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

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

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

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

Â matter what the other bidders might be doing.

Â So how do we understand this somewhat counterintuitive result?

Â We look at this from three different angles.

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

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

Â intuitive. But if you think about the price that you

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

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

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

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

Â Unless you overbid much more than the minimum increment.

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

Â Now the second angle we look at is mathematical argument.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â Actually, take my new advice of increasing the bid.

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

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

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

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

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

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

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

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

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

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

Â minus B2. But B minus B2 is negative.

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

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

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

Â either. >> So the key point in this mathematical

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

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

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

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

Â is you should bid exactly the same as the evaluation.

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

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

Â For example, why not the third price? Okay.

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

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

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

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

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

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

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

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

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

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

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

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

Â Now what about the other folks in the system?

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

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

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

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

Â externality. Last lecture note we talked about the

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

Â Modify that. Or what we call the internalized

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

Â the lost evaluation to other folks in the system.

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

Â Now, internalizing negative externality sounds like, abnormal English.

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

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

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

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

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

Â 8:27

Today, eBay at least in the United States, is the largest auction based online store.

Â And it sells all kinds of stuff and some of you may have used eBay or similar

Â versions of it grown out of your native country.

Â It's effectively a second price auction, except there are four important twists.

Â The first twist is that there is also a reserve price, which is optional you don't

Â have to sell it and it is not released to the public.

Â So you can't have a star price of say ten bucks for those Elvis Presley autographed

Â CD andsay wow that's a great deal but you may have a actual reserve price which is

Â one million dollar unless, the final price is above one million you'll reserve the

Â right to just cancel the option and not sell it.

Â Okay? And this is useful because you can make a

Â price very low to attract eyeball attentions, and traffic, and bidding.

Â And then you still can reserve the right to cancel the auction if it's a very

Â disappointing sales. The second important difference is that

Â there are some information that is displayed.

Â Okay. Let's just, what is the information

Â displayed? What would happen is that each time that

Â there is a new price, Ebay would update an ask price, or the announce price.

Â And this price is the following. So suppose the auction concludes right now

Â at this moment. Okay.

Â So what will the price that the winner need to pay?

Â It's going to be the current highest bid, or the second highest bid, B2 plus some

Â basic increment Delta. What is delta?

Â Delta basically says, you cannot pick an arbitrarily small kind of increment each

Â time. You'll make the auction very inefficient.

Â Let's pick, say, the delta is one or something.

Â Each time you bid, you have to bid at least one more than the current one.

Â So, the second price is basically the driving force, but with an increment

Â minimum. But, in case this second price, ones with

Â the little increment, is higher than the highest bid.

Â Then you would simply pick the highest bid.

Â So this whole operation is all just complicated, because there's a minimum

Â increment. So had the auction concluded right at this

Â moment, you'ld be paying the smaller of the two numbers.

Â Your own highest bid or the second highest bid, plus the increment.

Â But the auction has not stopped yet, so in order to go further you need to at least

Â bid one more delta above this number. So the announced ask price is this

Â formula, the mean of these two numbers plus another delta.

Â And I say then this is not a private valuation anymore, cuz you have some

Â glimpse of what other bidders are thinking about.

Â And that's what makes things a little more fun.

Â 12:13

So you may observe that lot of times you think it's yours, until the last five

Â minutes or two minutes, and suddenly a lot of people jump in with very high bid, and

Â you can't even react to that in time because people know that it's the final

Â few minutes that matters in Ebay auctions. But there's also another behavior which is

Â where some bitter comes in with a ridiculously high price and to scare away

Â all the potential new competitors off the bet.

Â But last to twist this note; in order to make this user interface more friendly

Â Ebay allows a proxy agent. That means you can enter a maximum of bid.

Â You can ever tolerate, and say, well, I don't want to sit in front of computer

Â screen all the time. Why don't you do the bidding for me?

Â You meaning the assistant computer proxy agent.

Â So, anytime you have to ask prices below the my maximum tolerance, just go ahead

Â and enter a bid for me. But if it's above it, then just give up.

Â Okay. So anytime when I'm not highest bidder,

Â but I can still take the displayed asking price, then go bid for me, otherwise you

Â stopped. So those are the four key differences,

Â between Ebay and Second Price. So let's take a quick look at a simple

Â made up example of an auction. So there is one seller who is trying to

Â sell something, like a lamp on Ebay. And on day zero, that's the

Â initialization. You set up the auction on Ebay.

Â It's the seller. You say, put the five days into the

Â duration field. 'Cause everybody knows we will conclude

Â five days. Asking price would be, let's say five

Â bucks. And the minimum increment, that's the

Â delta. And it stays one dollar.

Â Okay? Just to make our example simpler to write

Â down. And you say the reserve price is just the

Â same as stock price, which means basically you give up the right to cancel the

Â auction if the sales is disappointing. You just want to sell it for whatever it

Â might be. Alright.

Â So, now, let's take a look at what would happen in the next five days.

Â So, day one let's say one potential buyer comes in, Alice.

Â And Alice uses a proxy agent and says, well, My tolerance, is twelve dollars.

Â Then what would happen? Well, Ebay's agent will therefore say,

Â good, current minimum price is still just starting.

Â It's five bucks. That's less than twelve, so I will enter a

Â bid for you. I will bid five dollars.

Â 14:55

And now, the ask price will be therefore, five+ delta, the minimal increment which

Â is six. So, in summary, the highest the bid at the

Â end of day one is five from Alice. And the ask price is six displayed, for

Â everyone to see. Now, day two, Bob's comes in.

Â And they say, oh, I'm willing to take that six.

Â In fact, I'm going to just make it eight bucks.

Â Maybe I'll just get item. >> Okay.

Â >> Now then what would happen? Anna's agent.

Â Will say well, I can actually take this because once Bob bids eight bucks, Ebay

Â will say, I announce eight+ delta, which is nine.

Â Anyone? And his agent says nine is not too bad.

Â I can go up to twelve. So I will take that.

Â So I will take nine. Okay.

Â So now they ask the price, becomes the following, is the minimum of the highest

Â bid now, nine, and the second highest bid, eight, plus the delta, which is one.

Â So, they are the same as nine, plus one more delta, okay, so, that's the ten.

Â So, ten is now the displayed ask price for everyone to see.

Â So at the end of day two, nothing further happens.

Â Highest bid is now nine from stil, Ellis, and the ask price displayed is ten.

Â 16:48

Remember the minimum delta increments already included in the sale price so bob

Â can pay exactly the ten or something little higher than ten; doesn't have to be

Â as big as another delta. It's already incorporated.

Â So now eBay says good, I'm going to display ten and a half plus a delta now

Â that's eleven and a half. Anyone?

Â And of course Alice Agent says, eleven and a half, it's getting close but still less

Â than twelve. Yes, I will go on to bid eleven and a

Â half. Now the ad's price becomes 11-1/2 or

Â 10-1/2 plus one. Whichever the minimum of the two, Plus

Â another one, that's 12-1/2. That's the display price.

Â So, at end of day three, the highest bid is 11-1/2 from Alice.

Â And ask price is twelve and a half for everyone to see.

Â And moving on to day four now. And Bob say what's going on here huh, I'm

Â going to bid a real big number, say seventeen and one-half.

Â Okay. So, maybe I'll get it this time.

Â Now the ask price. What is it?

Â It would be the min of two numbers. The highest bid, seventeen one-half, or

Â the second highest bid, eleven one-half, plus one dollar.

Â Plus another increment. That would be twelve and one-half, plus

Â another one. That would be thirteen one-half.

Â Now Alice agent says, I want to go in, but.

Â I was only authorized to go up to twelve. 13-1/2, I can't take it.

Â So, no action, nothing. And therefore, at the end of day four, the

Â highest the bid is now 17-1/2, finally. It's from Bob.

Â And, the ask price is 13-1/2. Notice, the ask price is actually not

Â telling you all the information. You don't know what's the highest bid.

Â Day five, the last day. It so happens, the third bidder comes in

Â at the last moment. It's a sniper.

Â And it so happens he looked at thirteen one-half, but somehow guessed a pretty big

Â number eighteen, to enter the auction. And now, the ask price becomes.

Â Eighteen or seventeen one-half plus one. This is highest bid; second highest bid

Â plus the delta whichever is the smallest, plus another delta and that is nineteen,

Â because it is the smallest of the two plus one.

Â Nineteen dollars the display price. So at end of this day, highest bid is

Â eighteen from Chris. Came in at the very last moment.

Â And the display prize is nineteen, if anyone want to take it.

Â Well, so happens that nobody want to take it, and the auction concludes.

Â At the end, who's the winner? Of course, Chris is the winner.

Â