And so here,

let's start with two bidders, each has a value uniformly distributed on 0,1.

And let's think of changing the auction in a very simple format,

what we're going to do is stick in a reserve price, okay?

And this is quite common in auction.

So, often when people are selling things of,

there'll be a minimum bid, reserve price and

you're not allowed to enter into the bidding unless you exceed that threshold.

And so we get a set of reserve price in here,

we're going to have a second price auction to make life simple,

keep with the dominant strategies, but just stick in a reserve price.

Okay, so let's think about how this works.

Once we've put in the reserve price,

then if both people end up bidding below that we're not going to get anything so.

And here's the risk now, if we didn't have a reserve price,

we would always sell the object, now there's a possibility of no sale, so

we're not going to get anything.

If one bidder bids above the reserve and one below it, then the reserve price is

going to kick in, that's going to be the second highest price.

Second highest bid we'll think of the reserve price as if it was a bid, and

the second highest bid is now the reserve price and

so we'll sell at reserve if somebody bids above it and the other below.

So again, second price auction that's going to be the second highest.

And if both of them bid above it, if both bidders happen to bid above it

then it will just be second highest bid that will be the price.

Okay so that's the setting.

Which reserve price is going to maximize the expected revenue, so

let's take a look at that.

So first thing to note, it's still a dominant strategy to bid your true value,

and in fact,

you could think of the reserved prices just like the third bidder, in this case.

And so from many bidders' perspective,

you're still want to be bidding your true value if it's a second price auction

where you think of this reserved prices that was just affixed in out there.

You just happen to know what one of the bids are,

it's still the dominant strategy to bid your true value.

So that makes our life easy in terms of the analysis.

So, if both bids are below R, that's going to happen now

with probability R squared, because people are bidding truthfully.

So the chance that they're both below R with a uniformed distribution,

independent draws of a uniformed distribution,

probability that any one of them is below is R.

So, the probability the both of them are below is R squared, in that case,

the revenue is 0.

The chance that one's above, and one below?

Well, the chance that one's above is one minus R, the other one's below is R, and

it could happen in two different ways, in terms of which bidder is above,

and which bidder's below.

In that case, the second highest price is the reserve price, that's the revenue.

And then, it indicates where they're both above,

that's going to happen with probability 1- R squared and in that case,

then we're going to get the expected minimum of the two values.

Conditional on both of the values being above R and if you just work out

the integral of what's the expected minimum of two uniformly distributed

random variables, conditional on those variables going between R to 1.

So now they fall both within R to 1.

Figure out what the expected minimum of that is I'll save you the algebra.

It's 1 +2R over 3.

Okay, so when we look at these,

then we do the overall expected revenue.

What do we get?

We've got this probability that we're going to get a revenue of R.

So multiply those out.

We get 2 R squared (1- R) here.

And then we've got this probability that both people are above and

then we have this expected revenue so

we end up with an overall expected revenue of this expression.

Because if they're both below, we don't give anything so

that's the overall expression as the expected revenue.

If we collect terms here, in terms of Rs, you can simplify this,

so the expected revenue here is given by this.

Expression 1 + 3R squared- 4R cubed over 3.

So now we have an expression for revenue.

Just maximize that with respect to R.

So let's take the derivative with respect to R and set that equal to 0.

What do you get?

0 = 2R- 4R squared.

Solving that, R = a half.

So what does that tell us?

If you're facing two bidders in a second price auction,

uniform values between zero and one.

You set a reserve price of a half, that's going to maximize the expected revenue.

Okay, you can do calculations.

So supposed let's do our calculation, when we set the reserve price of a half,

what do we earn in terms of revenue?

Well, you can stick your half in here, do the calculation, you end up with 5/12.

If you set a reserve price of 0, then the expected revenue,

you get 0 for the R's, you get one-third here.