1:27

Right. You have some kind of intrinsic view by

Â yourself. Subjective, but you know it.

Â Vi for the Ith buyer as the valuation. I would assume that these valuations are

Â private, meaning that the auctioneer or the other potential buyers do not know

Â your valuation, neither do you know the others.

Â And furthermore, it's independent. So VI, it just depends on I.

Â It doesn't depend on the VJs where J is not = to I.

Â And of course, in many cases the valuation is not completely private, for example, on

Â eBay as we have seen in an example, you get a glimpse of other people's potential

Â valuation, just by watching the announcement of the asked prices.

Â And whenever there is a secondary market. For example, houses, for closure houses on

Â auction block. The secondary market existence means that,

Â when you think about evaluation of a house, is influenced by other people's

Â evaluation, if you want to ever sell it again in the secondary market.

Â But, having said that, lets assume private and independent evaluation for what we

Â need to do with add space auctions. So in order to compare one auction

Â mechanism with another, one allocation with another, we'll have to think about

Â the metrics to compare them with. So what does each party in this ecosystem

Â want? For the seller obviously the most

Â important one is the revenue. How much do I generate by this particular

Â way to allocate and price? For the buyers, each of them, it is the

Â payoff that matters. It's the difference between the valuation

Â of getting this item, and the price they have to pay as the Ith buyer.

Â And later we'll look at the difference between happiness and the price.

Â We call that net utility. In this case we call this a payoff.

Â Now of course valuation is determined by you and as I said, we assume it's

Â independent of others, but the price is not completely determined by you, it's

Â determined by what other buyers do and the rules of this auction.

Â So the auction will decide what prices you have to pay.

Â So clearly if you know the auction rule changed, you may change your bidding

Â behavior because your payoff is the difference between the valuation and the

Â price. Now the auction designer, which could be

Â the seller itself will like to be sure the auction induces an efficient and fair

Â outcome. Now how do we define efficiency and

Â fairness. We'll see some example in other contexts

Â later in the course. But for today we will take proxy, as a

Â proxy the truthful bidding property that says if you view this item with a

Â valuation of VI then just bid, that's your bid exactly the I.

Â You think that sounds pretty intuitive. Actually, you will see that, in quite a

Â few cases, that's not necessarily true. So today, we will simplify the picture,

Â and say that truthful bidding is so desirable that we want to maintain it.

Â But actually, it may not lead to revenue maximization or payoff maximization in all

Â cases. So that's how we're going to compare

Â auctions. And then, we're going to analyze auctions

Â as games. Just like last lecture, when we talked

Â about disputed power control, and look at the power control game in cellular

Â networks. We can view auctions as games.

Â As mentioned, each game is defined by three tubbles.

Â Set of players, strategy space per player, and a payoff function per player.

Â So who are the players in the auction game?

Â Simple. It's just the set of buyers and their end

Â of them.. We'll assume that it's a fixed seller.

Â And who are they, what are the strategy spaces?

Â For each player I, each buyer I, basically, there is a set of bids that

Â she's willing to submit. I'll make this simple by saying it can be

Â any positive real number. What is interesting is the payoff

Â function. First of all there are two possible

Â outcomes. You may get the item in the single item

Â case. In a multiple item case you may get some

Â item okay, or maybe you don't get it. Now this outcome of allocation is

Â determined clearly by the collection of all the bits submitted by everyone.

Â So this defined by the vector b, not just your own, not just bi, but all the b's.

Â B1, B2 up to Bn. And in the case that you get the item.

Â I don't know, congratulations, now let's take a look at your payoff.

Â It is your valuation minus the price you pay.

Â Again, the price is a function of the entire b vector.

Â And the auction designer will determine the shape of this function.

Â How do you map the whole vector bids into a single number called the price to the

Â winner? We'll come back to that in a minute.

Â But whatever that might be, this difference is your payoff.

Â [inaudible] UI, the notation for payoff, which clearly is a function of co-vector b

Â as well. That's the coupling of all the actions by

Â the buyers. But in case you don't get it, then you get

Â nothing. You get zero, because you pay nothing, you

Â receive nothing. Now, vi minus pi could be positive, could

Â be negative, could be zero. We don't know, depends on what is the

Â 7:55

But the prize clearly depends on others' behavior, the others' bidding behavior.

Â And what is interesting is that, by deciding a different function that maps

Â bidding behavior to prize, you will, in turn, induce different bidding behavior.

Â Okay. Different auction rules will induce

Â different bidding strategies in this auction game.

Â For example, lets take a look at a, a simple example.

Â Coming back to this. This is the definition of payoff function,

Â branching to two possibilities, determined by the b vector in one case, you look at

Â the difference. This part of difference determined by the

Â b vector. So, let's just look at this part.

Â P I as a function of vector B. And maybe I should just say let it be B I.

Â Okay, your own bid so if you win it then [inaudible] is the first prize.

Â You pay the first prize, and that is exactly what we saw.

Â And, it sounds quite intuitive. Okay, see what envelope, then just, you

Â know, you give it to whoever is the highest bidder and that decides the

Â allocation. As to the pricing, just pay the price that

Â person bid it. Then in this case your payoff, UI, is just

Â VI minus BI. Right?

Â Your own valuation minus your own business, if you get it.

Â So you may think what should I bid. Well, if I bid bigger than my valuation,

Â I'll get a negative utility or payoff. That's not good.

Â If I bid exactly as my valuation that's not good either cuz I get zero.

Â I may as well just not bid the item. So, I'm going to bid a little below the I,

Â but if I bid too low, I may not get item, so how should I bid in order to maximize

Â my payoff? And was impact on Google's revenue that's

Â actually pretty complicated question. We won't have time to talk about that

Â today. Instead we're going to look at the

Â intuitively not making sense but actually very sensible second price this says.

Â The price you pay is actually somebody else bid.

Â And this J user is exactly the second highest bidder.

Â And your utility or payoff, trying to differentiate U from V is your V minus

Â this BJ somebody else J. And it so happens that by decoupling the

Â winning and losing decision from the actual price, you pay.

Â Remember, utility depends on both allocation and the pricing.

Â Decoupled allocation decision and the pricing decision actually induces truthful

Â bidding behavior from all the potential buyers.

Â