Traditionally in a Double Auction (also known as Bilateral Trade) a seller and a buyer interact to sell an object. Earlier literature had shown that in such a situation no mechanism will guarantee efficiency, incentive compatibility, individual rationality, and balanced budget condition. In this note we will argue that if we â€œsufficientlyâ€ increase the number of buyers then there is a two stage mechanism which satisfies all the four conditions stated above.Bidding Strategies, Double Auctions

Rittwik Chatterjee

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 2 
  
Multi-stage Double Auctions With Many Bidders 
  
 
 
Rittwik Chatterjee  
National Institute of Science, Technology and Development Studies 
Abstract 
Traditionally in a Double Auction (also known as Bilateral Trade) a seller and a buyer interact to sell an object. Earlier 
literature had shown that in such a situation no mechanism will guarantee efficiency, incentive compatibility, individual 
rationality, and balanced budget condition. In this note we will argue that if we “sufficiently” increase the number of 
buyers then there is a two stage mechanism which satisfies all the four conditions stated above.
I am thankful to Krishnendu Ghosh Dastidar and Srobonti Chattopadhyay and an anonymous referee for their useful suggestions. 
Citation: Rittwik Chatterjee, (2011) ''Multi-stage Double Auctions With Many Bidders'', Economics Bulletin, Vol. 31 no.2 pp. 1902-1909. 
Submitted: Mar 29 2011.   Published: June 25, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
1 Introduction
Consider a situation where there is a seller and a buyer bidding for an indivisible good to be sold.
The valuations of both the seller and the buyer are both private information. The valuation of
the seller is denoted by C and is distributed over the the interval [c;c]. Similarly the valuation
of the buyer is denoted by V and distributed over the interval [v;v]. Both the valuations are
independently distributed. The distributions are common knowledge and have full support on the
respective intervals. Also assume that v < c and v > c , i.e. there may be a case where it is
e￿cient not to trade. The question we are interested in is : Is there a mechanism that ensures
trade when V > C? The next impossibility proposition is due to Myerson and Satterthwaite
(1983) who showed that it is impossible to design a mechanism under the above circumstances
which guarantees trade whenever V > C holds and simultaneously satis￿es three other conditions
viz. incentive compatibility, individual rationality and balanced budget.
Proposition. In a double auction problem, there is no mechanism that is e￿cient, incentive
compatible, individually rational, and at the same time balanced budget.
Here we will assume weakly balanced budget 1. Myerson and Satterthwaite (1983) showed that
under the above situation all the e￿cient mechanisms that satisfy incentive compatible and indi-
vidually rationality run in budget de￿cit. So, here we are interested in non-negative budget surplus
only.
2 The Large Double Auction
In this section, ￿rst we will show that the above impossibility proposition holds even if we assume
weakly balanced budget rather than balanced budget, for a situation where there is a single buyer
and a single seller. Next we will design a mechanism that ensures e￿ciency, incentive compatibility,
individual rationality and weakly balanced budget conditions under ￿su￿ciently￿ large competition.
This means to say that if the number of buyers are ￿su￿ciently￿ large, then the mechanism satis￿es
all the four conditions stated above.
Proposition 2.1. In a double auction problem, there is no mechanism that is e￿cient, incentive
compatible, individually rational, and at the same time weakly balanced budget.
Proof. Note that in this case the VCG mechanism runs in de￿cit. But since the VCG mechanism
runs in de￿cit, every other mechanism also runs in de￿cit 2. Thus, there does not exist an e￿cient
mechanism that is incentive compatible, individually rational and simultaneously weakly balanced
budget.
Now we formulate a mechanism that will ensure e￿ciency, incentive compatibility, individual
rationality and weakly balanced budget conditions under su￿ciently large number of buyers. This
1A mechansim satsi￿es weakly balanced budget condition if and only if the expected gain of the mechanism
designer is non-negative in that mechansim.
2See Krisna (2002), page 76, proposition 5.5
1
1903Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
is a two stage mechanism where in the ￿rst stage each buyer bids for the object and pays his/her
own bid and an entry fee. The mechanism designer selects the highest bidder. In the second stage,
the highest bidder in the ￿rst stage and the seller bid. The buyer gets a particular amount from
the mechanism designer depending on his/her bid. If in the second stage the bid of the buyer is
more than that of the seller then only trade takes place, where the buyer pays what he/she bids
to the seller and gets the object.
The idea behind such a mechanism is to compensate for the de￿cit of the mechanism designer
by charging all the bidders in the ￿rst stage. Formally the mechanism is discussed below. For
simplicity we assume that both the seller’s and the buyer’s valuations are distributed over the
same interval, i.e. v = c and v = c.
2.1 Assumptions
 There are N buyers, where N > 1 and a single seller.
 Seller has an object and wants to sell that object.
 The valuation of the seller is V0. Valuation of the ith buyer is Vi for all i = 1;:::;N.
 All the buyer’s valuations and the seller’s valuation lie in the interval [0;!] i.e. 8j =
0;1;:::;N Vj 2 [0;!]. The valuations are distributed with a distribution function F(:) which
is increasing. F(:) admits a continuous density f(:)  F 0(:) and has full support.
 Both the seller and the buyers are risk neutral i.e. they want to maximize their respective
expected pro￿ts.
 Buyers are not subject to any liquidity constraint.
 All components of the model other than the valuations are assumed to be commonly known
to all the bidders, the seller and the mechanism designer.
2.2 The Mechanism
Stage I:
 The mechanism designer sets a reserve price of amount R, for the good, all the buyers have
to submit a sealed bid more than or equal to that reserve price.
 The mechanism designer collects the bids from all the bidders (it is an all-pay auction with
reserve price).
 The mechanism designer locates the highest bidder, and all the bidders except the highest
bidder are out of the game.
Stage II:
2
1904Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
 The highest bidder bids again for the object. Let he/she bid bII
H in this stage.
 The mechanism designer pays the amount
R bII
H
0 F(t)dt to the highest bidder.
 The mechanism designer asks the seller that whether the seller is interested in selling the
object at a price bII
H . If the seller says ￿yes￿ then the highest bidder pays the seller an amount
bII
H and takes the object from the seller, otherwise no transaction takes place.
We will ￿rst derive the expected payo￿s of the bidders, the objective of each bidder is to maximize
her own payo￿s. We will, then, derive the equilibrium bidding strategy for the seller, and ￿nally
we will derive the equilibrium bidding strategies of each bidder. Let the mechanism designer set a
reserve price R. Let ith bidder be the highest bidder in stage I. Also let her bid bI
H in Stage I and
bII
H in Stage II. The payo￿ functions of the bidder are as follows:
Payo￿ function in Stage II:

II
i =
(
(Vi   bII
i ) +
R bII
i
0 F(t)dt if thesellersaysY ES
R bII
i
0 F(t)dt Otherwise
Payo￿ function in Stage I:

I
i =
8
> > > > <
> > > > :
(Vi   bII
i ) +
R bII
i
0 F(t)dt   bI
i if bI
i > max
j6=i
bI
j andthesellersaysY ES inStageII
R bII
i
0 F(t)dt   bI
i if bI
i > max
j6=i
bI
j andthesellersaysNOinStageII
 bI
i if bI
i < max
j6=i
bI
j
The expected payo￿s of the bidder is given below:
Expected payo￿ in Stage II: (Vi   bII
i )F(bII
i ) +
R bII
i
0 F(t)dt
Expected payo￿ in Stage I:
h
(Vi   bII
i )F(bII
i ) +
R bII
i
0 F(t)dt
i
G( 1(bI
i))   bI
i
Note that G( 1(bI
i)) = F( 1(bI
i))N 1, i.e. the probability that bI
i is the highest bid at Stage I.
Let us assume, VR is a valuation, such that no bidder with a valuation less than VR will participate
in this auction. We will derive VR formally after we calculate the equilibrium bidding strategies of
a buyer in both the stage.
We will derive three propositions below, with proposition 2.2 telling us the equilibrium bidding
strategy of the seller in stage II, proposition 2.3 focusing on the equilibrium bidding strategy for
the highest bidder of Stage I in Stage II, and ￿nally the proposition 2.4 deriving the equilibrium
bidding strategy for each buyer in Stage I.
Proposition 2.2. In the Stage II the equilibrium strategy of the seller is to accept the o￿er of the
mechanism designer if V0 6 bII
H and reject the o￿er otherwise.
Proof. Trivial.
3
1905Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
Proposition 2.3. It is always optimal for the highest bidder of Stage I, to bid her true valuation
in the Stage II.
Proof. See appendix I.
Proposition 2.4. In stage I the equilibrium bidding strategy of the ith bidder, whose valuation
isVi, is given by
(Vi) =
Z Vi
0
F(t)dtG(Vi)  
Z Vi
VR
F(y)G(y)dy
Proof. See appendix II.
Note that in Stage I
(Vi) =
Z Vi
0
F(t)dtG(Vi)   (Vi)
=
Z Vi
0
F(t)dtG(Vi)  
Z Vi
0
F(t)dtG(Vi)  
Z Vi
VR
F(y)G(y)dy

(2.1)
=
Z Vi
VR
F(y)G(y)dy > 0
as, at Stage II, we have already shown that, bidding her own valuation is the equilibrium bidding
strategy for the highest bidder at Stage I.
Note that proposition 2.4 implies that the bidding strategy is incentive compatible 3, and the last
equality of equation 2.1 guarantees that the bidding strategy is individually rational 4.
So, VR is unique because 0(Vi) = F(Vi)G(Vi) > 0 as Vi > VR > 0. Now, once VR is unique, we
can derive an expression which guarantees all the four conditions stated above.
Proposition 2.5. If N satis￿es the expression below then the auction would satisfy all the four
conditions.
N >
R !
VR
R t
0 F(y)dyf(t)dt
h
R[1   F(VR)] +
R !
VR
R z
VR
R y
0 F(t)g(y)dtdyf(z)dz
i (2.2)
Proof. See appendix III.
Below we are going to provide an example.
3For a bidder, in Stage I, it is optimal to bid according to her own valuation since it maximizes her expected
payo￿. So the equilibrium bidding strategy is a function of her own valuation alone (note that as R is given to
all the bidders VR is also given to all the bidders) and the bidder has no incentive to submit a bid which is not a
function of her true valuation.
4The expected gain from participating in the auction is strictly positive.
4
1906Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
Example 2.6. Let us now assume that all the buyer’s valuations and the seller’s valuation are
uniformly distributed over the interval [0;!]. First we will derive the equilibrium bidding strategy
of a buyer in Stage I when her valuation is Vi.
(Vi) = R +
R Vi
VR
R y
0 F(t)g(y)dtdy
= R +
R Vi
VR
R y
0

t
!
	 (N 1)
!
 y
!
N 2 dtdy
= R +
(N 1)(Vi VR)N+1
2(N+1)!N
Let us now calculate the expected revenue of the mechanism designer from this auction.
R[1   F(VR)] +
R !
VR
R z
VR
R y
0 F(t)g(y)dtdyf(z)dz
= R[1   F(VR)] +
R !
VR
n
(N 1)(z VR)N+1
2(N+1)!N
1
!
o
dz
= R

1  
VR
!

+
(N 1)(! VR)N+2
2(N+1)(N+2)!N+1
The expected loss of the mechanism designer is given by the equation below:
R !
VR
R t
0 F(y)dyf(t)dt
=
R !
VR
R t
0
y
!dy 1
!dt
=
!3 V 3
R
6!2
Therefore, the required number of buyers N must satisfy the following condition:
N >
!3 V 3
R
6!2
R
h
1 
VR
!
i
+
(N 1)(! VR)N+2
2(N+1)(N+2)!N+1
=
!2+!VR+V 2
R
6R!+
3(N 1)(! VR)N+1
(N+1)(N+2)!N 1
Note that if the above condition holds then the mechanism will satisfy all the four conditions stated
above. For example let ! = 1 and R = 0:1, so for any N > 4 the above condition is satis￿ed.
3 Conclusion
In this note we ￿rst show that even if we consider a weakly balanced budget condition as against
a balanced budget condition, and if there is a single buyer and a single seller, no mechanism
can guarantee e￿ciency, incentive compatibility, individual rationality and weakly balanced bud-
get simultaneously. Finally we design a mechanism with su￿ciently intense competition (i.e. if
the number of buyers is su￿ciently large) ensures e￿ciency along with incentive compatibility,
individual rationality and weakly balanced budget.
5
1907Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
References
Krishna, V. (2002) ￿Auction Theory￿, in Academic Press, California, USA.
Krishna, V. and M.Perry (1998) ￿E￿cient Mechanism Design￿ , mimeo, Penn State University,
April.
Myerson, Roger B. and Satterthwaite, M. ￿E￿cient Mechanism for Bilateral Trading￿, in Journal
of Economic Theory, 28, 1983, 265-81
Appendix I
As we derived above, the expected payo￿ of the highest bidder of Stage I, in Stage II is given by:
(Vi   b
II
H)F(b
II
H) +
Z bII
H
0
F(t)dt (3.1)
The objective of the highest bidder is to maximize this expected payo￿ by choosing bII
H. Di￿eren-
tiating the equation 3.1 with respect to bII
H and setting that is equal to zero we get the ￿rst order
condition.
(Vi   bII
H)f(bII
H)   F(bII
H) + F(bII
H) = 0
OR; Vi = bII
H
The second order condition can be checked routinely.
Appendix II
Note that, the expected payo￿ of the highest bidder of the Stage I, is
R Vi
0 F(t)dt at Stage II , where
let the ith bidder be the highest bidder in Stage I. As we have stated earlier that we de￿ne VR in
such a way that any bidder whose valuation is less than VR will not enter into the auction and
because the equilibrium pro￿t function of a bidder is continuous, we must have
Z VR
0
F(t)dt

G(VR)   R = 0 (3.2)
In other words we must have (VR) = R and VR > 0. In stage I, therefore, at equilibrium the
expected payo￿ of the ith bidder is
"Z bII
i
0
F(t)dt
#
G(
 1(b
I
i))   b
I
i
The objective of the ith bidder is, therefore, to maximize this expected payo￿ by choosing bI
i such
that bI
i > R . We already know that at the equilibrium if ith bidder is the highest bidder then bII
i =
6
1908Economics Bulletin, 2011, Vol. 31 no.2 pp. 1902-1909
Vi . Therefore at Stage I, the objective of the ith bidder is to maximize
hR Vi
0 F(t)dt
i
G( 1(bI
i)) bI
i
such that bI
i > R holds. Maximizing this with respect to bI
i yields the ￿rst order condition:
Z Vi
0
F(t)dt

g( 1(bI
i))
0( 1(bI
i))
  1 = 0
where g  G0 .
At a symmetric equilibrium, bI
i = (Vi) and as we assume (for the time being) that  is a strictly
increasing function, we have  1(bI
i) = Vi, and since (VR) = R, we have
(Vi) = R +
Z Vi
VR
Z y
0
F(t)g(y)dtdy
Note that this bidding strategy is increasing in valuations.The second order condition for the
minimization can be checked routinely. Finally note that
R +
R Vi
VR
R y
0 F(t)g(y)dtdy
= R +
R Vi
0 F(t)dtG(Vi)  
hR VR
0 F(t)dt
i
G(VR)  
R Vi
VR F(y)G(y)dy
=
R Vi
0 F(t)dtG(Vi)  
R Vi
VR F(y)G(y)dy
The ￿rst equality is obtained by integrating by parts and as
hR VR
0 F(t)dt
i
G(VR) = R
Appendix III
The expected revenue of the mechanism designer is given by the expression below
N
R !
VR (z)f(z)dz
= N
h
R[1   F(VR)] +
R !
VR
R z
VR
R y
0 F(t)g(y)dtdyf(z)dz
i
The expected payment of the mechanism designer is given by the expression below
R !
VR
R t
0 F(y)dyf(t)dt
The weekly balanced budget condition for the mechanism designer will be satis￿ed if the condition
below holds.
N

R[1   F(VR)] +
Z !
VR
Z z
VR
Z y
0
F(t)g(y)dtdyf(z)dz

>
Z !
VR
Z t
0
F(y)dyf(t)dt
7
1909

Multi-stage Double Auctions With Many Bidders

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Multi-stage Double Auctions With Many Bidders

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