>> Yuval Peres: Welcome. We're are very happy to have Professor Eyal Winter
from the Hebrew University who will tell us about multi-agent initiatives with type
dependent externalities.
>> Eyal Winter: Thank you, Yuval. Thanks for your invitation. I'm really happy
to be here. I'm going to talk -- this is part of my more extended research agenda
which I usually identify as the engineering of economics. I mean, there are two
types of -- two types of ways of doing economics. There's a more conventional
one, which is a descriptive approach where you design models to explain stylized
fact. Okay? And that's been very popular. Game theory has been a very
important tool within this domain of questions people you know you encounter
something, some phenomena in the economy and you try to explain it, and you
design a game hoping that the equilibria of the game will sort of match what you
see in real life.
This is a descriptive approach to economics. And there is the more the
normative or the engineering approach in which you go the -- basically you go
the other way around. You identify and outcome which has good properties in
the economy. For instance, efficiency. Okay? And you ask yourself, you know,
how -- what kind of games, what kinds of strategic environment will give rise to
this good outcome?
And in doing so, you are actually acting as an engineer, not -- not only I think
reverse engineering would be the right word. So you are asking -- you're not
taking -- you're not building the game in order to explain something, you're asking
what should be the game that will give rise to these desirable outcomes, given
this property.
This has a long -- this part of economics has also quite a bit of tradition starting
with social choice theory. I just spoke with Noga about, you know, fundamental
papers like error impossibility theorem. And recently, and this was my interest of
speaking to people like -- to people like those working in Microsoft Research has
been very popularized in computer science in algorithmic game theory, and I
want -- one of the purposes of presenting these things is to deliver to people here
some ideas that come from economics hoping that they slowly will penetrate into
the literature of algorithmic game theory.
Okay. So I'm going to talk about multi-agent initiatives with type dependent
externalities. So what is it? I'm going to think about situations in which people
do things together. I'm going to give you some example. And I'm going to
emphasize the idea that the utility, the games they acquire from these activities
are going to depend on who are the other people who participate in this initiative.
Okay?
And then I'm going to introduce a mechanism designer who will ask the question
how do I incentivize optimally these individuals taking into account that who are
the other people plays a major roll in the utility function. Okay.
So for instance, I'm going this think about the activity that relies on agent
participation. There are going to be externalities that arise among the agents,
okay? So it's going -- I'm going to either enjoy or suffer the fact that somebody
else participated in the same activity.
And then the principle offer is a set of optimal incentive for participation. He
wants to get people participating, and he wants to do it with minimal expenses,
okay, and the question how to do this.
So I'm going to stick to the terminology of throwing a party. Although the
application are much broader as you see in a minute, but this is a terminology
which seems to be very useful. There is a group of participants whom you would
like to come to the party that you throw, and I'm going to denote by WIJ is how
much person I enjoys the fact of person J attending the party. Okay? Where
WIJ could be negative. You could, you know, suffer a major disutility by seeing
your ex-girlfriend in the party for instance. Sorry? It's non-symmetric, right?
Sorry?
>>: Is she enjoys ->> Eyal Winter: She may enjoy seeing me. Okay. And the question is how
should you lure this group of individual effectively taking into account these
externalities?
Okay. I'm going to give you some example, some applications of this model to
convince us it's not only about parties, it's going to be very important to questions
in industrial organization. Think about firms that make acquisition offer for
several owners of other. A decision of an owner whether to sell or not depends
of whom are the rivals that are expected to be purchased, right? And how should
the acquiring firm -- the question is how should the acquiring firm go about
making offer to these firms taking into account that there is -- that these
externalities exist?
Another application which I think the most relevant for instance when I talk with
Yuval a bit for instance for Microsoft is network technologies, okay? Think of
network technologies like software. So the buyers were agents, and their
willingness to pay for the technology is affected not only by the number of buyers
of course by the number of buyers who are expected to adopt it, but also by the
identity of the buyers. Okay? Think of a seller of phone. We want to make sure
that perhaps he has some technology, network technology. We want to make
sure that your partners, your friends, your family, relatives are using the same
technology you are using a certain software. When you think about the software
it doesn't real matter what most people are using, but it's much more important
about what your coauthor are using in terms of a software. Okay?
So these are externalities. And if you want to market on product in an effective
way, you have to ask yourself what should be -- what are these -- what is this -how to take into account this structure of externalities. And perhaps I can levy
different prices. Perhaps I could discriminate users or buyers based on who they
are in this network of externalities. Okay?
And this is what I'm going to discuss. This is what I'm going to discuss.
A mall is another example. Mall owner tries -- in a minute, ma'am, I'm going to
address your question. Tries to lease stores. Big department stores or brand
name, attract consumers more than other stores. Thus induce positive
externalities on these stores.
And the question is how should the mall owner price leases? Okay? And
perhaps I'm going to bring you a startling evidence that my colleague Eric Guld
[phonetic] at the Hebrew University collected some data about this, and he
actually showed that these externalities play and enormous role in the way
leases are prices in mall. What he found out is that brand names, these -- and
what people call anchor stores like JC Penney, like Macy's and so forth, they
occupy more than 90 percent of the total leases spaces in malls in the United
States, more than 90 percent are brand stores.
At the same time when they pay the total lease that they pay, it's about 15
percent. Okay? So they are getting excellent -- these big department stores,
which is not only the richer, much more richer than the other store ->>: [inaudible].
>> Eyal Winter: Huh?
>>: Something didn't sound ->> Eyal Winter: They occupy. The question is how do you define an anchor
store. It's not just JC Penney and Macy's, there are lots of other stores. You
know, if you look -- if you go -- in you go to a mall at Bellvue, you'll notice that
there are lots of anchor stores there. It's not just JC Penney and Macy's.
>>: [inaudible].
>> Eyal Winter: Huh?
>>: [inaudible].
>> Eyal Winter: Huh? So they have some -- so 90 percent is okay. But thing
that started whatever definition you have for an anchor store, it's still quite
startling that this group of stores, this group of brands is occupying 90 percent of
the total space and are paying only 15 percent of the total lease.
>>: [inaudible].
>> Eyal Winter: Huh?
>>: [inaudible] they don't believe this data and we're going to ->> Eyal Winter: Okay. We're going to continue with your skepticism. Yes.
>>: [inaudible] factor of 10 between what one store pays ->>: Maybe one store. But don't believe there's 90 percent that extends that
[inaudible] top 90 percent of the space that they pay 15 percent. I think the top
store may [inaudible] but I think [inaudible].
>>: [inaudible].
>>: I'm sorry for the interruption.
>> Eyal Winter: Sorry?
>>: How do you -- 15 or 50?
>> Eyal Winter: 15. 15.
>>: So sorry. Can you clarify the model? Because what you said ->> Eyal Winter: You'll see the model in a minute.
>>: WIJ is this realized only if I is [inaudible] as well or not?
>> Eyal Winter: Here is the model. First of all some main assumption. And you
will see the formal model in a minute. They are heterogenous agents.
Externalities arise between agents. And when externality induced on agent one
due to the participation on agent two is the utility of benefit or loss of agent one
due to mutual participation. The principal maximizes profit. And let me skip this,
go ->>: [inaudible].
>> Eyal Winter: Huh?
>>: [inaudible] it's only [inaudible].
>> Eyal Winter: So here is -- I'm going to go directly to the model. Okay? I'm
going to -- this is a model. I have some extensions in my -- that we're not dealing
with, partly dealing with in the paper. And one of the purposes of this is some of
them simply are left open because of our limited mathematical skill. And one of
my hopes is that people here will be able to address these questions better than
me. Okay?
So here is the model. There is a group of agents and a metrics of externalities,
WIJ, okay. And WIJ as I said, that could be either negative, positive, it could be
mixed. I'm going to start. And most my discussion will adhere to the case in
which these externalities are positive. Okay?
So WIJ is how much agent I enjoys the presence of agent J. And I'm going to
stick, as I said, to the party terminology. And there is a vector of outside options.
So if you don't go to the party, you get CI. And it's really without loss of
generality to assume that these are identical for all the players. Nothing is -nothing is really gained too much. Okay?
Now, a solution is an incentive mechanism which is simply a vector of rewards.
This is what the principal offers these agents. So if you come to the party, these
are the goodies that you are going to get. And this is the ultimate payoff the
players are gaining. So if they come to the party with the set C of agents who
attend to the party, the utility for player I from being in the party is the sum of
externalities they experience from the group of people that attend, plus the
goodies that he's being offered by the principal. And if this exceeds his outside
option he's going to come to the party. Otherwise he will stay at home. Okay?
That clarifies your question? Okay.
>>: [inaudible].
>> Eyal Winter: Huh?
>>: [inaudible].
>> Eyal Winter: I said without loss of generality they are going to be identical.
It's really -- you gain really nothing by assuming differential outside options.
>>: [inaudible].
>> Eyal Winter: Sorry?
>>: You can put it noticed the VI.
>> Eyal Winter: You can put it inside the VI. Okay.
So agent -- decision are simultaneous, okay? So given the incentive mechanism
this vector of Vs, I'm going to say that a certain vector of V is incentive inducing
mechanism. If full participation is a unique Nash equilibrium of the normal form
game. Once you set up these V1 to VN, you have a normal form game which
has equilibria.
And I'm going to think about the principal as somebody who wants to get all
these agents together. So he wants to bring all these agents to the party. And
he wants to do it in a unique equilibria. So that may be multiple equilibria and
some which all of them attend the party, some of them all stay at home or some
of them stay at home. He wants to -- he wants to make sure that there's -- he's a
worst case scenario guy. He wants to make sure that there's no equilibria in
which some of them stay at home. And indeed there is some evidence,
experimental evidence that show that in these types of games, you know, people
will have preferences to play these bad equilibria. So he really needs to make
sure that he screams out these bad equilibria in which some people are staying
home. Okay?
>>: [inaudible].
>> Eyal Winter: This is not much different than saying the principal wants to
maximize profit. So this is indeed the more general case was that the principal
for each subset of agents, the principal gained some utility, call it V of S. S is a
subset of N again so utility V of S from getting players of S together and then we
wants to maximize net profit, which is V of S minus the expenses that he pays.
Okay?
But in order to solve this problem you first need to solve the full participation
because once you solve full participation then to solve this profit maximization,
you simply have to take each set and see what does it take to fully make all
these agents in S to participate and then you select the S which maximizes your
net pair. So indeed this is only a part of the overall maximization problem, but
this is the more -- most difficult, okay.
Now, let me warn you about something. Because I gave it -- I gave it several
audiences of computer scientists and what I'm not going to talk about, what is
really not interesting for me as an economist is to talk about complexity issue,
okay? So there are going to be lots of question I assume by people here. You
know, this is MP complete, this is exponential time, this is -- who cares? Okay.
Obviously people care. But not me. So my -- a but indeed I realized the
importance of this. And perhaps there are going to be -- I'm going to come with
the mechanism with an algorithm that indeed was going to be MP but maybe
there's interesting aspects in special cases, maybe people can identify special
cases in which their simplified structure for the mechanism in which computation
is possible, okay?
But this is not going to be the main computation is not going to be the issue. The
issue is going to be to identify this -- and to characterize the mechanism
mathematically -- a basically mathematically and conceptually, characterize the
optimal mechanisms in these problems.
Okay. So the optimal mechanism or the solution is the incentive so -- an INI
mechanism is a mechanism that is V that makes the game GV have only one
equilibrium in which -- in which all participate. And then I'm going to say that V's
optimal mechanism. If it boasts INI mechanism, if it's incentive inducing, has a
unique equilibrium which all of them participate and it's also -- it minimizes the
sum of rewards in terms of the expenses of the principal.
I'm going to start with positive externalities. And the first thing to note that has -which in fact comes up in a different -- in a -- not directly related paper of mine,
but somewhat related. I had papers about the incentive in the American
Economic Review that has a similar feature which is the first thing to note is the
optimal mechanism must be of a sort that I'm calling divide and conquer, okay?
So here is how the optimum looks like. But then I'll have to characterize it
further. It's not the ultimate characterization, it's just a step, okay? So what the
principal has to do, he has to select some agents, some agent, I don't know who,
but there will be some agent that will be selected. Remember the externalities
are all positive. So basically he wants to maximize the extraction rather than
minimize the payment. You know, it all depend on the outside option. If the
outside option is zero, then just extract money from that, okay?
So he has to identify some player for which he extract no money, he just pay him
-- pay him his outside option. Okay? One agent -- there must be one agent who
is getting -- is paid this outside option without any extraction. Why that?
Because if he paid all the other -- all the people less than their outside option,
there would be an equilibria in which all of them stay at home. Right? It's clearly
an equilibrium because no player alone can make himself better off by deviating
knowing that all the rest are staying at home.
So one player has to be given -- has to be given sufficiently high incentive so as
to make him to come to the party even if he believes that he's the only guy who is
going to come to the party. That's kind of a boring party. You have to
compensate sufficiently in order to get at least someone to attend the party
alone. But once you pay this guy, let's call him player one, once you pay this guy
that much, you can extract something from player two. How much can you
extract from player two? Exactly how much player two enjoys the presence of
player one. Once you pay this to player two, you can extract something from
player three. How much can you extract from player three? The amount he
enjoys, the fact that both player one and player two are there, and so forth.
Okay?
So this is -- once you set up this mechanism, if you write down the normal phone
game you see that the normal phone game has a unique equilibrium, okay in
which is not only unique equilibrium, it's a equilibrium which is attained by the this
is called iterative elimination of dominated strategies, right? And this unique
equilibrium gets everybody to attend the party. Okay?
One consequence of this thing is that if you -- if for instance the metric -- think of
metrics being completely symmetric, all agents like others in the same way, it's
all symmetric, WIJ equal WJI and all the WIJs are the same. Okay? Except that
on the diagonal you have zero. Then you would assume intuitively up front you
would assume that the optimal mechanism would have to pay them the same,
because they are all identical. Turns out what I showed you here is that in spite
of the fact that they are all identical and you want all of them to attend the party,
there exists no optimal mechanism that pay all of them the same. Not even two
of them the same. Right? You have to discriminate among all the M players.
And indeed you have to substantially discriminate.
So this is some interesting feature that these type of mechanism have, and it
comes -- it arises -- sorry?
>>: [inaudible] first guy also. Because he ->> Eyal Winter: You pay all of them this much, so you extract -- you extract from
them too much. There's going to be ->>: [inaudible].
>> Eyal Winter: You want to guarantee that there exists a unique equilibria.
Indeed if you pay all of them that much, you will have a Nash equilibrium all of
them attend, but there will be also a Nash equilibrium which none of them attend.
Okay?
Okay. So now the entire exercise boils down to the question, you know, in order
to identify the optimal mechanism the entire exercise boils down to the question,
what should be the optimal order here? So the question involves selecting an
order. And I skipped -- I skipped a graph here, but what turns out to be -- and
this is -- what turns out to be the case that this question which is a classical
mechanism design question, is answer to this question relates to a completely
different literature coming from operation research. In fact, operation research of
I think 10 years ago where people ask a question, an axiomatic question about
how to select an order based on a tournament. I mean, this was -- this literature
was motivated by the American obsessive to rank sport teams, okay.
So you get an outcome of a tournament. A tournament is let's say there are N
teams. Each two of them match. And let's assume for simplicity that one of them
wins, another one loses so you can construct out of the tournament a directed
graph, a complete -- a complete directed graph. And now people want to see at
the sport paper some ranking of the team. How do you work out -- how do you
work out a ranking out of the -- out of a directed graph?
>>: So all your data is whether there was win or lose? You don't -- there isn't a
score associated ->> Eyal Winter: Suppose that you also have a score, okay?
>>: [inaudible] more information. That's more information.
>> Eyal Winter: That's more information. In fact, the way they are doing it is
ignoring the score. And this is what happened. They are ignoring the score. I'll
show you what they do, but it turns out that the optimal solution for this thing
maps itself in some interesting way to this literature about ranking, about ranking
and also, it also it maps itself to a different type of literature which is selecting -which is basically identical but it has -- it's a different application which I'm going
to mention later also, which is Condorcet idea about how to rank [inaudible]
based on election outcome. Okay?
Okay. So let me explain how we select the order. So I'll start in this case. So
first proposition is V is optimal mechanism, then it must be a DAC mechanism.
Okay? So positive externality. Okay?
So what is a tournament? A tournament is a simple and complete directed
graph. So you have N nodes and you'll have arcs, and it may be cyclic or acyclic
but there is a connection between any two, it's complete any two nodes in this
tournament. In this tournament. Okay?
And so here is how you select the order. So first of all -- so first of all, you have
to define a graph out of these metrics. Okay? So the way I'm going to define the
graph is I'm going to look at all matches between agents and think about it -think about these metrics as a result of a tournament. More specifically I'm going
to announce that player I's win against player J if I likes J less than what J likes I.
Okay in so if WIJ is smaller than WJI, then I'm going to say that I wins J in the
tournament. Okay. If they are the same, then it's going to be a tie, okay in so if I
wins J, I'm going to -- if this is I and this is J, I'm going to run an arc from I to J.
Okay? This is the way I'm going to instruct the tournament. Okay. And it's
complete and everything. Okay.
Now I'm going to two cases. Case one in which the tournament is acyclic, okay?
If the tournament has no cycles, then it's a pretty trivial thing to prove that there
must be an order which is consistent with the graph. Not only their exists an
order which is consistent with the graph, there is a unique order which is
consistent with the graph. Right?
>>: [inaudible]. Cycle like one equality ->> Eyal Winter: Equality is also a cycle, yes. Okay. So if there is no cycles, in
particular the graph is asymmetric, right, then there exists a unique order which is
consistent with the graph, so for instance here you would go this way. This is I
think this is not cyclic, so for instance you will -- it's not a tournament, okay? But
there is -- so I won't complete it. But there exists a unique -- exists a unique
order I1, I2, IN starting with I and ending with J, okay, where I1 beats I2, I2 beats
I3 and so forth.
Now, the first lemma is that in this case this is the optimal order. Okay? This is
the order that the principal has to use for the divide and conquer mechanism,
right, in order to minimize his expenses in the optimal mechanism. Remember.
So this is the guy who is getting this C. Okay? The intuition here is that there is
some popularity contest here. But the way you define popularity here in terms of
the standard way we think about popularity is not only combination of how much
people admire you or want you to attend the party, but also take into account
what is your interest in being with the others. Okay?
And basically the better you perform in this tournament, the higher you are going
to be in this ranking. The higher you are going to be in the ranking means that
anything else equal you are going to be paid better. Okay? So this is what
happens in case of acyclicity. Okay?
But now the question is what happens with this, if this graph is cyclic, if this graph
is cyclic, their exists no consistent order or there may be more than one
consistent order with the tournament. The question what to do then. Okay?
So now I'm going to switch a little bit and do an intermission and tell you about
what the people in operation research suggest to do. And then connect it to the
mechanism design problem. Okay? So what they do is the following. I don't
need to write anything. I can simply say it. Okay? So again, they look at the
tournament that arise from the games and check again if this tournament is cyclic
or not cyclic. If the game is acyclic, if the tournament is acyclic, they to exactly
what we do. Okay? They take the order which is consistent with the tournament
and say that's the order that represents the best -- the tournament outcome.
If the graph is -- if the graph is cyclic, then what they try to do is they try to flip to
chit. That's Manchester United one against Chelsey, so they try to see whether if
they flip the outcome they're going to make the graph from cyclic to acyclic. If
they manage to do it by flipping only one arc, then they take the consistent order
with respect to the acyclic graph. If they can't do it with one arc, they try to flip
two arcs. So they are looking at the minimal number of arcs they can flip making
the graph from cyclic to acyclic, and then taking the order which is consistent to
the acyclic graph.
Of course as I'll see with example, this doesn't provide you with the unique
ranking. The ranking could be multiple. And then arbitrarily take one -- huh?
>>: You said you didn't care. But that's indeed MP hard.
>> Eyal Winter: That's MP hard, yes. That's MP hard.
>>: It's called a [inaudible].
>> Eyal Winter: Feedback arcs ->>: [inaudible].
>> Eyal Winter: Huh?
>>: It's not even -- the minimal one is not unique necessarily.
>>: Yes, that's what he said.
>>: [inaudible].
>> Eyal Winter: So let me now show you what's the parallel to these in our case.
Okay? First of all -- first of all I'm going to ask your guess. You already hinted it.
And also Noga about how to improve what I suggested now, knowing that each
game can result with scores, right?
>>: [inaudible].
>> Eyal Winter: Right. So in the context of games you could put on each arc
you could put the amount by which team A won against team B, and then what
would you try to minimize? The total -- the total -- so you are looking at the
minimizing the total violation, the total violation. This is what -- in fact, they call it
also in operation research minimal violation ranking.
But they are not looking at the weights, they are looking only at the number of
arcs. They are only concerned with the number of arcs. Okay?
So here is now what we have to do in our case. So this is acyclic. So I can skip
-- I can continue. I see that you are following me. I can continue without the
slide and then go back to the slide, okay.
So what we have to do, we have to define the weight. We need the weight for
the arc. For each arc. And the weight is going to be simply the difference
between WIJ and a WJI. Okay? So here is the result. The result says the
following. In order to get the optimal mechanism what you have to use is the
divide and conquer mechanism, and you have to select the order in the following
way. If the resulting tournament is acyclic you simply take the order which is
consistent with the tournament. If the -- in the graph is cyclic, then you have to
take the order which is consistent with the acyclic order with some acyclic order
that was obtained by minimizing the violations, the sum of violations where the
violations are defined by WIJ by WJI. Okay in.
So for me it's somewhat surprising that a solution to a mechanism design
problem is related to this -- huh? By the way, the treatment to the -- the
treatment in the operation recently the fully axiomatic, so the idea there is to
come up with a set of axiom and say I'm looking to a ranking that represent the
tournament outcome, and I want it to satisfy axiom A, B, and C, and then usually
you should come up with a result that says there is a unique way of summarizing
the tournament outcome with the rankings that satisfy these axiom and the
unique way is what I suggested. Okay?
So how am I doing with time? When did I start? How long do I still have?
>>: [inaudible].
>> Eyal Winter: Okay. Okay. So let me see if I want to give you.
>>: [inaudible]. [laughter].
>>: And you have to get 90 percent of the content.
>> Eyal Winter: Okay. There are, as you say, getting the ->>: So this was so far just all the WIs are positive.
>> Eyal Winter: This was -- okay. This was when all the WIs are positive.
Now, let's think about the case where -- so let's now think about the -- let me first
finish the positive case and then we -- I hope I still have time to talk about the
other one. Okay?
Now, as you mentioned, finding out the optimal order is a different computational
problem, hard computational problem, but turns out that there is an easy way to
summarize the cost of the principal. For the cost of the optimal mechanism
which is an important aspect when you get to the selection part. Remember,
eventually you have to select the target group that you want to attend the part in
giving your preferences about these people, and so it's important to be able to
say how much it cost me to get these people together as opposed to getting this
group together, okay?
And it turns out that it also -- this [inaudible] also gives some intuitive explanation
about how the preferences of the principal would be affected by the structure of
externalities. So there are two terms here that are relevant. One of them which
I'm going to call K aggregate which is simply half of the sum of the total
externalities. And the second one I'm going to call K asymmetry which is half of
the total sum of differences between WI and WJ. Okay? So the large
asymmetry you have between -- or the larger ereciprocity that you have between
players, right, like player one likes a lot player two but player two opportunity
really like watch player one, this thing here will become larger, right in and the
result says the following: Take any participation problem and look at the -- that
will be the sum of incentives of the optimum mechanism. This is the total that the
principal has to pay if the corresponding tournament is acyclic but we are later
where I'm going to show you a similar result for the cyclic case, then the total
payment is just N times C. This is the outside option, minus K aggregate, minus
K asymmetry. Which in particular means that the principal is better off in terms of
the cost, better off incentivizing the group where there is a lot of asymmetry
among the players, all right? He benefit from a large -- controlling the amount of
course the largest the total externality is the better it is for him as well.
But if we look at two matrices which have the same total externalities then he
would obviously prefer the -- it would be in a better situation with the externalities
that represent more asymmetry. Because he can sort of use -- take advantage
of these asymmetry in the optimal mechanism. Okay.
So I'm going to skip these. Here is an example. Now I'm going to -- no, that's
not interesting. I have limited time. Usually it's a 90 minutes talk. So I have to
make selection.
>>: [inaudible].
>> Eyal Winter: Yes.
>>: [inaudible] is that just the sum over all periods of the maximum?
>> Eyal Winter: Sorry?
>>: [inaudible] maximum of WIJ and WJI.
>> Eyal Winter: Why maximum?
>>: [inaudible].
>>: [inaudible].
>>: [inaudible].
>>: [inaudible] sum over all a pairs --
>> Eyal Winter: Sum of the all pairs?
>>: One had a [inaudible] and one was ->>: Half the difference is half the maximum.
>> Eyal Winter: Yes. But it's ->>: Are you just saying that you can simply ->>: But it makes [inaudible] from your formula, right?
>>: I'm trying to understand because you have -- you ever have these quantities
separated or are they always together, adding together? I'm just trying to
understand what is the ->> Eyal Winter: No, in trying to figure out what the total cost they are coming
together.
>>: Right.
>> Eyal Winter: So it may be ->>: [inaudible] more natural to look at this, the sum of [inaudible] for each pair
[inaudible].
>> Eyal Winter: But in terms of the interpretation, you know, in order to see how
asymmetry affect the pair of the principal, it's more revealing to have them
separate. But I get your point. Okay. So here is -- I'm going to skip -- I'm going
to show you the corresponding result for the -- for the cyclic case. The cyclic
case it also NC minus K aggregate minus K asymmetry, but then you have to
add these extra costs which arises from these violations. So you have to pay -you pay extra those links which you have to flip in order to get the tournament
from a cyclic tournament to acyclic tournament. Okay? But in terms of -- okay.
So here is an example. Sorry. All right. Here is an example just show you how
it works. Okay? And about the uniqueness. So this is the metrics. This is the
metric. This is how it translated into the graph, okay? And as you see, there are
two ways to do this. There's one way -- this is a cyclic graph. You see it can go
from three to one to two to four, to three and so forth. Okay?
There's one way -- one way to optimally incentivize this people is to flip two and
four, the cost of two and four, flipping two and four is a total cost of two. Okay?
And if you flip two and four, you end up with an acyclic graph. And the
corresponding order is going to be player four is the first, player three is the
second, one is third and two is fourth. And the corresponding payoff is 20, 13,
13, 12. But you have another way to -- you have an alternative way to do it,
which is to flip both one and two and three and four. If you flip one and two, it's
going to cost you one. If you flip only one and two, it's still going to be cyclic.
You are not -- you are not -- you can't make it by flipping only one and two, you
have to flip also three and four, which also costs you one. And if you flip one and
two and three and four, the corresponding order is different now, which player
three is the first, two is the second, four is the third and one is the fourth. And the
payoffs are different. 20, 16, 10, and 12. But if you sum these up, this is 20 plus
20, 6, 46, 58. And this is also -- the total is also 58. Okay? So they sum up to
the same cost.
Okay. So now negative externality. With negative externality the picture
completely totally fails and the mechanism -- the optimum mechanism is much
simpler. Basically the optimum mechanism which you have to pay each agent is
C plus the sum of absolute value of all the negative externalities. Right?
Because here you can't -- you can't avoid -- you can't avoid paying them that
much. Because you have to reimburse these agents for the disutility that you
caused them in getting all these people together. If you pay them epsilon less,
it's not a matter of getting multiplicity of equilibria here. If you pay them epsilon
less, then there won't be an equilibrium which all of them attend.
>>: Now, what if externalities are negative.
>> Eyal Winter: All externalities are negative.
>>: Okay.
>> Eyal Winter: So graph play no role here. There's no -- there's no role for
graphs. What happens -- what happens with mixed externalities? Okay. So
there's a slight error here, so I prefer to have you stick with this. So with mixed
externality things become very complicated. But we have one condition under
which things are more tractable, okay? And this condition requires that the
relation, that's defined by now a relation between I and J. We say I -- we say I
hates J. It's a reasonable definition. I hates J. If WIJ is smaller than zero.
Negative externality. Okay.
So if the binary relation hate is transitive, if I hates J and J hates K means that I
hates K, what you can think about environment in which that makes sense, not
every environment, but you can think about environment in which that makes
sense.
>>: [inaudible].
>> Eyal Winter: Sorry?
>>: [inaudible].
>> Eyal Winter: And enemy of my enemies is my enemy. This is sufficient
condition.
>>: What is it [inaudible].
>> Eyal Winter: Huh?
>>: What is an example of this?
>> Eyal Winter: Either this or the other one which is more -- more reasonable.
The friends of my friend is my friend. Okay?
>>: Is there an example of this transitivity?
>> Eyal Winter: Sorry?
>>: Can you give us an example where ->> Eyal Winter: Where it makes sense to assume it?
>>: Where it makes sense.
>>: If you hate -- everybody hates everybody [laughter].
>> Eyal Winter: No, if you -- if hating somebody ->>: We are mixing ->> Eyal Winter: No, if hating somebody involve a certain characteristic of a
person, right, so for instance -- so -- huh?
>>: [inaudible].
>>: [inaudible]. Because one animal and the animal is eaten up by another one,
so that means the enemy and the third one eats both of them? [laughter].
>> Eyal Winter: Anyhow, if this -- this is a sufficient condition. You know, that
may be improved to a -- you know, there may be much weaker condition under
which it holds. But we couldn't track really the sufficient and necessary condition.
But under this condition what happens is that you can decompose every mix
problem into two problems. One which is fully positive and the other which is
fully negative, and the optimal mechanism for the mixed problem is simply the
sum of the optimal mechanism for these two negative and positive problem. This
may apply for a much larger set of metrics only with these, okay. So this is one
thing.
And the other thing which I'm really curious about and seem to me we tried to
lead -- we have some results but not really -- not sufficiently in general is the
model where you look at the -- at more general externalities. Note that the
structure of externalities that I put on the board here was basically bilateral, okay,
so you either like [inaudible] the amount he likes player J is completing
independent who else is coming to the party. Okay? So for instance, for me,
you know, you can imagine, you know, if I'm invited to a party, and, you know, I
like to be with smart people but not with too smart people, you know, that
becomes embarrassed so the Yuval and Noga are coming, both of them, to the
party, then it's not really attractive for me. But any one of them separately is
going to be very attractive for me to have. This makes a lot of sense in many --
in many application that my -- the externality that player I and player J have
between themselves is going to depend on who other -- who are other who
attend.
So a model for that would be a model in which instead of writing things in terms
of metrics, the way we have to write this is the function VJ, VJ is a function from
2 to the power M minus J into R. Okay? This is -- okay. This VJ -- okay. So VJ
asks is how much player J enjoys is presence of the subset S of agents. Okay?
So then we have V1 to V -- V1 to V -- VN. These are functions, okay? A special
case of these functions are these metrics. But the question is can one design tell
me what the optimal mechanism should look like? Or find interesting properties
of the optimal mechanism as a function of these N tuple functions.
Can the optimal mechanism in some way represented also in terms of graphs?
Is there some perhaps more general concept than graph that perhaps hyper
graph that can sort of can -- that one can adopt or somehow generalize the result
from the metrics to these more general. Yuval, I think I'm done. I'm not going to
the -- you know, there's perhaps one minute. I mean, there are let's see. So
there are lots of for instance you can -- in terms of example you can look at the
examples like desegregation. Think of two groups, right. Let's call them, you
know, there are two groups rivaling groups or you know, you think of each person
in group A likes to associate only with members in his group, get the positive
utility out of it and get the negative utility of zero from associating with the other
group. This give rise to externalities. Okay? And then you can ask a question,
what should be the optimal mechanism to get all these N agents together. This
is one example that we look at.
Another example is what we call a status example. A status example -desegregation, desegregation is the opposite. Desegregation is a good model
for a desegregation is a good model for single club -- single clubs, okay? So
man likes to associate themselves -- the more women there are in the club, the
better. For women the more men there are in the club, the better. Okay? So
think about you know you get this utility -- up to a point. So you get this utility
where the more people from your own group are, and you get the positive utility
from the more people from the other group are. So this is the desegregation
example.
Perhaps status example is the third one.
>>: [inaudible].
>> Eyal Winter: Sorry?
>>: [inaudible].
>> Eyal Winter: A ratio ->>: [inaudible].
>> Eyal Winter: Right. So a status example is again where you have group A
and group B where group A is more -- is more attractive than group B and then
people gain from -- whether you are from group B, you gain more when you are
associated with people from group A. This is the status then. You can ask
question. How do you -- you know, how do you get optimally these people?
Okay.
I'm going to tell you, I'm going to -- this is -- I'm going to tell you another story,
you know, for an example is I heard from my nephew. My nephew was -- my
nephew finished a course of navy which is considered a very prestige course in
Israel in the military. And you get girls very easily if you finish this course. It's
very recommended for people who want -- and when they finish the course, they
were -- they decided they want to throw a party as you know, to celebrate the
end of the course. This is a true story.
And they started getting and form themselves in different night clubs in Tel Aviv.
And gradually they noticed the offers are becoming more and more attractive.
Once they mentioned that this is about celebrating the end of navy course, offers
became more and more attractive. At some stage the club owner said you know
what, if you allow me to open the club for other visitors, I throw the party for free.
Okay? You get -- you pay nothing. You -- all the drinks and food are on me as
long as you open it for other people.
They were impressed but not enough to strike a deal. Huh?
>>: [inaudible].
>> Eyal Winter: Sorry? And they continued searching. The end result was that
at some club they were willing to throw the party for free and give each cadet on
this course a DVD player as a present. There were lots of girls during this night
for -- you know, they cashed a lot from other people who. This is -- huh? Huh?
>>: [inaudible].
>> Eyal Winter: They got the DVD. They got the DVD. Thank you.
[applause]