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. Today, eBay at least in the United States, is the largest auction based online store. And it sells all kinds of stuff and some of you may have used eBay or similar versions of it grown out of your native country. It's effectively a second price auction, except there are four important twists. The first twist is that there is also a reserve price, which is optional you don't have to sell it and it is not released to the public. So you can't have a star price of say ten bucks for those Elvis Presley autographed CD andsay wow that's a great deal but you may have a actual reserve price which is one million dollar unless, the final price is above one million you'll reserve the right to just cancel the option and not sell it. Okay? And this is useful because you can make a price very low to attract eyeball attentions, and traffic, and bidding. And then you still can reserve the right to cancel the auction if it's a very disappointing sales. The second important difference is that there are some information that is displayed. Okay. Let's just, what is the information displayed? What would happen is that each time that there is a new price, Ebay would update an ask price, or the announce price. And this price is the following. So suppose the auction concludes right now at this moment. Okay. So what will the price that the winner need to pay? It's going to be the current highest bid, or the second highest bid, B2 plus some basic increment Delta. What is delta? Delta basically says, you cannot pick an arbitrarily small kind of increment each time. You'll make the auction very inefficient. Let's pick, say, the delta is one or something. Each time you bid, you have to bid at least one more than the current one. So, the second price is basically the driving force, but with an increment minimum. But, in case this second price, ones with the little increment, is higher than the highest bid. Then you would simply pick the highest bid. So this whole operation is all just complicated, because there's a minimum increment. So had the auction concluded right at this moment, you'ld be paying the smaller of the two numbers. Your own highest bid or the second highest bid, plus the increment. But the auction has not stopped yet, so in order to go further you need to at least bid one more delta above this number. So the announced ask price is this formula, the mean of these two numbers plus another delta. And I say then this is not a private valuation anymore, cuz you have some glimpse of what other bidders are thinking about. And that's what makes things a little more fun. The third important twist is there is also a fixed and publicly announced time horizon usually you know five days, seven days, nine, eleven days, and so on. This means that everybody knows when this auction will end and, therefore a lot of people wait until the very end in what is called a sniper action. So you may observe that lot of times you think it's yours, until the last five minutes or two minutes, and suddenly a lot of people jump in with very high bid, and you can't even react to that in time because people know that it's the final few minutes that matters in Ebay auctions. But there's also another behavior which is where some bitter comes in with a ridiculously high price and to scare away all the potential new competitors off the bet. But last to twist this note; in order to make this user interface more friendly Ebay allows a proxy agent. That means you can enter a maximum of bid. You can ever tolerate, and say, well, I don't want to sit in front of computer screen all the time. Why don't you do the bidding for me? You meaning the assistant computer proxy agent. So, anytime you have to ask prices below the my maximum tolerance, just go ahead and enter a bid for me. But if it's above it, then just give up. Okay. So anytime when I'm not highest bidder, but I can still take the displayed asking price, then go bid for me, otherwise you stopped. So those are the four key differences, between Ebay and Second Price. So let's take a quick look at a simple made up example of an auction. So there is one seller who is trying to sell something, like a lamp on Ebay. And on day zero, that's the initialization. You set up the auction on Ebay. It's the seller. You say, put the five days into the duration field. 'Cause everybody knows we will conclude five days. Asking price would be, let's say five bucks. And the minimum increment, that's the delta. And it stays one dollar. Okay? Just to make our example simpler to write down. And you say the reserve price is just the same as stock price, which means basically you give up the right to cancel the auction if the sales is disappointing. You just want to sell it for whatever it might be. Alright. So, now, let's take a look at what would happen in the next five days. So, day one let's say one potential buyer comes in, Alice. And Alice uses a proxy agent and says, well, My tolerance, is twelve dollars. Then what would happen? Well, Ebay's agent will therefore say, good, current minimum price is still just starting. It's five bucks. That's less than twelve, so I will enter a bid for you. I will bid five dollars. And now, the ask price will be therefore, five+ delta, the minimal increment which is six. So, in summary, the highest the bid at the end of day one is five from Alice. And the ask price is six displayed, for everyone to see. Now, day two, Bob's comes in. And they say, oh, I'm willing to take that six. In fact, I'm going to just make it eight bucks. Maybe I'll just get item. >> Okay. >> Now then what would happen? Anna's agent. Will say well, I can actually take this because once Bob bids eight bucks, Ebay will say, I announce eight+ delta, which is nine. Anyone? And his agent says nine is not too bad. I can go up to twelve. So I will take that. So I will take nine. Okay. So now they ask the price, becomes the following, is the minimum of the highest bid now, nine, and the second highest bid, eight, plus the delta, which is one. So, they are the same as nine, plus one more delta, okay, so, that's the ten. So, ten is now the displayed ask price for everyone to see. So at the end of day two, nothing further happens. Highest bid is now nine from stil, Ellis, and the ask price displayed is ten. Moving on to day three. Now Bob says what, ten dollars? Alright. I'm going to! So clearly my eight bucks didn't work so I'm going to pay ten and one-half. Remember the minimum delta increments already included in the sale price so bob can pay exactly the ten or something little higher than ten; doesn't have to be as big as another delta. It's already incorporated. So now eBay says good, I'm going to display ten and a half plus a delta now that's eleven and a half. Anyone? And of course Alice Agent says, eleven and a half, it's getting close but still less than twelve. Yes, I will go on to bid eleven and a half. Now the ad's price becomes 11-1/2 or 10-1/2 plus one. Whichever the minimum of the two, Plus another one, that's 12-1/2. That's the display price. So, at end of day three, the highest bid is 11-1/2 from Alice. And ask price is twelve and a half for everyone to see. And moving on to day four now. And Bob say what's going on here huh, I'm going to bid a real big number, say seventeen and one-half. Okay. So, maybe I'll get it this time. Now the ask price. What is it? It would be the min of two numbers. The highest bid, seventeen one-half, or the second highest bid, eleven one-half, plus one dollar. Plus another increment. That would be twelve and one-half, plus another one. That would be thirteen one-half. Now Alice agent says, I want to go in, but. I was only authorized to go up to twelve. 13-1/2, I can't take it. So, no action, nothing. And therefore, at the end of day four, the highest the bid is now 17-1/2, finally. It's from Bob. And, the ask price is 13-1/2. Notice, the ask price is actually not telling you all the information. You don't know what's the highest bid. Day five, the last day. It so happens, the third bidder comes in at the last moment. It's a sniper. And it so happens he looked at thirteen one-half, but somehow guessed a pretty big number eighteen, to enter the auction. And now, the ask price becomes. Eighteen or seventeen one-half plus one. This is highest bid; second highest bid plus the delta whichever is the smallest, plus another delta and that is nineteen, because it is the smallest of the two plus one. Nineteen dollars the display price. So at end of this day, highest bid is eighteen from Chris. Came in at the very last moment. And the display prize is nineteen, if anyone want to take it. Well, so happens that nobody want to take it, and the auction concludes. At the end, who's the winner? Of course, Chris is the winner. He holds the highest bid and that conclusion point. And what's the price, he's got to pay. What is the mean of the highest bid and the second highest bid plus the delta. Because the delta set is a little big in this example, so it actually exceeds the highest bid. You would never be asked to pay bigger than your own a highest bid. So you will be smaller of the two which is eighteen. So, Chris pays eighteen and receives the lab.