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Game Theory Lecture IV (4/5) Auctions with Reserve Price: Optimal Auctions - YouTube
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In previous videos, we introduced
two popular auction formats,
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the second-price auction and
the first-price auction.
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In this video, we're going to
see how the auctioneer can
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improve her revenue
with a reserve price.
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From the quiz in video 3,
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you can find that the seller can
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get a higher revenue from
the fixed price system
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than actions.
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So you may wonder,
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fixed price system is a better
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than auction in terms of
seller's profit.
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The answer is no.
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The seller can increase her revenue
with a reserve price.
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In many cases, sellers reserve
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the right to not sell
the object if the price
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determined in the auction
is lower than a certain level.
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Importantly, it is
also known that
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the revenue equivalence theorem
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holds with a reserve price.
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That means, in order to see how
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much auctions can improve
the seller's revenue,
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we can just focus on
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the second price auction
with a reserve price.
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So now we're going to see
the seller can improve her
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revenue with a reserve
price in this 2-bidder case.
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So we are going to
focus on
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each bidder's expected payment
to get seller's revenue.
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You can see in equilibrium,
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each bidder must bid
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their true value regardless of
this reserve price.
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So we have both
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bidders' true value, x1 and x2.
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If your value is lower than
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the reserve price,
then you are going to pay nothing.
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And if your value is higher
than the reserve price,
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then it depends on
your opponent's value.
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So if your value is lower than
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your opponent's value
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then you lose and
you pay nothing.
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But if your true value is
higher than the reserve price
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and your true value is also
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higher than your opponent's
value, then you win.
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And what's the payment?
You pay the maximum of
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x2 and r. So if
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your opponent's value is
lower than the reserve price,
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you have to pay the reserve price.
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Given this condition,
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your value is higher
than the reserve price.
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And what is your
expected payment?
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If your value is lower
than your opponent's value,
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then you pay nothing, you lose.
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And if your value is higher
than your opponent's value,
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then you have to pay
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this maximum of
your opponent's value.
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and the reserve price,
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conditional on you are going to
obtain the item, right?
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So we cancel out this 0 term.
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And this probability is
going to be F(x1).
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And we have this
expectation of a max of
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x2 and r. So this is nothing
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but the conditional expectation.
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So we can represent that
as this integration form.
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And then we are going to split
this integration into two parts.
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So first part is if x2
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is somewhere between
0 and r.
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In that case, you
are going to pay r.
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If x2 is somewhere between
r and your true value x1,
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then you have to pay
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this amount,
your opponent's value, right?
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So we have split
that into two part.
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But this part is simple,
because it is a constant term.
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So this r is coming out
and r and
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the remaining part is
going to be F(r). And
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this remains here, right?
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So we have one particular
bidder's expected payment.
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If the bidder's true value
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is bigger than the reserve price,
this is the expected payment.
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And of course, if
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it is lower than
the reserve price than his
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expected payment is actually 0.
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Now, in the seller's perspective,
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we are going to see
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the expected revenue
from that bidder.
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So take an expectation over
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this payment, expected payment.
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So it is an integration from 0 to one,
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but 0 to r,
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this m1 is going to be 0.
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There is m1. So we just
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consider integration
from r to 1, right?
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And put this part,
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this part into the middle of
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this integration.
We split this into two parts.
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This is the first part
and this is the second part.
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And this first part is a simple.
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So this is constant
with respect to x1.
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And so coming out and
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the other part is going
to be 1-F(r).
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And the second term, again,
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we're going to use the
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interchanging the
order of integration.
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So we have this and that is
1-F(x2), right?
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Alright, so now we have
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the seller's expected
revenue from one
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particular bidder.
Our main concern is
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what's the optimal reserve price.
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So this expected revenue is
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a function of r.
The seller can actually choose
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determine this reserve press r.
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So what is this
optimal reserve price?
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So we're going to get
the derivative of
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this expected revenue from one bidder
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and set that to be 0.
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So your choice variable is r
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and just take its derivative.
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So you are supposed to use a chain rule.
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Also, you have to use the
derivative of integration.
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So you can simplify this
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derivative into here
and set this to be 0.
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So your optimal reserve price
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r* should solve
this equation.
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So for any distribution function,
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the seller should set
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this optimal reserve price r*?
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So we have a very interesting
observation here.
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First of all,
the optimal reserve
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price does not depend
on the number of bidders.
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It only depends
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on each bidder's
distribution function.
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Also, that is
exactly the same to
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the optimal price for one buyer.
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Please recall the Video 1.
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So we derived the optimal price
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for fixed price system.
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And this optimal reserve price
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is exactly the same
to the optimal price.
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So here we have a quiz.
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And again, two bidders with
the uniform distribution.
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Please find the optimal
reserve price and find
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the seller's maximum
expected revenue
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from the second-price auction.
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And please compare
the revenue of
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the second price auction
with the optimal price.
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and the revenue of
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the second price auction
with no reserve price
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so how much the seller can
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improve her revenue
by charging a reserve price.
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Also, compare the revenue of
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the second-price auction with
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the optimal reserve price
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and the revenue from posting
an optimal fixed price.
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So you can see how much the seller
can improve her
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revenue by selling through an auction
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instead of posting a fixed price.
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