Right. You have some kind of intrinsic view by
yourself. Subjective, but you know it.
Vi for the Ith buyer as the valuation. I would assume that these valuations are
private, meaning that the auctioneer or the other potential buyers do not know
your valuation, neither do you know the others.
And furthermore, it's independent. So VI, it just depends on I.
It doesn't depend on the VJs where J is not = to I.
And of course, in many cases the valuation is not completely private, for example, on
eBay as we have seen in an example, you get a glimpse of other people's potential
valuation, just by watching the announcement of the asked prices.
And whenever there is a secondary market. For example, houses, for closure houses on
auction block. The secondary market existence means that,
when you think about evaluation of a house, is influenced by other people's
evaluation, if you want to ever sell it again in the secondary market.
But, having said that, lets assume private and independent evaluation for what we
need to do with add space auctions. So in order to compare one auction
mechanism with another, one allocation with another, we'll have to think about
the metrics to compare them with. So what does each party in this ecosystem
want? For the seller obviously the most
important one is the revenue. How much do I generate by this particular
way to allocate and price? For the buyers, each of them, it is the
payoff that matters. It's the difference between the valuation
of getting this item, and the price they have to pay as the Ith buyer.
And later we'll look at the difference between happiness and the price.
We call that net utility. In this case we call this a payoff.
Now of course valuation is determined by you and as I said, we assume it's
independent of others, but the price is not completely determined by you, it's
determined by what other buyers do and the rules of this auction.
So the auction will decide what prices you have to pay.
So clearly if you know the auction rule changed, you may change your bidding
behavior because your payoff is the difference between the valuation and the
price. Now the auction designer, which could be
the seller itself will like to be sure the auction induces an efficient and fair
outcome. Now how do we define efficiency and
fairness. We'll see some example in other contexts
later in the course. But for today we will take proxy, as a
proxy the truthful bidding property that says if you view this item with a
valuation of VI then just bid, that's your bid exactly the I.
You think that sounds pretty intuitive. Actually, you will see that, in quite a
few cases, that's not necessarily true. So today, we will simplify the picture,
and say that truthful bidding is so desirable that we want to maintain it.
But actually, it may not lead to revenue maximization or payoff maximization in all
cases. So that's how we're going to compare
auctions. And then, we're going to analyze auctions
as games. Just like last lecture, when we talked
about disputed power control, and look at the power control game in cellular
networks. We can view auctions as games.
As mentioned, each game is defined by three tubbles.
Set of players, strategy space per player, and a payoff function per player.
So who are the players in the auction game?
Simple. It's just the set of buyers and their end
of them.. We'll assume that it's a fixed seller.
And who are they, what are the strategy spaces?
For each player I, each buyer I, basically, there is a set of bids that
she's willing to submit. I'll make this simple by saying it can be
any positive real number. What is interesting is the payoff
function. First of all there are two possible
outcomes. You may get the item in the single item
case. In a multiple item case you may get some
item okay, or maybe you don't get it. Now this outcome of allocation is
determined clearly by the collection of all the bits submitted by everyone.
So this defined by the vector b, not just your own, not just bi, but all the b's.
B1, B2 up to Bn. And in the case that you get the item.
I don't know, congratulations, now let's take a look at your payoff.
It is your valuation minus the price you pay.
Again, the price is a function of the entire b vector.
And the auction designer will determine the shape of this function.
How do you map the whole vector bids into a single number called the price to the
winner? We'll come back to that in a minute.
But whatever that might be, this difference is your payoff.
[inaudible] UI, the notation for payoff, which clearly is a function of co-vector b
as well. That's the coupling of all the actions by
the buyers. But in case you don't get it, then you get
nothing. You get zero, because you pay nothing, you
receive nothing. Now, vi minus pi could be positive, could
be negative, could be zero. We don't know, depends on what is the
