Menezes, Flavio Marques

Monteiro, Paulo Klinger

English

Research Papers in Economics

Ensaios Econ￿micos
Escola de
P￿s-Gradua￿ªo
em Economia
da Funda￿ªo
Getulio Vargas
N 367 ISSN 0104-8910
Strategic Behavior in Repeated Trials of
Vickrey Auctions
Flavio Marques Menezes, Paulo Klinger Monteiro
Fevereiro de 2000
URL: http://hdl.handle.net/10438/393Os artigos publicados sªo de inteira responsabilidade de seus autores. As
opiniıes neles emitidas nªo exprimem, necessariamente, o ponto de vista da
Funda￿ªo Getulio Vargas.
ESCOLA DE P￿S-GRADUA˙ˆO EM ECONOMIA
Diretor Geral: Renato Fragelli Cardoso
Diretor de Ensino: Luis Henrique Bertolino Braido
Diretor de Pesquisa: Joªo Victor Issler
Diretor de Publica￿ıes Cient￿￿cas: Ricardo de Oliveira Cavalcanti
Marques Menezes, Flavio
Strategic Behavior in Repeated Trials of Vickrey
Auctions/ Flavio Marques Menezes, Paulo Klinger Monteiro ￿ Rio
de Janeiro : FGV,EPGE, 2010
(Ensaios Econ￿micos; 367)
Inclui bibliografia.
CDD-330Vwudwhjlf ehkdylru lq uhshdwhg wuldov ri Ylfnuh|
dxfwlrqv
Iodylr P1 Phqh}hv dqg Sdxor N1 PrqwhlurW
IJY2HSJH
Sudld gh Erwdirjr 4<3/ 44r1 dqgdu
Ulr gh Mdqhlur/ UM/ 555860<33
Eud}lo
Mdqxdu| 46/ 5333
Ylfnuh|*v lqvljkw wkdw lq d vhfrqg0sulfh vhdohg0elg dxfwlrq rqh*v grplqdqw vwudw0
hj| lv wr elg rqh*v ydoxdwlrq  zkhq frpshwlqj iru d vlqjoh remhfw xqghu wkh lq0
ghshqghqw sulydwh ydoxhv dvvxpswlrq  lv zhoo xqghuvwrrg e| prvw hfrqrplvwv1
Ohvv zhoo xqghuvwrrg lv lqglylgxdo ehkdylru lq uhshdwhg dxfwlrqv1 Iru h{dpsoh/
wkhuh duh vhyhudo frqwh{wv zkhuh elgglqj rqh*v wuxh ydoxdwlrq lv qrw dq htxlole0
ulxp elgglqj vwudwhj| lq d vhfrqg0sulfh dxfwlrq 1 Frqvlghu/ iru lqvwdqfh/ wzr
lghqwlfdo remhfwv wkdw duh vrog vhtxhqwldoo| e| vhfrqg0sulfh vhdohg0elg dxfwlrqv
dqg wkuhh srwhqwldo ex|huv zkr zdqw rqh remhfw hdfk1 Dowkrxjk lw lv d grplqdqw
vwudwhj| wr elg rqh*v wuxh ydoxdwlrq lq wkh odvw dxfwlrq/ lq wkh uvw dxfwlrq elg0
ghuv elg ohvv wkdq wkhlu ydoxhv  lq htxloleulxp sduwlflsdqwv elg lq vxfk zd| wkdw
wkh lqglylgxdov zlwk wkh wzr kljkhvw ydoxdwlrqv duh lqglhuhqw ehwzhhq zlqqlqj
wkh uvw ru wkh vhfrqg remhfw1 +Vhh/ iru h{dpsoh/ Zhehu +4<;6,,1
Lq wklv sdshu zh dqdo|}h elgglqj ehkdylru lq uhshdwhg wuldov ri vhfrqg0sulfh
dxfwlrqv1 Wklv lv prwlydwhg e| vhyhudo h{shulphqwdo ghvljqv lqyroylqj pxowlsoh
wuldov ri vhfrqg0sulfh dxfwlrqv dqg d udqgrp gudz wkdw ghwhuplqhv zklfk urxqg
frxqwv1 Iru h{dpsoh/ dprqj rwkhuv/ Vkrjuhq/ Vklq/ Kd|hv dqg Nolhehqvwhlq
+4<<7, dqg Frxuvh| dqg Vfkxo}h +4<;9, xvh uhvshfwlyho| yh uhshwlwlrqv dqg
xs wr whq uhshwlwlrqv ri vhfrqg0sulfh dxfwlrqv wr dqdo|}h wkh glvsdulw| ehwzhhq
zloolqjqhvv wr sd| +e| ex|huv, dqg zloolqjqhvv wr dffhsw +e| vhoohuv,1 Lq wkh
iruphu vwxg|/ d udqgrp gudz ghwhuplqhg zklfk urxqg zrxog frxqw dqg lq wkh
odwhu vwxg| d yrwlqj phfkdqlvp zdv xvhg wr ghwhuplqh wkh elqglqj urxqg +xs
wr wkh whqwk urxqg zkhq wkh jdph qhfhvvdulo| hqghg,1 Lq wkhvh h{shulphqwv/
w|slfdoo| vrph lqglylgxdov lq d jurxs duh hqgrzhg zlwk fhuwdlq remhfwv +fdqg|
eduv/ frhh pxjv/ shqv/ hwf1, zkloh rwkhuv duh hqgrzhg zlwk prqh|1 Dq
h{shulphqw lv wkhq rujdql}hg zkhuh vhoohuv vxeplw dvn sulfhv dqg ex|huv
vxeplw elg sulfhv1 Sulfhv duh ghwhuplqhg e| d Ylfnuh| dxfwlrq phfkdqlvp1 Li
rqh remhfw lv ehlqj wudghg/ d vhfrqg0sulfh dxfwlrq lv xvhg1 Li wkuhh remhfwv duh
WZh zrxog olnh wr wkdqn Vlprq Judqw iru xvhixo frpphqwv1
4wr eh wudghg/ wkhq d irxuwk0sulfh dxfwlrq lv xvhg/ dqg vr rq dqg vr iruwk1 Dw wkh
hqg ri hdfk urxqg elgghuv ohduq wkh rxwfrph ri wkh dxfwlrq +qdpho|/ sulfh dqg
doorfdwlrq, dqg dw wkh hqg ri wkh h{shulphqw ohduq zklfk urxqg zloo eh fkrvhq
wr ghwhuplqh wkh elqglqj rxwfrph1
Lw lv frpprqo| dujxhg wkdw uhshdwhg wuldov duh qhhghg iru lqglylgxdov wr ohduq
wkdw elgglqj rqh*v ydoxdwlrq lv d grplqdqw vwudwhj|1 Iru h{dpsoh/ Frxuvh|/ Krylv
dqg Vfkxo}h +4<;:, dujxh wkdw dq lpsruwdqw revhuydwlrq wr eh gudzq iurp
h{shulphqwdo hfrqrplfv lv wkdw lqglylgxdov sduwlflsdwlqj lq d Ylfnuh| dxfwlrq
gr qrw lqlwldoo| uhyhdo wuxh ydoxhv1 Rq d sxuho| wkhruhwlfdo hfrqrplf edvlv/
wkh| vkrxog uhdol}h wkdw wklv lv wkhlu grplqdqw vwudwhj|1 Krzhyhu/ d qxpehu ri
wuldo lwhudwlrqv duh uhtxluhg wr doorz lqglylgxdov wr ohduq wkdw uhyhdolqj wuxh
ydoxhv lv wkhlu ehvw vwudwhj|1 Vkrjuhq/ Vklq/ Kd|hv dqg Nolhehqvwhlq +4<<7, dovr
dujxh wkdw wkh| xvh yh wuldov/ zlwk wkh elqglqj wuldo ehlqj vhohfwhg dw udqgrp/
wr doorz ohduqlqj rq krz wr sod| wkh jdph41
Dqrwkhu h{dpsoh lv Ndkqhpdq/ Nqhwvfk/ dqg Wkdohu +4<<3, zkr udq d vhulhv
ri h{shulphqwv wzr ri zklfk duh uhsruwhg khuh1 Lq wkh uvw h{shulphqw/ lq0
gxfhg ydoxh wrnhqv  zhuh wudghg1 Lq vxfk pdunhwv doo vxemhfwv duh wrog krz
pxfk d wrnhq lv zruwk wr wkhp/ zlwk wkh dprxqwv ydu|lqj dfurvv vxemhfwv1 Kdoi
wkh vxemhfwv zhuh doorfdwhg wrnhqv/ wkh rwkhu kdoi zhuh qrw1 Vxemhfwv dowhuqdwhg
ehwzhhq ehlqj d ex|hu dqg ehlqj d vhoohu lq wkuhh vxffhvvlyh pdunhwv/ dqg zhuh
dvvljqhg d glhuhqw lqglylgxdo uhghpswlrq ydoxh lq hdfk wuldo1 Diwhu hdfk pdu0
nhw/ erwk wkh pdunhw0fohdulqj sulfh dqg wkh qxpehu ri wudghv zdv dqqrxqfhg1
Vxemhfwv zhuh vshflfdoo| wrog wkdw lw lv lq |rxu ehvw lqwhuhvw wr dqvzhu wkhvh
txhvwlrqv wuxwkixoo|1 Iru dq| txhvwlrq/ wuhdw wkh sulfh dv {hg1 Lqghhg/ dv
ydoxdwlrqv zhuh lqghshqghqwo| gudzq diwhu hdfk urxqg/ wkhuh zdv qr urrp iru
vwudwhjlf ehkdylru1 Wkh uhvxowv frquphg wkh wkhruhwlfdo suhglfwlrq1
Lq wkh vhfrqg h{shulphqw/ zklfk zdv udq lpphgldwho| diwhu wkh uvw h{0
shulphqw/ vxemhfwv rq dowhuqdwlqj vhdwv zhuh jlyhq frhh pxjv iurp Fruqhoo
Xqlyhuvlw| +dydlodeoh iru '9/33 dw wkh orfdo errnvkrs,1 Vxemhfwv zhuh lqiruphg
wkdw irxu pdunhwv iru pxjv zrxog eh frqgxfwhg xvlqj wkh vdph surfhgxuhv dv
wkh sulru lqgxfhg ydoxh wrnhqv pdunhwv h{fhsw wkdw rqo| rqh ri wkh irxu pdunhw
wuldov zrxog vxevhtxhqwo| eh vhohfwhg dw udqgrp dv wkh elqglqj wuldo1 Pruhryhu/
doo wudghv zrxog eh lpsohphqwhg rq wkh elqglqj pdunhw wuldo/ xqolnh wkh vxevhw
lpsohphqwhg lq wkh lqgxfhg ydoxh wrnhqv pdunhwv1 Wkh lqlwldo dvvljqphqw ri
ex|hu dqg vhoohu urohv zdv pdlqwdlqhg iru doo irxu wudglqj shulrgv1 Wkh fohdulqj
sulfh dqg wkh qxpehu ri wudghv zhuh dqqrxqfhg diwhu hdfk shulrg1 Wkh elqg0
lqj wuldo zdv lqglfdwhg diwhu wkh irxuwk shulrg/ dqg wudqvdfwlrqv zhuh h{hfxwhg
lpphgldwho|1 Li wkhvh pdunhwv zhuh lqghshqghqw/ wkh wkhru| zrxog suhglfw wkdw
kdoi ri wkh pxjv vkrxog wudgh1 Lq wkh irxu pxj pdunhwv wkh qxpehu ri wudghv zdv
7/ 4 / 5 dqg 5 uhvshfwlyho|1 Krzhyhu/ jlyhq wkdw wudqvdfwlrqv zhuh qrw lqghshq0
ghqw/ xqolnh lq wkh uvw h{shulphqw/ wkhuh lv qr fohdu wkhruhwlfdo suhglfwlrq 
4Lw lv zruwk qrwlqj wkdw Fr{/ Urehuvrq/ dqg Vplwk +4<;5, lv xvxdoo| flwhg dv vxssruwlqj
wklv ylhz wkdw uhshdwhg wuldov duh qhhghg iru ohduqlqj krz wr sod|1 Krzhyhu/ lq Fr{/ Urehuvrq
dqg Vplwk hdfk wuldo zdv elqglqj dqg jhqxlqho| lqghshqghqw ri hdfk rwkhu vr wkdw vwudwhjlf
ehkdylru e| elgghuv ehwzhhq urxqgv zdv yhu| xqolnho| dqg qrqh{lvwhqw iurp d wkhruhwlfdo
ylhzsrlqw1
5h{fhsw iru wkh odvw wuldo zkhuh wkhuh lv lqghhg qr lqfhqwlyh iru plvuhsuhvhqwdwlrq
ri ydoxhv15 Lw lv rxu frqwhqwlrq wkdw wkhuh pd| eh vwudwhjlf uhdvrqv iru ex|huv
wr xqghuvwdwh wkhlu wuxh ydoxdwlrqv rq hduolhu wuldov1 Wklv lv vshfldoo| uhohydqw
lq dq hqylurqphqw zkhuh vhoohuv kdyh wkh rssruwxqlw| wr ehkdyh vwudwhjlfdoo|
dqg wr xvh lqirupdwlrq uhyhdohg e| srwhqwldo ex|huv rq hduolhu urxqgv dv lq wkh
h{shulphqwdo ghvljq ri Ndkqhpdq/ Nqhwvfk dqg Wkdohu +4<<3,1
Lq wklv vkruw sdshu zh ghprqvwudwh wkdw lw lv qrw d elgghu*v grplqdqw vwudwhj|
 lqghhg lw lv qrw hyhq dq htxloleulxp  wr elg rqh*v wuxh ydoxdwlrq lq wkh uvw
ri wzr urxqgv zkhuh wkh elqglqj rxwfrph lv fkrvhq e| d udqgrp ghylfh1 Lq
rxu h{dpsoh/ wkh vhoohu*v ehkdylru lv {hg> kh xvhv wkh lqirupdwlrq uhyhdohg e|
elgghuv lq wkh uvw urxqg wr vhw wkh uhvhuyh sulfh iru wkh vhfrqg dxfwlrq1 Lq
sduwlfxodu/ wkh vhoohu vhwv wkh vhfrqg shulrg uhvhuyh sulfh htxdo wr wkh kljkhvw
elg lq wkh uvw dxfwlrq1 Lqghhg zh vkrz wkdw zkhq ex|huv eholhyh wkdw wkh vhoohu
iroorzv wklv vhfrqg0shulrg uhvhuyh0sulfh srolf| dqg wkh vhoohu lv frqvwudlqhg wr
vhw d uhvhuyh sulfh qr juhdwhu wkdq wkh kljkhvw elg lq wkh uvw dxfwlrq/ wkhq lw lv
dq htxloleulxp iru wkh vhoohu wr vhw wkh uhvhuyh sulfh dw wkh kljkhvw uvw0shulrg
elg1
Ri frxuvh/ li wkh vhoohu lv iuhh wr vhw dq| uhvhuyh sulfh kh olnhv dqg li wkhuh lv
d v|pphwulf htxloleulxp elgglqj vwudwhj| lq wkh uvw dxfwlrq wkdw lv lqfuhdvlqj
lq ydoxhv/ wkhq wkh vhoohu zloo lqyhuw uvw0shulrg elgglqj vwudwhjlhv/ frpsxwh wkh
kljkhvw ydoxdwlrq dqg vhw wkh vhfrqg0shulrg uhvhuyh sulfh dw wkh kljkhvw ydoxd0
wlrq1 Krzhyhu/ zh vkrz wkdw zlwk d ixoo| vwudwhjlf vhoohu wkhuh lv qr lqfuhdvlqj
v|pphwulf htxloleulxp elgglqj vwudwhj| lq wkh uvw shulrg1 Lq sduwlfxodu/ elg0
glqj rqh*v wuxh ydoxdwlrq lv qrw d v|pphwulf htxloleulxp elgglqj vwudwhj| lq wkh
uvw shulrg hyhq zlwk d ixoo| vwudwhjlf vhoohu1
Wklv qrwh lv phdqw wr looxvwudwh rqh ri vhyhudo uhdvrqv iru ex|huv qrw wr uhyhdo
wkhlu wuxh ydoxdwlrqv lq uhshdwhg lqwhudfwlrqv1 Wkh srvvlelolw| ri vwudwhjlf ehkdy0
lru kdg douhdg| ehhq srlqwhg rxw e| Juhjru| dqg Ixue| +4<;:,/ zkr hpskdvl}h
 lq wkh frqwh{w ri Frxuvh|/ Krylv dqg Vfkxo}h +4<;:,/ Frxuvh| dqg Vfkxo}h
+4<;9, dqg uhodwhg uhvhdufk  wkdw pruh vrsklvwlfdwhg lqglylgxdov pljkw xvh
wkh irxu qrq0elqglqj wuldov vwudwhjlfdoo|/ fuhdwlqj edujdlqlqj srvlwlrqv wr eh xvhg
lq wkh odwhu +elqglqj, urxqgv1 Khuh zh h{soruh d vshflf uhdvrq iru ex|huv wr
plvuhsuhvhqw wkhlu ydoxdwlrqv rq hduolhu urxqgv1
Wklv surylghv dq dowhuqdwlyh h{sodqdwlrq iru wkh glvsdulw| ehwzhhq zloolqj0
qhvv wr sd| dqg zloolqjqhvv wr dffhsw lq dgglwlrq wr h{lvwlqj h{sodqdwlrqv vxfk dv
wkh hqgrzphqw hhfw +Ndkqhpdq dqg Wyhuvn| +4<:<,,  lqglylgxdov ydoxh wkh
jrrg pruh kljko| diwhu wkh| hqgrz wkhp> Kdqhpdqq*v +4<<4, vxevwlwxwdelolw|
dujxphqw  glhuhqfhv ehwzhhq zloolqjqhvv wr sd| dqg zloolqjqhvv wr dffhsw
uhodwh wr pryhphqwv dorqj glhuhqw lqglhuhqfh fxuyhv> ru glhuhqw vhjphqwv
ri wkh vdph fxuyh> ohduqlqj derxw ydoxhv +Vkrjuhq hw do1 +4<<7, dqg Pruulvrq
+4<<:0d/e,, dqg lpsuhflvh suhihuhqfhv +Gxerxuj/ Mrqhv0Ohh dqg Orrphv +4<<7,
dqg Pruulvrq +4<<;,,1
5Ndkqhpdq/ Nqhwvfk dqg Wkdohu ixuwkhu whvw iru vwudwhjlf ehkdylru e| vhohfwlqj wkh pdunhw
sulfh e| d udqgrp gudz1 Wkh| ￿qg vlplodu uhvxowv wr wkh vhfrqg h{shulphqw ghvfulehg deryh1
64 Elgglqj Ehkdylru zkhq wkh vhoohu*v vwudwhj| lv
{hg
Zh dvvxph wkdw wkhuh lv rqh remhfw iru vdoh dqg wkhuh duh q elgghuv zlwk lq0
ghshqghqw sulydwh ydoxhv yl 5 ^3>4` gudzq iurp d glvwulexwlrq I zlwk ghqvlw|
i +, A 3= Wkh dxfwlrq lv rujdql}hg dv iroorzv=
Shulrg 4= Elgghuv vxeplw elgv lq d vhfrqg0sulfh vhdohg0elg dxfwlrq dqg wkh vhoohu
vhwv d uhvhuyh sulfh U4 @3 >
Shulrg 5= Elgghuv vxeplw qhz elgv lq dqrwkhu vhfrqg0sulfh dxfwlrq nqrzlqj wkh
uhvxow ri wkh uvw dxfwlrq +sulfh dqg doorfdwlrq, dqg wkh vhoohu vhwv d qhz
uhvhuyh sulfh U5 @ zlqqlqj elg lq wkh uvw dxfwlrq>
Shulrg 6= D udqgrp ghylfh lv xvhg wr ghwhuplqh zklfk rxwfrph +wkdw lv/ sulfh dqg
doorfdwlrq, zloo eh elqglqj1 Wkh ghylfh dvvljqv suredelolw| 4
5 wr hdfk ri
wkh dxfwlrqv1
Wkhvh uxohv duh frpprq nqrzohgjh sulru wr wkh vwduw ri wkh elgglqj surfhvv1
Lq sduwlfxodu/ wkh vhoohu*v srolf| rq vhwwlqj wkh uhvhuyh sulfh lv frpprq nqrzohgjh
dowkrxjk wkh vshflf ydoxh ri U5 lv rqo| nqrzq dw wkh hqg ri shulrg 41
Zh zloo qrz frpsxwh wkh htxloleulxp elgglqj vwudwhjlhv lq erwk urxqgv1
Vxssrvh +el>f l,
q
l@4 lv dq htxloleulxp ri wklv dxfwlrq1 Lq wkh vhfrqg dxfwlrq lw
lv d grplqdqw vwudwhj| wr elg fl +yl,@yl  wkh vhfrqg dxfwlrq lv vlpso| d
rqh0vkrw vhfrqg0sulfh vhdohg0elg dxfwlrq= Lq wkh uvw dxfwlrq/ krzhyhu/ lw lv qrw
dq htxloleulxp wr elg rqh*v wuxh ydoxdwlrq dv zh ghprqvwudwh qh{w$
Wkhruhp 4 Ex|huv elg ohvv wkdq wkhlu ydoxdwlrqv lq wkh uvw dxfwlrq v|pphwulf
htxloleulxp zkhq wkh vhoohu*v vwudwhj| lv {hg1
Surri1 Wr fkdudfwhul}h wkh v|pphwulf htxloleulxp elgglqj vwudwhj| el+,@
e4 +,>5  l  q> zh dvvxph wkdw e+, lv d glhuhqwldeoh lqfuhdvlqj vwudwhj| dqg
qg d glhuhqwldo htxdwlrq wkdw e+, vdwlvhv1 Ohw*v ghqh ]4 @p d {iyl>l  5j=
Li elgghuv l @5 >===>q elg dffruglqjo| wr e+,> Elgghu 4 kdv d vljqdo y dqg
vxeplwv d elg {  3 klv h{shfwhg xwlolw| lv jlyhq e|
K +{,@H

4
5
"{￿e+]4,

y  e

]4
.
4
5

y  pd{

{>e

]4
>]4.

=
Qrwh wkdw pd{

{>e

]4
lv wkh vhoohuv uhvhuyh ydoxh iru wkh vhfrqg dxfwlrq1
+z,. lv wkh vwdqgdug qrwdwlrq iru pd{i3>zj1 Wr pd{lpl}h K Elgghu 4 qhyhu
fkrrvhv {Ayvlqfh kh zloo qhyhu zlq wkh vhfrqg dxfwlrq dqg zloo zlq wkh uvw
dxfwlrq rqo| zkhq hlwkhu {Ae

]4
Ay o r v l q jy  e

]4
r ul iy  e

]4
zkhq e| elgglqj { @ y kh zrxog kdyh zrq dv zhoo1 Wkhuhiruh zh pd| vxssrvh
7zlwkrxw orvv ri jhqhudolw| wkdw {  y= Dv e+, lv dq htxloleulxp/ zh kdyh e+y,  y=
Elgghu 4*v h{shfwhg xwlolw| fdq wkhuhiruh eh uhzulwwhq dv=
5K +{,@H
k
"{￿e+]4,

y  e

]4
.

y  pd{

{>]4.l
@
] e￿4+{,
3
+y  e+},,i]4 +},g} .
] 4
3
+y  pd{i{>}j,
. i]4 +},g} @
] e￿4+{,
3
+y  e+},,i]4 +},g} .+ y  {,I]4 +{,.
] 4
{
+y  },
. i]4 +},g}=
Wkh uvw rughu frqglwlrq lv
5K
3
+{,@

e￿4 +{,
3
+y  {,i]4

e￿4 +{,

 I]4 +{,.+ y  {,i]4 +{,  +y  {,i]4 +{,@3 =
Diwhu fdqfhoolqj wkh odvw wzr whupv zh rewdlq

e￿4 +{,
3
+y  {,i]4

e￿4 +{,

 I]4 +{,@3 =
Lq htxloleulxp { @ e+y, dqg wkxv zh rewdlq
+y  e+y,,i]4 +y,
e
3 +y,
 I]4 +e+y,, @ 3=
Uhduudqjlqj zh rewdlq wkh glhuhqwldo htxdwlrq wkdw fkdudfwhul}hv e+, dv iroorzv
e
3
+y,@
+y  e+y,,i]4 +y,,
I]4 +e+y,,
= +4,
Wkh glvwulexwlrq ri ]4 lv I]4 +x,@Iq￿4 +x, dqg lwv dvvrfldwhg ghqvlw| lv
i]4 +x,@+ q  4,Iq￿5 +x,i +x,= Vxevwlwxwlqj wkhvh h{suhvvlrqv lqwr +4, zh re0
wdlq
e
3
+y,@
+y  e+y,,+q  4,Iq￿5 +y,i +y,
Iq￿4 +e+y,,
= +5,
Qrwh wkdw e+y,@y lv qrw d vroxwlrq ri +5,1
Iru wkh xqlirup glvwulexwlrq/ iru h{dpsoh/ h{suhvvlrq +5, uhgxfhv wr=
e
3
+y,@
y  e+y,,+q  4,yq￿5
eq￿4 +y,
=
Lw iroorzv wkdw wkh ixqfwlrq e+y,@y vroyhv wkh glhuhqwldo htxdwlrq li
j+,@
q  +q  4,+4  ,@3 =
8Vlqfh j+3, @ q .4dqg j+4, @ 4 wklv htxdwlrq kdv d vroxwlrq u 5 +3>4,=
Pruhryhu vlqfh j +, lv vwulfwo| lqfuhdvlqj lw lv wkh xqltxh srvlwlyh vroxwlrq1 Li
q @5zh kdyh wkdw u @ ￿4.
s
8
5  3=95= +Zh rplw wkh surri wkdw wkh vroxwlrq
ri +5, lv uhdoo| dq htxloleulxp1, Wkdw lv/ elgglqj rqh*v wuxh ydoxdwlrq lv qrw
dq htxloleulxp lq wkh uvw urxqg1 Wkh uhdvrq lv wkdw e| uhyhdolqj rqh*v wuxh
ydoxdwlrq lq wkh uvw urxqg/ rqh frxog ehfrph slyrwdo lq wkh vhfrqg dxfwlrq dv
wkh sulfh lq wkh vhfrqg dxfwlrq lv vlpso| wkh pd{lpxp ri U5 dqg wkh vhfrqg
kljkhvw elg lq wkh vhfrqg dxfwlrq1
Qrwh wkdw li ex|huv eholhyh wkdw wkh vhoohu vhwv U5 @ e+y+4,, dqg wkh vhoohu lv
frqvwudlqhg wr fkrrvh d uhvhuyh sulfh U5  e+y+4,,> zkhuh y+4, lv wkh h{shfwhg
ydoxh ri wkh kljkhvw ydoxdwlrq/ wkhq klv h{shfwhg surwv duh
4
5
e+y+5,,.
4
5
pd{
q
y+5,>U 5
r
Fohduo|/ vhwwlqj U5 @ e+y+4,, pd{lpl}hv wkh vhoohu*v h{shfwhg surwv dqg
wkhuhiruh zh kdyh ixoo| ghvfulehg dq htxloleulxp iru wkhvh uhshdwhg wuldov ri
Ylfnuh| dxfwlrqv zlwk frqvwudlqhg vhoohuv16 Lq wkh qh{w vhfwlrq zh h{dplqh wkh
fdvh zkhuh wkh vhoohu pd| vhohfw dq| uhvhuyh sulfh1
5 Ixoo| Vwudwhjlf Vhoohu
Lq wklv vhfwlrq zh doorz wkh vhoohu iuhhgrp lq vhwwlqj dq| uhvhuyh sulfh kh olnhv1
Fohduo|/ li wkhuh lv d v|pphwulf htxloleulxp elgglqj vwudwhj| lq wkh uvw dxfwlrq
wkdw lv lqfuhdvlqj lq ydoxhv/ wkhq wkh vhoohu zloo lqyhuw uvw0shulrg elgglqj vwudwh0
jlhv/ frpsxwh wkh kljkhvw ydoxdwlrq dqg vhw wkh vhfrqg0shulrg uhvhuyh sulfh dw
wkh kljkhvw ydoxdwlrq1 Krzhyhu/ zh vkrz qh{w wkdw zlwk d ixoo| vwudwhjlf vhoohu/
wkhuh lv qr lqfuhdvlqj v|pphwulf htxloleulxp elgglqj vwudwhj| lq wkh uvw shulrg1
Lq sduwlfxodu/ elgglqj rqh*v wuxh ydoxdwlrq lv qrw d v|pphwulf elgglqj vwudwhj|
lq wkh uvw shulrg zlwk d ixoo| vwudwhjlf vhoohu hlwkhu1 Iru vlpsolflw| zh dvvxph
wkdw wkhuh duh rqo| wzr ex|huv dqg rqh vhoohu1
Wkhruhp 5 Wkhuh lv qr lqfuhdvlqj v|pphwulf htxloleulxp elgglqj vwudwhj| lq
wkh uvw shulrg zlwk d vwudwhjlf vhoohu1
Surri1 Wr rewdlq d frqwudglfwlrq vxssrvh wkdw wkhuh lv d v|pphwulf htxl0
oleulxp vwudwhj| e+, wkdw lv lqfuhdvlqj lq ydoxhv= Wkh vhoohu*v uhvhuyh sulfh iru
wkh vhfrqg dxfwlrq zloo eh U+e+y4,>e+y5,, @ pd{iy4>y 5j= Wr qg wkh ehvw
uhvsrqvh ri elgghu 4 zkhq elgghu 5 sod|v e+, vxssrvh elgghu 4 elgv {  3 zkhq
6Doorzlqj wkh vhoohu wr vhw wkh ￿uvw0shulrg uhvhuyh sulfh grhv qrw prgli| wkh sureohp ixqgd0
phqwdoo| dv ￿uvw0shulrg elgglqj vwudwhjlhv uhpdlq wkh vdph iru wkrvh elgghuv zlwk ydoxdwlrqv
deryh wkh uhvhuyh sulfh dqg ehfrph }hur rwkhuzlvh1 Zlwkrxw orvv ri jhqhudolw|/ zh dvvxph
wkdw wkh vhoohu vhwv wkh ￿uvw0shulrg uhvhuyh sulfh dw -￿ ’f 1
9klv ydoxh lv y= Klv h{shfwhg xwlolw| lv
k+{,@
4
5
H
k
+y  e+y5,,"{￿e+y5,
l
.
4
5
H
k
+y  U+{>e+y5,,,
.l
@
U e￿4+{,
3 +y  e+y5,,gy5
5
.
H
k
y  pd{

e￿4 +{,>y 5
.l
5
=
Elgghu 4 zloo qhyhu elg {Ae+y, vlqfh k+{, lv ghfuhdvlqj lq wklv uhjlrq1 Wkhuh0
iruh/ zh fdq vxssrvh zlwkrxw orvv ri jhqhudolw| wkdw {  e+y,= Wkxv
k+{,@
U e￿4+{,
3 +y  e+y5,,gy5
5
.
U y
3

y  pd{

e￿4 +{,>y 5
.
5
@
U e￿4+{,
3 +y  e+y5,,gy5
5
.
e￿4 +{,

y  e￿4 +{,

.
U y
e￿4+{, +y  y5,gy5=
5
=
Li zh ghqh $ @ e￿4 +{,> Elgghu 4*v sureohp lv wr qg $ wkdw pd{lpl}hv
5j +$,@
] $
3
+y  e+y5,,gy5 . $ +y  $,.
] y
$
+y  y5,gy5=
Wkh uvw0rughu frqglwlrq lv
5j +$,@y  e+$,.y  5$  +y  $,@y  $  e+$,@3 =
Vlqfh j +y,@e+y, ? 3 zh vhh wkdw wkh rswlpdo $?ydqg wkh rswlpdo
{?e+y,=
Uhpdun 6 Iru h{dpsoh li e+y,@y wkhq wkh ehvw uhso| ri elgghu 4 lv { @ y
5=
6 Frqfoxvlrq
Lq wklv vkruw sdshu zh surylgh d vlpsoh h{dpsoh ri uhshdwhg wuldov ri Ylfnuh|
dxfwlrqv zkhuh lw lv qrw dq htxloleulxp iru d elgghu wr elg klv ydoxdwlrq lq hyhu|
wuldo1 Wklv h{dpsoh looxvwudwhv wkdw dq| frqyhujhqfh  ehwzhhq zloolqjqhvv wr
sd| dqg zloolqjqhvv wr dffhsw pljkw qrw eh wkh frqvhtxhqfh ri ohduqlqj derxw
rqh*v ydoxdwlrq  dv vxjjhvwhg e| vrph hfrqrplvwv  exw udwkhu wkh uhvxow ri
vwudwhjlf ehkdylru1 Wklv vxjjhvwv wkdw shukdsv rqo| wkh h{lvwhqfh ri d zhgjh
ehwzhhq zloolqjqhvv wr sd| dqg zloolqjqhvv wr dffhsw dw wkh odvw wuldo vkrxog
eh vhhq dv dq dqrpdo| dqg qrw dw hduolhu wuldov zkhuh erwk ex|huv dqg vhoohuv
pd| eh jdplqj1
Uhihuhqfhv
^4` Frxuvh|/ G1 O1/ M1 O1 Krylv dqg Z1 G1 Vfkxo}h/ 4<;:/ Wkh Glvsdulw| Eh0
wzhhq Zloolqjqhvv wr Dffhsw dqg Zloolqjqhvv wr Sd| Phdvxuhv ri Ydoxh/
Txduwhuo| Mrxuqdo ri Hfrqrplfv 435+6,/ 9:<0<31
:^5` Frxuvh|/ G1 O1 dqg Z1 Vfkxo}h/ 4<;9/ Wkh Dssolfdwlrq ri Oderudwru|
H{shulphqwdo Hfrqrplfv wr wkh Frqwlqjhqw Ydoxdwlrq ri Sxeolf Jrrgv1
Sxeolf Fkrlfh 7</ 7:09;1
^6` Fr{/ M1 O1/ E1 Urehuvrq/ dqg Y1 O1 Vplwk/ 4<;5/ Wkhru| dqg Ehkdylru ri
Vlqjoh Sulfh Dxfwlrqv/ Y1 O1 Vplwk/ hg1/ Uhvhdufk lq H{shulphqwdo Hfr0
qrplfv LL1 Juhhqzlfk/ FW= MDL Suhvv1
^7` Gxerxuj/ Z1 U1/ P1 Z1 Mrqhv0Ohh/ J1 Orrphv/ 4<<7/ Lpsuhflvh Suhihu0
hqfhv dqg wkh ZWS0ZWD Glvsdulw|/  Mrxuqdo ri Ulvn dqg Xqfhuwdlqw| </
44804661
^8` Juhjru|/ U1 dqg O1 Ixue|/ 4<;:/ Dxfwlrqv/ H{shulphqwv dqg Frqwlqjhqw
Ydoxdwlrq/ Sxeolf Fkrlfh 88/ 5:605;<1
^9` Kdqhpdqq/ Z1 P1/ 4<<4/ Zloolqjqhvv wr Sd| dqg Zloolqjqhvv wr Dffhsw=
Krz Pxfk Fdq Wkh| GlhuB Dphulfdq Hfrqrplf Uhylhz ;4+6,/ 968097:1
^:` Ndkqhpdq/ G1/ M1 O1 Nqhwvfk dqg U1 K1 Wkdohu/ H{shulphqwdo Whvwv ri wkh
Hqgrzphqw Hhfw dqg wkh Frdvh Wkhruhp/ Mrxuqdo ri Srolwlfdo Hfrqrp|
<;+9,/ 46580467;1
^;` Ndkqhpdq/ G1 dqg D1 Wyhuvn|/ 4<:</ Survshfw Wkhru|= Dq Dqdo|vlv ri
Ghflvlrq Xqghu Ulvn/ Hfrqrphwulfd 7:+5,/ 59605<41
^<` Pruulvrq/ J1 F1/ 4<<;/ Xqghuvwdqglqj wkh glvsdulw| ehwzhhq ZWS dqg
ZWD= Hqgrzphqw Hhfw/ Vxevwlwxwdelolw|/ ru Lpsuhflvh SuhihuhqfhvB Hfr0
qrplfv Ohwwhuv 8</ 4;<04<71
^43` Pruulvrq/ J1 F1/ 4<<:0d/ Uhvroylqj Glhuhqfhv lq Zloolqjqhvv wr Sd| dqg
Zloolqjqhvv wr Dffhsw= Frpphqw/  Dphulfdq Hfrqrplf Uhylhz ;:+4,/ 5690
5731
^44` Pruulvrq/ J1 F1/ 4<<:0e/ Zloolqjqhvv wr Sd dqg Zloolqjqhvv wr Dffhsw=
Vrph Hylghqfh ri dq Hqgrzphqw Hhfw/  Dssolhg Hfrqrplfv 5</ 744074:1
^45` Vkrjuhq/ M1 I1/ V1 \1 Vklq/ G1 M1 Kd|hv dqg M1 E1 Nolhehqvwhlq/ 4<<7/ Uh0
vroylqj Glhuhqfhv lq Zloolqjqhvv wr Sd| dqg Zloolqjqhvv wr Dffhsw/ Dphu0
lfdq Hfrqrplf Uhylhz ;7+4,/ 58805:31
^46` Zhehu/ U1 M1/ 4<;6/ Pxowl0Remhfw Dxfwlrqv/  lq U1 Hqjhoeuhfkw0Zljjdqv/
P1 Vkxeln/ dqg U1 P1 Vwdun/ +hgv1,/ Dxfwlrq/ Elgglqj/ dqg Frqwudfwlqj=
Xvhv dqg Wkhru| +49804<7,1 Qhz \run= Qhz \run Xqlyhuvlw| Suhvv1
;

Strategic Behavior in Repeated Trials of Vickrey Auctions

http://bibliotecadigital.fgv.br/dspace/bitstream/10438/393/2/1215.pdf

Strategic Behavior in Repeated Trials of Vickrey Auctions

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics