## English auction

The word "auction" is
derived from the Latin **augere**,
which means "to increase"

This is the most known kind of auction and is commonly used for selling
goods, such as antiques and artwork. In this auction, the bidding price
is raised until only one bidder remains. There are many ways this can
be done, which essentially are equivalent. Either the auctioneer
announces prices in an increasing manner, or bidders call their bids
themselves, or bids are submitted online having the current highest bid
posted at any time (see tangent). All of these methods have one common
characteristic, at any point in time, all bidders know the current
highest bid. We point out once more that we will assume in what follows
hypotheses A1-A4 from the

previous
lesson.

## Example

Assume that an item is being offered in an English auction with only

*two
risk-neutral bidders*, bidders 1 and 2. The seller and bidder

*j*
(

*j=1,2*) know that bidder

*i*'s
valuation of the item is either

*v*^{i}_{l}
with

probability
*p*^{i} or

*v*^{i}_{h}
with probability

*1-p*^{i} (

*i=2,1*).
Suppose that

*0< v*^{i}_{l}
< v^{i}_{h}
for

*i=1,2*. Assume further that the valuation of the
item is independent among bidders (this corresponds to assumption A2,
independent-private-values). Also, assume that

*v*^{1}_{l}=v^{2}_{l}=:v_{l},

*v*^{1}_{h}=v^{2}_{h}=:v_{h}
and

*p*^{1}=p^{2}=:p
(this corresponds to assumption A3, symmetric information). The payoff
function for bidder

*i* (=1,2), is either 0 if he
never bids over bidder

*j* (=2,1), or

*v*_{l}-(i's
bid), or

*v*_{h}-(i's bid),
depending on his/her valuation of the item, if he bids over bidder

*j*
(=2,1). Clearly the payoff function in this case is a function of the
bids alone and assumption A4 holds (see figure). It seems to be clear
that in this case bidders do not bid over their own valuation, and

the winner of the auction is the bidder with
the highest valuation who calls his/her bid first. The price he/she
will pay for the item is equal to the valuation of the other bidder
which is equal to the lowest valuation.
**Activity 1**

Organize an English auction like the one explained above with two of
your friends. You will act as the seller and they will act as two
bidders. Explain the rules of the game to the bidders in a clear
manner. What is the outcome of the auction? Does this outcome agree
with the one explained before? How would the outcome have changed if
some of the hypotheses A1-A4 are relaxed?

The outcome of the previous auction can be generalized to any English
auction, assuming that A1-A4 hold. The highest valuation individual
wins the bidding and pays a price equal to the valuation of his/her
last remaining rival. However, the seller and the other bidders never
know the true valuation of the winner of the auction. In what follows
we will explain how to obtain, from the perspective of the seller, the
average price paid by the winner of the auction. In order to do so we
introduce the concept of

**order statistic**.

## Order statistic

Suppose that

*X*_{1}, ..., X_{N}
are independent random quantities equally distributed, i.e for any
value

*x* the

probability
that

*X*_{i} is less than or
equal to

*x* is the same for all

*i*.
We define the

*k*-th order statistic,
denoted by

*X*_{(k)}, as the

*k*-th
smallest value among

*X*_{1}, ..., X_{N}.
For instance, in the example presented above if

*v*_{i}
is the valuation of player

*i* for

*i=1,2*,
and player 1 has a highest valuation than player 2, then

*v*_{(1)}=v_{2}
and

*v*_{(2)}=v_{1}.

## Expected price paid to the seller

Consider the example presented above. If

*v*_{i}
is the valuation of player

*i* for

*i=1,2*,
the price paid to the seller by the winner of the auction is equal to
the lowest valuation,

*v*_{(1)}.
The

expected
value of this quantity is calculated below.

**Activity 2**
- Where have we used the
independence of the valuations among bidders?
*Hint:*
The concept of independence is defined here.
- If the seller considers that
the valuation of each player can be
determined by flipping a fair coin, with tails making the valuation
equal to $10 and heads equal to $20. What is the expected price
received by the seller in this case?

## Continuous distributions

In the example discussed above we have assumed that the distribution of
the valuations is discrete, in the sense that the valuations are equal
to specific values with positive probability. Nice results are achieved
when one considers continuous distributions instead of discrete
distributions. In this case the probability of having a valuation equal
to

*x* is equal to 0 for all

*x*.
What is possible to determine is the probability of having a valuation
in a certain interval. The simplest of such distributions is the

**uniform
distribution**. We say that a random quantity

*X*
is

*uniformly distributed* over the interval

*[a,b]*
if

*Pr(X≤ a)=0*,

*Pr(X>b)=0*
and for any

*x* in

*[a,b]*,

*Pr(X≤x)=(x-a)/(b-a)*.
For instance, it can be proved in this case that if the valuations of
the bidders

*v*_{1},v_{2}
are independent and uniformly distributed on [0,1] then the expected
amount received by the seller is

A more general result is stated in

lesson
4.

## The second-price sealed-bid auction

William Vickrey (1914-1996) was a Canadian professor of economics who
was awarded with the Nobel Memorial Prize in Economics for his research
on games with asymmetric information.

Under this auction, bidders submit sealed bids with the knowledge that
the highest bidder wins the item but pays a price equal not to his/her
own bid but the second highest bid (in case of a tie between

*m*
bidders, a random number between

*1* and

*m*
is picked to decide the winner). This type of auction was created by
William Vickrey, reason why is also known as

**the
Vickrey's auction** (see tangent). A modification of
this type of auction is used by eBay's system and Google's and Yahoo!'s
online advertisement programs.

## Example

Suppose that there are only two bidders in a second-price sealed-bid
auction. Suppose that bidder 1's valuation of the item is $5 and bidder
2's valuation is $7. They both know their own valuation and also that
the other bidder has a valuation of $5 with

probability
*0.5* and $7 with probability

*0.5*.
If both bidders bid their own valuation, bidder 1 wins the item and has
to pay $5, a price equal to the lowest valuation (see figure). It turns
out that

bidding his/her own
valuation is an equilibrium strategy for both bidders and therefore the
amount of money received by the seller in this case is equal than the
one obtained in an English auction.

**Activity 3**
- What is the expected
price paid to seller in the auction described above?
- Organize a second-price sealed
bid-auction with two of your friends.
Assume that their valuations satisfy the same properties as the ones
listed in the example presented at the beginning of the lesson. You
will act as the seller and they will act as two bidders. Explain the
rules of the game to the bidders in a clear manner. What is the outcome
of the auction? How would the outcome have changed if some of the
hypotheses A1-A4 are relaxed?

## Incentive equilibrium

We claimed before that bidding their own valuation was an equilibrium
strategy for every player in the second-price sealed-bid auction. The
following reasoning shows why this holds. Suppose that bidder i submits
a bid

*b*_{i} higher than its own
valuation of the item, i.e

*b*_{i}>v_{i}.
Call

*M* the maximum bid among the other
bidders.

- If
*M
< v*_{i}, then the bidder wins the
item and his/her payoff would be equal to *v*_{i}-M.
The same payoff would be obtained by submitting his/her own valuation.
- If
*M
> b*_{i} then the bidder does not
win the auction and obtains a payoff of 0. The same payoff would be
obtained by submitting his/her own valuation.
- Finally if,
*v*_{i}<
M < b_{i}, the bidder would win
the auction but would have to pay a price higher than his/her own
valuation. The payoff is less than 0, which is the one obtained by
submitting his/her own valuation.

A similar argument shows that submitting the bidder's valuation is
better than submitting

*b*_{i}< v_{i}.
This reasoning shows that revealing their own valuation is optimal for
all bidders and the second-price sealed-bid auction is an incentive
mechanism (see

the revelation
principle). This auction is different than the English
auction in that the valuation of the winner is known to the seller.
However, in both auctions the price paid for the item is the same and
equal to the second highest valuation among bidders.

**Activity 4**

Explain why, in a second-price sealed-bid auction, it is not optimal to
submit a bid less than the valuation of the item.