The version of the Revenue
Equivalence Theorem that we present in this lesson was first proved by

Vickrey
in 1961. This result
was generalized 20 years later by

Myerson,
and independently by

Riley
and Samuelson.

In this lesson we will state the first important theorem of auction
theory, commonly known as the

*Revenue Equivalence Theorem*.
Before doing so, we will generalize some results obtained in the
previous lessons to the case when valuations are

continuous random quantities
with certain properties. In order to do this it is important to recall
the concepts of

**distribution and density functions**
for a continuous random variable.

## Distribution and density function

Given a continuous random variable

*X*, the

**distribution
function of X**, denoted by

*F*, is
defined by the formula

When this function is differentiable, we say that

*X*
has a

**density function** which is
defined by the formula

For instance, if

*X* is

uniformly
distributed on

*[a,b]* the density function
of

*x* is

*f(x)=1/(b-a)* if

*a≤x≤b*
and

*f(x)=0*, otherwise (see figure).

**Activity 1**
The definition of distribution function for discrete random quantities
is exactly the same as above. Find the distribution function of the
valuations considered in the first example of

lesson
2.

## Expected revenue in the English and Vickrey's
auctions

We observed in lesson 2 that under assumptions A1-A4 the revenue of the
seller in the English and second-price sealed-bid auctions was equal to
the second highest valuation among bidders. In terms of the

order statistics this revenue
is equal to

*v*_{(N-1)}, where

*N*
is the number of bidders. Assuming that the valuations have a density
function,

*f*, with some nice properties, it can be
proved that

In fact, we already have used this fact in

lesson
2 for the specific case of the uniform distribution.

## Expected revenue in the Dutch and first-price
sealed-bid auctions

In

lesson 3 we observed
that assuming a Dutch or first-price sealed-bid auction has two bidders
whose valuations are uniformly distributed on [0,1]. Under

hypotheses A1-A4, the
Bayesian Nash equilibrium for each bidder is to bid half of his/her
valuation. This result can be generalized to the case of

*N*
bidders whose valuations have a common density

*f*,
with some nice properties. It can be proved in this case that a

Bayesian Nash equilibrium for
player

*i* is to bid

where

*v*_{l} is the lowest
valuation each bidder can have (if the bidder bids less than this
quantity he/she has zero surplus).

**Activity 2**
By using the formula above, find the Bayesian Nash equilibrium of a
dutch or first-price sealed-bid auction if there are

*N*
bidders whose valuations are

uniformly
distributed on [0,1]. Observe that for

*N*
big, each bidder's bid approaches his/her own valuation.

*Hint*
Use the fact that the integral between

*a* and

*b*
of

*x*^{M} is

*(1/(M+1))(b*^{M+1}-a^{M+1}).

Hence in terms of the

order
statistics the expected revenue of the seller in this case is
equal to

It can be shown that this quantity is equal to the revenue of the
seller in the English and Vickrey's auctions (formula above), which
leads to the following theorem.

## The Revenue Equivalence Theorem

For the benchmark model, with hypotheses

A1-A4,
each of the English auction, the Dutch auction, the first-price
sealed-bid auction, and the second-price sealed-bid auction yields the
same price on average.

**Activity 3**

In the activities from the previous lessons, where the outcomes from
the different types of auctions that you organized with your friends,
almost equal on average? If not, could you explain why you think this
happened?

There exists however a fundamental difference bewteen the equilibrium
in the English and Vickrey's auctions and the equilibrium in the Dutch
and first-price sealed-bid auctions. In the latter, the equilibria are

**dominant
equilibria** in the sense that each bidder has a well
defined bidding strategy regardless of how high his rivals bid. In the
former, the equilibrium for each bidder is optimal given that his/her
rivals are using the same decision rule (

Bayesian
Nash equilibria). Also, recall that in the English auction
the seller never discovers the winner's valuation, but in the
second-price sealed-bid auction the winning bid is actually equal to
the highest valuation. Hence, although the average revenue in all the
auctions is the same, there exist properties that are different for the
different types of auctions. Also, if some of the assumptions

A1-A4 are dropped the
outcome of the auctions could change substantially. For instance, when
assumption A2 is dropped we have the following result.

## The winner's curse

Suppose A1, A3, A4 hold but instead of having hypothesis A2
(independent-private-values assumption), there is a common value of the
item that is unknown for all bidders. In this case, for a sealed-bid
auction each bidder makes an estimate about the true value of the item.
The bidder with the highest estimate will win the auction (why?). But
this means that the winner of the auction overestimated the value of
the item, since everybody considered it to be less, and hence he/she
might end up overpaying. A more formal statement of this result uses
conditional probabilities and it is out of the scope of these lessons.
Hence by dropping A2, the winner of the auction might lose the rent
he/she would win if A2 were satisfied.

**Activity 4**

Have you noticed this phenomenon in the auctions organized in the
previous lessons?

## A double auction

Double auctions are used for example in the stock market as described
in

lesson 1. Here we
present a very simple version of a double auction where there is one
seller and one buyer who have private information about the value of
item. The seller names an asking price

*p*_{s}
and the buyer an offer price,

*p*_{b},
simultaneously. A transaction occurs if and only if

*p*_{b}>p_{s},
and in this case the price paid for the item is equal to

*p=(p*_{s}+p_{b})/2.
Assuming that the valuations of seller and buyer, are uniformly
distributed on [0,1], a Bayesian equilibrium for this auction is given
by the formulas

As a consequence, in this equilibrium trade occurs if and only if

*v*_{b}≥v_{s}+(1/4)
(see figure). It can be shown that there is no Bayesian Nash
equilibrium that is efficient in the sense that trade occurs if and
only if

*v*_{b}≥v_{s}.
This auction is different than the ones presented in the previous
lessons in that the seller also takes part in the auction by submitting
an asking price.

## Further reading

The results presented in these lessons are merely introductory. Further
results are obtained by relaxing hypotheses

A1-A4. Also auctions are a
particular example of games with incomplete information, additional
results are obtained for sequential games of incomplete information
when for instance, bargaining is possible. In this regard, we refer the
interested reader to the literature below.

## References

- Gibbons, R. Game theory for applied economists,
Princeton University Press, 1992.
- McAfee, P. and McMillan, J. Auctions and
bidding, Journal of Economic Literature 25:699-738, 1987.