In common value auctions the value of the item for sale is identical amongst bidders, but bidders have different information about the item's value. This stands in contrast to a private value auction where each bidder's private valuation of the item is different and independent of peers' valuations.[1]
A classic example of a pure common values auction is when a jar full of quarters is auctioned off. The jar will be worth the same amount to anyone. However, each bidder has a different guess about how many quarters are in the jar. Other, real-life examples include Treasury bill auctions, initial public offerings, spectrum auctions, very prized paintings, art pieces, antiques etc.
One important phenomenon occurring in common value auctions is the winner's curse. Bidders have only estimates of the value of the good. If, on average, bidders are estimating correctly, the highest bid will tend to have been placed by someone who overestimated the good's value. This is an example of adverse selection, similar to the classic "lemons" example of Akerlof. Rational bidders will anticipate the adverse selection, so that even though their information will still turn out to have been overly optimistic when they win, they do not pay too much on average.
Sometimes the term winner's curse is used differently, to refer to cases in which naive bidders ignore the adverse selection and bid sufficiently more than a fully rational bidder would that they actually pay more than the good is worth. This usage is prevalent in the experimental economics literature, in contrast with the theoretical and empirical literatures on auctions.
Common-value auctions and private-value auctions are two extremes. Between these two extremes are interdependent value auctions (also called: affiliated value auctions), where bidder's valuations (e.g.,
\thetai=\theta+\nui
\theta
\nui
In the following examples, a common-value auction is modeled as a Bayesian game. We try to find a Bayesian Nash equilibrium (BNE), which is a function from the information held by a player, to the bid of that player. We focus on a symmetric BNE (SBNE), in which all bidders use the same function.
The following example is based on Acemoglu and Özdağlar.[3]
There are two bidders participating in a first-price sealed-bid auction for an object that has either high quality (value V) or low quality (value 0) to both of them. Each bidder receives a signal that can be either high or low, with probability 1/2. The signal is related to the true value as follows:
This game has no SBNE in pure-strategies.
PROOF: Suppose that there was such an equilibrium b. This is a function from a signal to a bid, i.e., a player with signal x bids b(x). Clearly b(low)=0, since a player with low signal knows with certainty that the true value is 0 and does not want to pay anything for it. Also, b(high) ≤ V, otherwise there will be no gain in participation. Suppose bidder 1 has b1(high)=B1 > 0. We are searching the best-response for bidder 2, b2(high)=B2. There are several cases:
The latter expression is positive only when B2 < V/2. But in that case, the expression in #3 is larger than the expression in #2: it is always better to bid slightly more than the other bidder. This means that there is no symmetric equilibrium.
This result is in contrast to the private-value case, where there is always a SBNE (see first-price sealed-bid auction).
The following example is based on.[3]
There are two bidders participating in a second-price sealed-bid auction for an object. Each bidder
i
si
vi=a ⋅ si+b ⋅ s-i
a,b
a=1,b=0
a=b
Here, there is a unique SBNE in which each player bids:
b(si)=(a+b) ⋅ si
This result is in contrast to the private-value case, where in SBNE each player truthfully bids her value (see second-price sealed-bid auction).
This example is suggested[4] as an explanation to jump bidding in English auctions.
Two bidders, Xenia and Yakov, participate in an auction for a single item. The valuations depend on A B and C -- three independent random variables drawn from a continuous uniform distribution on the interval [0,36]:
X:=A+B
Y:=B+C
V:=(X+Y)/2=(A+2B+C)/2
Below we consider several auction formats and find a SBNE in each of them. For simplicity we look for SBNE in which each bidder bids
r
r ⋅ X
r ⋅ Y
r
In a sealed-bid second-price auction, there is a SBNE with
r=1
PROOF: The proof takes the point-of-view of Xenia. We assume that she knows that Yakov bids
rY
Y
Z
Z\geqrY
V-rY=(X+Y-2rY)/2
Z<rY
| Z/r | |
| \int | |
| Y=0 |
{X+Y-2rY\over2} ⋅ f(Y|X)dY
f(Y|X)
By the Fundamental theorem of calculus, the derivative of this expression as a function of Z is just
{1\overr}{X+Z/r-2Z\over2} ⋅ f(Z/r|X)
X=2Z-Z/r
Z={rX\over2r-1}
In a symmetric BNE, Xenia bids
Z=rX
r=1
The expected auctioneer's revenue is:
=E[min(X,Y)]=E[B+min(A,C)]
=E[B]+E[min(A,C)]
=18+12=30
In a Japanese auction, the outcome is the same as in the second-price auction,[4] since information is revealed only when one of the bidders exits, but in this case the auction is over. So each bidder exits at his observation.
In the above example, in a first-price sealed-bid auction, there is a SBNE with
r=2/3
PROOF: The proof takes the point-of-view of Xenia. We assume that she knows that Yakov bids
rY
Y
Z
Z\geqrY
V-Z=(X+Y-2Z)/2
Z<rY
G(X,Z)=
| Z/r | |
| \int | |
| Y=0 |
{X+Y-2Z\over2} ⋅ f(Y|X)dY
f(Y|X)
Since
Y=X+C-A
f(Y|X)=Y-(X-1)
X-1\leqY\leqX
f(Y|X)=(X+1)-Y
X\leqY\leqX+1
Substituting this into the above formula gives that the gain of Xenia is:
G(X,Z)={1\overr3}(XZ2r/2+Z3/3-Z3r)
Z={rX\over3r-1}
Z=rX
r=2/3
The expected auctioneer's revenue is:
=E[max(fX,fY)]=(2/3)E[B+max(A,C)]
=(2/3)(E[B]+E[max(A,C)])
=(2/3)(18+24)=28
Note that here, the revenue equivalence principle does NOT hold—the auctioneer's revenue is lower in a first-price auction than in a second-price auction (revenue-equivalence holds only when the values are independent).
Common-value auctions are comparable to Bertrand competition. Here, the firms are the bidders and the consumer is the auctioneer. Firms "bid" prices up to but not exceeding the true value of the item. Competition among firms should drive out profit. The number of firms will influence the success or otherwise of the auction process in driving price towards true value. If the number of firms is small, collusion may be possible. See Monopoly, Oligopoly.