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Game Theory Lecture IV (3/5) First-Price Auctions - YouTube
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In the previous video,
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we introduced the
second-price auction,
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which is a quite simple because
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each bidder bids their true value.
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But it turns out the
second-price auction
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this is quite common
in real life.
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In this video,
we're going to study
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another popular auction format,
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which is the first-price auction.
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So it's quite straightforward.
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Each bidder submits their bids
in a sealed envelope.
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And the highest bidder wins
and pays his or her bid.
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So here we have
some observations on
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reasonable bidding strategy.
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Truthful bidding is not optimal.
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So if you bid your true value
then your payoff is always 0.
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If you win, then you
pay exactly your value.
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So your net payoff is 0.
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If we lose, then your payoff is
again 0.
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You also have reason to overbid.
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In case of your win,
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you have to pay more than
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your value and you will
get negative payoff.
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The first-price auction
is a sealed auction,
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but there is an open format
of the first-price auction,
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which is the Dutch auction.
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So the Dutch auction is strategically
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equivalent to the
first-price auction.
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In Dutch auction,
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the auctioneer begins by
calling a price from high enough.
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The price is going
down until some
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bidder indicates her interest.
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So in Dutch auction or
Tulip auction,
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there is a clock. And the clock
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is ticking down and
the price is changing.
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And bidders should press the bid button.
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The first bidder
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who presses the button wins
the item at that price.
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So this Dutch auction is used
for like flowers and
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fresh foods, because Dutch auction
can proceed trade very quickly.
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So what should be your
bidding strategy?
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You should wait until
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the price is lower
than your true value.
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Theoretically, to
maximise your payoff,
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you should bid right
before others bid.
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But you don't know others' bid
until the auction ends.
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And so you have to some kind of
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estimation on others' bid.
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So that's quite
complicated indeed.
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So we are going to
consider a two-bidder case.
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And in this first-price
auction, again, your payoff.
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Depends on your true value
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and also your bid and
your opponent's bid.
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In case of your winning,
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you are going to pay
your bid, b1.
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So the difference between
your true value and
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your bid is your payoff
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in case of your winning.
If you lose,
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then again, your pay will be 0.
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So we are going to have some
reasonable assumptions
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on bidding strategy.
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First, a bidding strategy
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is a function of your true value.
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So you remember that
a strategy is a plan of action.
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So you're going to
have a plan of bid
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depends on
your true value.
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Okay? So this beta is
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your bidding function,
or a bidding strategy
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that is a function of your true value.
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And for simplicity, we
assume two players
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play the same bidding strategy.
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Of course there are two values,
which might be different,
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but they are going to employ
the same strategy
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beta, alright? And you
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are not going to overbid
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because it just hurts your payoff.
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So beta(x1) should
be lower than x1.
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And the biding
function is monotone.
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That means if you have
a higher value then
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you should bid higher.
Makes sense?
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And no negative bidding.
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Non-negative bidding is obviously
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you cannot be a negative number.
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So even if your value is 0,
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you should be 0.
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So what we have to do is
find an equilibrium.
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So suppose beta is the
equilibrium bidding strategy.
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And both players
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play this same equilibrium
bidding strategy.
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And each bidder
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bids without knowing the other's bid,
right?
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Because it's private information.
You don't know
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your opponent's bid
before the auction ends.
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So what we concern is
each bidder's expected payoff
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without knowing opponent's
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true value or opponent's bid.
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So your expected payoff
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is just a function of
your true value and
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your bid because
you don't know your opponent's
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true value or your opponent's bid.
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But, you can take the
expect value over
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this your opponent's bid.
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You don't know your opponent's bid,
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but that is a random variable.
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And that depends on
your opponent's true value and
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your opponent is going to play
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this equilibrium
bidding strategy.
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So we are going to take this
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beta(X2) and we are going to
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have expectation over this X2.
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Alright. So you are going to win,
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if your bid is bigger than
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your opponent's bid.
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With that probability,
you are going to win and you are
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going to get that amount
of payoff.
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So similarly, we can have
your opponent expected payoff.
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And in equilibrium,
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the bidding strategy is
mutual best responses.
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So that means your
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bidding strategy -- that you follow
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this biding strategy.
That maximises your expected payoff.
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And similarly your opponent
follows
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this biding strategy
and that maximises
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your opponent's expected payoff.
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Okay, so now we are going to
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derive a condition
for equilibrium.
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So given x1, given
your true value x1,
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your expected payoff.
We've already derived this.
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That is the probability of winning
and your payoff,
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in case of your winning.
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Since we assume that this
beta is an increasing function.
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So we can define its
inverse function.
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So we take this inverse function,
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we take this into
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here. We have this.
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And then we can rewrite
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this probability in terms
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of this cumulative
distribution function, right?
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And you are going to choose
b1 to maximise this payoff.
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So what you have to do is get
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the derivative
and set that to be 0.
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And you are going
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to get the derivative
with respect to your b1,
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because b1 is your
choice variable.
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And here it looks complicated,
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but it's just of the chain rule
of this. And this is
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the derivative of this one,
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dF/Db1.
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So please recall the derivative
of the inverse function.
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And set that to be 0.
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Okay? So your bid b1 should
satisfy this equation.
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And then now we're going
to simplify this equation.
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That's the condition
for equilibrium.
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As you are going to play,
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the equilibrium biding strategy.
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So b1 should equal to
beta of your true value x1.
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So replace b1
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with beta(x1) and
beta-inverse(b1) with x1.
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So make this as a equation of x1
instead of b1.
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Alright? So just replace
this x1 and x1.
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And this is beta(x1)
and this is x1.
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Alright? And then.
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We can just simplify
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this by multiplying
beta-prime(x1).
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So we have this equation,
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that is a condition
for equilibrium.
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Alright? So now we're going to
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derive equilibrium strategy
from that condition.
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Ok. So recall this is
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the derivative of this part
using the chain rule.
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And then integrate
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both sides.
We are going to have this equation.
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We have this bidding strategy,
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equilibrium bidding strategy that
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satisfy this
equilibrium condition.
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That means beta(x1)
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is some integration
divided by F(x1).
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Please recall that
this equals to
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your expected value on
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your opponent value
conditional on your winning.
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Now we have
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this equilibrium
bidding strategy.
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Your probability of
winning is F(x1).
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And your expected payment is
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the probability of winning
times your bid, actual bid.
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So the product of F(x1)
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and beta(x1).
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Actually, this F(x1) cancelled out.
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So what you have is
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this integral, integration term.
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We just changed this variable to z.
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Now we can find your
expected payoff.
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You are going to bid
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based on the
equilibrium strategy,
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beta,
given your true value x1.
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And that is the winning probability
times your payoff.
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And again, this term,
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the first term, x times F(x1).
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And the second term
is going to be
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F(x1) times beta(x1).
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That is nothing but your
expected payment, right?
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So expected payment,
subtract this.
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Again, you need to
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use integration by parts
and the first is cancelled out.
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Then you have this integration form.
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Finally, we are going to derive
the seller's revenue.
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So again, as we did
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in the second-price auction,
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we start from each bidder's
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expected payment and
the seller's expected revenue
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from one particular bidder
without knowing x1.
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So we have to take
this expectation
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over true value of one
particular bidder.
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And what's this m1?
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m1 is here. From here,
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we use the
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interchanging the order
of integration, right?
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So this comes first.
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This is going to be 1-F(z),
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and that remains here.
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So again, the seller's
expected revenue from
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two bidders, it is just a
twice of this formula.
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So you can notice
the seller's revenue is
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exactly the same with
the seller's revenue
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in the second-price auction.
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Actually, this is not
just a coincidence.
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So we have a quite
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surprising result which is
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called the Revenue Equivalence Theorem,
under
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some conditions,
the expect revenue
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is the same in any
standard auction.
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So this is a quiz.
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Based on the same situation:
Two bidders with uniform distribution.
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please try to find
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the equilibrium bidding strategy,
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the seller's expected revenue.
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And also please compare the revenue
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of the first-price auction
and the revenue of the fixed
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price system from the video 1.
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So please check whether
the seller can improve her
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revenue by selling through an auction
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instead of posting
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a fixed price.
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