We say that it is precisely because of the decoupling between the allocation result

and the pricing result of an second price sealed envelope auction for a single item

that leads us to this desirable property of truthful bidding being a dominant

strategy for each bidder. That is, you should bid your valuation no

matter what the other bidders might be doing.

So how do we understand this somewhat counterintuitive result?

We look at this from three different angles.

The first one is to look back to the more intuitive, open options.

Most of us would agree that increasing price in oak open option is very

intuitive. But if you think about the price that you

get to pay as the winner of a increasing price open option.

Your own bid determines how long you stay in the price war.

But when it stops, when you pay the price, you effectively pay the bid of the next

highest bidder, plus a small amount capped by the minimum increment per bid.

Unless you overbid much more than the minimum increment.

So effectively, you're actually paying the second price.

Now the second angle we look at is mathematical argument.

Suppose as an adviser to you, I suggest you don't bid your evaluation.

Well then I can only tell you two things: One is, please bid below your evaluation,

or please bid above your evaluation. Let's take a look at these two cases one

by one. In this case I suggest you bid say, B2,

okay, not the original B, which is the same as evaluation.

So now B2 is less than B. Now, for such a change in your behavior to

make any difference, either in the allocation or the pricing of the auction,

it must be such that there is another bidder: The second-highest bidder.

Who bid less than b but more than b2. In other words, your taking my advice into

account, and lowering bid from b to b2, would only make a difference in the actual

outcome of the auction, only if there is a b2 in between b and b2.

Now let's see what happens then before and after taking my advice.

Now after take, you take my advice you actually lose the auction, so your payoff

is zero. But you could have, if you could have bid

V = B. And then what kind of payoff would you

get? You're going to get B-P, of course.

And in this case, P would be the second price.

B-b2, which = B-B2 because you are bidding the same as your valuation.

And we know V is bigger than B2. So this is a positive number.

So you could have got a positive number, not gotten zero.

So you're saying nope, I'd rather not take your advice.

And lower my bid below my valuation. What about the other case?

Suppose I suggest, hey, don't listen to this advice of lowering.

Actually, take my new advice of increasing the bid.

Okay, when you increase the bid above the valuation then what would happen?

We must look at before and after again. Again, in this case by changing your

bidding behavior taking a [inaudible], going from B to B2, you know.

It will only make a difference if there is some other bid, B2, in between.

Then, in this case, the chain of inequality errors reverses the direction.

And in that case what would happen? Well, before you got my advice that you

would bid here, and the other bidder would have outbid you and you have got zero

utility. Now that you've listened to my advice and

raised your bid from B to B2, then what would you get for your payoff?

B minus P, which is B minus B2, and that's the price you pay now, which equals B

minus B2. But B minus B2 is negative.

So in other words, before this advice is taken into account, then you get zero.

After you get less than zero, and that is worse.

And therefore you would also say, look, I won't take your advice to raise the bid,

either. >> So the key point in this mathematical

argument is that, in both cases, this switch in behavior would only make a

difference in the outcome of the auction if there is another bidder in between

these two values, and we have demonstrated that you would rather not take my advice

to either lower or to increase your bid. And therefore the only logical conclusion

is you should bid exactly the same as the evaluation.

>> Theo, this sounds like a mystery to me. It's like a cloud there.

I don't know, intuitively, what's the right way to think about this.

For example, why not the third price? Okay.

[inaudible] third prize option, you would charge the winner based on not the one

below it, but the one, the second one below it.

Okay. What's so special about the second prize?

Actually what's so special about the second prize is it depicts the lost.

Evaluation to others in the system. So, if you were not in this auction system

as one of the, a potential buyer then the, what's the second highest bidder now would

have been the highest bidder and should would have gotten this item and received

evaluation. So, but now you jump into the system, and

she becomes the second highest bidder. And you've got the item.

So the damage that you cause to the system, is basically the price that this

person is willing to pay for. And that's the third and intuitive

explanation of why second rather than third, fourth, fifth price.

Now what about the other folks in the system?

Well, even if you were not there, they wouldn't be able to get that item anyway.

So, they do not matter, as far as calculating damage is concerned.

Your damage is all inflicted, on what becomes now, the second highest bidder.

So this acts as a recurrent theme. That, there's something called a negative

externality. Last lecture note we talked about the

negative externality of interference in a wireless cellular network, and a way to

Modify that. Or what we call the internalized

[inaudible] is by power control. In this case, the negative externality is

the lost evaluation to other folks in the system.

And the way to internalize that is to charge you accordingly.

Now, internalizing negative externality sounds like, abnormal English.

So a more commonly understandable term would be simply pay for what you damaged.

That's how we charge. Well, let's take a look at an example now

of second price. Sealed envelope, it's not entirely sealed

envelope, actually, we'll see the difference and examples around eBay.

I found it in 1995, I think with over 40 million users.