A COMPARISON BETWEEN SIMULTANEOUS AND ...ex-ante expected profits to the bidders in Ex-ante expected...

A COMPARISON BETWEEN SIMULTANEOUS ANDSEQUENTIAL AUCTION PROCEDURES: FROM SYMMETRIC TO ASYMMETRIC INFORMATION SETTING

Aleesha MOHAMUDALLY Paul PEZANIS-CHRISTOU

BETAUniversité Louis Pasteur Strasbourg I

Conference of the French Economic Association (AFSE) Conference of the French Economic Association (AFSE) 2323--25 May 2007(GATE, Lyon)25 May 2007(GATE, Lyon)

From single unit to multi-unitauctions

Several issues arise when focusing on multi-unit auctions: one of them is how to auction all the units? simultaneously or sequentially?Sequential auctions differ from simultaneous ones in terms of information released in between the sales which leads to two opposing effects: information and deception effects.Our interrogations:

o What would be buyers bidding strategies ?o Which auction type generates more revenue to the

seller?

Existing Literature: Symmetric information setting (1)

Hausch D.B. [1986], “Multi-object auctions: Sequential v/s Simultaneous sales”Symmetry implies that bidders have same quality of information about the object. General frameworkCommon-value environment: The actual value of the objects is the same to the bidders but unknown prior to the sale.Discriminatory auction format :Winners pay the amount equal to their own bids2 identical units are auctioned off to two bidders (1 and 2)Random variable V: True but unknown value of units prior to the sale


Each bidder receives a private and independent signal Sidrawn from the same distribution conditional on the true value.

o signals can be “High” or “Low”o Bidders have correct information if they receive a high

(respectively low) signal when the value of the objects is high (respectively low)Notations:

V(s,t)=E[V/(S1 , S2) =(s,t)] for i=1,2p(s,t)= probability that (S1 , S2) =(s,t)p(s/t)= probability that Si=s conditional on Sj=t for i,j=1,2i≠j


Theoretical predictionsSimultaneous auctions

o Repetitions of 2 isolated single unit auctionso In each auction equilibrium bidding strategies is such that:- Players with signal L bid V(L,L) in both auctions- Players with signal H draw one bid b in both auctions from the

interval derived from the cumulative distribution

with

[ ]( , ), ( , ) ( / ) ( , ) ( / )b V L L V H H p H H V L L p L H∈ +( )( )

( , ) ( / )( )

( , ) ( / )b V L L p L H

J bV H H b p H H−

=−

0 ( ) 1J b≤ ≤


o Ex-ante expected profits to each bidder in each auction:

o Seller’s expected revenue for the simultaneous auctions:

( )( , ) ( , ) ( , )V H L V L L p L H−

[ ] ( )2 2 ( , ) ( , ) ( , )E V V H L V L L p L H⎡ ⎤× − −⎣ ⎦14444444244444443

.

Seller’s expected revenue in a single auction


Sequential auctions:o Two possible forms of equilibria:

---->Equilibrium in which the 1st auction is non>Equilibrium in which the 1st auction is non--revealing:revealing:-In the first auction both players bid V(L)

In the second auction a low signalled player bids b=V(L,L)and a high signalled player draws a bid b from the cumulative distribution J(b)

-This occurs under the condition that(1)2( ( , ) ( , )) ( / ) ( ) ( )V H L V L L p L H V H V L− ≥ −


- The seller’s expected revenue is non revealing sequential auctions is given by:

If condition (1) is satisfied sequential sales generate higher revenue for the seller otherwise we consider the second form of equilibrium in whicha player with a high signal randomizes between bidding like a low signalled player and revealing his high signal in the first auction.

( ) ( )12 ( ) 2 ( ) ( ) ( ) ( , ) ( , ) ( / )2

E V V H V L p H V H L V L L p L H⎧ ⎫⎡ ⎤× − − + −⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭

Objectives of the paperExtend to multi-unit auctions with asymmetric information setting

Carry out laboratory experiments

To test theoretical predictions in terms of bidding behavior and revenue generated for simultaneous and sequential auctions in both informational settings

To asses the impact of change in informational setting on bidding strategies and seller’s revenue

Two-unit auction model with two asymmetrically informed bidders (1)

General framework:Same as in symmetric setting except that bidder 1 is better informed than bidder 2.This means that each bidder draws his signals from different distributions conditional on true value V.We consider a particular case of asymmetry in which the less informed bidder (player 2) receives completely uninformative signals. This situation may be appropriate in auctions where all bidders have access to the same public information but some bidders who are not specialists are not able to interpret and make use of the available information


Under this particular case the following simplifications occurp(H,L)=p(H,H), p(L,L)=p(L,H), p2(H)= p2(L), p1(H/L)= p1(H/H), p1(L/L)= p1(L/H), V(L,L)=V(L,H)=V1(L), V(H,L)=V(H,H)=V1(H) and

V2(H)=V2(L)=p1(H) V1(H)+p1(L) V1(L)


Theoretical predictionsSimultaneous auctions

o player 1’s (better informed) equilibrium strategies for each auction are as follows: If he receives L signal he bids V1(L)

If he receives H signal he draws his bid b from the bid distribution

( )( ) ( ]1 1

1 1 1 1 1 11 1

( ) ( )( ) . ( ), ( ) ( ) ( ) ( )

( ) ( )H

b V L p LG b for b V L p H V H p L V L

V H b p H−

= ∈ +−


o As far as the player 2 (uninformed bidder) is concerned, hewill not take into consideration his own signals but he willuse information about the better informed bidder’s equilibrium strategies to develop his own. In fact the best strategy by player 2 (uninformed bidder) with high or low signals will be to draw his bid b from a uniform distribution

[ ] [ ]1 12 2 2 1 1 1 1 1 1

2 1

( ) ( )( ) ( ) ( ) ( ) . ( ), ( ) ( ) ( ) ( )

( ) ( )H L

b V L p HG b G b G b p L for b V L p H V H p L V L

V L V L−

= = = + ∈ +−


o The ex-ante expected profit to player 1 in each auction =

o The ex-ante expected profit to the uninformed player 2 =0o The expected revenue for simultaneous auctions:

2×{E(V)- }

[ ]1 1 2( ) ( ) ( )p H V H b G b− [ ] [ ]1 21 1

2 1

( ) ( )( ) ( ) 1

( ) ( )p H V L b

p H V H bV L V L

⎡ ⎤−= − −⎢ ⎥−⎣ ⎦

[ ]1 1 22 ( ) ( ) ( )p H V H b G b× −


Sequential auctionso We again distinguish between 2 possible forms of

equilibium: the non-revealing first auction andthe equilibrium in which the better informed player with a

high signal will randomize between bidding like a low signalled bidder and revealing his high signal.

o In this paper with we will analyse the non-revealing first auction equilibrium


o The non–revealing equilibrium strategy will be of the form:1. In the first sale

Player 1 always bids V1(L).Player 2 always bids V1(L) .

2. In the second sale: Player 1 (better informed) with a low signal bids V1(L ) and if he receives a high signal he draws his bid b from the bid distribution G1H(b) and Player 2 draws

his bid from the distribution G2(b).


o Sequential sale schemes with a non-revealing first auction will be an equilibrium under the condition

≥ (2)

o Ex-ante expected profits to player 1 is equal to

o Ex-ante expected profits to player 2 is equal to

[ ]1 2( ) ( )V H b G b− [ ]1 11 ( ) ( )2

V H V L−

[ ] [ ]1 1 2 1 11( ) ( ) ( ) ( ) ( )2

p H V H b G b V H V L⎧ ⎫− + −⎨ ⎬⎩ ⎭

[ ]1 1 11 ( ) ( ) ( )2

p H V H V L−


Ordering of seller’s revenueWe notice that

For and

Under the non revealing equilibrium, simultaneous auctions will be favored by the seller.

( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )1 1 2 1 1 1 1 1 2ex-ante expected profits to the bidders in Ex-ante expected pnon revealing equilibrium sequential auctions

p H V H b G b p H V H V L p H V H b G b⎡ ⎤ ⎡ ⎤ ⎡ ⎤− + − ≥ × −⎣ ⎦ ⎣ ⎦ ⎣ ⎦144444444424444444443rofits to the

bidders in simultaneous auctions

1444442444443

[ ]1 1 1 1 1( ), ( ) ( ) ( ) ( )b V L p H V H p L V L∈ + 2 ( ) 1G b ≤

Experimental Framework (1)

Discrete version of the models:V takes the value of 10 with probability θ and 0 with a probability 1- θ.

Signals received: High(which means that the value is 10) or Low( which means that the value is 0)

The probability of receiving correct signal (information) is

( / 0) ( / 10)p L V p H V p= = = =


Symmetric information: Both bidders draw their signals from same distribution. p1=p2=p

Condition (1) for non revealing equilibrium in sequential auctions becomes

(3)2

2 2

(1 ) (1 ) (1 )2(1 ) (1 ) (1 )(1 ) (1 )(1 ) (1 ) (1 )

p p p p pp p p p p p p p

θ θ θθθ θ θ θ θ θ θ θ

⎛ ⎞− − −− ≥ −⎜ ⎟− + − + − − + − − − + −⎝ ⎠


Asymmetric information settingPlayer 1 receives better signal than the completely uninformed player 2

p1>p2=0.5Condition for non revealing equilibrium becomes

( )1 11 1 1

11 1 1 1 1 1

1 1

(1 )(1 ) (5 )10 (1 )1 510 (1 )(1 )(1 ) (1 )(1 ) (1 ) (1 )5(1 ) (1 )

p p bp p pb pp p p p p pp p

θ θ θθ θ θθθ θ θ θ θ θθθ θ

⎡ ⎤⎢ ⎥+ − − −⎡ ⎤ ⎡ ⎤−⎢ ⎥− − ≥ −⎢ ⎥ ⎢ ⎥−+ − − + − − − + −⎢ ⎥⎣ ⎦ ⎣ ⎦−⎢ ⎥− + −⎣ ⎦

(4)


Choice of parametersθ =0.8 The value of the objects is 10 with 80% of chance.For the symmetric setting we consider 2 treatments p=1/2 :Signals are uninformativep=2/3 :signals are correct 2 times out of 3parameters are such that condition (3) is satisfied with strict equalityFor the asymmetric setting: We choose p1=2/3>p2=0.5 such that condition (4) is satisfied


Treatments

(2/3,1/2)(2/3,2/3)(1/2,1/2)Sequential auctions

(2/3,1/2)(2/3,2/3)(1/2,1/2)Simultaneous auctions

Asymmetric auctions

AuctionsSymmetricθ=0.8 (p1,p2)


Conduct of the experimentExperiments conducted in LEES.Each session lasted about 80 minutes students gained on average 25 eurosIn each treatment:18 participants (3 groups of 6)60 periods:Every period, the software determines

o Pairs randomly within the groups, o draws the true value for each pair and o sends independent private signals.

Theoretical predictions (1)

Bid Distributions

SYMMETRIC AUCTIONS: Theoretical bid distribution of J(b)

0

0 .2

0 .4

0 .6

0 .8

1

1.2

0 0 .5 1 1.5 2 2 .5 3 3 .5 4 4 .5 5 5.5 6 6 .5 7 7.5 8 8 .5 9 9 .5 10

b id s

p=2/3


ASYMETRIC AUCTIONS:Theoretical bid distributions

0

0.2

0.4

0.6

0.8

1

1.2

0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7 7.7 8.4 9.1 9.8

bids

Proba

bility

G1H(b) G2(b)


Seller’s average revenue

[12.03;12.26]13.3316Sequential auctions

[13.39;13.85]13.3316Simultaneous auctions

Asymmetric

(2/3,1/2)

Auctions

(2/3,2/3)

Symmetric

(1/2,1/2)θ=0.8 (p1,p2)

RESULTS: Bidding strategies(1)Symmetric settingSimultaneous auctions:

p=1/2: of equality of bids between high signaled andlow signaled bidders (p-values=0.588 and 0.089 forthe first and second unit respectively)

p=2/3: of equality of bids between high signaled andlow signaled bidders (p-values=0.0004 and 0.0001 forthe first and second unit respectively)

RH0

RH0

RESULTS: Bidding strategies(2)

Bid for first unit

0

1

2

3

4

5

6

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57periods

bids

High signal Low signal

bid for the second unit

0

1

2

3

4

5

6

periods

bids

high signal low signal

RESULTS: Bidding strategies(3)

Sequential auctions with p=2/3o Sale of the first unit

we test H0:equality of bids between high signaled and low signaled bidders.

if all data considered (p-value=0.0019)if all last 20 periods considered(p-value=0.0559)

Bidders tend to bid optimally towards the end of the experiment

RH0 RH0

RESULTS: Bidding strategies (4)

o Sale of the second unitRH0 equality of bidsbetween winners andlosers of the first unit. (p-values=0.0263 and0.0620 for H and L signaled bidders resp.)

Results: Bidding strategies (5)Asymmetric setting

(pvalues=.0545 for 1st unit & .1143 for2nd unit)

(p-values=.1325 for 1st unit & .2027 for 2nd unit)

Uninformed bidders

p-values=.0235 for 1st unit & .0087 for 2nd unit)

(p-values=0.0235 for 1st unit & 0.0087 for the 2ndunit

Better informed bidders

Sequential auctionsSimultaneous auctionsH0: Equality of bids between H and L signaled bidders

RH0

RH0

RH0

RH0

Results: Seller’s revenue (1)Symmetric setting

o RH0: Equality of revenue between simultaneous and sequential auctions For p=1/2 (p-value=0.7355) an for p=2/3 (p-value 0.4841)

Experimental results are consistent with predictions

o In The case where p=1/2 the level of revenue significantly less than predicted revenue of 16.

This could be explained by attempts of collusion between bidders pushing bid levels downwards

Results: Seller’s revenue (2)

Asymmetric settingo Prediction: seller’s revenue higher in simultaneous auctions

than in sequential auctions.

But of equality of revenue between simultaneous andsequential auctions

seller's average revenue in asymmetric information setting

0

2

4

6

8

10

12

14

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58

periods

av

era

ge

re

ve

nu

e

sequential auctions simultaneous auctions

RH0


Comparison between symmetric and asymmetric settingDoes asymmetry of information increases or decreases seller’s revenue ?

o General notion in auction theory: increasing (respectively decreasing) asymmetry decreases (respectively increases) seller’s revenue

o There exists some cases where theory predicts opposite results: that is reducing asymmetry reduces seller’s revenue.

Results: Seller’s revenue (4)Case 1: player 1’s signal gets better

(1/2,1/2) ----------- (2/3,1/2)Revenue decreases from 16 to a range [13.39;13.85] in simultaneous auctions and [12.03;12.26] in sequential auctions:

Case 2:player 2’s signal gets better(2/3,1/2) ----------- (2/3, 2/3)Revenue decreases from [13.39;13.85] to 13.33 in simultaneous auctions and increases from[12.03;12.26] to 13.33 in sequential auctions:


Case 1Sim(1/2,1/2)=Sim(2/3,1/2) p-value=0.2171Seq(1/2,1/2)=Seq(2/3, 2/3) p-value=0.0079

Case 2Sim(2/3,1/2)=Sim(2/3,2/3) p-value=0.3714Seq(1/2,1/2)=Seq(2/3, 2/3) p-value=0.0254

RH0RH0

RH0RH0

Summary and conclusion (1)PredictionsExtension from symmetric setting to a particular case of asymmetry of information .

o We derived equilibrium bidding strategies for simultaneous auctions (2 isolated one shot auctions)sequential auctions with non revealing first auctions

o We determined seller’s revenue and found that under the condition in which the first auction is non revealing the seller favors simultaneous auctions over sequential auctions

Summary and conclusion (2)

ExperimentsBidding strategies

In symmetric information setting experimental data are consistent with theoretical predictions in both auctions procedure

In asymmetric information setting, we find that in the first sale of sequential auctions high signaled bidders with informative signals submit bids that deviate significantly from the optimal strategy of non revealing their signals

Summary and conclusion (3)

Seller’s revenueo Symmetric settings

Revenue equivalence between simultaneous and sequential auctions in both cases

o Effect of change in information setting on seller’s revenue:our experiments tend to confirm the general notion that increasing (respectively decreasing) asymmetry of information decreases (respectively increases) seller’s revenue. The potential for asymmetry of information to increase seller’s earnings that may result in some cases is not explicitly observed on the basis of our data

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