>> Yuval Peres: Hi all. Welcome this morning. It’s my pleasure to introduce Vasilis Syrgkanis from
Cornell University. He’s done some really interesting and fundamental work in the analysis of simple
mechanisms for complex markets. So let’s hear about it.
>>: Thank you, Nikhil, and thank you for coming.
>> Vasilis Surgkanis: One primary example being electronic markets for ad impressions, such as search
ads or ad exchanges. Other examples being electronic markets for goods like ebay, auction
marketplaces; or even markets for information made primarily in the context of, say, targeted
advertising; and even crowdsourcing for user-generated content or even effort. My research addresses
mainly game-theoretic and algorithmic questions in these markets, motivated primarily by the interplay
between incentive … by individual incentives and social efficiency. The main technical work that I will be
presenting today is motivated by the following observation: most of these markets, you can think of
them as being markets that are composed of simple mechanisms, where players participate in many of
these mechanisms at the same time. So for example, you can think of an ad auction marketplace as
being composed of several generalized second-price auctions, each second-price auction selling an ad
impression, and then advertisers, at the same time, participate in many of these simultaneously, or they
might be interested in several different listings or keywords, so they might be participating in many
auctions at the same time, and they need to decide how to bid on these different auctions. You can also
think of ebay marketplaces as also being composed of simple mechanisms: you have different sellers,
each selling his item via a single-item second-price or ascending-price auction, and then buyers in this
market are participating in many of these mechanisms at the same time and trying to acquire the goods
that they want by one of these auctions.
Moreover, these electronic markets have some very distinct characteristics that have not been the focus
of classical mechanism design. So for instance, in most of these markets, thousands of mechanisms are
run at the same time; like almost seven thousand search queries happen per second, and most of these
queries trigger an ad auction. So we need the mechanisms that are used in these markets to have very
simple rules with a very fast implementation. So we cannot base our mechanisms on very hard
combinatorial problems. Moreover, these are markets, as I said, that … where players participate in
many mechanisms at the same time, either simultaneously or sequentially, and so designing
mechanisms under the assumption of that they are going to be run in isolation is not a good feat for
mechanism design in this electronic auction market context. Moreover, these are environments where
the decision-making process is far too complex to believe that the advertisers—or the participants—are
going to optimally behave and reach an accord … a classic economical equilibrium, and it’s more … it’s a
better feat to think that because of the repeated nature of these markets, that players instead are going
to be use simple learning rules to learn how to play in the game, and they’re going to try to adapt, and in
the limit, learn how to play almost optimally. Last, most of the participants in these markets have very
incomplete information about the environment, so they don’t even know the valuations of their
opponents, and they might not even know who they’re playing against. So when designing mechanisms
for these markets, we need to take into account all of these distinct characteristics, and so the main
question then becomes: how should we design efficient mechanisms for such markets and how efficient
are existing mechanisms?
>>: So efficiency here is economic or operational?
>> Vasilis Surgkanis: Economic, yes. I mean, like the allocation should be close to the optimal allocation
of resources. The main technical question I’m going to address in this talk is going to be: how should we
design mechanisms such that the market composed of such mechanisms is approximately efficient at
equilibrium? And the high-level outline of the talk is: I’m going to give the … I’m going to introduce the
notion of a smooth mechanism—of the class of smooth mechanisms—and then I’m gonna argue that a
market composed of smooth mechanisms is globally approximately efficient at equilibrium, and by
equilibrium, I mean a robust notion of equilibrium that even involves learning behavior by the agents
and incomplete information.
Okay. Now before moving to the introduction of the notion of a smooth mechanism, let me give you a
simple example of how statistic inefficiency can arise in a market that is composed of many mechanisms
and how it compares to classical mechanism design. So suppose that you’re a seller and you have a
single item to auction, and you ask an auction theorist how should I sell my item such that the highest
value player gets the item—such that the allocation of the item is efficient? The auction theorist would
tell you that you should simply run a Vickrey auction, a second-price auction. You should solicit bids;
you should award the item to the highest bidder; and he should pay the second-highest bid. Now the
classic result by Vickrey says that this mechanism is going to have a dominant strategy equilibrium that
is going to be efficient and truthful, meaning that no player is gonna have any incentive to misreport his
true value, and under such truthful report, the highest-value player is going to win. So in this example,
the highest-value player is going to win the item, and he’s going to pay the second-highest valuation.
However, suppose now that we move this auction into a market where you have several such
mechanisms that are running at the same time. So suppose now that we are in a market where, for
example, two second-price auctions happen at the same time, and you have buyers that are
participating in these auctions, and maybe they just want, for example, only one camera, and they might
have different preferences of their cameras. So in this particular example, let’s assume that the top
player … both players want only one camera, and the top player prefers the top camera; the bottom
player prefers the bottom camera. So here, the top player has a value of two if he gets the top camera,
a value of one if he gets the bottom, and if he gets both of them, he only values the highest-value item
that he got; so he gets a value of two. Now in such a market, it doesn’t even make … even the notion of
truthfulness doesn’t even make sense, so the fact that each market is truthful—for example, the fact
that each auction is truthful, that it’s a second-price auction—doesn’t even make so much sense,
because the value of the player for each auction depends on what happened on the other auction. So
the value of the player for the top camera—if he doesn’t win the bottom camera—is two, but if he wins
the bottom camera, the marginal extra value that he gets is just one. So there’s no … it doesn’t make
sense to say that the player’s going to truthfully report his value in the auction, because the player
needs to understand what’s gonna happen in the other auction that is happening simultaneously. So
instead, the players in this situation are gonna play a game. So they’re gonna play a game where they
need to decide how to bid on each of the cameras, and then we need to analyze the situation at
equilibrium.
So it’s easy—relatively easy—to see that one equilibrium of this situ … of this setting is going to be
where players mismatch and go to the cameras that they value less. So here, suppose that they bid
their true value on the camera that they value less, and bid zero on the other thing. Now, this bidding
strategy is an equilibrium, because even if a player wants to go and get his best camera, he has to pay
the price that the other player is bidding. So he has to pay at least one, and he’s already getting a utility
of one, because he’s getting his least favorite camera, but he’s paying zero. So not only isn’t dominant
strategy truthfulness a relative concept in such a market composed of many mechanisms, but the
allocation that might arise when you have such a composition of such mechanisms might even be
inefficient. So we need another theory to tell us when can we get composable efficiency guarantees in a
market that is composed of such mechanisms? And we need a theory that will tell us when our auctions
and mechan … and games that arising from such auction game … from such auction mechanisms are
going to lead to approximately efficient equilibria. And this is exactly what the smooth mechanism
framework’s trying to achieve. So it’s a mech … a framework analyzing efficiency at equilibrium, leading
to robust and composable efficiency guarantees. Okay.
Now as a first point, as a … judging from the previous example, we see that an equilibrium analysis is
necessary, so we need to have a theory that is going to allow us to quantify the inefficiency or the
efficiency of a market at equilibrium, and the reason is that truthfulness is a concept that doesn’t
compose. There is no coordinator to run a centralized—in most of these markets—there’s no
coordinator to run a centralized global mechanism, or even if there is, this centralized mechanism might
be too complex or costly to implement. So for instance, you can gather all the ad impressions that
happen in an hour and run a centralized—that happen in a second—and run a centralized mechanism
for all the allocation, but that might be too costly to implement. But on the other hand, we need our
efficiency guarantees and our predictions to be robust under rationality assumptions and formation
assumptions. So for example, two things that I’m going to focus in this talk: we need our guarantees to
extend, even if our players use no-regret learning strategies to learn how to play the game, and even if
they have incomplete information about the game that they are playing, and I’m going to model that as
a Bayesian incomplete information. So the players are going to have Bayesian beliefs about the
environment that they’re playing.
Now, to portray how we can quantify the inefficiency of such auctions at equilibrium, let me give you a
very simple example—arguably the simplest auction that is not dominant strategy truthful, and we need
to analyze its equilibrium—which is the single-item first-price auction. So I’m going to give an analysis of
how can we claim efficiency guarantees for the single-item first-price auction, which is also going to
uncover some structures which motivate the definition of a smooth mechanism. So let’s assume that
you are … have a single item, you have bidders which have a value for the item, and that you’re running
a first-price auction. So you solicit bids for the item; the highest bidder wins and pays his bid. Let’s
assume that the preferences are quasi-linear, so the player has some value for the item, and if he gets
the item, his value … his utility is his value minus his payment, which is his bid. And the objective that
we want to analyze is that of social welfare, which is simply the value of the resulting allocation.
Now before moving to the robust solution concepts that we want to analyze, let’s first visit … try to
argue about efficiency this single-item first-price auction in the most vanilla setup, which is a pure Nash
equilibrium and complete information. So let’s assume that all the players know each other’s
valuations, and let’s also assume that they are all going to use deterministic strategies so that they all
going to submit a deterministic bid, and this bid is going to constitute a pure Nash equilibrium, meaning
that each player is going to maximize his utility conditional what everyone else is doing.
Now, there is an easy theorem that says that any such pure Nash equilibrium has to be efficient. So it
has to be that the highest-value player is going to get the item. Now, the proof is very simple, but let me
just go through the proof, because it’s going to give rise to the property that is gonna motivate the
definition of a smooth mechanism. So the equilibrium … it has to be efficient because highest-value
player—so in this case, player one—can always deviate and bid just above the current price of the item.
So suppose that you have some equilibrium b—some bid profile that is an equilibrium—then there is an
increasing price of the item, which is the maximum of all these bids, then the highest-value player can
always bid just above this current price. By doing that, his utility from this deviation is going to be equal
to his value minus this … just above the current price, because he’s deterministically winning, and
because in equilibrium, it must be that his utility at this equilibrium is greater than his utility from this
deviation. Now all the rest of the players can also always deviate to zero, so you … we know that their
utilities are always non-negative. And so by these two properties, we get that the total utility at
equilibrium has to be at least this deviating utility, which is equal to the value of the highest-value player
minus the current price. And because of quasi … because of co-quasi-linearity of preferences, we’re
gonna get that the total utility is total allo … the value of the allocation at equilibrium minus the prices,
and hence the prices are gonna cancel out, and we’re gonna get that the total welfare at equilibrium is
at least the value of the highest-value player. So the total welfare is optimal, right?
But the pure Nash equilibrium and complete information setting is a very brittle setting, so guarantees
for such a scenario might … most probably will not carry over to a practical application, like electronic
markets. The reason might be that a pure Nash might not always exist. In a first-price auction, it might
always exist—or at least an epsilon pure Nash exists—but the goal is to study more complicated auction
scenarios, and there, a pure Nash is not always guaranteed to exist. Moreover, the game might be
played repeatedly, and players might be using learning strategies to play the game, so that would lead
to correlated and randomized behavior. The players might not know the other valuations, and so we
need to take into account probabilistic beliefs about the values of the opponents.
So the two extensions, for example, you can say that we want is guarantees for no-regret learning
behavior … so suppose that you’re in a market—in an electronic market—where this first-price auction
is happening over time—so you have repetition of this first-price auction—and then the players are
bidding on this auction time after time; they are observing the history of play; they are adapting their
bidding strategy based on the history of play. And in the beginning, they might not know how to play
the game, but you would assu … you would believe that as time goes by, they are playing almost
optimally. We don’t want to assume how they are going to play this repeated game, but to the … at
least, we want to assume that their regret for any fixed strategy vanishes to zero. So their average
utility of this sequence of first-price auctions is going to be at least as good as if they had switched to a
fixed bid throughout the time. So they might doing some very complicated learning rule, but to the
least, they should be achieving at least as much utility as a fixed strategy over time would achieve. And
what’s nice about this property is that there are several … there are many simple rules that can achieve
this property in such a game-theoretic scenario, such as the multiplicative weight updates algorithm or
the regret-matching algorithm, and what’s nice about these algorithms is that in fact, the player doesn’t
even need to know the game that he is playing, he can simply … he just needs to be able to get his utility
from any bid, and then he doesn’t even need to argue about how the opponents are behaving, and so
on and so forth, so that this is a very robust solution concept, even in an electronic market scenario. So
we would want to say that the average welfare of such a no-regret sequence is at least as good as the as
the optimal welfare. The other …
>>: What kind of feedback are you assuming the players [indiscernible]
>> Vasilis Surgkanis: So … in the most … in the simplest setting, I’m assuming that players know the
utility from any bid. So for example, they might be getting statistics about their opponent distribution in
an auction marketplace, and then they can calculate the average cost per click or the average … you
know, clicks per bid. But there are even algorithms that can work in such a scenario when if players can
only—you know—get the utility from the bid that they submitted, not from any bid … but they might
take longer time than this setting.
Moreover, we want to analyze settings where players have probabilistic beliefs about their opponents.
So let’s assume, for example, that the value of each opponent is not common knowledge, but rather, it’s
drawn from some distribution, independently for each bidder, and then the players in this setting are
going to be playing a Bayes-Nash … a Bayesian game, where the equilibrium is going to be a Bayes-Nash
equilibrium, meaning that it’s going to be a mapping from values to bids such that each player is
maximizing his utility in expectation over the values of his opponents. And then we want to say that the
expected equilibrium welfare of any such Bayes-Nash equilibrium is at least as good as the expected expost optimal welfare under such a value distribution. And we might even want to combine both
learning rules and Bayesian settings. The theory works even for such combinations, but I’m not going to
be going into this combination of these two settings in the talk.
So the nice … so idea is: what if the conclusions that we drew for the pure Nash equilibrium and
complete information setting directly extended to these more robust solution concepts. So wouldn’t it
be very nice if we could just study the pure Nash equilibrium and complete information, and then
whatever guarantee we derived directly extends to these more robust solution concepts? Now, it’s
obvious that the full efficiency theorem that we proved for the first-price auction doesn’t carry over, and
it’s known, for example, that under Bayesian beliefs, a first-price auction can be inefficient. However,
it’s possible to have such a direct extension as long as we restrict the analysis that we do for the pure
Nash complete information setting, and that’s the approach that we’re gonna take. So we’re gonna say:
just focus on the pure Nash complete information setting, prove an efficiency guarantee based on some
restricted type of analysis, and then, whatever guarantee you’re gonna get, it’s going to directly extend
to no-regret learning and to Bayesian beliefs and incomplete information.
Now let’s revisit the pure Nash complete information proof and see where that proof is going to break
when we try to move to the more robust solution concepts. So recall that the reason why the pure Nash
equilibrium is efficient is because at this equilibrium, the highest-value player doesn’t want to deviate to
bidding just above the current price. But the challenge is that if you are … want to apply this reasoning
to a no-regret learning sequence or to a mixed Nash equilibrium, let’s say, or even to Bayesian games of
incomplete information, then there is no such thing as the current price. The bid of the players is a
randomi … comes from some randomized distribution; the price is going to be a random variable, so
then the player doesn’t know what is going to be the realization of the price. In games of incomplete
information, he doesn’t even know what the value of the opponents are, so he doesn’t even know
whether he’s the highest-value player or not. So both of these things break when we try to prove it for
more robust settings, and the idea is to … let’s restrict the type of analysis to not depend on the current
price of the auction. So let’s try to argue about efficiency of the pure Nash equilibrium setting and
complete information setting, but without using—when we’re doing this deviation analysis—without
using the current value of the price.
So let’s try to deviations, bi prime, for all the players that don’t depend on the current price. Now, is
that even possible? So, at least for the first-price auction, that’s actually possible. So you can think that
the highest-value player can always deviate to bid half of his value, right? So this is a price-ignorant
deviation—a price-oblivious deviation—it doesn’t depend on the current price, and we can argue that
either the current price of the auction is going to be at least the value of the highest-value player, or—if
that … if the current price is below, then the player is going to win, and his utility from this deviation is
going to be equal to half of his value. So in any case, the utility from this deviation plus the current price
is going to be at least half of that player’s value, right? So we’ve got some type of inequality similar to
what we got for the pure Nash equilibrium setting, and similarly, also all the other players can always
deviate to zero and hence their utility is going to be non-negative. And by combining these inequalities,
we’re gonna get that because at the equilibrium, the utility of the player at the pure Nash equilibrium is
going to be at least this deviating utility, we’re gonna get that the utility at equilibrium—at the pure
Nash equilibrium—is greater than half of the highest-value player’s value minus the current price. And
so, again, by quasi-linearity of utilities, the prices are gonna cancel out, and we’re gonna get a worse
theorem that says that the social welfare at the pure Nash equilibrium is at least half of the optimal
welfare. But now we did it using deviations that are price-oblivious, and the whole point is that this
guarantee—the half approximation guarantee—is going to directly extend to learning outcomes and to
Bayesian beliefs. So we proved a weaker theorem, but because we proved it using price-oblivious
deviations, it’s going to directly extend to other solution … to more robust settings.
So for example, how does it … it’s easy to see how it would extend to no-regret learning sequences: we
know that every player is going to have a utility that is at least his utility from any fixed strategy, so plug
in here the fixed strategies that we used in the proof, which is half his value for the highest-value player.
We’re gonna get that the average utility for the highest-value player is at least this aver … at least the
utility from this fixed strategy, and by the analysis that we did previously, this utility’s gonna be at least
half of his value minus the current price—whatever the price of each iteration is. And so we’re gonna
get that the average utility of the highest-value player is at least the … half of the optimal welfare minus
the average price, and so, also using that the other players are gonna get non-negative utility, and prices
again are gonna cancel out, you’re gonna get that the average welfare of this no-regret learning
sequence is at least half of the optimal welfare minus some term that vanishes to zero as time goes to
infinity.
Now, for the case of Bayesian beliefs, it’s slightly more subtle of how it directly extends, mainly because
the deviation that we use depends on the opponent values. So we said that the highest-value player
should bid something, and the other players should bid zero, so we kind of used that people know what
is the … who is the highest-value player. So we need to instead construct deviations that are feasible in
the Bayesian game of incomplete information. So we need to construct other deviations, based on this
deviation that we used in the complete information setting, that depend only on what the player knows,
which is his value, and on the distributions of other players. So there’s actually a black box reduction of
how to do it, which I cannot go through the technical details, but essentially what’s going to happen is
that the deviation of the Bayesian game is going to be: simply random sample the value of your
opponents, and then play the complete information deviation that you used in the complete
information setting, assuming that the true value of your opponents are this random sample that you
drew. And you—because of independence of value distributions, similar analysis is gonna work out—
and you’re gonna get that even the Bayesian Nash equilibrium is going to be at least half of the …
achieve half of the optimal welfare. ‘Kay?
So going back to the proof, the main property that allowed us to prove all these extensions was that we
could find the price-oblivious deviations for each player such that the sum of these deviating utilities is
at least half of the highest player’s value minus the current price. And so this is the core property that
enabled all of the efficiency guarantees: that there exist deviations that don’t depend on the current
price—or more generally, on the current bid profile … or the current equilibrium bid profile—such that
this property holds: that the deviating utilities is at least half of the optimal welfare minus the current
price—or more generally, half of the optimal welfare minus the current revenue of the auctioneer. And
this is exactly the approach that we want to generalize to any mechanism design scenario. So this is the
property that we would try to prove for any mechanism to get robust efficiency guarantees.
So what do I mean by a general mechanism design scenario? So I mean any mechanism that solicits
some action by the players—let’s call them bids, but it could be some arbitrary, abstract action that they
could submit to the mechanism. The mechanism then, given these actions that the players submitted, is
going to decide an allocation and some payment for each player. The allocation can come from some
abstract allocation space, which can have some weird feasibility constraints—whatever you want. And
then, the players have quasi-linear utility, meaning that their utility from an allocation and the payment
is their value for the allocation minus the payment that they were asked to pay. And we’re interested in
analyzing the social welfare of the equilibria of this game that is defined by this mechanism, whereby
social welfare is simply the value of the resulting allocation.
So this, for example, could capture combinatorial auction settings, where the mechanism is trying to
split the items to the bidders; it could capture public projects, where the mechanism has to pick a single
project to build, which is gonna be shared among all the participants; or—even more abstract—
bandwidth allocation scenarios, where you have a divisible resource—a divisible capacity—which you
want to split among the participants. And in this general setting, we say that this mechanism is going to
be lambda, mu smooth if there exists special deviations for each player that don’t depend on what … on
the current bid profile, such that, whatever that current bid profile is, the sum of these deviating utilities
is at least lambda of the optimal welfare minus mu the current revenue. So in the first-price auction, we
had the property that there always exists such deviations that you get at least half of the optimal
welfare minus the current revenue, but in other scenarios, you might get … it gives you more freedom to
track … to quantify the inefficiency in more general mechanism design settings if you put some arbitrary
lambda and mu there—some quantifiers. And this definition is closely related to the notion of smooth
games by Tim Roughgarden, but the definitions are slightly … are kind of orthogonal, but the intuition is
the same. And also this definition … it’s more of … tailored to a mechanism design setting with quasilinear preferences and gives a much more market-clearing interpretation, like an envy-free type of
interpretation of smoothness.
Now, the main theorem is that—one of the main theorems—is that if you have a mechanism that is
lambda, mu smooth, then every pure Nash equilibrium with the complete information setting is going to
achieve at least essentially lambda over mu of the optimal welfare. And this is going to extend directly
to no-regret learning outcomes and to the Bayesian setting of incomplete information, assuming the
value … the distribution of valuations of the players are independent, and even to no-regret outcomes
under incomplete information. And this direct extension theorem generalizes some previous result that
Tim Roughgarden and I had that was for more general game-theoretic scenarios, but used a stronger
smoothness property … and so—you know. But that stronger smoothness property wor … also implied
guarantees for more general games rather than games arising from mechanism design settings.
Now, you can see that this condition is kind of artificial. Maybe there are not so many mechanisms that
satisfy it, or that the … maybe we were lucky just with the firs-price auction. But in fact, almost all of the
recent literature on quantifying the analysis of simple mechanisms can be cast as showing that the
corresponding mechanism is actually smooth. So you can view all of these recent results in the
algorithmic game theory literature that says that simple auctions are approximately efficient as showing
that the corresponding mechanism is smooth for some constants of lambda and mu, and some of this
can actually be thought—of these papers—can actually be thought as compositions of smooth
mechanisms, and this will become even clearer in the second part of the talk. So … for instance … there
were … there are several applications of smooth mechanisms, and in fact, by using the smooth
mechanism framework—because it’s a cleaner type of argument—you can optimize over the lambda,
mu parameters, and you can improve a lot of the previous results in the literature. So for example, you
can show better bounds for the first-price auction, or for combinatorial auctions that are based on
greedy algorithms, or even for uniform-price or marginal-price multi-unit auctions. It also enabled so
new results. So for example, we got results for the all-pay auction, which is useful in modelling contest
scenarios; we got results for first-price position auctions that applied to more general valuations that
were … than what was studied before in the literature for position auctions—so for example, you can
even have values per impression or values … different values for different slot, and you can still get that
there exists a simple mechanism that is going to be approximately efficient. Or … we even revisited
older games that were studied in the operations research literature, such as the proportional bandwidth
allocation of Johari-Tsitsiklis, and by applying the smoothness framework, we were able to get
guarantees that extend to incomplete information and to learning outcomes—something that was not
analyzed before. So the smoothness … the smooth mechanism framework is quite applicable, and it can
improve previous results, generate new results, or even extend previous results to incomplete
information or learning outcomes. Okay.
So I hope I convinced you that smooth mechanisms yield robust efficiency guarantees, and under very
general scenarios. And now I want to move to the second part of the talk, which is to argue that this
smooth … these guarantees are also composable in a market that is composed of such smooth
mechanisms.
>>: Maybe before you move on, can you again state in which settings you think the … actually the goal
of maximizing social welfare is the right one? Because you didn’t mention how the—you know—
sometimes the prices will mean that the auctioneer will have to invest in achieving this.
>> Vasilis Syrgkanis: Yes. You mean, like, how it compares to revenue guarantees and …
>>: Right. I mean, have you looked at what kind of revenue results from the mechanism?
>> Vasilis Syrgkanis: Yes. So I haven’t looked at the revenue in my work. So the reason why I’m … I was
started with welfare … so I think … so the … both welfare and revenue are kind of like correlated: if you
have a mechanism that as the market grows large, the welfare goes to zero, then you go … then the
revenue you can achieve also is going to go to zero, so it’s going to … it’s a …
>>: That implication doesn’t go the other way.
>> Vasilis Syrgkanis: True. Yes, but it’s one—you know—at least it’s one benchmark that you can say
that your mechanism is … that your market is working well. It’s also that if you are in a market—the
classical reasoning of why you would care about welfare—if you’re in a market with several
competitors—you know—if your market does not yield good allocations, then your participants are
gonna go to the other market which achieves better efficiency. And also, welfare is a measure of if
there are money left on the table; like, if there is a lot of value that is left on the table, this is bad for—I
guess—for all the participants of the market. So we shouldn’t be very greedy in focusing about
Microsoft or Google, we should be also focusing on advertisers also in some sense.
But I also want to argue that … so there are … there is some recent work by Jason Hartline that tries to
apply the smoothness framework in trying to get revenue guarantees for non-truthful mechanisms, and
I think that’s a very nice direction. So that’s something that I wanted to look at. Jason—you know—did
it before me, but I think there’s a very … there are many good questions to explore in that direction,
especially in settings where things are multi-dimensional. So this paper that Jason has is mostly in
simple single-dimensional settings, but can we get some efficiency guaran … some revenue guarantees
for multi-dimensional scenarios? Maybe not with respect to the optimal mechanism, which is very
complicated, but maybe with respect to some other benchmark. Sure.
>>: Yeah … if you go back a slide, you had e listed. How do you go about … it seems like if you have a
really complicated mechanism, it might be hard to prove that it’s smooth. How did … how do you
prove—I mean …
>> Vasilis Syrgkanis: So the only thing … okay, so you need to say—most of these, the idea is kind of
similar: you use some random deviation that tries to capture the fact that you don’t know the price … so
… and then you are trying to say that if I—so for example in the … in all of these mechanisms, we can
even think of deterministic ones, like, you can say in the combinatorial auction setting—so suppose that
I am a player and I bid half my value, and I have some, let’s say, single-minded bidders, and I bid half my
value for the set that I am interested, then either the threshold for winning is high, or otherwise, you
win and you get at least half of your value. So it’s a similar reasoning as what we used in the first-price
auction: you argue that by deviating to something that is sufficiently high with respect to your value,
then either you’re going to win and then you’re gonna get good utility or the current price is higher than
this sufficiently high fraction of your value, and then you get some—you know—the interplay between
smooth … price and utility, right? So if you—for example—if you reverse this … if you take this on this
side, essentially what you want to say is that either the utility of this deviation is high, or the current
price of some item is high—higher than this lambda fraction of my value. So you want to say that there
is always a good deviation that a player can grab his optimal set at approximately the current prices that
this set is being sold at to some other player.
But … so, and I’m now also looking at: can we get more algorithmic characterizations of smoothness? So
for example, can we say—so one characterization is something like that: so if you have a combinatorial
auction, it’s based on some greedy algorithm, you get smoothness—can we get more general
characterizations saying that if your allocation is based on such an algorithm and the feasibility
constraint is such and such, then you get smoothness with these parameters? So, some more general
smoothness characterization based on the algorithm and the feasibility constraints that you use for the
allocation. Okay.
Okay, so … going into the composition, let me start with a simple example. So we’ve revisited the
guaran … the efficiency guarantees of a single-item first-price auction, and now let’s look at a market
that is composed of such first-prices auctions. So suppose that you have a market has several first-price
auctions that are being run by different sellers, and the players have uni-demand valuations, and so they
want only one of these cameras, but they might have different values for the different cameras. So their
game that they’re playing is that they’re submitting simultaneously bids for each of these … in each of
these auctions, and the highest bidder is going to win on each of these auctions, and he’s going to pay
his bid. And now we want to claim that … can we … whether … we want to analyze the global efficiency
guarantees of this market. So can we claim that the allocation of this game that they’re playing is good,
based solely on the local smoothness property of each individual first-price auction? So for example,
can we say here that the allocation that arises at equilibrium of this simultaneous first-price auction
game is going to be a good fraction of the optimal matching allocation? And we want to do it as a … in a
black-box way, based solely on the smoothness property of each individual first-price auction.
So the idea is that we’re going to try to … we’re going to view this global market as another mechanism
that allocates resources, and we’re going to try to prove smoothness of this global mechanism. And to
prove the smoothness of this global mechanism, we need to construct a good deviation for each player.
And the idea is that we’re going to construct a good deviation for each player based on the local
deviations that smoothness implies for each of the first-price auctions. So for example, in the unidemand valuation setting, the good deviation for each player is going to be: go to the item that you’re
matched in the optimal matching allocation, and bid the smoothness deviation that we used in the firstprice auction, so bid half your value for that item. Now here, the Bayesian extension theorem is even
more important, because for example, the deviation here—very crucial—depends on knowing the other
players’ values. So the optimal matching is a function of the … all players … of all players’ values, and
the deviation that I used is: go to your item in your optimal matching allocation—so in order to know
this item, you need to know what the values of your opponents are, but the direct extension theorem
say that: simply do an analysis on the complete information setting, it’s gonna directly extend, through
the random sampling trick, to the Bayesian game of incomplete information. So you don’t need to
worry about the fact that players don’t know other players’ valuations.
Now, smoothness, locally, of each individual first-price auction is gonna give you that the utility of a
player from this deviation is at least half of his value for that item minus the current price of that item.
And then, by summing this property over all players, you’re simply going to get that the utility of a
player from this deviation—the sum of the utilities—is going to be half of the optimal matching
allocation minus the current revenue of the whole of the global mechanism, because here you’re
summing the prices of all the items, essentially. So, you proved smoothness of the global mechanism
based on the one half comma one smoothness of the local mechanism. And so then, directly using the
guarantees from the first part, you’re gonna get that every no-regret learning outcome or even
incomplete information is gonna achieve half of the optimal welfare. And now, this is not specific to the
first-price auction or to uni-demand valuations, this applies to any composition of mechanisms. So
suppose more generally, you have m mechanisms, each—and the game that the players are playing is
that they are submitting an action simultaneously in each of these mechanisms—each mechanism has
its own allocation space, allocation function, and payment function, and then the players have some
complex valuation over the outcomes of these mechanisms. So they have some valuation that is a
function of the allocation that they get from each of these mechanisms.
>>: What independence assumption is made now about the valuations?
>> Vasilis Sargkanis: That it’s simply independent across different players, not across different
mechanisms. So in the matching, it could be an arbitrary … on the product of …
So the composability theorem that we’ve showed is that if you have simultaneous composition of m
mechanisms, and each is lambda, mu smooth, and assuming that these complex valuations here satisfy a
no-complement assumption across mechanisms, then the composition is also going to be a lambda, mu
smooth mechanism, and so the …you’re gonna have good global efficiency guarantees. Now there is
some technicality here of what do I mean by no complements across mechanisms, because we have
these abstract allocation and feasibility spaces, so one other technical aspect of this work is trying to
extend the definitions of complement-freeness in this more general composition of mechanisms setting.
So we know that no-complement valuations are fairly well-understood when you have values defined on
sets of items, and we need to define natural generalizations of these complement-freeness assumptions
when you have a valuation defined on products of allocation spaces.
And the idea is to try to define what we mean by complement-freeness across mechanisms, and not
within each mechanism. So we need to have a … to allow for a, like, a left and a right set to be sold
within the same mechanism, but we don’t want to allow such complement goods to be sold across
different mechanisms. And the more concrete definition is: we want to assume that the marginal value
that you get from any … from the allocation of any mechanism can only decrease if more and more
mechanisms come into the market and give you some nonempty allocation. And this assumes no
structure about the valuation within each mechanism. So, under such a complement-freeness
assumption, we get that if you have a market that is composed of the lambda, mu smooth mechanisms,
then this market is going to achieve at least lambda over mu of the optimal welfare at no-regret learning
outcomes, even under incomplete information, when players have these submodular valuations across
mechanisms.
There are several other extensions of this smoothness framework, that I didn’t manage to go through,
that I did in my research. So for example, smooth mechanisms have implications even when you have
sequential composition. So suppose that you played this game, but now instead of these mechanisms
being run at the same time, you have mechanisms run one after the other. So you can an ebay-like
setting where the auctions have a very different ending time, and assuming that most of the bidding
happens towards the end, it’s gonna look more like a sequential auction, rather than a simultaneous
auction. So we also get that smooth mechanisms compose well sequentially, but now, for the more
restricted class of uni-demand valuations—or a generalization of uni-demand valuations when you talk
about more general mechanism design scenarios—and this is in fact tight, through a recent … another
recent work that we did, which basically showed that when you have sequential auctions, you cannot
hope for global efficiency guarantees for valuations that are more general than uni-demand valuations.
So the moment you step out of uni-demand valuations, the inefficiency of a sequential auction process
can actually degrade very badly with the size of the items, for example.
>>: You mean uni-demand over the whole sequential …
>> Vasilis Syrgkanis: Yes.
>>: Okay.
>>: Vasilis Syrgkanis: So you have many items, and you—for example—you want a … [indiscernible]
many items, and you just want one. And then you have good guarantees, but for example, if you have
some players being uni-demand, other players being additive—so they want … they could get a value for
every item—then you could get a very high inefficience.
There are also extensions in trying to capture second-price type of auctions; I mostly talk about firstprice auction schemes, but a similar adaptation … a very similar adaptation of the smoothness
framework—a more generalized smoothness framework that we have in the paper—says that you can
also have very similar guarantees for second-price payment rules—like generalized second-price—but
now you need to make assumptions that players don’t over-bid, and this is, like, a standard assumption
in the literature. For example, you can think of in a single-item second-price auction, if you … if people
can over-bid—bid more than their value—you can get high inefficiency at equilibria. So under this
assumption, which for … all of the guarantees are going to extend.
We also have extensions for hard budget constraints on payments, which is even more realistic in ad
auction scenarios. So there, we al … though there, the guarantees are not with respect to the optimal
welfare, but we get that the … the exact same guarantees with respect to a natural benchmark of
efficiency, which is what is the optimal welfare that can be achieved if you cap each player’s value by his
budget.
So in sum—in brief—we defined the notion of a smooth mechanism, and so that many simple
mechanisms are smooth or you … can be thought of as being smooth mechanisms and that their
guarantees compose well and they lead … and they are quite robust under learning behavior and
incomplete information—and for this, I think that smooth mechanisms are a good … a useful design and
analysis tool for efficiency in electronic markets. I also have some related on-going projects that are
related to the smooth mechanism framework. So in one project, I’m trying to see how these guarantees
degrade with … when you have complementarities in your market—so when you have actually
complement goods that are transacted across different mechanisms. So for example, think of an ad
auction where you have two slots on the same page, and then you … people might want … might have
complementary valuations, because they want both slots, and they … for an impression effect. And then
we show that the efficiency—the global efficiency—degrades smoothly with the size of the
complements. So for example, if you have pair-wise complements, you’re gonna get half of what you
would get for each mechanism in isolation for the global market, and so on and so forth.
Another very interesting direction that we’re exploring with Nikhil and with Jamie Morgenstern trying to
find alternatives to sequential auctions that have good guarantees, even beyond the uni-demand
valuations. So that respect, we studied an alternative to sequential auctions, where instead of
auctioning each item sequentially, you auction the right to choose items sequentially—inspired by draft
auctions in sports—and for there, we saw that we get an exponential improvement for more general
valuations when compared to the sequential auction processes. Another direction that I am exploring
is: what is the efficiency of—when you assume that players can cooperate—so what is the worst-case
efficiency when players can cooperate? Maybe cooperation can help efficiency in games, because you
can think that players are going to get out of bad equilibria by group deviations, for example. So there,
you might get that in games where equilibria have high inefficiency, if you restrict to coalitionally-stable
equilibria, you’re gonna get much better inefficiency, and that inefficiency …
>>: Much better inefficiency?
>> Vasilis Syrgkanis: I’m sorry, much better efficiency. And this efficiency might be more … a more
reasonable bound than the worst-case Nash equilibrium bound. So towards that direction, we introduce
the … a coalitional smoothness framework that quantifies the efficiency of strong Nash equilibria, which
is a class of equilibria that are robust to coalitional deviations, and we also show that these guarantees
extend to some sort of dynamic cooperative gameplay—which is useful when strong Nash equilibria
actually don’t exist, and these are many games where they don’t exist.
So more generally, I’m interested in understanding incentives and efficiency in these electronic markets,
which is one of these two works that I’m mainly focused today, but I’m also interested in other
questions in such markets. So for example, how does information affect bidder behavior in online ad
auctions? And how does this extra … third-party information markets affect bidder behavior in ad
auctions? So we studied a model of informational symmetries in this paper, and we analyzed the
equilibria of common-value auctions with informational symmetries. I’m also interested in
crowdsourcing questions. So for example, how do you incentivize user-generated content using virtual
rewards in such user-generated content websites? Or more generally, how do you create mechanisms
for incentives that are not money-based, but rather virtual-reward-based, like attention or social status
and so on and so forth?
More generally in the future, I’m interested in exploring how to design large-scale markets, and … both
from a theoretical perspective and from a … and a data perspective. So from the theoretical side, I’m
interested in understanding efficiency when you have distributed market design such as the work that I
presented today. Other questions that I’m very interested is what happens if players have uncertainty
about their own valuations, which is very realistic in these electronic markets, where players don’t even
know what their own value is, and most of the work that I presented today assumes that players know
what their value for each impression. So how does this theory break or doesn’t break for when players
have uncertainty about their valuation? What about large-game approximations of these games? And
how does the efficiency change when you go to large market limit of these games? What about other
out-of-equilibrium dynamics other than no-regret play and other suboptimal models of user behavior?
And with that respect, I’m most interested in the practical side of how can we do out-of-equilibrium
econometrics in such markets? So if you assume that players use such no-regret learning sequences,
how can you le … what way can you leverage that and infer their values, or even if they have data that
they’re using, based on data that we have. And so how—if we … if you look at a sequence of bid data in
ad auctions—can you infer what … an abstract model of learning behavior of the player, or can you even
infer his value under the assumption that he’s using learning update rules, and not that he has reached
an equilibrium, which is the standard assumption in econometrics? And that’s it. Thank you. [applause]
>> Yuval Peres: Are there any more questions?
>>: So … these things sound exciting. One thing I wanted to [indiscernible] composition. Are you
assuming in the compositions that they are simultaneous or they can be sequential?
>> Vasilis Syrgkanis: So in the main part, I assume that they are simultaneous, and there is this
extension for sequential composition, but for more restricted classes of bid … of valuations across
mechanisms.
>>: So how do you deal with fact that in sequential, say the second-price auction may no longer be
truthful, because the value transmitted in the first stage might be useful to opponents in more
[indiscernible]
>> Vasilis Syrgkanis: Yeah, so the things that … so the nice fact about this analysis, which is like this
implicit price of anarchy analysis, is that you don’t even need to argue about equilibrium—how do
bidders behave? You just need to argue that, no matter how bidders behave, there’s always a good
strategy for me to play. And for example, in sequential auctions, a good strategy for you to play is to
play as you would play at equilibrium until your optimal item arrives, and then try to get this optimal
item. So, I don’t care what you did at equilibrium—maybe the equilibrium is a very weird fixed point of
this game—you cannot compute it in general scenarios, but no matter what equilibrium is, you can
always find such a deviation that will give you good deviating utility. And yeah … so it gives guaran … so
the good thing is that it gives guarantees for more general scenarios than what you could get by solving
an equilibrium and looking at the properties of the equilibrium, but you can think of a handicap as that it
doesn’t tell you how bidders behave at equilibrium, it just gives you implicitly what the … the final result.
>>: I mean, one more reason is usually given, and that you gave, for focusing on social welfare is
because the existence of alternative markets or auctioneers. But somehow … so that’s using social
welfare as a proxy for what you really want to understand, which is this larger market [indiscernible].
Part of your analysis was exactly looking simultaneously at multiple markets or auctions going on at the
same time, and then maybe this argument for using this proxy of the social welfare goes away, and you
really want to focus on revenue of an auctioneer by taking into account alternative competitive
auctioneers.
>> Vasilis Syrgkanis: So there’s an—actually, that’s a very interesting direction indeed—there has been
some papers on trying to—like, for example—compute equilibria of competing auctioneers, and
assuming that all their strategy is setting a reserve price. So that’s very nice, but it’s very limited: like
only uni-demand bidders, same valuation, and things like that. So yeah …
>>: So you mean it’s hard.
>> Vasilis Syrgkanis: It’s hard, it’s hard. So it’s hard to find equilibrium, and the implicit analysis that
we—all this implicit analysis of price of anarchy currently mostly works for welfare guarantees. So I view
this direction of, like, revenue guarantees as one of the most exciting future directions, but currently all
we have is for … mostly for welfare. Yes.
>> Yuval Peres: That it? [inaudible] [applause]