Counterspeculation, Auctions, and Competitive Sealed Tenders William Vickrey
The Journal of Finance 16, no. 1 (March 1961) Presented by Art Shneyerov ECON 695S
January 14, 2009
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Written by a Nobel-prize winning economist William Vickrey, this remarkable paper laid a foundation for all subsequent theoretical and empirical work on auctions It contains a number of important results. We will focus on auction-theoretic results.
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Progressive aucitons
This is what is commonly understood by the word "auction" by non-economists. Also known as the English auction. I I
I
The bids are made until no one is willing to bid any further. Assuming the price can vary continuously, Vickrey shows that in equilibrium, the price will be equal to the second-highest valuation. The object is assigned to the highest valuing bidder. The allocation is therefore Pareto e cient.
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Dutch auctions
In practice, this is a less common auction. I
The price is lowered continuously (by a "clock") until one of the bidders accepts the price and buys the object.
I
Used for wholesale ower sales in the Netherlands.
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First-price auctions
Commonly known as sealed tender by non-economists In the rst-price auction, bidders submit bids in sealed envelopes and the object is awarded to the highest bidder, who pays his bid. Many examples in government procurement and sales I I I I I
Highway construction Milk purchases by schools O shore oil sales Municipal bond sales etc. etc.
Vickrey shows that rst-price and Dutch auctions are strategically equivalent.
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First-price auctions We model an auction as a Bayesian game. We assume that bidders draw their valuations vi independently from some commonly known distributions Fi ( ) and submit bids Bi (vi ) without observing the valuations of other bidders If a bidder with valuation vi bids b, then his expected pro t (payo ) in both auctions is given by (vi ; b) = (vi
b) Pr fBj (vj )
b for all j 6= ig
Note that this payo function is the same in both Dutch and rst-price auctions, which explains why they are strategically equivalent.
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Second-price auctions
Vickrey devised a mechanism, the second-price auction, that is outcome equivalent to the rst-price auction. The object allocation rule as well as the expected payo s are the same. The object is still allocated to the bidder who sent in the highest bid, but the winner pays the second highest bid. This auction has a remarkable property that each bidder has a dominant strategy to bid his valuation.
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Second-price auctions Theorem In the second-price auction, each bidder has a weakly dominant strategy Bi (vi ) = vi
Proof. Suppose for a moment that the bidder knows the bids of his rivals. In particular, suppose that the maximum rival bid is equal to z. A weak best response in this situation is to bid his valuation vi . If vi < z, i.e. it is optimal to loose, then bidding vi is obviously a best response. If, on the other hand, vi > z, then once again it is optimal to respond by bidding vi . This is because in this case, it is optimal to win, while the price is xed (at z), so any bid above z will do the job. Finally, note that the argument does not depend on the value of z, so bidding own value is a weakly dominant strategy. Q.E.D. 8/29
Summary: auction equivalence
First-price and Dutch auctions are strategically equivalent Second-price and (continuous clock) English auction are outcome equivalent
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Equilibrium bidding strategy for rst-price auctions: uniform case Suppose, following Vickrey's example, that each bidder has a privately known valuation vi for the object. The valuations are distributed uniformly on [0; 1]. Bidders draw their valuations independently from each other. This is also called the IPV assumption. The bidders do not know valuations of each other, only the (uniform) distribution from which these valuations are drawn. Vickrey shows that an equilibrium bidding strategy is given by B (v) =
N
1 N
vi
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Proof Let's check that this is indeed a (Bayesian-Nash) equilibrium bidding strategy Suppose that the bidder who has valuation v bids as if his valuation was x, while all other bidders follow their equilibrium bidding strategy Since strategies are increasing functions, the probability that the bidder wins is equal to xN 1 . The expected pro t is therefore equal to (v; x) =
v
N
1 N
x xN
1
:
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Proof
In equilibrium, a bidder doesn't have an incentive to deviate from the prescribed bidding strategy. Therefore the necessary FOC of a maximum should hold: @ (v; v) @x
=
0
=) (N
1)v v N
2
N
1 N
N vN
1
=0
Observe that the last line is an identity. Since the pro t function is This means that the strategy is in fact a Bayesian-Nash equilibrium one.
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Revenue Equivalence
One of the most surprising results in Vickrey's paper is that English and Dutch auctions yield the same expected revenue. While it is true that the seller gets the highest bid in the rst-price auction, while only the second-highest in the second-price auction, bidders in the rt-price auction bid less than they would in the second-price auction. The two e ects balance each other precisely. To compute the expected revenue from the English auction, we need the CDFs and densities of the rst- and second order statistics. These statistics are commonly denoted by subscripts (1) and (2)
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First-Order Statistics
The CDF of the rst-order statistic is F(1) (x) = F (x)N Taking the derivative gives the density f(1) (x) = N f (x) F (x)N
1
For the uniform distribution, the density is f(1) (x) = N xN
1
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Second-Order Statistics The CDF of the second-order statistic is F(2) (x) = F (x)N + N (1
F (x))F (x)N
1
Taking the derivative gives the density f(2) (x) = N f (x) F (x)N +N (N = N (N
1) (1 1) (1
1
N f (x) F (x)N F (x))F (x)N
F (x))F (x)N
1
2
2
For the uniform distribution, the density of the second order statistic is f(2) (x) = N (N 1) (1 x)xN 2 :
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Revenue equivalence continued
The expected revenue from the English auction is Z 1 RE = xf(2) (x) dx 0 Z 1 = N (N 1) (1 x)xN 1 dx 0
= (N =
1)
N (N 1) N +1
N 1 N +1
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The expected revenue of the Dutch auction is Z 1 Z N 1 1 N 1 RD = xf(1) (x) dx = xN xN N N 0 0 Z 1 N 1 N 1 N xN dx = = N N +1 0
1
dx
Therefore, in this example, RE = RD !
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Bidding strategies in general symmetric case
Let's generalize the derivation of equilibrium strategies Suppose that bidders draw valuations independently from some commonly known distribution with cdf F (v) and density f (v). We assume that the support of the valuations is a bounded interval, [v; v]
Theorem The unique symmetric equilibrium bidding strategy is given by Z v 1 v~dF (~ v )N 1 B (v) = F (v)N 1 v
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Proof
We rst show that equilibrium strategies must be increasing Denote as Q (b) bidder i'th probability of winning the auction if he bids b while all other bidders follow their equilibrium strategies B (v) Q (b) is a nondecreasing function: bidding higher cannot reduce one's probability of winning
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Proof Fix two particular valuations, v and v 0 < v We use a revealed-preference argument. In equilibrium, bidder with value v must prefer bidding b = B (v) to bidding b0 = B (v 0 ), and the bidder with value v 0 must prefer bidding b0 to b: (v v
0
b) Q (b) 0
b Q b
0
b0 Q b0 ;
v v
0
(1)
b Q (b) :
Manipulating this gives the following inequality (v
v 0 ) Q (b)
which implies that Q (b) b b0 , only that v > v 0 ]
Q b0
0;
Q (b0 ) [Note: we haven't assumed that
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Proof We now show that b inequality (1), v0
and therefore b
b0 . From the second revealed preference b0 Q b 0 v0 b v b0
v 0 b Q (b) Q (b0 ) 1; Q (b)
b0 .
Can b = b0 ? No. This would imply that B (v) is " at" over the range of values [v 0 ; v], and therefore bid b it tied with a positive probability. A bidder can best-respond by bidding slightly above to break the tie and win with a substantially higher probability. A contradiction. This observation completes the proof that B ( ) is an increasing function. 21/29
Proof
We now derive the equilibrium bidding strategy Suppose that a bidder whose valuation is v deviated to bidding as if his valuation was x, i.e. to bidding B (x) His expected pro t would then be (v; x) = (v
B (x))F (x)N
1
:
The rst-order equilibrium condition is that locally this deviation is not pro table @ (v; v) = 0: @x
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Proof Taking the derivative, we obtain the following di erential equation for B (v): ( B 0 (v))F (v)N
1
+ (v
B (v))
dF (v)N dv
1
= 0:
It is easier to re-arrange this equation B 0 (v)F (v)N
1
+ B (v)
dF (v)N dv
1
=v
dF (v)N dv
1
:
The l.h.s. can be recognized as a full derivative, we have d h B(v)F (v)N dv
1
i
=v
dF (v)N dv
1
:
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Proof
Integrating this from v to v and noting that F (v)N Z v B(v)F (v)N 1 = v~dF (~ v )N 1 :
1
= 0 gives
v
Solving for B (v), B (v) =
1 N 1
F (v)
Z
v
v~dF (~ v )N
1
v
Q.E.D.
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Revenue Equivalence Recall that F (~ v )N the N 1 rivals.
1
is the cdf of the highest rival value among
Thus: A bidder's equilibrium bid is equal to the expectation of the maximum of his rivals' values conditional on it being less than his own, B (v) = E max vj j max vj j6=i
j6=i
v
Note that, conditional on the event that bidder i is the winner, the expected revenue is equal to the expected second-highest valuation v(2) . The last statement is true also unconditionally. Recall that the second-highest valuation is equal to the price in the second-price auction. Therefore, the revenue equivalence theorem holds even when bidders draw their valuations from an arbitrary distribution F() 25/29
Alternative expression for bidding strategy We can Integration by parts gives N 1
B(v)F (v)
N 1
= vF (v)
Z
v
F (~ v )N
1
d~ v;
v
which nally gives a neat expression for the bidding strategy Z v 1 B(v) = v F (~ v )N 1 d~ v: F (v)N 1 v The bidding strategy decomposes into the true valuation and the competitive markup. As the number of bidders N increases, the markup decreases, so that in the limit bidders bid truthfully.
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Extension: Reserve prices and entry fees
In real life, it is common for the seller to set a minimal acceptable price, the so-called reserve price. If a reserve price r is imposed, then only the bidders with valuations above the reserve price can make a pro t, so only they submit (serious) bids. The derivation of the bidding strategy requires only minor changes. We have for v > r, B (v) =
F (r)N N
F (v)
1
r+ 1
1 N 1
F (v)
Z
v
v~dF (~ v )N
1
r
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Extension: Reserve prices and entry fees
Another common device is an entry fee c > 0. This fee is paid by each bidder regardless whether the bidder wins or not. We call the bidders that have entered active bidders. We need to determine the lowest participating type v . By de nition, this type must be indi erent between entering or not. [One can show that the pro t function is continuous in the type of the bidder.] A bidder with the lowest type can only win if all other bidders have drawn valuation below v . This is because, given that bidding strategies are increasing, the bidder with type v is certain to loose against any active bidder
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Extension: Reserve prices and entry fees
It then should be clear that a v type bidder will nd it optimal to bid r. Therefore, the zero pro t condition for v is r)F (v )N
(v
1
= c:
The bidding strategy is similar to the previous ones, with the lower limit of integration replaces with v [Check!] B (v) =
F (v )N
1
F (v)N
1
r+
1 F (v)N
1
Z
v
v~dF (~ v )N
1
v
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