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.
A classic example of a pure common values
auction
An auction is usually a process of Trade, buying and selling Good (economics), goods or Service (economics), services by offering them up for Bidding, bids, taking bids, and then selling the item to the highest bidder or buying the item from th ...
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
In economics, insurance, and risk management, adverse selection is a market situation where Information asymmetry, asymmetric information results in a party taking advantage of undisclosed information to benefit more from a contract or trade.
In ...
, 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.
Interdependent value 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.,
) can have a common value component (
) and a private value (
) component. The two components can be correlated so that one bidder's private valuation can influence another bidder's valuation.
These types of auctions comprise most real-world auctions and are sometimes confusingly referred to as common value auctions also.
Examples
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.
Binary signals, first-price auction
The following example is based on
Acemoglu and
Özdağlar.
There are two bidders participating in a
first-price sealed-bid auction
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest b ...
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:
* If at least one bidder receives a low signal, then the true value is 0.
* If both receive a high signal, then the true value is V.
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 other bidder bids B2 < B1. Then, his expected gain is 1/2 (the probability that bidder 2 has a low signal) times −B2 (since in that case he wins a worthless item and pays ''b2''(high)), plus 1/2 (the probability that bidder 2 has a high signal) times 0 (since in that case he loses the item). The total expected gain is −B2/2 which is worse than 0, so it cannot be a best response.
# The other bidder bids B2 = B1. Then, his expected gain is 1/2 times −B2, plus 1/2 times 1/2 times
− B2(since in that case, he wins the item with probability 1/2). The total expected gain is (V − 3 B2)/4.
# The bidder b2 bids B2 > B1. Then, his expected gain is 1/2 times −B2, plus 1/2 times
− B2(since in that case, he wins the item with probability 1). The total expected gain is (2 V − 4 B2)/4.
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
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest b ...
).
Independent signals, second-price auction
The following example is based on.
[
There are two bidders participating in a second-price sealed-bid auction for an object. Each bidder receives signal ; the signals are independent and have ]continuous uniform distribution
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that li ...
on ,1 The valuations are:
:
where are constants ( means private values; means common values).
Here, there is a unique SBNE in which each player bids:
:
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).
Dependent signals, second-price auction
This example is suggested as an explanation to jump bidding in English auction
An English auction is an open-outcry ascending dynamic auction. It proceeds as follows.
* The auctioneer opens the auction by announcing a suggested opening bid, a starting price, or a reserve for the item on sale.
* Then the auctioneer accepts ...
s.
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
In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that li ...
on the interval ,36
* Xenia sees ;
* Yakov sees ;
* The common value of the item is .
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 times his/her signal: Xenia bids and Yakov bids . We try to find the value of in each case.
In a sealed-bid second-price auction, there is a SBNE with , i.e., each bidder bids exactly his/her signal.
PROOF: The proof takes the point-of-view of Xenia. We assume that she knows that Yakov bids , but she does not know . We find the best response of Xenia to Yakov's strategy. Suppose Xenia bids . There are two cases:
* . Then Xenia wins and enjoys a net gain of .
* . Then Xenia loses and her net gain is 0.
All in all, Xenia's expected gain (given her signal X) is:
::
where is the conditional probability-density of Y given X.
By the Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of derivative, differentiating a function (mathematics), function (calculating its slopes, or rate of change at every point on its domain) with the concept of integral, inte ...
, the derivative of this expression as a function of Z is just . This is zero when . So, the best response of Xenia is to bid .
In a symmetric BNE, Xenia bids . Comparing the latter two expressions implies that .
The expected auctioneer's revenue is:
: