>> Nikhil Devanur Rangarajan: Welcome everyone. It's my great pleasure to
welcome back Shuchi Chawla from the University of Wisconsin-Madison. Shuchi
is no stranger to people here. She spent a year as a visiting researcher
between Europe and Microsoft and we missed her very much since she left. I
hope she missed us too.
>> Shuchi Chawla:
I came back.
>> Nikhil Devanur Rangarajan: Yeah. So we're glad to see her back, and
she's going to tell us something about auctions with interdependent values.
And this is something she did during her visit here. So over to Shuchi.
>> Shuchi Chawla: Thanks, Nikhil. Definitely a pleasure to be back. So
this is joint work with Anna Karlin, who is right there in the audience and
Hu Fu who is just finishing a post doc at MSR in New England and is going to
be in the neighborhood soon at UBC. And also I wanted to very much
acknowledge Microsoft's support. This was work done when all three of us
were visitors here at Redmond. So it's been really great. So let me start
by talking about the classical auction design problem. So here's the kind
of setup that I want you to keep in mind. There's a seller who wants to
sell some items or some services and there are a number of buyers that
arrive interested in buying these goods or services and each buyer has some
value for this item which is the maximum amount of money that this buyer is
willing to pay for this item. And so what we like to do here is to come up
with a mechanism or a protocol that looks at these values that the buyers
have and figures out who to give the item to or how to allocate the service
to and also figures out how much to charge each person. And one of the
things that this protocol should satisfy is that here, the buyer's values
for the item are like an input to this auction and the buyers can game the
system by misreporting their values. So we're going to ask for a truthful
auction, namely one that incentivizes people to report their values
truthfully. And you can go beyond this but for this talk, we'll just think
about truthful auctions. And subject to these constraints, the seller might
have some objective that he wants to optimize. For example, the seller
might want to maximize amount of revenue he gets or the economic efficiency
of allocation meaning that people that desire the item or service the most
are the ones to get it. This is also called social welfare and there may be
other different kinds of objectives. These are the two objectives that we
will look at in this talk. So this is the classical setting of auction
design. And we make an important assumption which is that the seller might
not know the values that the buyers assign to the item or service but does
know the distribution from which these values are drawn. And there's a lot
of work on the setting. It's a very well understood problem. Now, what
often happens in the real world is that buyers may not in fact know how much
value they want to assign to the item. Okay. So they look at the item.
They try to get as much information as they can about it and this informs
their value. And so they might have some sort of guess or estimate as to
what their true value is but not know what the value is. And in fact, their
value might depend on how other people value this item so if they don't have
full information about the item, looking at other buyers' behavior might
change their estimate of their own value over the length of the protocol or
the auction. And so the question that we want to ask is how do we design an
auction in such a setting where buyers don't know their values precisely.
Let me give two examples of where this kind of setting arises. The first is
a classic example by Milgrom and Weber. And this is a setting where
multiple companies are competing for the rights to drill at a particular
site for oil. And the value that any particular company would have for this
depends on the amount of oil that they would find at this particular site
but until they get the rights to drill and they actually do the drilling,
they cannot know the exact amount of oil at that particular site. So
typically, what happens is different companies have different ways of
estimating what the total amount of oil is, and different people might get
different estimates but their values are all the same and it's the total
value of oil at that particular place. So here's a formalization of this
sort of setting so let's say that V is the common value that is the true
value of the resource and every bidder or every company gets some noisy
signal of what these values are. Now, these signals are what they get to
see and they don't get to see this value and so over the course of a
protocol or an auction, if they get to see some information about other
people's signals, then their best guess of what they value might look
something like an average of these signals. Okay. So this is a setting in
which every everyone's best guess as to their own value is a function of all
of the information that not just they have on their own but also what
everyone else gets. Okay. Here's another more contemporary example of
where such a situation may arise. So imagine that you are a user who is
searching for something on a search website. And one of the things that
search engines do when you search for a term is to figure out what adds to
display on this page and we do this by running an auction over different
advertisers. Okay. And so when they run this auction, they use cookies to
figure out what kind of user is performing the search. And they share this
information with the advertisers. And having this information better
informs the advertisers how much value they see from this particular
customer viewing their ad. Okay. And this is a setting where the values of
these advertisers are correlated in this manner by having been informed by
the same piece of information. Okay. So here, the advertisers look at this
information and derive their value. We will think of this as a setting
where the values are private. They're known to the advertiser but they're
correlated. Okay. So this is already a little bit different from the
classical setting where everyone independently obtains their value from
something. Okay. In contrast, consider the following sort of setup where
again the search engine obtains some information about the user that is
searching on this web page, but now sends different pieces of information to
the different advertisers. And once again, based on these signals that the
advertisers get, they form some estimate of their value for this particular
consumer. However, now, their true value is a function of this entire
information put together but at the beginning of the auction, they only have
a limited amount of information. So again, formalizing this, we can think
of the advertisers as obtaining some signals from a joint distribution, and
then their true values being a particular function of all these signals and
not just their own. Okay. And this is the kind of setting that we call an
interdependent value setting, values of all of the buyers in the system
depend on each other's information.
>>
[Indiscernible].
>> Shuchi Chawla: That's right. Yeah. They get to see the signal. Then
they place the bid. But they get to place a bid without knowing what their
true, you know, without knowing their exact value because this depends on
information that they haven't received so far. Yeah. Okay?
>>
[Indiscernible]?
>> Shuchi Chawla: So it is true that the search engine shares some
information and not necessarily all, but I'm not sure that in practice what
happens is that they share different information. So just a hypothetical
example of how it might be.
>> [Indiscernible]. It seems like the value is some function of the
signals and something that happens eventually.
>> Shuchi Chawla: Right. So the value eventually might also be a random
variable, but here we're thinking of the value as being the buyer's expected
value given all of the information they could possibly obtain through the
auction. Yeah. Yeah. So at any point of time, the buyer can think about
their expected value given all they know and this could evolve as they
gather more information.
>> So this is kind of most accurate estimate of the value [indiscernible]
auction.
>> Shuchi Chawla:
>>
That's right.
Before you actually discover --
>> Shuchi Chawla: Before you discover more, that's right. Yeah. Yeah. So
the question you want to ask is how should we design auctions or run
auctions within this setting? Is this setting similar to the classical
setting in how we approach auction design or not? Okay. So I'm going to
start with a quick primer on classical auction theory just to make sure
we're all on the same page and then we'll talk about the two objectives of
welfare maximization and revenue maximization. As I mentioned before.
Okay. So let's talk about the classical setting where buyers know their
values. And these values are independent of each other's values. First
we're going to talk about the welfare maximization objective. Recall that I
very vaguely defined this before. So here, we are trying to maximize the
total value that buyers achieve by the allocation that the auction
generates. So in other words, we want to give the goods or services to
people that value them the most. Okay. So how can we maximize this
objective in the context of auction design? Well, we could, to begin with,
we could figure out what is this allocation that maximizes the value that
the buyers get in total. Okay. And the question is: Is it possible to
come up with payments that the buyers make in such a way that this
encourages buyers to report their values truthfully. And it turns out that
the VCG mechanism is a classic mechanism that gives a payment scheme for
supporting precisely these welfare maximizing allocations. Okay. And it's
based off of this informal statement like any truthful auction, any auction
that aims to incentivize buyers to report their values truthfully must be of
the following form. It offers the item or service to each agent at a price
that is independent of the agent's own value. So it comes up with a price
that depends on other people's reported values and then just offers the item
or service to the agent at that value. Okay. And so in the context of the
VCG mechanism, it turns out that every buyer pays the minimum value that he
needs to bid in order to win the item. And if he can afford to pay this,
then he wins the item. This is also called the critical value. And let me
illustrate through a quick example. So let's say we have three buyers
competing for a single item. And these are the values. Then we can figure
out for each particular buyer, we want to give the item to the buyer that
desires it the most, that has the largest value for this item. So for each
buyer, we can ask the question of how high he needs to bid for the item in
order to win the item. So this guy clearly needs to bid $100 to beat this
other guy. So we offer the item to him at $100. And this guy can't afford
to pay that much. This guy needs to bid $100 to win the item, can't afford
it. This guy only needs to bid $15 to beat the other two. So we offer the
item to the third guy at $15 and he accepts and in fact, the way that we set
up these offers, the buyers can make this accept or reject design without
having to -- these offers are independent of their own values and so the
buyers will behave in a manner that the auction wants them to behave in. So
in that sense, this is truthful. And achieves what we want to achieve,
mainly giving the item to the person that values it the most. Okay. So in
fact, this is an example of what's called a Vickrey auction. This is
precisely the Vickrey auction. The highest bidder wins the item and pays
the second highest value that was reported. Okay. So this is the classic
auction here. Very beautiful result. You can do welfare maximization in
this nice manner. What about revenue maximization? What if the seller
wanted to earn as much as possible? Well, in this case, the seller might
ask for as much as a hundred dollars in revenue but it turns out that this
is not something that the seller can hope to obtain in a truthful manner.
So for revenue maximization, it turns out that we need to make the
assumption that the seller knows the distribution from which the values are
drawn and then designs the auction based on that distribution. And in this
case, it turns out that one can slightly generalize the VCG mechanism that
we described here in order to maximize the seller's revenue and expectation
over the value distribution. And here's what it looks like. So you ask the
buyers to report their values. You look at their value distributions and
use those to apply a certain transform to the values giving you virtual
values. And then we run the VCG mechanism over these virtual values. And
this turns out to be optimal for revenue maximization. Don't worry if you
don't quite get this fully. The one point that I want to make about this
mechanism is that these virtual value functions can be strange functions and
so the mechanism can look strange and complex for that reason. It can also
give the appearance of being unfair in the sense that sometimes a person
with a small value wins the item whereas a person with a much larger value
doesn't. So it can look like a strange mechanism. It turns out that in
fact there's a tighter connection between revenue maximization and welfare
maximization of the following sort. So if the buyer can set result prices,
meaning that if everybody bids below a certain price, then nobody wins the
item, then welfare maximization with these result prices is near optimal.
Doesn't give you the optimal mechanism but it brings you very close to
optimal. So for example, in the case where the seller has a single item to
sell, we saw that welfare maximization can be done with this Vickrey
auction. Here, if you set an appropriate result price in the Vickrey
auction, this gives you a two approximation to revenue, to the expect
revenue. So this is nice. It nicely aligns revenue maximization with
welfare maximization, gives a simple mechanism that has nice fairness
properties. So we also understand revenue maximization in this respect very
well. Now, let's move on to a more general setting where values of
different people are correlated. So we didn't use anything about the
distributions of values when we talked about the VCG mechanism or the
Vickrey auction. So indeed, it doesn't matter what distribution values are
picked from. These mechanisms continue to be optimal for welfare
maximization. But revenue maximization relied on looking at the value
distributions of the agents and in fact, the mechanisms that I talked about
earlier don't work anymore when we go to a setting with correlated values.
So what happens here, so intuitively, the objective of revenue maximization
is difficult for the seller because of this lack of information that the
seller has about the agent's value. And but when the values are correlated,
that means that the values of other agents give you some information about
the value of any particular agent. And this extra information should only
help the seller to obtain more revenue from the buyer. Okay. And so indeed
this happens. So under some very mild assumptions, it's known that you can
in fact extract the entire value of the buyers as revenue, meaning that this
is the most possible, the largest possible price that a buyer might be
willing to pay for the item and so you can do the best possible thing by
exploiting this correlation. The caveat is that sometimes the buyers pay
more than they value the item for and sometimes they pay less and so at the
end of the auction, a buyer might be unhappy if they paid more than how much
they value the item. And it also tends to be a very complicated mechanism.
Okay. So you might ask, well, what can we do if we impose the restriction
that the buyer never pays any more than how much they value the item and
then this problem becomes NP-hard. Okay. So in terms of trying to optimize
for revenue in this correlated setting, we understand a little bit less than
our understanding for the independent case. But fortunately, the connection
between approximate revenue maximization and welfare maximization that I
mentioned earlier continues to work even in this correlated private value
setting. So in the single item special case, once again, one can show that
the Vickrey auction with appropriately set result prices is 2-approximation
to expected revenue. I'm going to talk about why this -- I'm going to show
proof of this result a little bit later on. Okay. So great. We understand
the independent value setting. We understand the correlated value setting.
>>
[Indiscernible].
>> Shuchi Chawla: Right. So this is NP-hard relative to the optimal -- the
revenue optimal mechanism. So meaning that the optimization problem, what
is the mechanism that maximizes expect revenue over the class of all
truthful mechanisms, that problem is NP-hard. So there is going to be a gap
between the total expected value in the system and the revenue that the
seller can extract but that's not what we are comparing against. We are
just comparing against the revenue optimal mechanism. What's the most
revenue you can obtain truthfully. So I'm going to go on to interdependent
settings now. And first describe what challenges we face when we talk about
welfare maximization in these settings. So here is the formal model that
we're going to assume. So for the next portion of the talk, I just want to
think about a setting where a seller has a single item to sell to the buyers
and the buyers are competing for this one item. So we're going assume that
the buyers receive some signals about their value. And these signals are
drawn from some drawing distributions, some arbitrary distribution. And
then as I said before, the value of every buyer is going to be a function of
all of these signals. And we're going to assume that the value is an
increasing function of the buyer's own signal. Now, here's the first
problem that arises in this setting. Before we talked about the notion of
truthfulness which is also called incentive compatibility for auctions
meaning that a buyer, once he knows what format the auction is going to
have, is going to be happy to reveal his true value for the item for this
auction. But here, if the buyer doesn't even know his true value, how do we
define this notion of truthfulness? In fact, indeed, the auction may have a
format where the buyer learns more and more than information through the
auction, so what he thinks about his value at the beginning of the auction
may be different from what he thinks about his value at the end of the
auction. Okay. So we side step this issue a little bit by looking at the
notion of ex-post incentive compatibility, which is the following: So we
require the mechanism to have the property that at the end of the auction,
once everything has been revealed, a buyer is satisfied with what he bid,
what strategy he followed through the auction, even after everything has
been revealed about his value. So for every possible setting of other
people's private information, when other people reveal their private
information and the buyer knows what his value is, he is still satisfied
with his strategy at the end of the auction. This is called ex-post
incentive compatibility. And with this notion of incentive compatibility,
the following generalized version, one can define the following generalized
version of the VCG mechanism. So recall what the VCG mechanism does for
private values. It asks people to reveal their values. Here we're going to
ask people to reveal their signals and then compute their values from their
signals. And then the VCG mechanism serves the welfare maximizing set.
Here again, we're going to serve the welfare maximizing set, meaning give
the item to the person with the highest value. And now the question is what
prices should we charge? Now, before, in the private values setting, we
said that the VCG mechanism should charge any person the minimum amount they
needed to have bid in order to win the item. Here, we're going to look at
the minimum signal that the buyer needs to report entered to win the item.
Okay. And then so let's call this the critical signal. This buyer, we're
going to compute the value of the buyer at this critical signal given the
signals that other people reported, and this is the price that we will
charge. Okay. So and in the single item case, this is the generalized
Vickrey auction. Again, we asked people to report their signals for the
buyer with the maximum value given these signals. We ask what is the
smallest signal this person should have reported entered to win, and then we
recompute his value at that critical signal and that's the price that we
offer the item at. So pictorially, let's imagine you have two buyers in the
system. Their signals are S1 and S2. Now, let's suppose we want to try
selling the item to buyer one. We're going to fix the second person's
signal, so S2 is fixed. And we're going to look at how the values of the
two buyers evolve when S1 changes. Okay. So this is the signal that the
first buyer reports. Now, note here that this is the point at which the
first buyer's value starts to become bigger than the second buyer's value.
So this is going to be his critical signal. He better report a signal
larger than this point here in order to have a higher value. Okay. And so
we're going to charge this person the price, the value corresponding to this
critical signal. That's the price that we offer the item at. Okay. And
this is the generalized victory auction. Does anyone see a problem with
this auction or with this description here?
>>
[Indiscernible].
>> Shuchi Chawla: Yes. So the buyer was revealing -- so let's again think
about the private value setting where everyone knows their value. They
reveal their values, but we want to charge prices, so to person one, I want
to allocate him the item at a price that's independent of his value. But
only if his value is the highest. So how am I going to do that? I'm going
to take the highest of everyone else's value and set that as the price for
this person. And if his value exceeds that price, he's going to buy and but
the price is independent of his own value. Okay. Likewise, here, now, I'm
going to ping in terms of signals. So everyone reports their signals to the
auction. And I look at -- I compete every person's value. And I want to
allocate the item to the person who has the highest value given these
reported signals. Now, let's say this is buyer one. And I want to ask,
well, what is the smallest signal that this person should have reported in
order to have the highest value? Okay. And if they wanted to lie to me,
and they knew what everyone else is going to report, they could have
reported that signal. So I better not charge them anything more than how
much I compute their value to be at that particular signal, critical signal.
And that's what this mechanism is going charge. Okay. So it's going to
pretend that they reported this critical signal. It's going to compute
their value at this critical signal and that's the price that we charge.
Yeah?
>>
[Indiscernible]?
>> Shuchi Chawla: I'm sorry? Yeah. So the value functions are known.
That's right. Yeah. Yeah. So we don't know the signals, but we know how
the values depend on the signals for instance in the oil field setting,
every person's value is an average of everyone's signals.
>>
This is not exposed.
[Indiscernible] the other signals [indiscernible].
>> Shuchi Chawla: So this is ex-post in the sense that given what other
people reported, now, assuming that to be the truth, that's my best
information about my value, I'm going to be happy to pay this amount. Yeah.
>>
And here showing as [indiscernible].
>> Shuchi Chawla: That's exactly right. So that is the problem with this
mechanism is that if let's say that the value functions as a function of S1,
looked something like this, well, now, the critical signal is not
well-defined. So what should the critical signal be? This one or this one?
It's challenging. These are both places at which V1 crosses V2 becomes the
higher value. In fact, it turns out that if value functions look like this,
they cross each other many times, then we don't know -- in fact, there's no
mechanism that is ex-post incentive compatible and is always going to
allocate the item to the person with the highest value. Okay? And so we're
going to make this assumption that for any particular signal, for any vector
of signals of other people, if I vary agent I's signal, then his value
function crosses any other at only one point. This is called the single
crossing and this is a pervasive assumption in all of the work on
interdependent value settings and given that we don't even -- we cannot
maximize welfare without this assumption exposed, it seems to be challenging
to work without this assumption. So we're going to assume this for the rest
of the talk. Okay? Okay. Once we have signal crossing then, the
generalized Vickrey mechanism as I define, it becomes well defined, it's
ex-post incentive compatible and it also always maximizes welfare. Okay.
So now, let's go on to revenue maximization. So for revenue maximization,
we might start out doing -- taking the same sort of approach that Meyerson's
optimal mechanism took for the independent private value setting. Okay. So
that is certainly possible to do here. You can define certain virtual value
functions such that maximizing welfare over virtual values would be
equivalent to maximizing revenue over the actual values. Okay. However,
this does not lead to a way of finding the optimal mechanism and that is
because these virtual value functions can be very strange looking functions
and so in Meyerson's setting, once you define these functions, there is a
natural optimal mechanism that comes out of that approach. Here, you need a
lot of very strong assumptions in order for this approach to result in an
optimal exposed incentive compatible mechanism. So assumptions such as
seeing that different buyers look symmetric, their values are drawn from -their signals are drawn from a symmetric distribution. Their value
functions are symmetric. There is a certain assumption on the distribution
that's called affiliation. I won't define this. There are other
assumptions on the signal distributions that some of you are familiar with
but don't worry, if you don't know what these are, regularity hazard rate
assumption but in fact, much stronger version of these assumptions that
apply to conditional values and conditional signals as well. And so
characterizing the optimal mechanism requires very, very strong assumptions
on the setting and we don't really understand this very well. Okay. What
about near optimal mechanisms? You might ask, well, what about, you know,
we talked earlier about VCG with reserves and this turns out to be near
optimal in the private value setting. So does the generalized Vickrey
auction with appropriately set result prices, is that approximately optimal.
And it turns out that it is. But again, you require a pretty strong
assumption this general mono tone hazard rate assumption, and you can get
this sort of a result also under some very strong symmetry assumptions on
this setting. Okay. And in general, so, again, I want to point out that in
private value settings this sort of Vickrey with results being approximately
optimal happens in the absence of any assumptions. But here, it turns out
that in general, this mechanism can be really, really bad, really far from
optimal if you don't have these assumptions and I'll show you an example
where this happens. Okay. And so to summarize what's going on, we don't
really have a good understanding so far of revenue maximization in these
interdependent settings in the absence of some very strong assumptions.
Question?
>>
[Indiscernible]?
>> Shuchi Chawla: So the generalized assumption is that conditioning on any
vector of signals of other buyers, if you look at the distribution of value
of any particular buyer, then this satisfies monotone hazard rate
assumption. So every conditional distribution of values satisfies this
property. And likewise, the generalized regularity assumption is similar.
Okay. So to summarize what we have so far in the private value setting, we
understand welfare maximization, revenue maximization, and nice property
that we achieve are that revenue and welfare are roughly aligned. In the
interdependent value setting, we don't understand the revenue objective very
well. And furthermore, so far, we don't know how to maximize revenue by
appealing to welfare maximization which we do understand. Okay. So what do
we do in this work? It's taken me a long time to get there. We've come up
with a new tool that we can exploit here. And we call this admission
control and I will describe the mechanism very shortly. And we show that
this slight modification to the Vickrey auction or the VCG mechanism leads
to approximate optimality and this is a way of reducing our aligning revenue
maximization with welfare maximization. So this alignment happens, just not
in the obvious way. Needs a little tweak to welfare maximization. And
furthermore, we require no assumptions on the value distribution. We do
need the signal crossing assumption and we need one more technical
assumption that I will come to. Okay. Okay. So before we go there, I'm
going to show you why revenue and welfare align in private value settings.
So why is it that the Vickrey auction reserves gives you approximately
optimal revenue and why it fails in interdependent settings and that will
naturally lead us to this modification that gives us an approximation.
Okay. So let's again recall this private correlated setting. We have a
single item to sell. We have a number of different buyers that are
interested in this item. Their values are drawn from some drawing
distribution. Okay. And we're interested in some variant of the Vickrey
auction for this setting, and so in particular, let's say these are the
values that the different buyers have. We're going focus on trying to sell
the item to the person with the highest value. That's what the Vickrey
auction tries to do. Now the question is how much should we charge this
person for the item? Okay. And so we're going to use all of the
information we have at our disposal in order to come up with a price for
this highest value agent. What do we know about his value? Well, we can
come up with a price that is based on everyone else's reported values
because we're not selling to them. We're only selling to this one guy. And
so we're going to use all of these other values. So this is all values
other than I star and we're going to use the fact that this guy that we're
selling to has the highest value. Okay. And putting all of this
information together gives us some conditional distribution over this
highest agent's value. Highest bidder's value. And we're going to look at
this conditional distribution and we're going to pick an optimal reserve
price from this distribution. We're going to pick an optimal price to sell
the site to this agent from this distribution. Okay. So that's the
mechanism that we're going look at. This is Vickrey with reserves. It's
also called lookahead auction and it was first proposed by Ronen in 2001.
Okay. So let me now convince you that this is a good auction. So let's use
R to denote the revenue of this lookahead auction and let's use opt to
denote the revenue of the overall optimal mechanism. And what I'm going to
do is to look at how much revenue this overall optimal mechanism gets from
all of the losers essentially. Everyone but the highest value guy. And
then I'm going to look at the revenue that the optimal auction gets from
this guy and L plus H gives me opt. Now, what did our mechanism do? Well,
it focused on selling to this guy. And it used all of the information it
had at its disposal in order to come up with the best price to charge to
this guy. Okay. And so that tells us that this I have can be no more than
R. Even optimal mechanism can get no more than how much we get because we
do the best possible thing knowing that we're going to sell to this guy.
Okay. On the other hand, how much money could the optimal mechanism extract
from all of these losers? Well, their value is no more than the value of
the second highest guy among everyone. So the highest among all of these
people. Okay. That's the maximum possible value that they have. The
mechanism can't possibly charge any more than that. But what did we do? We
came up with a price to charge to this guy conditioned on his value being
larger than this max. So one of the things we could have done is to
basically sell the item at this value at this price to this guy and he would
have always accepted at this price. And so our revenue is at least as large
as the second highest value. Okay. And so that tells us that two times R
is at least as large as opt, gives us 2-approximation to optimal revenue.
And this is a simple argument but it turns out to be tight. There are
settings where this mechanism will be a factor of two off from the optimal
revenue. Okay. So what happens if we try to apply this mechanism to
interdependent settings? Let me show you an example where it might go
wrong. So now let's imagine that you have two buyers and you're selling one
item. And recall that in an interdependent setting, people just get signals
about this item and their value depends on their joined signals somehow. So
here, I'm going to assume that only the first buyer gets any information
about the item. So S1 is drawn from some distribution. And his value is
equal to his signal. And the second buyer gets no information but his value
is equal to the first guy's signal minus some tiny epsilon. Okay. So the
first guy always has the higher value. In fact, their values are very close
to each other, but let's imagine that we wanted to use some where end of the
Vickrey auction, meaning that we wanted to sell to the guy with the higher
value. Okay. What do we do then? Well, knowing the second guy's signal
doesn't help us at all. The second guy is not getting any information. And
so we try to price the item in the best possible way for the first guy. And
let's say that the first guy has a signal distribution which does not get
very good revenue for the seller. So this is called equal revenue
distribution regardless of the price that you charge this guy. Let's say
you pick a price of X. The probability that this buyer is going to buy the
item is one over X. Your expected revenue is going to be one. So any
auction that only sells to this highest value guy is going to get a revenue
of one. But in fact, you could get much higher revenue by ignoring the
first guy, selling to the second guy at a price equal to his value. Okay.
The point here is that once I decide to sell to the second guy, I know his
value precisely. It's what the first guy told me minus epsilon. Okay. So
there's some informational asymmetry. I can get much better information
about the second guy's value from the first guy's signal than I can about
the first guy's value from the second guy's signal. And somehow my
mechanism, my auction should take that into account. Okay. And so that's
the reason that it's not always best to sell to the highest value agent.
Pictorially, one of the things that's going on in this example is that we're
selling to the first guy at his value at his critical signal. Okay.
Whereas at his true signal, which could be here, even the second guy's value
might be much larger than the price at which we are selling item to the
first guy. So we would have been better off trying to exploit the second
highest value. Okay. So this suggests the simple modification to the
Vickrey reserve price mechanism. Here's what we're going to do. We are
going to ask everyone to report their signals. And then we're going to flip
a coin for every buyer and we're going reject them out right if the copy
comes up tails. Okay. So everyone is rejected outright, probably one-half.
Independently. And then so we retain about half of the guys in the auction
and now we're going to run the Vickrey auction just as Ronen's mechanism did
before. But of course, we have a bunch of rejected buyers. They did report
some information to us. They reported their signals. So we are going to
use those signals to compute the conditional value distributions of the
remaining buyers. Okay. So I'm going to now give you a proof of the fact
that this modified auction that we call the generalized Vickrey auction with
reserves and admission control, so this first step is like admission
control, I'm going to convince you that this gives an approximation to the
optimal revenue. Okay. So again, we're going to focus on the case of two
buyers which essentially has all the complexity that we need to worry about.
And so let's say that at the reported signals, this is V1's value as a
function of S1 once we fix S2 and this is V2's value. Okay. So once again,
we are going to assume that the optimal mechanism gets some amount H from
this highest value guy and some amount L from the other buyer. Okay. And
we're going to think of the value of the second highest value guy as some of
these two components. One is his value at the first guy's critical signal.
Okay. And so that's L2. And L1 is this difference, this increase in the
second guy's value from the signal S1. Okay. So what happens in our
auction. Well, at the first step, with probability one-half, we reject
buyer one outright. With probability one-half, we reject buyer two
outright. And so with some probability, both of them are going to survive
the first step. Okay. In which case we will run this Vickrey auction with
reserve and our auction is going to get at least H condition on both of them
surviving. Now, that takes care of this component that opt gets. In fact,
it also takes care this have quantity L2 because the price that we charge to
the first guy is conditioned on this guy having the higher value out of the
two. And that price is going to be at least as large as this critical price
here. So that takes care of this L2, our auction, this L2 can be no more
than H. Okay. Now, all we need do is to make up for this quantity L1. And
here, we're going to argue that our auction is going to get at least L1 in
the case where two survives our admission control and one doesn't survive.
Okay. And so if this happens, then the optimal auction is getting at most
2H plus L1 and we're getting at least H over 4 plus L1 over 4. Okay. And
this gives an eight approximation and in fact, you can add just the
probabilities in this first step I think the optimal thing to do is to
reject everyone with probability one-third independently and then that gives
a four and a half approximation. So you can equalize these terms to get a
better approximation. We have this claim that we left and we need to prove
this. So the claim once again is that there's this amount L1 which is the
difference between V2's actual value and the two signals and the critical
value, the value at the signal where the two values are equal. There's this
difference L1 that we want to make up. And we argue that if two survives
the first step and one doesn't, then we can get this value in revenue. So
why is that? This requires another assumption. And the assumption is the
following. So what's happening over here? We fixed the second guy's signal
and this quantity L1 is the increase in the second agent's value from an
increase in the first agent's signal. Okay. Now, let's say that the two
signals had independent contributions to V2, for instance hypothetically
let's say V2 was S1 plus S2. Okay. So here, we're just looking at the
contribution to V2 from the first guy's signal, S1. The first guy's signal
went up, S2 didn't change, and there's the improvement in V2's value. Now,
what we are going to do for the second guy, here's one thing that we could
do for the second guy, we could charge him V2 of S1, his own signal being
held zero. Okay. So this value of the second guy, when his own signal is
zero and the other guy's signal is S1, is something that's independent.
This quantity is independent of his own signal. So we can imagine charging,
you know, trying to allocate the item to the person at this price. And he's
going to accept this price. Okay. And so our mechanism could get at a
minimum this revenue from the second guy. Okay. And if we assume that the
responsiveness of V2 to S1 is at least as large when S2 is zero as when S2
is something else, then this quantity is at least L1. Okay. So what is
this assumption again? Let me restate it. So this is saying that I get
some increase in my value from an increase in the other person's signal and
this increase depends on what my own signal is. But I see more of an
increase when my signal is low than when my signal is high. Okay. So the
responsiveness of my value to someone else's signal decreases or does not
increase as my own signal increases. Okay. Other people's signals become
less important to me as my own signal goes up. That's this assumption and
with this assumption, then the mechanism gets at least this amount of
revenue. And so that put together with the proof I described earlier gives
this eight approximation. Okay. So to summarize what I said so far, let me
just compare the result that I just described against two other recent
results on the interdependent settings. So there is a paper by Rough Garden
and Talgum Cohen that characterized optimal mechanisms using a Meyerson type
theorem for these settings. And there's another paper by Lee which gives an
e-approximation via a Vickrey with reserve prices mechanism and we get this
4.5 approximation by adding admission control to that mechanism. So what
are the assumptions that these results require? We all require some
assumptions on the value functions themselves. Single crossing as I
described earlier, all of the results need that. We need this assumption on
how my value changes as a function of someone else's signal and how that
change is affected as my own signal increases. This is a concavity type
assumption that we need. The e-approximation does not need any such
assumption but the rough garden and L paper needs several assumptions of
this sort as you can see they are quite similar but in fact more restrictive
than the kind of assumption that we have.
>>
[Indiscernible].
>> Shuchi Chawla: Yes. Yes. So this is exactly opposite, except when both
are equal to zero, then both are satisfied. So yeah.
>>
[Indiscernible] very commonly?
>> Shuchi Chawla:
like convexity.
>>
Right.
This is
The first one is [indiscernible].
Yes.
[Indiscernible].
>> Shuchi Chawla:
>>
So concavity to me seems somewhat natural.
[Indiscernible] two conditions.
>> Shuchi Chawla:
>>
Yes.
So decreasing differences is concavity, right?
The second condition, yeah, that is concavity.
>> Shuchi Chawla: Yeah. Yeah. So this is measured along a different
direction. Yeah. But it's -- the second derivative along a certain
direction is less than equal to zero. Yeah. And it's a long and different,
a certain direction. So yes. So these are all conditions on different -second derivatives and different directions and partials, second
derivatives. So it's a little hard to compare, yeah. Does that make sense?
>>
Yeah.
[Indiscernible].
>> Shuchi Chawla: Okay. Okay. So yeah, one thing I want to mention is
that here's a sort of value function that this condition captures. So if
the value of an agent is a sum over some functions of different people's
signals or even a concave function of a sum of different people's signals
but as long as each individual signal affects a different term, that
satisfies this condition for instance. So those are the assumptions on the
value functions. As I mentioned earlier, these works also require
assumptions on the value distributions. So the rough garden and Al work for
instance requires generalized [indiscernible] hazard rate affiliation
symmetry. The e-approximation also requires generalized monotone hazard
rate and our work does not need any assumptions on the value distribution.
>>
[Indiscernible] regularity?
>> Shuchi Chawla: Not even regularity. Ronen's mechanism also doesn't need
regularity, so that is really where this is coming from. Okay. Let me
quickly mention that this approach extends to broader settings. So one way
to extend this is to again think of every buyer desiring 1 item or 1
service. But then let's say that the seller has a feasibility constrained
on which subsets of people can be allocated their items simultaneously. So
in particular, there is a kind of feasibility constraint that is natural.
This is called matroid constraint and some of you know what matroids are,
and if you don't, that's fine, but let me give you two examples. One
example of a matroid is just K-unit auctions. Where you have K units of the
same item 1 is one side of a matching. So let's say you have items.
Everyone has some subset of the items that they'd be satisfied by, and the
people that you can serve feasibly are those that can simultaneously be
matched to some items. Okay. So that's an example, another example of a
matroid. So we show that VCG with reserves and admission control works also
in matroid settings and here is what the mechanism looks like. You again
take the buyers, ask them to report their signals, and then independently
reject each one with probability one-half. And then over the remaining set
that survives, we run this generalized VCG mechanism where we compute prices
optimally, conditioning on all of the reported signals. And we show that
this gives a constant factor approximation to the optimal revenue. So this
is the result for matroids. You could ask, well, what happens if we have a
more general feasibility constraint when we go beyond matroids? So it turns
out that generalized VCG is no longer exposed incentive compatible so in
that case, and the reason is somewhat similar to what happens if you don't
have the single crossing assumption. So it's the same sort of reasoning
that makes generalized VCG be no longer truthful. And so again, going
beyond maitroids, we don't even know how to maximize welfare. Okay. And so
to summarize, we give these natural mechanisms for interdependent settings
that get constant factor approximations to optimal revenue without
assumptions on value distributions and there are some very natural open
questions going forward. One is to improve approximation factors,
approximation 18 doesn't sound so great so can you do better? Remove the
concavity type assumption that I showed that's a technical and it's possible
that we don't need that assumption in order to be able do well. And then of
course can we go beyond matroids, go beyond single crossing. How would we
maximize welfare and expectation, let's say, and would that lead to revenue
maximization and expectation? And finally, here's a very interesting open
question. We kept talking about ex-post incentive compatibility and you
could also define a notion of Bayesian incentive compatibility here and
independent private value settings it turns out that there's no difference
between the power of Bayesian IC and exposed IC mechanisms for single
parameter settings and so you could ask, well, are Bayesian IC mechanisms
more powerful in the interdependent settings? And if so, how much more
powerful? And that appears to be a challenging open question. Okay. I'll
take questions.
[Applause]
>> Nikhil Devanur Rangarajan:
>> Shuchi Chawla:
Thank you.
>> Nikhil Devanur Rangarajan:
>> Shuchi Chawla:
[Applause]
Questions?
Yeah.
[Indiscernible] all of next week?
I'll be happy to drop more [indiscernible].