>> Max Chickering: So it's my pleasure to introduce Yagil to the packed audience here. Yagil
and I worked together in Live Labs in 2007, and we did a bunch of work with paid search, and so
Yagil is going to talk about pricing display ads now. He's in our research lab in Israel.
>>Yagil Engel: Thank you Max. I will talk about the posted prices exchange for display
advertising contracts. This is joint work with Moshe Tennenholtz, with MSR Israel. Display ads
are those banner ads that are shown on webpages and more and more on apps. What they let
us do is effective segmentation of the target audience according to criteria that are some
subset of large number of available criteria which can be demographic, they can relate to the
browsing history of the user, and they can relate to the current content that the user is
accessing. I will start with some preliminaries about the current display advertising market with
motivation and some basic notation, and afterwards I will get to the actual description of the
market that we propose, equilibrium, and how we calculate demand and supply.
The atomic good in the display ad market is called an impression, which is one view of one user
of one ad. Impressions are sold in roughly 2 ways: today, either through a spot market, which
takes place while the user is browsing, or through contracts for future impressions; a contract
specifies quantity of impressions of specific segments and specific time framing in the future
which is those contracts should be delivered. Those impressions should be delivered. And the
market today of contracts is very dispersed, very decentralized. Contracts are sold by bilateral
negotiation between sales force of the publisher. In our case, our salespeople and the
advertisers. The problem with this is that, beside efficiency, in terms of economic efficiency
that is of course not, does not exist in this market at all, it is, it will also not scale to a large
number of publishers. So, today we are selling our own inventory and the inventory of some
large publishers that are, for which we are providing these sales as a service like Wall Street
Journal. But it will not let us take a bite of them, of the larger market where lots of little
publishers of apps or webpages are selling their own inventory.
Looking at the literature, we see there are, I know of two kinds of works. One is talking about
the pricing of contracts under the framework of the current decentralized market, these are
works that came from Yahoo and Google, and there is also work that came from Yahoo, which
we can say today that came from MSR that, in which a publisher is selling fixed inventory via an
auction. Our goal is to introduce a market that is, at the same time continuous, so trade can
occur at any time, not as [inaudible] auction event that, to which advertisers and publishers
have to wait, but something that exists all the time. But on the other hand, it also brings the
economic efficiency into this picture so we can achieve some global optimality of a location.
And it is also two-sided. So many advertisers, many publishers can participate easily. And the
approach we chose for that is that of posted prices which are based on supply and demand. So
prices that are will equate supply and demand on each impression type.
Besides the advantages that I just mentioned, it will also let us elicit the demand in response to
prices, rather than asking advertisers to report their willingness to pay for various types of
impression, we give them prices and they tell us what they want to buy in these prices. And
this lets us do effective utility elicitation, even if the utility structure that we assume is pretty
expressive. And the second thing it let us do is to assume, it lets us be sloppy on the incentive
side by assuming that the players, the advertisers and the publishers, are price takers, which
means that they are not aware or they just cannot affect the price by their behavior because
there are just too many players. So one player cannot, will not have any incentive to change
his behavior, not to be behave optimally because it will not change the price anyway. So it will
just optimize their behavior with respect to current prices.
The challenges with this approach is that first, how do we define the prices? We cannot define
price every atomic good because there are just too many of them. We have thousands of
criteria, so the number of impression types is exponentially in the thousands. We also don't
want to take some subset of the atomic groups and price them because then it will be too
specific, the chance that it will get trades on those exact goods are slim. And the third
challenge is that in order to find prices that equate supply and demand we have to do some
kind of depend elicitation, and we have to consider how we are going to represent supply.
Technically, our approach would be to price, we’ll solve this price structure problem by pricing
sets of impressions. Sets that the demand does not distinguish between various items in the
set. And this will, actually the set of prices that we’ll have is small enough to be polynomial in
the input that the system gets in the demand that the system gets, but will still be expressive
enough so that the demand cannot be distinguished between elements of the same price of the
same set.
The problem, the technical problem with this is will be that those sets of impressions that have
the same price overlap. And then we’ll have to think about what to do with the impressions
that can fit into two sets. I hope this will become clearer later. And then we will obtain
demand quantities with respect to current prices, so if we have some initial set of prices, we
elicit demand based on these current prices, from that we will be able to extrapolate the
demand or the utility functions as functions of any price, so we will be able to assess the
demand that is going to be in any other price. And this will let us calculate demand and supply
for any price, and with this we can calculate equilibrium prices.
Okay. So little bit of notation. We have a set of attributes which are those segmentation
criteria. For example, gender, income, sports, location, the space of impressions is the
Cartesian product of the domain of these attributes. The size of this domain, like I said, is the
size of A is exponentially in the thousands. So what we do when we have a size that is too large
to deal with, we define an even larger space, and this is going to be this space F which includes
not only all of the fully specified impressions, but also all the partially specified impressions. So,
for example, high income male would be a statement in F because it is a partial assignment to
the attributes. Actually F is not that larger than A, it’s just that we can think about it as adding
the null value to the domain of each attribute and then taking the Cartesian domain of those
new attribute domain, the Cartesian product on those on those new attribute domains. But
what this means is that every statement in F refers to a set of atomic impression types. For
example, a high income male will refer to all the impression types of for each, if income is high
and the gender is male, and it also gives rise to a satisfaction relation where we say that f prime
satisfies f double prime if all the impressions in the set f prime are also in that set f double
prime, and here's an easy example, if f prime is high-income male, then it satisfies the
statement high income because any a high income male is also just high income person.
>>: Can you explain more of what the [inaudible]?
>>Yagil Engel: Yeah. Maybe this should have been domain prime. So this is, so I defined a new
domain which includes, for each attribute I defined a new domain which includes the original
domain [inaudible] plus the null value. Now I take the Cartesian product of this new domain
and this is it. So any, so F includes all the full specification of all of the impressions that are fully
specified, but also impressions that are partially specified, like high income male I'm only
specifying the value of this attribute and of this attribute, so this refers to any full specification
for which G is male and I is highly.
>>: Okay.
>>Yagil Engel: So these were the preliminaries, now I will define how this market is going to
look like, what are the market goods, the demand, supply and the equilibrium? So we have
some n advertisers, a1 through a, n. And we assume that each advertiser A, J cares about
some set of demand statements. Those statements in F. So one advertiser wants to advertise
to high income females and to male scientists. And the second advertiser wants to advertise to
high income females and to cyclists and to males. And the third advertiser was advertise to
female cyclists. Then we will just take all of those statements and define them as a set of
market goods. So the second market goods here would be high income females, male cyclists,
cyclists, males, and female cyclists. And this is of course, linear in the size of those, of the input
that the system gets from advertisers which are those demand statements.
And we will, just price those goods. So the market will assign a price vector of [inaudible] that
assigns a price to each good in D. Now these are somewhat nonstandard market goods. Like I
mentioned, each market good is a set of atomic goods. Cyclists refers to all the impressions in
which sports equals cyclists. And males refers to all the impressions which genders equals
male. And the problem is that we might have male cyclists and to each one of those goods a
demand or supply of male cyclists will refer. So this is something we'll have to answer. For
now, we'll just say that the semantics of this good is that if with we are purchasing a male
cyclist and we're getting a guarantee to get an impression of a male cyclist, if I'm just
purchasing an impression in which sports is cyclists, then I don't know anything about the
gender, for example. So it might be a male or female, so it's not guaranteed to be random,
male or female, it's just what it is so it just provides me the guarantee on those attributes that
are mentioned in the statement that I'm purchasing. We'll see more examples so it will, it
might be more clear.
What does it mean in terms of demand? So we have this set of market goods, which we call F1
through F Omega. This is D. We know that advertiser A, J gets utility from some subset. Some
subset of these goods. Those goods that he actually mentioned, that he cares about, which are,
let's say it's we’ll index them F one through F Omega J. So he has utility function that is defined
over quantities of those Omega J goods. So he has utility function U, J that is defined over Q1
through Q, Omega J. So some of the market goods are not mentioned in the utility function of
these advertiser A, J, which normally this is okay because we can just assume that if the market
good is not referred to by consumer’s utility function, that it’s just getting zero value from that
good. But this is not the case here because maybe advertiser A, J wants to purchase cyclists, for
example, so his utility function refers to cyclists. But then there are other market goods which
are cyclists from Michigan or high income cyclists. And they are also cyclists. So they are part
of this market good that is called cyclists. So it's not, and we assume that advertiser A, J will get
the same utility from getting a cyclist for which Michigan is also specified, or high income is also
specified, because, merely because they are also cyclists. So it's not true that [inaudible]
getting zero value from those other market goods that are not mentioned by utility function.
But what we will do, we'll just refer to the economic definition of demand and we’ll just leave it
at that for now. And we’ll show later how this can be easily overcome.
So the demand at prices pi prime given some budget constraint that each advertiser has, so A, J
has budget constraint W, J. It's just the vector of quantities of the market goods, so we'll just
stick to the economic definition of demand, is a vector over the market goods, which maximizes
the utility of that advertiser, the utility that is defined over this subset of goods under prices pi
prime and budget constraints W, J. And will just have to, we'll just assume that the utility
maximization procedure will know how to deal with those market goods that are satisfying the
goods that appear in our utility function. We’ll see later how it is done. And will also define the
aggregate demand as the sum over the demand of all the advertisers.
On the supply side, we have an analogous problem in which supply, one supplying impression
type may satisfy multiple market goods. So for example, a supply may have high income
cyclists, and there is one market good which is high income and one which market good which
is cyclists. So to which of these do we assign that good? Well this is actually completely falls
into the, was called in economics, production market. Because this advertiser just, it can be
seen as a production choice, whether I produce with what I have I can the either produce one
unit of the market good that is called high income are one unit of the market good that is called
cyclists. And it's actually a very simple production problem because we don't need any inputs.
And now we can define the supplier pi prime as profit maximizing production plan. So we'll
have S, J, pi prime will be the quantities will also be a vector of quantities over the market
goods which maximizes the profit of the supplier, and what we actually do, we are also pushing
here the question of how to assign the goods, we are pushing it into this problem of profit
maximizing production plan. And the aggregate supply is also, is again the sum over supply of
all the publishers.
So now we can define market equilibrium, which is just a vector of prices that equate supply
and demand. And what we can also do is easily show that an equilibrium exists in this market
because we can take the market model that we just defined and easily, pretty easily reduce it to
was called an Arrow-Debreu production economy, which is a model where we have consumers,
we have producers, and they also, there are a few things that we do not need from this more
general model, but we can easily make the reduction. So an equilibrium is going to exist under
some mild assumptions like convexity and continuity of the advertiser’s utility and we’re going
to have that. And in addition, there are previous results, previous literature that show that
equilibrium prices can be computed in polynomial time under some additional assumptions,
mainly on the utility function and we're going to make those assumptions too, later. But we’ll
also have to show, in order for this result to hold, that we can calculate, we can actually find
out what is the demand and what is the supply in polynomial time for any given vector of
prices. So this will be the key here, to be able to calculate, to find equilibrium in polynomial
time.
So this would be the next section where we’ll describe those demand and supply oracles. So
the, in order to be able to describe how we calculate demand, we need to first come up with
some model for the utility function of advertisers. How this utility function look like. So we
have this set of market goods D, we know that advertiser A, J cares about some subset of this
good, which is called B, J and these are F1 for F Omega J, so we have a utility function that is
defined over quantities Q1 through Q Omega J, and there is some budget W, J to these
advertisers. Now the first assumption that we are going to make is what called in economics
constant elasticity of substitution. In the next slide I will describe what this assumption means.
In the meantime, I will just give the mathematical interpretation of that, which is this, under
this assumption, the utility function must looks like that which is a summation over weighted
quantities where each quantity is also taken to the power of row which is constant that is
related to the elasticity of substitution. And those weights sum up to one. Now what elasticity
of substitution means is, elasticity of substitution is denoted by Sigma, which is actually one
over one minus row, so this is how these two parameters relate to each other. It refers to how
flexible the budget allocation of, how flexible the advertiser is in his allocation of budget
between impression types.
So on one extreme, if Sigma equals one, meaning that the elasticity is very small, is the smallest,
and this happens when row is taken to zero at the limit, what you get is the Cobb-Douglas
utility function, and under Cobb-Douglas, the budget is firmly allocated between the goods,
which are impression types in this case. So if we have some budget of hundred thousand
dollars and we decided to spend, and given one set of prices, we are spending 20,000 on
cyclists from Michigan, and 85,000 on high income females, then even if prices will change of
any of these goods, it will not change this budget allocation. So if the price of cyclists from
Michigan will go up, we’ll just purchase less of these, and we will not transfer, it will not cause
us to transfer budget between those impression types.
And this is actually completely not how advertisers behave. They substitute all the time
between different impression types according to the prices. So this is very limiting and on the
other extreme, the elasticity is infinite, so Sigma equals infinite, and this happens when row
equals one, what we get is the linear utility function. And you can see if we replace row here by
one we get linear function. And with the linear utility function, there is no notion of allocating
budget to goods. So it is, we'll just get, we’ll have some activity from each impression type and
we will just get the combination of, or the collection of goods that maximizes our linear utility
function. So if a price of one good becomes really, really low, then we’ll just purchase lots and
lots of this impression and we might not purchase other impression types at all. So we might
allocate all the budget to that cheap impression type. And this is also not how advertisers
work, they do care about some balance between the types of audience that they are exposed
to.
So what our structure means is that we cover all this spectrum between those two extremes.
So we can take Sigma anywhere between one and infinite. Or row anywhere between zero and
one. So the elasticity of substitution can be anywhere, can be anything as long as it is constant.
So we only assume that this elasticity is the same for any pair of goods, but we do not assume a
particular value. And as you can see, it is much more general than the linear utility function
which is taken an assumption in most of the previous literature. At least when we are talking
about such a huge space like the display advertising space. The second assumption that we
take is that the goods are gross substitute. This means that we do not allow advertisers to
express complementarities between different impression types. So we will never, so an
advertiser will never be willing to pay for getting two types of impressions more than he will
pay, more than the sum of what he will pay for each one separately. And none of this
assumption will strictly hold in practice, but we believe is that it is close enough, what this gross
substitute assumption let’s us, this is exactly the assumption that is required to achieve this
polynomial computation in previous literature. I will also show where, why this assumption is
required for computation of purposes. In terms of the mathematical structure, what this
assumption means is that row is between zero and one, if row is less than zero, then what we
get are complements.
>>: So the constant elasticity, is that for advertiser or for [inaudible]?
>>Yagil Engel: It doesn't have to be the same elasticity for everybody. So next is the actual
elicitation of demand. How we figure out what is the demand of an advertiser. And let's first
assume that we know the elasticity of a given advertiser, row, and we assume that the input
that we get is a demand vector at some price vector. So we have prices pi prime out there and
advertisers coming and purchasing Q1 through Q Omega, some quantity of each market good.
So now we use what's called first order condition, meaning we take the derivative of the utility
and equate it to zero, so this must hold at the utility maximizing demand. And it can, this
equation can be pretty easily obtained. It's called the tangency condition of the constant
elasticity utility. It tells us about this ratio between prices and the weighted goods. And we are
getting, we have Omega minus 1 equations like that, so when I equals one and K equals two,
and I equals two and K equals three, and so on. And we also have, we also know that the betas
sum up to one, so what we have are Omega linear equations with Omega unknowns, so we can
solve them and get the betas, remember that what the knowns here are the pi’s and the Q’s
because we are getting the demanded price at price pi prime. Actually, we don't even have to
solve the system of linear equations, we can just start with any beta and instantiate the rest of
the betas and then normalize it so that those betas sum up to one. And we also assume that
we have the budget now because we assume that the advertisers are exhausting their budge by
stating their demand vector. So the budget is just the sum of the expenditure of the advertiser
in these prices.
Now, initially we do not know what the elasticity role. So in order to be able to calculate it, we
need two demand factors at two different prices. And then what we can do is some binary
search over the values of row between zero and one for each one, we can find the betas fast
and we’ll actually have too many equations when we come to solve, to find this vector of betas,
and most likely we’ll not get an exact solution, but we’ll just find the best set of betas and find
the row that gives us the smallest error over the betas. And how do we get two demand
vectors at two different price points? This is going to be a market which is dynamic, which is,
prices are out there, prices change, and even though, so when an advertiser comes and gives us
the first demand vector, we will not have a very good assessment of the utility function because
we'll have to guess some elasticity, maybe based on other advertiser’s elasticity, but then once
price changed and we get another demand vector, we can get it much better assessment.
>>: So how important, so I assume that you can solve for row or you can fit row, if only a
subset of the [inaudible]. If you could change just a few of them, then you’re going to get kind
of evidence of this>>Yagil Engel: Yeah, because actually because all we need is one more equation. So even if one
price changes this would be enough. But we can't, I mean, we do not want to assume that only
one price change. So the demand oracle now, first we’ll show that it's actually not a problem
that we have those overlaps that I showed in the beginning, so we have several market goods
that may satisfy a single demand statement. We'll define this set as Delta f. So, for example, if
a statement is, if an advertiser wants to advertise to cyclists from California, and then there
might be two market goods, one is cyclists from California and the other is female cyclists from
California, they both satisfy this demand statement. And what we want to do is just minimize,
so first, remember that we are looking for the demand set of an advertiser given the prices.
This is completely greedy thing. So the advertiser wants to, in equilibrium, the demand set of
each advertiser must be, must give that advertiser the highest profit or the highest gain. So
what we want to find is the set of impressions that satisfy the demand that provides the highest
utility to that advertiser at the lowest price. What I'm saying is that it we can safely pick the
cheapest impression type of this set Delta f. And we call this f hat k, so for the impression type
that is the cheapest way to satisfy the demand statement F, K is F, K hat. And now we can get
utility maximizing demand factor in the current prices, which means it will just do maximization
of this utility function given the budget constraint, and this is also pretty easy to solve. Again,
we take first of the condition, we get that equation from a previous slide, this tangency
condition, and now the betas are known and the pi’s are known and what we are looking for
are the Q’s, but we can find it exactly the same way that we did before instead of the last
equation, the normalization will not be the sum of betas because the betas are known, but will
be that budget constraint. So after we instantiate the Q’s, we’ll just normalize it into the
budget constraint.
>>: So I’m a little bit confused. So suppose I’m the producer and I have a female cyclist from
California. In dropping the female is cheaper, aren’t I never going to drop the female?
>>Yagil Engel: You will. First of all, this is the demand side. We'll talk about the production in
the next slide. But you will, definitely. The answer is yes. But we just didn't get there yet. This
is on the demand side. If I want to advertise to a cyclist from California, and it’s cheaper to
advertise to a female cyclist from California, then I will pick female cyclists from California
because it satisfies my demand. This is on the demand side. What you said is true for the
production, for the publisher side.
>>: So when you're picking the cheapest, it’s probably the case that female isn’t going to be
cheaper because if it were, it wouldn’t have been [inaudible].
>>Yagil Engel: Right. Not in equilibrium. So this can happen on the way to the equilibrium
prices. In equilibrium we will not even have to. Yeah. Now I see where your question came
from. So now looking at the supply side, the problem that we have here first is to match the
supply with the market goods. The market goods came from the demand. The supply can look
very differently. And what we are, what we need in order to answer this question is some way
to represent the supply. The first representation that we considered is the proposal that came
from MSR, to represent supply by Bayesian network. I think I will skip the, this is a very
technical part, for which the result is that we do not, we can't or we don't know how to use
Bayesian networks in this context, so I will skip this part. What we need in order to be able to
match the supply with the market goods is what we called the more explicit supply
representation as opposed to the Bayesian network which is a very, a concise representation.
Then with an explicit representation, a publisher will submit a set of triplets where each triplet
mentions an impression type, the quantity that he has from that impression type, and the
impression type is in f, right? It can be a statement. It doesn’t have to specify all the attributes.
So the impression type, the quantity, and the minimum price that you want for these
impression types, for this impression type, the reason I think that minimum price is needed
here is that it's not true that the costs of putting an ad is completely zero, at the very least, we
have to consider the disturbance to the user.
And the key about this representation is that those quantities are assumed to be disjoined. So
we’ll say that the total number of impressions in the inventory of that supplier is the sum over
those quantities. We can see it in order to understand it we can look at that tree. We have, at
each node we are branching on one of the attributes. So at the top here, we branch over the
location. And now we say that for those impression types that are from Michigan, we branch
over the income. So we have 100,000 impressions of high income people from Michigan, and in
addition, we have 300,000 impressions of low income people from Michigan, and then in
addition to that, we have 150,000 impressions for which we don't know the income, 50,000 of
which, are cyclists, and another hundred thousand we do not know whether they are cyclists or
not or what kind of sports they like. So here we have those hundred, actually here we have
another hundred thousand for which we know they are from Ohio, but we don't know anything
else. And in addition, we have 200,000 for which we don't know the state but we know that
they are females. So you get the point, each leaf here has a quantity which is disjoined from
the rest of the quantities. So we can just to sum up over the all the quantities that we have and
this is the total number of impressions.
The key, whether this is a valid supply representation or not is whether supplies can even come
up with such a representation. And we believe they can even though I forgot to mention that is
a work in progress, and some things are not completely nailed down yet, but we believe that
this is, that this representation can be learned from data. This actually looks like a decision tree
and it can be learned like a decision tree where the class that we are, that this decision tree is
classifying on is the price. Because the supply actually doesn't care, doesn't need to distinguish
between different goods if they have the same or roughly the same price. So we can assume
that we are going through the logs of the type of impressions that the advertisers have seen in
the past. And we can obtain a price for each such type of impression just by querying our own
price structure, or if we don't have it yet, we can look at bids in the spot market. So we have a
price for each type of impression, and we learn this decision tree and I will was told that this is
because the class is continuous, it is the price, then what we're getting is actually a regression
tree. So we learn this tree from this past data, and what we get is that we only distinguish
between a different; we only expand the tree as long as it can provide the supplier more
revenue to do so. You will not expand it more if it doesn’t have the potential to give more
income. If, at some point we are, it's possible that some advertisers will not get the type of
impressions that they need even though such supply exists because it has the same price as a
different kind of impression. But in that case, the price of that impression type that the
advertiser really wants will go up. So there will be difference in the price at some point, and
then the publisher will have a reason to distinguish between these impression types. Does is
make sense? This is pretty new, the idea to build this as a decision tree, so this [inaudible]
requires more thought.
>>: The last point was, the question I had is I understand you’re representing all of the
inventory [inaudible]. And the claim is that if the supplier doesn't care about distinguishing
inventory, that means that neither, there is no demands [inaudible] as well.
>>Yagil Engel: Right. Well, initially, it’s possible that you have two types of different goods
which have the same price. And somehow, so let's say that one advertiser wants high income
people from Michigan and another advertiser wants low income people from Michigan. But
they have the same price. So what the supply does, so if they have the same price, the supplier
does not need to branch here. But then, prices will go up because one of these goods will be in
over demand. So at least in the next round, hopefully the supplier will have reason to branch
over those, over the income.
>>: But do you need to know Q for these two distinctions I don't care about in order to get to
the [inaudible]?
>>Yagil Engel: Well, what will happen in this scenario that I described is that you will have some
quantity of people from Michigan. You will not have high income people from Michigan. And
then there will be a demand statement for high income people from Michigan, which there will
not be enough for supply for. So the price of that, of high income people from Michigan will go
up.
>>: So the supply is zero for that in that case until>>Yagil Engel: From this publisher. You have lots of publishers. Maybe the supply is zero.
>>: So how do you, so are you going to need to take the M into account in your split criteria?
>>Yagil Engel: I don't think so. No. M will be used in the next slide. But the fact that I have
different minimum price for these two impression it types, I don't think really matters because I
will, in any case, I will only sell an impression if its price is at least M. But maybe it makes sense
to branch so I will supply those impressions that are, but it's not M, if it's considered cost then
maybe, yeah, because I will want to supply the impressions that have, that are cheaper for me.
But at least in our current semantics, we don't assume that M is cost. We do not in the
optimization here, the profit maximization, we just take maximum price, we do not take price
minus M. If we do, which might make sense, then yes. Then we should maybe branch, instead
of the price, we should branch over price minus cost.
So for each, so the supply oracle, when we have this explicit representation, would be pretty
simple, again, we find the set of matching market goods for each supply good, so this is Delta f
here. This is reversed to what we did in the demand. Here we’re looking for all the market
goods that are satisfied by our supply goods. So if we have female cyclists from California, then
we might have, this satisfies market good like cyclists from California or someone from
California and so on. And of these, we picked the one with the highest price, and here we have,
the reason this is so simple is that our utility function is linear here. We just look for, we'll just
assign the quantities of our good, of our inventory in a way that will maximize the sum of the
prices that we get. And we assign a good to a market good, we assign an inventory impression
to a market good at the current price of that market good, this cheapest price that we got here
is at least our minimum price.
Now, what's left is finding the equilibrium, and I already mentioned that there is a result, that
there is an algorithm for finding the equilibrium in polynomial time. And we have the supply
oracle, the polynomial supply and demand oracle, and we have the gross substitution
assumption, so we are fine there. But the problem is that algorithm is relying on the ellipsoid
algorithm that is known not to perform well in practice, so we are also proposing this simple
[inaudible] like procedure to find equilibrium. It's not strictly polynomial, it’s pseudopolynomial. But it might be the case, and of course it has to be tested against the ellipsoid
algorithm. It's possible that this one will actually perform better. We start with zero prices,
then we iterate, we query over the next iteration, we query what is the demand and what is the
supply in the current prices, and once we hit a good for which the demand exceeds supply, then
we increase the price of that good by an epsilon. The reason that this will work is that, first of
all, when we increase the price, it will never reduce the demand of another good. This is
actually the formal definition of gross substitute. So we’re fine there because we assume a
gross substitute. This is exactly definition. It roughly means no complements, but not exactly.
This is a definition that is exactly came to help computer scientists when they try to find an
algorithm for calculating equilibrium. And this exactly matches, this definition exactly matches
the, this range of elasticity between of row between zero and one.
The second observation is that when we increase the price of one good it will never increase
the supply of another good because if a price increase will cause any change in supply, it will be
for more supply of that good. Of the good whose price increased. It will never cause any
publisher to move his supply, to increase supply for different good. So these together means
that this prices never have to be decreased because we are only increasing prices, it will never
be the case that we get, that the supply somehow, it will never be the case that the demand
will suddenly, that the demand was higher than the supply and suddenly it is the other way
around that we need to decrease prices. So it's possible to show that this will converge to an
equilibrium.
So I have one more slide which is very not technical. And this is the part of the work that we
believe we can do better. But for now, this is how we propose that this market will work in
practice. We divide the time into steps. Into periods. And the contracts are sold throughout
the period for the next period at this period’s prices. And then prices can be recalculated at the
end of every period according to the supply and demand of this past period. So we have some
assumption here that the demand and supply will not change that much from one period to
another, so we can use the prices that we calculated based on the period that just ended. Now
this is not going to be exactly the case, there will be excess demand or excess supply. The way
to handle excess demand is we’ll have to, when we calculate equilibrium prices, we will ignore
some fraction of the supplies, so it's like we'll set aside some of the supply and we’ll
compensate advertisers for unsatisfied demand, so if they purchase something in the current
prices and we actually don't have supply of that, for that they will get some kind of
compensation. And we have excess supply, then we can still set it in the spot market. But like I
said, it's probably too simplistic. It is probably the case that if you want more flexible periods,
not fixed periods, but this is what we have for now.
To summarize, what we are trying to obtain is a market in which we have posted prices for
impression types. So this will give us both continuity and efficiency. It will accommodate two-
sided markets where any advertiser, any publisher can join at any time. It's completely
transparent, everybody knows what are the prices, there are no, it's not, we do not need the
effort in which is done today to close a contract for each advertiser separately. We propose a
compact price structure that is based on the actual demand. And we show that in this market
there exists an equilibrium, and it can be computed in polynomial time. We allow a rather
expressive demand in comparison to previous literature. And we can elicit this demand just by
observing the behavior of advertisers. We do not need to explicitly ask advertisers or go
through some kind of demand elicitation process with advertisers. We still have some open
points with respect to the inventory, to the representation of inventory, especially whether we
can or cannot use Bayesian networks in some acknowledgments. And I'll be happy to take if
there are additional questions.
>>: I have another question about the supply trade. So how is it that you, suppose the goods
are all the time. Right. So I have somebody wants cyclists, someone wants Michigan, someone
wants income, right? So all of>>Yagil Engel: So it's not how I use the word atomic. Meaning they're all>>: Singular. One-dimensional demand. But I have demand for all of the thousand bands. And
they'll have separate prices. They all have different prices. Does the tree blow up?
>>Yagil Engel: That's an interesting question; I need to think about it. So you’ll have, let's say
you first branch over state. And you have some states that people mention and then you have
the unknown. And then, actually I don't think so because you will first branch on state, but
then any of the other, the only branch that you will need to keep extending is the unknown
because for, once you know that someone is from Michigan, you don't need to know anything
else. This is only sold to this advertiser that wants to purchase impressions from Michigan.
Now advertiser that cares about gender, that would fall into the branch under the unknown
state. So you'd, so you always have the branch for each attribute where you say that the value
of this attribute is not known, right?
>>: So what if I have a Michigan high income impression? And I'm interested in high income?
Don't I have to follow the, I know it's not unknown because I know what it is>>Yagil Engel: I see what you’re saying. You know what it is but it satisfies the other branch.
The branch of unknown. So the question really is how would the decision tree or regression
tree algorithm work, whether it will try to blow up everything or it will do the right thing and I
can't tell that. I thought sufficiently about that in order to tell you that. It will do the right
thing. But it's possible to express it with a compact tree. Whether the algorithm that will
actually achieve that or not, I need to think about because-
>>: But if it does the right thing, then you no longer have the property that you sum the leaves,
you get the total inventory, right? So if you have a Michigan high income, it's going to, that's
impressions going to appear to>>Yagil Engel: No, no. If you expanded both. Well, if you expanded both, then the Michigan
high income will appear under Michigan high income. It would not appear under the unknown
high income. The question is whether, but what I was saying is that you can assign those
Michigan high income to a branch where you don't know the state because no one cares about
Michigan high income. But I'm really not sure that this is what will, I'm just saying that you can
represent it through a compact tree where you don't even expand Michigan further because
no one cares whether someone from Michigan has high income or not. So it doesn't have to
appear in the tree. But it's a good point. I'll have to see whether the algorithm will do the right
thing or not. Okay. Thanks for coming.