Revenue and Efficiency in Revenue and Efficiency in Uniform-price Private Uniform-price Private

Value AuctionsValue Auctions

Michal.Bresky@Michal.Bresky@cergecerge-ei.-ei.czcz

Michal BresMichal Breskyky

(Summer 2007)

Properties of Private-value Uniform-price Auctions with the Reservation Price

Literature:

Dasgupta and Maskin (1986) RES, Reny (1996) Econometrica, Simon and Zame (1990) Econometrica, Matthew and Swinkels (1999) Econometrica, Amman and Leininger (1996) GEB, Krishna and Morgan (1997) JET, Engelbrecht-Wiggans and Kahn (1998a,b) GEB, Engelbrecht-Wiggans and Chakraborty (2005) ET,

Ausubel and Cramton (2002,2004) ET, Back and Zender (1993) RFS, Noussair (1995) ET.

Seller's goals:• Revenue - to raise high revenue.• Efficiency - to distribute units to those who value them the most.• Fair competition - to prevent collusion among bidders and attract large number of bidders before auction.• Market structure - to prevent monopoly in using the sold units after auction.

Results:

• The equilibrium strategy is strictly increasing.

• The reservation price increases the bidder strategy.

• The proper reservation price improves the efficiency of final allocation ex ante.

Properties of Private-value Uniform-price Auctions with the Reservation Price

THE GAME)1kk ( )1nn (

.];,[ 2,1,2,1, iiiii vvvvv where values sBidder'support compact the on values of PDF the - )( ii vG

The ex post payoff when losing the unit is 0.

Vi j 1l 0,v j

identical units of goods for sale.bidders in an auction.

Independence is required between a pair of values of two different bidders.bi,1 , ,bi,k 1, B bids submits bidder Each k

Uniform-price auction - the auction price for winning the unit is the 1st rejected bid, submitted. were above bids enough not if bid nreservatio or RR

.

.

A tie occurs when kth and k-1st highest bids are equal (random tie-breaking).

Ex post payoff is: iv i,b,R

j 1

J i

v i,j maxR,c, #

bids. submitted all of bid highest st the is

and units, winningnumber the is where

1,max 11, kcbc

J

ii JkJi

i

is bidding with bidder to payoff anteex the then , strategies opponents Given bvib

v,b|b E v,b,c,R. #

,, vbvEb

bi

: is payoffstrategy pure anteex his then

, strategy pure uses bidder the When

Shape of equilibrium strategies

b2v 1 ,v 2 ,R b2v 2 ,R for v 2 R,v2 , b2v 1 ,v 2 ,R 0 for v 2 0,R. #

b1v 1 ,v 2 ,R v 1 for v 1 0,b2

v1 for v 1 b2 ,v1 where b 2

v1 if 1 k n

b2v 2 ,R if k n

v2 if n k 2 n

. #

In equilibrium the first unit bid strategy is:

In equilibrium the subsequent unit bid strategy is:

No Ties in Equilibrium with Positive Probability

jkc 1

unit th the on bidder a ofStrategy j

jb

11

jkcF

in point Mass

jkc 1 ondistributi Inverse

Tie

jb

vb j

vb j

vb j

jkc 1

jv00

exists. strategy response better a then ),( value his bid not does he

point massany for andy probabilit positive withtied is unit th on bid a that such uses bidder the If

. strategies opponent givenany and defined game auction price-uniform the Consider

Bbvvb

jb

BbLemma

jj

:

v V

0. is revenue seller´s and efficient not is units of allocation Final

if

if and isstrategy mequilibriu Nash Then

otherwise and if i.e. uniform, is

value subsequent and initial the of functiondensity The . Suppose

.0

,,,,

.0,0,

2:

2

22121211

21212

21 2

Rv

RvRRvvbvRvvb

vvgvvvvvg

nkAExample

v

.,,)(

)(,,

.3:

112112

22

212 RvvRvvbRvRRv

RvRvvb

kExample B

for and for isstrategy mequilibriu Nash Then

i.e. sale, for unit additionel one offers seller but A,example in as settings same the Suppose

above values for abovestrictly bid bidders for therefore and ties, no are there then If .021 RRRjlk

y.probabilit positive with bid bidders whichin mequilibriusymmetric no is there Then price nreservatio at value a of

ondistributiy probabilit in point mass a have bidders the of none and satisfies wherebids most at

submit to restricted is bidder each positive, is price nreservatio valid, is C1 that Suppose

RR

lklnll

RCorollary

.

,212

No Bidding at Positive Reservation Price with the Number of Bids Restriction

Note: In my game l=2. Then the inequality is 2.n>k>2

The Effect of Reservation Price on Continuous Equilibrium Strategy

. any and any For

:C2 strategies continuous for Condition

0,,,,

,, 221222

22122

2

222222

vbPvbPb

vbPbv

b

bvvRbvRv kk

k

.332

)1.

,012

1

22221

1

nkg

nkgbgbgk

gnk

price nreservatio the above are bids all when winnot does bids threeexactly and increasing weakly is If 3.

.( bidders than sale for units more not is there and increasing weakly is and If 2

and price nreservatio the above are bids all when winnot does bid oneexactly If 1.

:continuous isstrategy mequilibriusymetric every hence and C2 conditionsatisfy that onsdistributi someshow me Let

This condition is necessary because subsequent bid increases both the probability of winning subsequent unit andthe price the bidder pays for the initial unit if subsequent bid is the first rejected bid.

.,,,

0,,

312,32,,33:

.ˆ,

,0ˆˆ

,3:

22222

2222

2

22

11

vRvRvbRvb

RRRvbRvb

nknknTheorem

bv

Rbvvbv

bvkTheorem

any for Then

. prices nreservatio different for and strategies mequilibriusymmetric two Consider

. or valid are C2 and C1 conditions and , that Assume

point through goingstrategy mequilibriusymmetric one most at is there

that such bid and valueany for then

valid, is C2 condition and use bidders that Assume

The Effect of Reservation Price on Continuous Equilibrium Strategy

.ˆ tojumpmust responsebest sbidder' then the

,ˆ of odneighborhoright on thestrategy his

decreasesopponent theif Therefore

positive. is payoff marginal the

,ˆ1ˆ2minˆ,ˆ and ,ˆ,ˆ

,ˆ2,0for Similarly response.best symmetric a

yields abovestrategy opponent no Hence

.ˆat ousdiscontinuor ,ˆ of odneighborhoright on the

constant bemust responsebest sbidder' the

then ,ˆ of odneighborhoright on the

strategy his increasesopponent theif Therefore

2222

2

v

v

avavvvbvvv

va

b

vv

v

2222

2222

,

,

bvbv

bvbv

2b 22 vb

2v 2v

2b

2b

.,ˆminˆ,ˆ and ˆ,ˆ ,,ˆ2for negative isIt

ˆˆ

1ˆˆ

12,

is payoff marginal Then the

.1,ˆfor strategy increasingly other weakany

ˆ,ˆfor ˆˆ

ˆ,0for

0 somefor e.g. ,01,0ˆ some of odneighborhoright in the

above ishat strategy t continuousanother usesopponent that theAssume .strategy its changes

opponent when thechanges responsebest mequilibriu thehowconsider and in as setting same the takeusLet :

22

222

222

22

222

22

2

22

2

22222

22

22

2

vvvvavvbva

va

vbv

a

vb

abv

b

bv

vv

vvvvvva

vvvvb

vb

av

bb

bBExample

The Effect of Reservation Price on Efficiency

.,

,,,,

..1

0

1211

PLossRv

SLossRvbRvb

DLoss

region pooling the over bid same the bidding from loss 2b)

- values subsequent and initial on strategies in difference the to due loss 2a)

:strategy of shape the from Loss 2.

- price nreservatio the above bids submitted of number small the to due Loss

SLoss

1v2v1v2v

2b2b

1b

1b

Rb 2 Rb 2

The Effect of Reservation Price on Efficiency

).,0(0

)2/1(

)2:

00 RRR

kn

kTheorem

interval an in for increases efficiency auction that such exist there then

,strategies mequilibriu same thefollow bidders the and demand sbidder' than greater

is units auctioned of number and ( auction the in units three least at be there Let

.strategies mequilibriu continuous ofzation characteri with problem a is e then ther2k if :Note

decreases. efficiency ante-ex hein which t ),0( interval exists always n there the

strategy, continous same play the bidders and 122 if 2.

y,efficientl allocated areauction in sold units then the,12or 1 if 1.

:into results thesummarizecan wegeneral,In

0

LossR

nk

nkk

Thank you for your attantion.