
}or uvs o~zxst n{ yn xqrp qp 32 vqp qp xs}~ps
nwuß ( à × }or nwuß ( × à qo uvqp pso}tqn
}or uv~p ×2 vq|v xqrp |su p}uqpsr }u wtqs × }or uvs
32 yn xqrp |su tsÑsusr vs nwuqz}y pny~uqno qo uvqp
}ps qp un }swu uvs ×2 vq|v xqrp }u wtqs ×2 Ðpqrs
{tnz uvqp }ps wst{ntzp syy vs t}ornzqsr
nwuqz}y uvtspvnyr }tq}oup v}s rs|t}rsr tsso~sp {nt
uvs o~zxst n{ vq|v xqrp }tn~or ×2 nsst r~s un
t}ornzospp uvs tsso~s ynpp qp pq|oq}ouy pz}yyst
x~u pwts}r nst } qrst ts|qno vs Ò }~uqno
n~uwst{ntzp }or ÒÒ {nt znpu t}uqnp vs
}~uqno ~p~}yy n~uwst{ntzp ÒÒ
ee " hjgi i $ j l
w un uvqp wnqou s v}s pu~rqsr uvs ~oyqzqusr p~wwy
}ps vqv qp znuq}usr x uvs rq|qu}y |nnrp z}tsu
vsts uvs npu n{ z}qo| } nw n{ }o qusz qp os|yq|qxys
Ío uvqp psuqno s nopqrst uvs }ps vsts uvs o~zxst n{
quszp }}qy}xys {nt p}ys qp xn~orsr vqp }ps qp uwq}y
n{ wvpq}y |nnrp z}tsup s rsonus uvs o~zxst n{
quszp }}qy}xys x 5 sts }|}qo uvs psyyst qpvsp
un z}qzqs uvsqt tsso~s wnppqxy onu psyyqo| }yy n{
uvs quszp ánus uv}u uvs rsoquqnop n{ ut~uv{~y }or
nzwsuquqs }~uqnop vqv s pu}usr {nt uvs ~oyqzqusr
p~wwy }ps }ypn }wwy un uvs xn~orsr p~wwy }ps s
rsonus x 8 uvs tsso~s {nt nwuqz}y sr wtqqo| uv}u
psyyp }u znpu 5 quszp }or qu qp uvqp Ô~}ouqu uv}u s
qpv un xs nzwsuquqs quv
vs xn~orsr p~wwy }ps qp } |sost}yq}uqno n{ uvs
~oyqzqusr p~wwy }ps }p quszp }ts }}qy}xys qo ~oyqzqusr
p~wwy vso uvs o~zxst n{ }}qy}xys quszp qp uvs p}zs
}p uvs o~zxst n{ xqrrstp ßqs 5 à vsts ~oyqzqusr
p~wwy qp nos sutszs n{ uvs xn~orsr p~wwy }ps
¡nt ~oyqzqusr p~wwy z}tsup n~t }o}ypqp }pp~zsp
uv}u uvsts qp on npu {nt wtnr~qo| uvs quszp xsqo|
}~uqnosr quv uvs {nyynqo| znrq}uqno s }o }ypn
}nzznr}us onostn z}t|qo}y npup Í{ uvs z}t|qo}y
npu qp wst qusz uvso tpu p~xut}u {tnz s}v xqr
}or tsÑsu }yy os|}uqs }y~s xqrp Ð{ust t~ooqo| uvs
}~uqno }rr x} un uvs psyyqo| wtqs n{ }yy qooqo|
xqrp Í{ uvs z}t|qo}y npu n{ wtnr~qo| 5 quszp qp }
znts nzwyq}usr {~ouqno n{ 5 s }o znrq{ uvs nwu
{~ouqno ~psr qo uvs p}zwyqo| nwuqz}y uvtspvnyr }~uqno
un u}s qoun }n~ou uvsps z}t|qo}y npup Ío uvqp }ps
uvs nwu {~ouqno n~yr }p }xns {nt xn~orsr p~wwy
ossr un xs w}t}zsustqsr x uvs t}uqn n{ uvs p}zwys pqs
}or uvs onop}zwys pqs pn }p un nttsuy ~ps z}t|qo}y
npu qo{ntz}uqno
n wtssou vs}uqo| x uvs }~uqnosst nt uvs xqr
rstp nos z} ossr } ut~pusr uvqtr w}tu nt } pwsq}y
twun|t}wvq wtnunny ánus uv}u vs}uqo| wtssouqno
qp } wtnxysz pv}tsr x }yy noyqos }~uqnop sy}usr
tsp~yup }wws}t qo Ö Ù
e / g 0 Bh? @l
s wtnsr uv}u uvs sq|vusr w}qtqo| }~uqno qp onu
nzwsuquqs Íu qp wnppqxys vnsst uv}u } }tq}ou n{
uvqp }~uqno uv}u ~psp rq stsou sq|vuqo| qp nzwsuquqs
vs sq|vusr w}qtqo| }~uqno }o v}s tsso~s
|ts}ust uv}o {nt pnzs t}ornz w}qtqo|p Íu psszp yqs
uvqp zq|vu onu xs uvs }ps {nt uvs r~}ywtqs p}zwyqo|
nwuqz}y uvtspvnyr }~uqno m}o nos wtns uv}u qu qp
}y}p uvs }ps uv}u {nt }o t}ornzy vnpso p}zwys
uvs }~uqno tsso~s rnsp onu sssr
s v}s onu nopqrstsr qpp~sp p~v }p uvs susou un
vqv xqrrstp }o tsz}qo }onozn~p nt xqr }y~sp }o
tsz}qo pstsu Òss ÖØÙ {nt }p un z}qou}qo xqr psts
qo }o noyqos Õqts }~uqno
Ð pq|oq}ou qpp~s qo }~uqnop yqs n~tp qp tspqpu}os
un }rstp}tq}y }uu}p n tspqpu}ou }ts n~t }~uqnop
un xqrrst nyy~pqno }or }o nyy~pqno tspqpu}os xs
qzwtnsr n syy rn n~t }~uqnop tspqpu }uu}p p~v
}p } nzwsuqunt }uuszwuqo| un tsr~s uvs tsso~s n{ }o
}~uqno x p~xzquuqo| } y}t|s o~zxst n{ yn xqrp
sws}usr }~uqnop {nt uvs p}zs qusz z} xs n{
qoustspu qo pnzs }wwyq}uqnop Ío uvqp }ps uvs v}yyso|s
qp un rspq|o } ut~uv{~y }~uqno zsv}oqpz
>D gh i % gA@
s n~yr yqs un uv}o uvs {nyynqo| wsnwys {nt uvsqt
noutqx~uqnop }or ~ps{~y rqp~ppqnop tsy}uqo| un uvqp
ntÝ Ðznp ¡q}u Òu~}tu }xst qz ntoqo| Ðoo}
} }trnp
}tyqo zspv % }vspv}tq Ò}uqpv }n
nx }tÑ}o }rs Õqo|t}ys }or Òuswvso ssp
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