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>> Yuval Peres: We're happy to have Elliot Anshelevich from RPI. His topic is very
topical these days and a lot of interest here. So I'm glad that the talk will be recorded.
So we can look more at these problems, because I think these problems are going to be
central for a long time.
Please, Elliot.
>> Elliot Anshelevich: Okay. So this is going to be a talk about games and social
networks. All right. So the idea is let's say we have some social network, we know there
are people, whatever, and the links between the people represent relationships or
collaborations or friendships, something like that.
Okay. So I mention there's a big social network, which I didn't draw, and we're this
green person. And these are my friends. So I just drew the links that are next to the
green person.
Okay. When the green person like all of us has a problem, that problem is that you can't
possibly put 100 percent effort into every single thing you're working on or 100 percent
into every friendship that you're participating in.
So we're going to model this by saying, okay, this green person has one unit of effort.
And they're going to allocate this one unit of effort amongst all their relationships that
they're involved in. So, for example, it might do something like this.
So here are maybe this green person puts 60 percent of its effort on this relationship or
this collaboration. It's working really hard trying to make this relationship succeed.
And maybe this person puts 20 percent on this one and this one. 20 percent is actually
a lot. Put some small amount on those two. So it means that the green person in this
relationship or this collaboration is sort of putting in just enough effort to push it forward,
not really working that hard but just putting in enough to make sure it keeps going.
And then these two friendships the green person is completely ignoring, letting them
take their own course and putting in no time at all. So this is a pretty common problem
that occurs to people. And, okay, so what will happen if let's say that this is a particular
link.
So this is a relationship between two people, orange and green. And what happens if
they both put in a lot of effort into this friendship.
Well, presumably what should happen is this relationship will succeed. If it's a
collaboration, publish lots of papers or something, or if it's a friendship, then they'll both
be happy and they'll both become good friends and get a lot out of this friendship. So
they'll both be happy because of it.
What happens if neither person puts in very much effort? Well, as unfortunately a lot of
us are familiar, if neither person works very hard at a relationship or collaboration, it will
just die.
Eventually, maybe not for a while, but usually it will fizzle out, nothing will happen if
nobody's trying, neither person will be very happy with it. Finally, what if one person
puts in a lot of effort, in this case the green person, and the other person doesn't put in
much at all, the answer depends, it depends on the kind of project or kind of relationship
or friendship that it is.
Sometimes, I mean, it's possible that actually they'll both still be quite happy because of
this. It's possible that this collaboration will succeed, even though this person is not
putting in much effort at all. They'll both reap rewards and they'll both be pretty happy
because of this. So it sort of depends.
We're going to try to model this as generally as possible or at least I'll try to be as
general as possible. So I'm going to say that -- okay, let's say that this edge is, let's call
it E.
So for every edge E in the graph we're going to have some reward function. So this
function F sub E. It can be different reward function for every edge. And it's going to
take an X and Y. So X and Y are the efforts that the two people are putting into this
project or into this relationship.
So this function takes X and Y and spits out how successful this friendship is. Spits out
how the success or the number of papers or whatever, how happy these people are with
this friendship.
So it's an arbitrary function of two variables. But we're not going to try to not assume too
much about it. We're going to say this function is nonnegative. And it's nondecreasing
in both variables.
If you put in more effort, you should only get a better relationship or better collaboration.
Not worse, right? I hope it's true.
>>: Symmetric X and Y.
>> Elliot Anshelevich: Doesn't have to be symmetric.
>>: You're assuming [inaudible].
>> Elliot Anshelevich: I was just about to get to that. This function doesn't do symmetric
X and Y. What happens is both efforts go into a function. The success of the project
comes out and both people get that benefit.
So I'm assuming exactly that the success of a project or benefit of a project is the same
for both participants. You can imagine, in fact maybe I'll talk about this a little bit at the
end of the talk, you can imagine if somehow there's a total amount of benefit and they're
dividing it amongst themselves or get to somehow unequally -- that's not what this talk is
going to be about. There's some extensions we can talk about but that's not what this
talk is about.
In this talk you think of it like they published some paper together and both people get
the credit. And if the paper is good, they both get the credit. Or you can think of it as
they're doing this project together and their boss sees the outcome of the project. The
boss sees you did well and you both get promoted or something. The boss doesn't
realize that this guy didn't do almost anything.
But that's how it is sometimes. So for this talk the assumption is they both get the same
benefit from the edge. Okay?
Okay. So this is sort of the introduction. This is a setup. Now let's actually talk about
what is the actual game. Define the game that I'm going to talk about.
So the game is as follows. So we're given an undirected graph. This is the social
network. We know exactly what it is for this kind of talk. So we're given the entire graph.
The players in the game are going to be just the nodes of the graph. So every node is a
player.
And so, for example, maybe here's a very simple graph with three players. Right? And
each player has a budget of effort. This is the budget they're going to allocate amongst
all the edges that participate in.
Now the budget can be different. So here we have budget of 5, 10 and 8. Why is the
budget for different players different? I think it's very natural. I can make them all the
same. Some nicer things happen if they're all the same, but I think it makes perfect
sense to be different I certainly know people have different levels of energy. People can
put in more effort than others in total.
If you talk about something like friendships, I certainly know people who spend more
time on their social lives than others. I think it's better to have their effort levels to be
very different and we're given the reward functions. In this case the reward functions are
very simple, 3 times X plus Y or 2 times X plus Y. In general they can be arbitrary
reward functions.
This is what we're given. And what are the strategies of the players? It's exactly what I
said. Each player is going to take their budget of effort and divide it amongst the
friendship it participates in.
So to put this in some notation, although I don't think we'll ever use this notation in this
talk, but just to make it formal. So each node is going to, okay, this node, for example,
takes us ten years of effort, say I'm going to put in four here and six here.
And, in fact, you're never going to really keep some effort to yourself. You're going to
use it all, right, because the functions are nondecreasing. So there's no reason to keep
it to yourself.
And to formally write or contribute some amount to each edge, total amount it
contributes at most to the budget. Okay. So is there a question?
>>: So there's no benefit to just relaxing?
>> Elliot Anshelevich: In this model, no. That's a very reasonable extension to have,
actually, which I'll try to mention at the end. Yes. But in this case, yes, it's a budget
constraint as opposed to an elastic constraint. Here it's just a total budget and you want
to spend it all.
Okay. So this is what happens. So people choose how to allocate their effort. And now
that they allocate their effort we know what their reward is going to be for every edge.
15 is just three times four plus one. So now we know what the rewards are going to be
for every edge.
And then what's going to be the utility of a person? Well, it's just going to be the total
reward they gather from their edges. So to put it another way, this person's going to get
15 plus 22 and to put it into formal notation, the utility node is going to be the sum over
all the edges sent to them of reward in those edges. So that's all that means.
Okay. So that's the game. So notice that the utility, the reward of a particular player
depends, okay, certainly depends on how they allocate their effort. It also depends on
how their neighbors allocate their effort. But it also sort of indirectly depends on
neighbors of neighbors and so on.
For example, you can imagine some nodes over here, this node over here decides to put
a lot of effort on the edge with this node. Because I put a lot of effort over here, this
node labeled eight might say I'm going to put a lot of effort over here too. Depends on
the reward function. Now that I put a lot of effort here means I'm subtracting the effort
here and now this person is worse off. Have this sort of behavior where what one
person does affects a neighbor, which affects a neighbor and propagates and various
dynamics and so on.
And another thing to know since we're looking at this picture is, well, the reward
functions are different, right? So here even though because the reward function here
has four and here has two, even though people are putting in less effort, you have 28
instead of 22. That certainly makes sense. Reward functions should be very different,
because what do reward functions represent?
One of the things they represent is compatibility of the pair of people. For example, if
two people are very compatible, then a little bit of effort might give them a lot of reward.
If they're not very compatible, a lot of effort won't give them very much reward here.
Makes sense the reward is high even though the effort is less. This is the game. Any
questions?
So that's the game. Now, what are we going to look at in this game? I'm going to look
at very stable solutions, think about their quality and how to find them where they exist
and things like that. So I'm going to look at the equilibria. We'll look at natural equilibria.
But I'm going to try to convince you that natural equilibria is not very interesting here.
What do we know about natural equilibria in this game. We know pure natural equilibria
exists. We know there's a potential function that matches best responses, which we
means we know people always converge to natural equilibria, by doing best responses.
We even know that the optimal solution, there's the one of maximal social welfare, is a
natural equilibrium. So we know a lot of stuff.
But they're not very interesting or they're not the right concept to use here. To give you
an example. Looking at this example, let's look at this edge. Imagine I have a thousand,
some big number, times X plus, X times Y.
So if that's the reward function here, then really what should happen is that both of these
people should put some effort into this edge, they'll get a huge reward.
But in the solution I drew here, this person is not going to move any effort into this edge,
right? Because for this person's point of view, if I move some effort into this edge, how
much reward am I going to get? Zero. Because the other person didn't.
>>: This shows there's some bad equilibrium.
>> Elliot Anshelevich: Shows there's some bad equilibrium.
>>: But the best.
>> Elliot Anshelevich: The best one is as good as social optimum.
So if you want to say, okay, imagine the single edge, right, where both people are
contributing nothing to this edge where the function is something like this. Then natural
equilibrium will contribute everything. That's natural equilibrium, social optimum. But to
contribute nothing is also equilibrium because you need both of them to do things
together.
>>: But this is something that often occurs.
>> Elliot Anshelevich: This is something that often occurs. Something that often occurs.
Some equilibria are really bad and some very good.
>>: Naturally will occur, I think, in practice, because ->> Elliot Anshelevich: That's the question.
>>: They'll be great collaborators without knowing each other.
>> Elliot Anshelevich: This is the way the links exist, right?
>>: Or they don't know each other.
>> Elliot Anshelevich: If they don't know each other, this link shouldn't exist. So feel
free to disagree. My view here is that I think for this kind of game, natural equilibria don't
quite make sense because in real life, whatever that means, I think these two people
who know each other, because there's a link between them, would talk to each other and
say if we both increase our effort a little bit, we'd both get a huge reward. So I think they
would do that.
Or to put it another way I'm going to look at coalitionnal equilibria. And if you like natural
equilibria in this, game I just told you all the results that I know. So quite a lot of things
are known about natural equilibria as well.
But here's what I'm going to do in this talk. I'm not going to look at natural equilibria. I'm
going to look at a more interesting concept for this game, and that's pairwise equilibrium.
What is pairwise equilibrium? Okay. So just to define some pretty usual terms.
Unilateral improving move means that a single player, a single person, can change their
effort to location and improve their utility. A bilateral improving move means that a pair
of players can both change their strategies, can both change their effort and locations at
the same time, and both of them strictly improve their utility.
Okay. So natural equilibrium is exactly something that has no unilateral improving
moves. No single [inaudible] but a pairwise equilibrium is something where, which has
no unilateral or bilateral improving move. Every pairwise equilibrium is also a natural
equilibrium but not vice versa.
And, in fact, I'm going to -- so most of the results that I'm going to tell you actually are
going to hold for strong equilibrium as well. A strong equilibrium is even the stronger
concept than pairwise equilibrium. In fact, that's why it's called strong. It's a really
strong concept. Means no -- equilibrium means no pair will deviate, no coalition, no set
of players including all the set of players will deviate such that every single person in the
set will improve.
>>: Core.
>> Elliot Anshelevich: What is a core? Core is you have a coalition and they pool their
money, transferrable utility. And they deviate such that the total utility goes up.
Strong equilibrium means that it's nontransferrable utility. So each person has to
improve. So strong equilibrium is the equivalence of the core for NTU games, for
nontransferrable utility games.
Okay? And the games I'm talking about are nontransfersable. It means that every
person has to improve. It's not somehow they get together and pool their money. I
guess it's not so easy to do with friendships or with credit from papers. You can't quite
say like if that -- oh, since you're better off and I'm worse off I wouldn't necessarily be
happy. You want that everybody improves.
Any more questions about transferrable utility and nontransferrable utility?
So in this talk I'm going to look at pairwise equilibria for this game where you partition
your effort amongst your neighbors. And I'm going to ask these kind of questions.
Okay. First of all, I'm going to ask: Do pairwise equilibria even exist here? So as I
mentioned pure natural equilibria always exists. That's easy to show. But a pairwise
equilibrium as you see doesn't always exist but I'll show you when it does.
I'll ask about questions about price of anarchy for this talk I will mean the relation of the
social welfare in the optimal solutions, maximum social welfare possible. And the worst
pairwise equilibrium. So that's what I mean by, that's what I mean by the price. Price
stability, I can also give balances for optimal solutions versus the best pairwise
equilibrium. You can ask me afterwards if you want.
>>: Compute, you're going to add up all these nontransferrable things.
>> Elliot Anshelevich: Right. Social welfare. Social welfare is the total utility of
everybody together. Which is in essentially the friction between total happiness and
happiness of particular people. Because social welfare might go higher but a particular
person might not like it which is why they would deviate. So yes. But I think I can see
your point also.
So price of anarchy, we'll look at price of computation, pairwise continuum, when it
exists, we can compute it. And we're going to look at basically can the players compute
it, meaning are there some nice reasonable dynamics where players just take turns
deviating or something which will actually converge to pairwise equilibrium, when they
exist, of course.
So these are the kinds of questions I'm going to look at in this talk. Let me mention very
quickly, let me mention related work. First of all, there's a lot. And many more.
Social networks in general and games and networks, all this is a huge area. There's lots
of stuff that's related. Ask me afterwards if you want to know more. But let me mention
a few of these things.
All right. So even things like stable matching. So utility maximizing version of stable
matching is actually an integral version of a very special case of our game.
So there's some relationship there which you'll actually see probably towards the end of
the talk.
Network creation games. There's a lot of network creation games of all kinds. Probably
the most relevant network creation game to this is the co-author model from 1996.
There's lots of differences in this model you don't spread effort among edges. You sort
of choose -- you construct edges. So you form edges there or not as opposed to having
some effort on the edge.
There they have a complete graph instead of some given graph. They have very
specific reward functions, notion of pairwise equilibrium. Lots of differences, but I think
this is the most relevant version.
And there they also look at pairwise equilibrium because they want people to be able to
form edges in pairs as opposed to sort of one person forming an edge and then later
saying okay I'll do it too.
Okay. So the game I mentioned, this contribution game, is technically atomic splittable
congestion games, for those who know congestion games. Congestion games are a a
very big field. Lots of results known.
Atomic splittable congestion games are sort of the most annoying ones to handle. And
there's lots of differences between what we're doing and various congestion game
results. For example, usually in congestion games people care about natural equilibria
and minimizing costs.
We're looking at pairwise equilibrium and maximizing utility versus maximizing utility
different when you look at price of anarchy, if you look at optimal solutions, it's the same.
>>: How are you splitting the atom here?
>> Elliot Anshelevich: So atomic splittable -- I didn't think of this term right. Atomic
splittable means that each person controls some big amount of flow, in this case effort.
And they can split it amongst different parts of a network.
>>: Why is it atomic?
>> Elliot Anshelevich: It's called atomic to differentiate from nonatomic. Nonatomic
means that each person controls an infinitesimal amount of flow, something that
approaches zero. Therefore nonatomic. I didn't make up the name, but yeah.
I mean, there's many other differences. Like, for example, usually the delayed function,
congestion games, correspond to reward functions. In this game are some function of X
plus Y. In our case they're arbitrary functions of two parameters.
Okay. I should mention in some public goods and contribution games because they had
the words contribution games in them. These are very different games. Although, I
think they're very interesting and there's lots of cool stuff in there but they're very
different. Here's how.
Those games, each person has one number as their strategy. So you just decide how
much you want to contribute to society as a whole. So you don't like say how much do I
want to contribute to each project or each relationship. We just figure out how much
they want to contribute to society. And that's your strategy.
Depending on how many people contribute, various nice things happen. But I think it's
actually very interesting area. Finally, last thing I'll mention is minimum effort
coordination games. So one of the reasons we started working on this topic is because
of minimum effort coordination games. These are in fact a bona fide special case of the
games I'm going to be talking about. Studied in various economic papers. Usually
experimental. So nothing was proven about them until I guess this work, or almost
nothing.
But some very interesting experiments were done. Some on actual people as opposed
to simulation, and I think they're pretty cool.
What are minimum effort coordination games? It essentially would be a game I
mentioned, but the reward functions are the form min of XY. It's the minimum of a
contribution that matters and that's all.
So you can see how they'd be much more structured because of that and many more.
So this is all I'm going to say about related work at least today.
So let me tell you then a little bit about the results that -- what do we know about these
contribution games, basically? So what I want to do is I want to talk about various
properties of these games, depending on the type of reward functions that are present.
So as you probably can see, what the reward functions are like can make a huge
difference. For example, if the reward functions are convex it kind of means I want to
put more of my effort on the same edge.
If they're concave you want to spread out your effort. So what I'm going to do I'm going
to classify various properties, specifically existence of pairwise equilibrium and price of
anarchy. Depending on the type of reward functions present in the network.
So, for example, suppose that all the reward functions are convex. They don't have to
be the same function. They can be different for every edge but suppose they're all
convex.
Okay. Then pairwise equilibrium always exists with a caveat. So pairwise equilibrium
always exists if both people have to contribute a little bit of effort to get non-zero reward
to every edge.
If this is not true, then it may not exist. In fact, it's NP hard to figure out given a graph if it
exists or not.
So that's existence. And I'll tell you a little bit about more about this result in a moment.
Let me just go into price of anarchy. Price of anarchy is always 2. Pairwise of equilibria
are factor of two with optimization. Except there's a caveat again which I'll tell you about
in a moment. But this is only true if a reward function all exhibit digit complements. I'll
tell you what this means on the next slide.
Annoying. I don't like these caveats, but that's the truth. So that's how it is.
Okay. What if all the reward functions are concave? No caveats here. So here it may
not exist. They may not be any pairwise equilibrium at all. But if it does exist, it's within
a factor of 2 of the optimum solution always.
The next thing is I like this special case. It's sort of a very nice structure special case.
This is when all the reward functions are X plus Y times a constant. And the constant
can be different for every edge. But other than that, all the reward functions are the
same.
>>: Why would you put all of it in this one edge?
>> Elliot Anshelevich: In this one? Yeah. In this one you would -- if the Cs are the
same, you would split it. This is actually a very easy case. Here's what happens, right?
So, first of all, price of anarchy is one, that's obvious.
Each person is always independent from the other person. Just put it all in the edges of
best C.
The only reason I'm bringing this up here is I think this is interesting. So pairwise
equilibrium may not exist given a graph. Natural equilibrium always exists, remember,
but pairwise may not.
I can convince you why, but maybe afterwards. Basic thing is the pair is deviating. So
even if they're getting a very high reward, they put it on the maximum CE, they might still
want to move their effort to each other and the reward will increase by twice this,
because I'm putting stuff here and they're putting stuff here. And the reward goes to
both.
That's my two-second version of why. If you didn't get that, that's okay. So it may not
exist. But figuring out if it exists or not is efficiently solvable. And I think it's kind of a
cute algorithm for getting out. That's why I put it up here.
Okay. Let's go on. Minimum effort games. Number of minimum effort games, what I
mean by this is all the reward functions are something like convex function of min of XY.
Then, okay, this is the last caveat, I promise. So here, again pairwise equilibrium exists,
price of anarchy is two if all the budgets are uniform, if they're all the same. It would be
NP hard otherwise. Notice this is not a special case of this. This 2 doesn't come from
here. And this doesn't come from here. So convex of min is not convex. So this is not a
special case of this.
Minimum effort concave is concave of min. Okay. Concave of min is concave. So this
just comes directly from here. So that part's easy. But now we have this. So the fact
that nonconcave of min gives us the very nice statement which I'll tell you about in a few
slides that now pairwise equilibrium always exists when it didn't before.
Maximum effort frankly is boring. I'll just include this for completeness. It's not very
interesting. It's not very hard. I think this part is more interesting. If you want to look at
approximate equilibrium, the optimal solution, the one of maximum social welfare, is two
approximate equilibrium, two approximate pairwise equilibrium. Meaning no pair can
switch and both increase their utility of more than a factor of two. That's it. So that
solution is nice.
So as a result, there's other things which I won't mention like price of anarchy bounds
are tight. There's price stability results. But I promised I would tell you something about
convergence. And I'll tell you very quickly the idea and try to give you the details if I
have time.
Okay. So here's what we know about convergence. Okay. Here there's two
checkmarks. What it means is things are really nice. In this case basically whatever
deviations or whatever dynamics you choose, they're going to converge to pairwise
equilibrium if it exists, of course, and they're going to do it really fast like N squared time.
Here there's only one checkmark. Things are not as nice. That means that things will
converge. But they might take a very long time they get to pairwise equilibrium. But
pairwise equilibrium always exists so they'll get there eventually.
Here things are sort of weird. The answer is kind of yes and no and I'll tell you about
that at the end, because I think it's sort of an interesting case.
So the remaining time, what I want to do then is give you an idea of why some of these
things are true.
I'm going to start with both this red box. So really there are two theorems here. The first
one is if the reward functions are convex for all edges, and if this thing holds, meaning
that for every edge both people have to contribute at least a little bit to get positive
reward, so I guess it should say FE of X0 equals FE of 0Y equals 0. If you don't
contribute anything, if one of the people doesn't contribute anything then there's no
reward. Pairwise equilibrium exists. Otherwise it may not. And in fact it's NP hard to
determine if it does. The second theorem is if all the functions are convex and exhibits
digit complement then the price of anarchy is two. What's digit complement? Let me tell
you the words first. It's going to mean this but why this.
Digit complement is a pretty standard term in economics. Probably most of you have
heard it before. Here's what it actually means. In words it means if I put in more effort,
then your incentive to put in more effort only goes up. The more I do the more you want
to do. The strategic complement.
Okay. So what does it mean for how do we write that for a function? It's just this. So
the second is nonnegative.
So the digit complement is opposite, substitute, where if I put in more effort your
incentive to put in more effort goes down.
So this is digit complement. This is some of the things -- do you have a question? So X
increases the derivative of Y.
>>: So why, some version of ->> Elliot Anshelevich: Yes. It's a version of -- this goes up, right?
>>: Similarity?
>> Elliot Anshelevich: Unfortunately, convex doesn't mean what you want it to mean.
Really if I had my choice, maybe this is what convex would mean, because it means like
what -- okay. So follows digit complement.
For example, these functions all exhibit digit complements. Any polynomial of positive
coefficients is going to exhibit digit complements. For all of these, this theorem will hold.
And for these two, they also satisfy this thing which means it will also pairwise
equilibrium will also exist. But not for these two, but in fact I can give -- if you look at a
triangle with reward function being 2 to the X plus Y pairwise equilibrium doesn't exist.
But when it does, price of anarchy is two. So let me give you an idea why these
theorems hold. First one is quite easy. Here's an algorithm for the first theorem which
creates a pairwise equilibrium. Always.
Take your edges. Sort them from highest weight to lowest weight, where by weight, I
just mean, okay, if both participants put all their effort on this edge, like all their budget
on this edge, then that's their reward of this edge.
So say that's the weight. Certain highest weight to lowest weight and now do a greedy
matching. That's it. That's a pairwise equilibrium. It's that easy. And it's not even hard
to see why it's a pairwise equilibrium.
So that's theorem one. Theorem two is going to use some of those ideas. So if you're
familiar, right, with -- if you look at the greedy maximum weight matching it's a 2
approximation to the actual maximum weight matching.
How many people here are aware of this? Hopefully a lot. So if you're familiar with how
that works, the idea is actually going to be the same. There's going to be a bunch of
details and stuff. But that's the basic idea. If you know that, this is going to seem quite
familiar.
So let's see why theorem two holds. So theorem two. The first thing to show, just allow
us to use this greedy matching argument, is that there always exists an integral optimum
solution, meaning that so by integral here I mean that every player spends all their
budget on a single edge. For example, this red solution is integral optimal solution.
So we can just conserve this solution instead of other optimal solutions. This is because
of convexity and because of strategic complements because you're going to have more
than one subject to put all of this stuff on one edge.
>>: You mean social welfare?
>> Elliot Anshelevich: In this talk by optimal I always mean the objective as well. In this
talk I will always mean highest social welfare, yes. So highest sum of utilities.
So this is the highest social optimal solution. Now let's look at pairwise equilibrium. So
there's no similar result for pairwise equilibrium. Pairwise equilibrium might look like this.
They might not put all their effort in a single edge. Let's take this pairwise equilibrium,
compare it to the optimal solution. We better use the fact it's a pairwise equilibrium.
So, for example, let's look at this edge. Conserve deviation where both of the players
take all of their effort and put it on this edge. So from the green solution, consider where
we take all the effort and put it on this edge. Since the pairwise equilibrium, it means
that one of them doesn't benefit. Otherwise it wouldn't be a pairwise equilibrium. So
which one doesn't benefit? Let's say it's this one. Let's say it's the one on the left.
Okay. So what does it mean they don't benefit? It means that their utility, their utility in
the green solution, which I'm going to draw like this, must be bigger than their reward
they would get if both of these people put all their effort in this edge. In other words, it's
the reward of the edge in the optimal solution.
So this green circle, the utility of a green person, the utility of this person in the green
solution is bigger than the reward of the optimal solution. Now we can do this for every
edge of opt. We're never going to have the same node being pointed to by two different
areas that's because of this claim, because pointed to by an arrow it means I'm spending
all my budget on that arrow and I'm only spending all my budget on one edge.
Because of this claim each node is pointed to at by most one arrow which means we
sum this up, what do we get on the left we get the sum of all these green circles. That's
the total utility of the pairwise equilibrium. Because we sum up all utilities. On the right
we get the sum of these red edges. This is the total reward of all the edges in the
optimal solution which is exactly opt divided by two because the reward of an edge goes
to exactly two people.
That's it. It's pretty easy. I'm leaving out some details, and this, but that's pretty much it.
Okay. So that's why this result holds. That's why the convex stuff holds. In the
remaining time what I want to do is tell you about this right here. Because I think it's
pretty equal. Basically if you have concave functions, they may not be pairwise
equilibria. But once you look at minimum concave, we'll always have pairwise
equilibrium. I'll show you why. Just to remind you what an minimum effort games, it
means all functions are of this form. There's some function H sub E could be different
for every edge of min of XY. H is a single variable function, because it takes min of XY.
So if -- and concave minimum effort games means all these H sub ER concave. If we
have general concave functions, like square root of XY, for example, is a general
concave function, there may not be pairwise equilibrium. In fact, this very simple game
there's no pairwise equilibrium. You can verify with yourselves if you want. But once
you look at minimum effort concave, for example, functions like square root of min XY,
there's always a pairwise equilibrium. In fact, for this game there's a very simple, just
spread everything out half, half. Every person puts half, half effort. The actual theorem
is the following: Pairwise equilibrium always exists if the functions, if there's a minimum
effort games and functions are concave and they're continuous and they're piece wise
differentiable. I don't think assuming continuity and piece wise differentability is too
much to ask.
So now I'm going to show you how to construct the pairwise equilibrium in this case. I'm
going to do basically, well, yes, I mention if we can compute it to arbitrary position, not
just prove existence, if the function is strictly concave, it's unique, price of anarchy is two
if it's proved. What is it about minimum effort games that really helps us?
The answer it's obvious. It's the min. Like specifically what is it about the min? And the
minimum effort game where the functions are something of min, and a pairwise
equilibrium people are never going to have unequal contributions, right? Because
suppose you're contributing more than somebody else, but your actual reward depends
only on the min, you might as well contribute the same amount as them.
And that's the obvious observation but it's going to turn out to be enough.
>>: Question.
>> Elliot Anshelevich: Yes.
>>: Differentiability, it's the assumption that certainly it's not needed.
>> Elliot Anshelevich: It's needed in my proof.
>>: But if you have solutions, stability question, if you have solved it for some function H
and HI and they converge to another one, H, isn't that implied that they converge?
>> Elliot Anshelevich: Me, if they're increasing, converging to a particular function? I
don't know.
>>: Concave function has only -- has only so many points of ->> Elliot Anshelevich: I used it, but I haven't, frankly, truly tried to get rid of that
assumption because I figured it was good enough, but I can try to think of ->>: Additional asking [indiscernible].
>> Elliot Anshelevich: It may not be too hard to get rid of. It might be worth thinking
about, but not at this second. Okay. So I'm just going to give you basically -- I'm going
to show you how the algorithm compute a pairwise equilibrium works on this and that
should also prove to you why pairwise equilibrium also -- okay.
So what do I mean by this? Now that I mentioned that the contributions of both sides
are always going to be the same, because it's a min, really what I mean by here this is
not three square root X, it's three square root of XY. But I'm going to write it three
square root X for compactness. But it's min three square root of min XY. Here's the
graph, budget. These are the budgets and the reward functions. What do we do? First
thing we do is for every node V, let's compute the best, its best response. So best
strategy could do. If it were able to control all the other nodes. So this is the best you
could possibly do ever.
There's no way you could get better utility than this. For example, for this node, this is
the best thing you could possibly do. Is it would put -- budget is one. This adds up to
one. But 414 from this edge and make this node match its offer. And then that's, then it
will get improved square root of 414. This is the best it could do. Ever
How do we figure out what this is? We can just, do it by solving convex program. This
is where we're maximizing the sum of concave functions due to a linear budget
constraint. Divide using convex program. But there's sort of a better more intuitive way
to think about what these numbers are.
And that is that well on all these edges, right, the derivative should be equal now. Of
course. Because if one derivative one of the edges is higher than the other, you would
just move some effort from the lower derivative edge to the high derivative edge and
improve your utility.
So they must be all the same. One way to compute it sort of intuitively, you look at the
concave functions. You start putting effort on the ones with highest derivative, keep
doing it until you come up there being the same every way. There's a few annoyances.
For example, here we're assuming this person will match the contribution. We can only
do that if they have enough budget. If you don't have enough budget, then they can't
match the contribution even if we make them. Just doesn't have enough budget. Those
are details.
Okay. So for every -- so for this person there's derivative value associated with them.
The derivative that it hasn't always three edges. The derivative is for every person. We
computed the best edges for every node on the graph and each node has a derivative
value now. You can believe me these are the best.
What do we do next? We pick the node with the highest derivative value. In this case
it's this one. And we fixed its strategy. This is going to be its strategy. This is going to
be its effort contribution in the final solution window.
And that works just fine. The problem here arises what if there's several nodes with the
highest derivative value and then you can't just pick an arbitrary one. You have to be
extremely careful about which one you pick. So this is what I mean by there's a crucial
tie-breaking rule. If several nodes with highest derivative value, you better pick the right
one and there's various things in there. But if there's only one node of highest derivative
value, you pick that one and you're fine.
So I won't really talk about this. I'll leave this out. But in this case this is the only one of
highest derivative value. So we fix its contributions. Okay. Now we do the same thing
again. We compute the best strategy for every nonfixed, for every gray node, if it were
able to control all the other gray nodes. So this one is fixed. It's done.
But we get the best strategy for each node if we're able to control all the nonfixed nodes.
For example, so here's what it becomes. These are the new strategies. So there is one
nice thing about this, right? Here, this person said I want to contribute, put one effort on
here and you're supposed to put one also.
And that's what happened. Well, if only that was so easy, here this is what happened,
but it's not true if we just compute the best strategy for this person, if it were able to
control all this, it wouldn't necessarily put one here. It might do something quite different.
But there's this lemma, which really just comes from the fact we're doing this highest
derivative and because we are doing the correct tie-breaking, which is it's true, not every
best response of this person was the control of all of these people actually matches the
contributions of the fixed nodes, but there always exists at least one. I don't think it's
surprising. This is, especially once you think about the highest derivative stuff. And so
we picked that one. So we can always pick contributions of these people which will
match the contributions that have already been fixed.
So that's what we do. And now we continue, fix the strategy of node of highest
derivative. In this case it's this one. So fix the strategy and we go on. So now we do
this again. Best strategy of this person if we control all the gray people, do this for
everyone, we get three strategies and we keep fixing them one by one and eventually
we get this solution.
And the lemma is not very easy to prove. The lemma is sort of where you have to use a
lot of stuff. But once you get this, I claim this pairwise equilibrium and this is actually
very easy to see. Here's why it's a pairwise equilibrium. So suppose it weren't. Suppose
maybe this pair of nodes would rather deviate. So it could deviate together and get
better utility, both of them.
Then I claim the node that was fixed first would not want to deviate. In this case this is
the node that was fixed first. Why wouldn't it want to deviate? It's very simple why
would it want to deviate together with this other person, why would you deviate as a pair
in general? Because by yourself you can't improve your utility but if you both do
something different, then my utility will actually go up.
But because of this lemma and because we formed this solution, this node is actually
getting the best utility it could possibly get if it could control these four people. So it's
getting the best utility if we can control all of these people. Which is certainly going to be
better than the best utility it could get if it controls these two people so it's already doing
really well. It's never going to want to deviate. That's why it's a pairwise equilibrium.
Okay. And that's it. That's the gist of why pairwise equilibrium always exists, and as I
said we can then use the results for general concave function to show what the price of
anarchy is at most two. So there's always a nice pairwise equilibrium. In fact, all
pairwise equilibria are within a factor of two.
So that's what I'm going to talk about here, and I guess I promised at some point that I
would talk about this question mark. So let me talk about this question mark and then I'll
mention some future work and stuff. As usual.
So here's what this question mark means. I already told Nikhil about this. So this
question mark means the following. In minimum effort convex games, even when a
pairwise equilibrium exists, like when these three stars hold, for example, if you do best
response, there may not be any path from starting point by doing best responses to
pairwise equilibrium. Let me define what I'm saying. If you look at the state graph, so
each node in the graph is a state, and a transition in the graph is a pair deviating, and
both of them improving their utility, so the path in this is just the path in best response
dynamics. So for here there are examples where there's a nice pairwise equilibrium
sitting right here but you start right here and there's no path to get from here to here by
using this response dynamics. It will just cycle. But not only will it cycle there's no path
to get from here to here so best response dynamics are not going to converge on the
fact you can't make them converge if you tell them how to deviate.
But here's why I think it's an interesting question. Why it's a question mark and not just a
know. If instead of doing best response dynamics, if instead of doing deviations where
both people improve strictly you allow deviations where one person actually gets a better
utility but the other person's utility stays the same, then at least so far I have no
examples where this doesn't converge.
I don't have a proof for anything. But so far every example I come up with, it always
seems to converge. Somehow it says for this kind of game, being sort of a little bit
altruistic, so saying I'm willing to switch as long as I don't lose anything but also if I don't
gain anything. Has the same difference causes convergence to happen. Which is
interesting. I don't know why yet because I have no proof. So that's what this question
mark is.
>>: [inaudible] this response on the [inaudible].
>> Elliot Anshelevich: No, it's a super set of the best response. More transitions. So
before we only had transitions where both people had to strictly improve there's no path
from here to here. Now I'm also adding transitions where only one of the people can
improve and the other staying the same. So I'm adding more transition toss the graph
right? There's more deviations possible now.
So there's more possibility for convergence. But you're right, there's still paths which will
cycle. But that's only if people choose those. Seems like there's always a path to go
from here to pairwise equilibrium.
So to say it one more time, it's not true that every path will now lead to a pairwise
equilibrium. There are cycles in the dynamics. But it is true that there seems to always
be a path to pairwise equilibrium when there wasn't before.
>>: A way to identify it?
>> Elliot Anshelevich: If I did I would have a proof. I don't know. So it's quite possible
doing some sort of random best response dynamics where you don't take -- in the cycle
you don't do the same one again, maybe that will converge, I don't know.
>>: Missing something here. A cycle ->> Elliot Anshelevich: So ->>: Utilities are going up.
>> Elliot Anshelevich: Utilities are going up for this pair but not for other people.
>>: Right.
>> Elliot Anshelevich: If it deviate ->>: Okay.
>> Elliot Anshelevich: Yeah. All right. So then let me just mention, last slide, let me
mention some extensions that I think are interesting open problems. So, first of all,
certainly I talked about convex and concave, I think it will be interesting to look at
specific class of reward functions especially ones motivated by some specific application
and see what behavior is like then.
Also I talked about these best response dynamics, but there are many kinds of dynamics
which I haven't really looked at. Imitation dynamics and all kinds of random dynamics
and no regret dynamics and all kinds of stuff.
So one thing that I have looked at is general contribution games. What I mean by this is
okay so far this was a graph. Each of these relationships is a pairwise thing. What if
instead of a graph it was a hypergraph? So, for example, what if you have projects
which with many participants, K participants get together and they all put in some effort
and the reward function takes in K inputs and spits out how good the project is.
So in other words what if it's exactly the same game but with a hypergraph. In that case
a lot of the same things actually still work out. So all -- forget about maximum effort.
Maximum effort doesn't work out. Maximum effort is kind of stupid anyway.
So all the existence results work out in the same way. The price of anarchy changes
unsurprisingly. So here and here it becomes K, where K is the sizable largest project.
Not surprisingly.
Here and here, this actually came from here, instead of two you don't even get K. You
get something like, think you might be able to get K squared but you get something
really annoying unless you assume strategic substitutes which is the concave, which sort
of means concave or supposed to mean concave and you get two again. So you don't
get K you actually get two but you have to make some extra assumptions.
So that's general contribution games. And here's a few things which they seem maybe
seem like minor differences but they actually cause pretty big differences to the results.
One thing is I want to look at the capacities in the maximum contribution to an edge.
You shouldn't be able to put all your budget in an edge. There's some maximum amount
of effort you can contribute to an edge.
That changes things. But you can still prove versions of results I mentioned. The really
interesting thing, which is the main thing I'm actually currently working on in this area is
cost functions for contributing functions. This is what I mean by this. It's what you asked
earlier. And that is instead of a budget constraint, sort of a hard budget constraint, what
if you have some cost function for how much effort you put in? So the more effort you
put in in total the more cost you experience.
So now there's a benefit for you for relaxing. Because now you don't have to pay as
much cost. And that changes things quite a bit. Sometimes. I mean in minimum effort
games, doesn't change things too much. If you assume everybody has the same cost
function then you could still prove some result, but in general games, totally different.
>>: Think of it as a loop.
>> Elliot Anshelevich: That's what you do. Technically if you think about it as a loop it's
slightly different. But ->>: Whatever constraints you imposed --
>> Elliot Anshelevich: Has to be here, too. But, yeah, that's what it means. It means
allowing loop edges. And minimum effort it's not that bad because you're both going to
be -- again you're both going to be putting the same amount and so it's going to be just a
function of this and basically the amount you're putting on the edge is going to be the
same for both of you, so if you have the same function then your difference because of
its edge is going to be the same you can use that symmetry but if you have different
functions on every edge, like different cost function on every edge or if you have over
here, that's what I'm working on. So that's pretty much open.
And I think it's very interesting. Finally, sort of the big thing that really changes
everything is what if instead of both people getting the same reward, what if somehow
you divide the reward or maybe the reward is divided proportionately to the effort you put
in or things like that, that we're only starting to work on and then there's lots of different
cost sharing schemes you can think of and I think it's a pretty cool question.
Anyway, if you have any more questions, please ask. I'll stop here.
[applause]