Revenue and Efficiency in UniformMPrice Private ... - Editorial Express

0 downloads 0 Views 375KB Size Report
reservation price above f in the uniform0price auction. ... of initial and subsequent units as GБ EvБ and G2 Ev2 , and their densities as. gБ EvБ and g2 Ev2 . ... If less than & n Б bids are submitted above or equal to the reservation price R, then ..... general formula for the mean value of Uloss is shown below (4.2) and can be.

Revenue and Efficiency in Uniform-Price Private Value Auctions By Michal Bresky∗ This paper shows that in a private value multi-unit uniform-price auction, the reservation price increases both the efficiency and revenue. In equilibrium the difference between the true value of a unit and the submitted bid is different for each unit; therefore, the seller cannot allocate units efficiently. When the seller increases the reservation price, the bidders increase their bids on the units with a larger difference between the value and bid more. Some bidders with a low value do not participate in the auction. Therefore the units are allocated among those with high initial and subsequent values that is efficient. For a low range of reservation prices is, the higher the reservation price the higher the expected efficiency and revenue. Keywords: multi-unit auction, multiple-object auction, market efÞciency, optimal selling mechanism, discriminatory and uniform price independent-private value auction with reservation price. JEL ClassiÞcation D44.



I would like to thank Philip Reny, Avner Shaked, Viatcheslav Vinogradov, Michal

Kejak and Jan Hanousek for their helpful comments. I would appreciate any comments that you can sent to [email protected]

1

41

LQWURGXFWLRQ

Dq dxfwlrq lv dq h{fkdqjh phfkdqlvp zlwk dv|pphwulf lqirupdwlrq1 Lw fdq eh wuhdwhg dv d jdph lq zklfk wkh vhoohu rhuv rqh ru pruh xqlwv +ri wkh vdph w|sh, wr wkh sduwlflsdqwv1 Wkh vhoohu grhv qrw nqrz wkh elgghu*v ydoxh ri dq| sduwlfxodu xqlw/ exw kh fdq vhw xs dq h{solflw vhw ri lqvwlwxwlrqdo uxohv ghwhuplqlqj uhvrxufh doorfdwlrq dqg sulfhv rq wkh edvlv ri elgv iurp wkh dxfwlrq sduwlflsdqwv1 Krzhyhu/ rqo| lq Ylfnuh| dxfwlrqv +Ylfnuh| +4 4> 5, elgghuv kdyh wkhlu lqlwldo ydoxhv deryh

l5

U

l4

lv wkh suredelolw| wkdw h{dfwo|

dqg vxevhtxhqw ydoxhv ehorz

h{dfwo| l5 +l5 @ 3>    > 5  l4 ,elgghuv kdyh erwk ydoxhv deryh erwk ydoxhv ehorz U=

45

U

/

/ dqg rwkhu elgghuv kdyh

U

uhvhuydwlrq sulfh/ exw wkh rwkhuv vxeplw kljkhu elgv dqg vkdgh ohvv ri wkhlu wuxh ydoxhv1 Wklv phdqv wkdw elgv pruh suhflvho| uhyhdo wkh wuxh ydoxhv/ dqg wkh vhoohu doorfdwhv xqlwv h!flhqwo| pruh riwhq1 Lq rwkhu zrugv/ pdq| lqh!flhqflhv fdxvhg e| wkh glhuhqfh lq lqlwldo dqg vxevhtxhqw vwudwhjlhv fdq eh holplqdwhg e| wkh surshu uhvhuydwlrq sulfh vhwwlqj1 L lghqwli| wzr uhdvrqv iru wkh orvv lq h!flhqf| +ru zhoiduh, rffxuv= 41 Orvv gxh wr xqdoorfdwhg xqlwv wkdw rffxuv zkhq ohvv wkdq & elgv duh vxeplwwhg deryh -/ LuJrr 51 Orvv gxh wr plvdoorfdwhg xqlwv wkdw rffxuv gxh wr wkh glhuhqfh lq vwudwh0 jlhv rq lqlwldo dqg vxevhtxhqw ydoxhv/ uJrr1

44

L uJrr lv/ exw uJrr lv qrw/ suhvhqw lq d vwdqgdug vlqjoh0xqlw dxfwlrq v|pphwulf htxloleulxp1 Lq jhqhudo/ L uJrr rffxuv li qrw doo & xqlwv duh vrog lq wkh dxfwlrq ehfdxvh ohvv wkdq & elgv duh vxeplwwhg deryh d wrr kljk uhvhuydwlrq sulfh1 Wklv phdqv wkdw li dq| ydoxh  +lqlwldo ru vxevhtxhqw, ehorz wkh uhvhuydwlrq sulfh lv rqh ri wkh & kljkhvw ydoxhv/ wkhq LuJrr rffxuv1 Orvv lq wkh vlqjoh0xqlw dxfwlrq zlwk wkh uhvhuydwlrq sulfh rffxuv iru wkh vdph uhdvrq1 Wkh h{shfwhg ydoxh ri wklv orvv/ ghqrwhg dv L uJrr E- c lv fkdudfwhul}hg e| wkh iroorzlqj wkhruhp +suryhg lq wkh Dsshqgl{,1 Wkhruhp 51 Ohw  / 2 eh udqgrp lqlwldo dqg vxevhtxhqw ydoxhv ri dq djhqw/ dqg ohw M E eh d wk kljkhvw rughuhg vwdwlvwlf ri 2?  2 rssrqhqwv* ydoxhv1  Edw prvw  rssrqhqw ydoxhv   ' M E ' [

'

2 2 n  $ 44

E?  E  C2 E2 EC2 E  C E C?  - E?       2   2 

+714,  2 3 

3 3

E 1

Lq sulqflsoh wkhuh lv dqrwkhu uhdvrq iru d orvv wkdw rffxuv li wkh wzr elgv iru glhuhqw

ydoxhv duh wlhg dqg wkh vhoohu udqgrpo| fkrrvhv zklfk ri wkhp zlqv1 Exw wlhv fdqqrw rffxu lq dq htxloleulp zlwk srvlwlyh suredelolw|1

46

Wkhq wkh phdq ydoxh ri LuJrr lv 3 ] L uJrr E- ' ?  C  M&

4

]3

 E  _C E 

n

f

2 M&

2 E2  _C2 E2 D 

3

f

+715,

L uJrr E- lv lqfuhdvlqj lq -c dqg LuJrr Ef ' fc dqg li C E dqg C2 E duh frqwlqxrxv/ wkhq L uJrr E- lv frqwlqxrxv lq - Wkh irupxod +614, h{suhvvhv wkh phdq ydoxh ri uJrr lq wkh Vhfwlrq 6 h{dpsoh1 Lq jhqhudo/ iru &  2/ frqvlghu wzr elgghuv dqg wkhlu rssrqhqwv lq wkh iroorzlqj fdvh1 Dw ohdvw &  2 rssrqhqw ydoxhv dqg erwk vhfrqg elgghu ydoxhv duh deryh wkh uvw elgghu lqlwldo ydoxh/ dqg wkh uvw elgghu lqlwldo elg zlqv d xqlw1 Lq dgglwlrq/ dw prvw &  2 rssrqhqw ydoxhv duh deryh wkh vhfrqg elgghu vxevhtxhqw ydoxh wkdw lv deryh wkh uhvhuydwlrq sulfh/ dqg klv vxevhtxhqw elg grhv qrw zlq d xqlw1 Wkhq wkh plvdoorfdwlrq orvv/ fdoohg uJrrc rffxuv1 Wkh h{whqw ri uJrr lv wkh glhuhqfh ehwzhhq wkh vxevhtxhqw ydoxh ri wkh uvw elgghu dqg wkh lqlwldo ydoxh ri wkh vhfrqg elgghu1 Wkh phdq ydoxh ri wklv orvv/ ghqrwhg dv uJrr E- c lv fkdudfwhul}hg e| wkh iroorzlqj wkhruhp +suryhg lq wkh Dsshqgl{,1 Wkhruhp 61 Wkh irupxod iru uJrr E- lv ]] uJrr E- ' E2    ? E?   l E2 c   _C E  _C2 E2  / 2 ME-c72 c  MEK2 E2 c-c2 

zkhuh l E2 c   '

[ f$ $e &2 f3

+716, E?  2l E2 c   - E?  & n  E&  2  2-

l E2 c   ' E  C2 E2  EC2 E2   C E &

2 3 2

3

C?

3

& n

E  

Iru & 2?   uJrr E- lv d ghfuhdvlqj ixqfwlrq ri - li wkh htxloleulxp vwudwhj| K2 E2 c - lv lqfuhdvlqj lq - iru dq| 2 5 E-c 72 o  Pruhryhu/ li wkh htxloleulxp vwudwhj| lv frqwlqxrxv lq -c dqg C E dqg C2 E duh frqwlqxrxv/ wkhq uJrr E- lv frqwlqxrxv lq - 47

Ohw ph qrz lqyhvwljdwh wkh hhfw ri - rq uJrr E- dqg L uJrr E- li pruh wkdq rqh vxeplwwhg elg fdqqrw zlq +& ' 2c    c 2?  2,1 Qrwh wkdw li uJrr rffxuv/ wkhq doo & xqlwv duh vrog lq wkh dxfwlrq exw li L uJrr rffxuv/ ohvv wkdq & xqlwv duh vrog1 Wkhuhiruh wkh| gr qrw rffxu vlpxowdqhrxvo| dqg wrwdo uJrr E- lv mxvw wkh vxp ri LuJrr E- dqg uJrr E-1 Wkh iroorzlqj wkhruhp +suryhg lq wkh Dsshqgl{, vkrzv wkdw wkh uhvhuydwlrq sulfh wkdw plqlpl}hv wkh wrwdo h{shfwhg orvv lv w|slfdoo| srvlwlyh1 Wkhruhp 71 Frqvlghu d v|pphwulf htxloleulxp vwudwhj| lqfuhdvlqj lq - iru - forvh wr f zkhq & ' 2c    c 2?  2 Dvvxph wkdw iru vrph 7f : f dqg hyhu| 2 5 dfc 7f o wkh elgghu elgv f li wkh uhvhuydwlrq sulfh lv f Wkhq uJrr Ef n L uJrr Ef : uJrr E- n L uJrr E-  Pruhryhu dvvxph wkdw iru - voljkwo| deryh f wkh vwudwhj| lv frqwlqxrxv lq -c dqg } E dqg }2 E h{lvwv dqg vdwlvi| wkh iroorzlqj frqglwlrq }2 E EC2 E  C E C& *4  fn } E

 

3 3

E




Wkhq

Wkdw lpsolhv ohvv uhvwulfwlyh J4

+y, @

K

5 +y ,

dqg

J5

+y, @

dqg wkh olplw whup lq frqglwlrq +717,

5+4K +y,,k+y,+5K +y,+4K +y,,,l K 5+n4l, +y,

K +y,k+y, K 5n6l +y,

5

@

wkdw lv 3 lq wkh olplw iru dq| K +y, =  K +y,, 5 Li  lv olpU3 $ . vxs Ce CU +U > U , dqg n @ 6> d vx!flhqw frqglwlrq wr jxdudq0 whh wkdw Orvv lv ghfuhdvlqj iru U voljkwo| deryh 3 lv olpU$ . J + UJ,4UJ U ? 4 dqg l.4

5l y +4 46

3

3

3

+

$

olpU

3.

J5 +U,J4 +U, J5 + U  ,J5 +U,

3

? 4=

48

5



,

5+

,

frqglwlrqv rq wkh glvwulexwlrq ri elgghu ydoxhv dqg doorzv xv wr dsso| wkhruhp 7 li wkh qxpehu ri elgghuv lv ohvv wkdq wkh qxpehu ri xqlwv iru vdoh1 Lq vxppdu|/ d srvlwlyh uhvhuydwlrq sulfh ghfuhdvhv h!flhqf| orvv lq wkh xqlirup0sulfh dxfwlrq lq pdq| fdvhv1 Lq Hqjhoeuhfkw0Zljjdqv dqg Ndkq +4

Suggest Documents