18256
>> Yuval Peres: So we're very happy to have Christos Papadimitriou with us today. And often
when I've given talks I wanted to give a white board talk, but in the last minute I had my laptop
with me and I just fell for the temptation and just plugged it in. So Christos has found the method
to avoid such temptation. And we're very glad that he'll tell us about computing Nash equilibria.
So the plot thickens.
>> Christos Papadimitriou: The way I found was I forgot my laptop at my apartment this morning.
So I know many of you have heard me speak before. And you have been wondering, can't you
do it without PowerPoint? And today you'll find out. [laughter] so today I'll find out, too, okay.
All right. The beginning of all this is you see what happened is that in 1928 John von Neumann
proved the min/max theorem. And this was sort of the beginning of the game theory. And 20
years later with Morgenstern, they published games on economic behavior. And that was sort of
the official sort of star of game theory as a discipline.
But there was a huge gap. I mean, if you read that book today, which it reads fantastically well,
I'll come back to the book for a different reason, there's a huge intellectual gap. And that's sort of
that we didn't know the stable concept like the solution concept that we know for zero sum games
did not exist for any other games. And this gap was closed four years later by John Nash, and
that's the theorem we're starting; that every game has a Nash equilibrium. He didn't call it Nash
equilibrium but...
Okay. And these completely changed the landscape in game theory. And all of economics,
actually. It's very interesting. So perhaps the most consequential results in mathematics
economics to date came three years after that when [indiscernible] proved that every market has
price equilibrium. That's a different problem I'm not going to define.
Incidentally, I'm doing a complete faux paus. So I'm omitting the two-minute definition of what's a
Nash equilibrium. But let me just tell you that an equilibrium basically means what it should
mean, that you have players each going their own way and they choose a strategy. In this case it
happens to be a probabilistic strategy, a distribution over all strategies. And it's a equilibrium if
none of them can improve upon the current situation. So it's assuming that everybody else will
stay the same.
So I was telling you that the price equilibrium is completely different. It's basically in a market
where you have market forces, consumers, sellers, producers. You have a price equilibrium if
there is a set of prices where they're announced the market completely clears. So supply equals
demand.
The amazing thing is under reasonable assumptions this always exists, and something that is
perhaps not very well widely known is [indiscernible], the guys who proved it, got a prize for this,
were inspired by Nash, methodologically, because the proofs are similar, they both exploit six
point theorems. So I'll show you one of these proofs later.
And in 1999, five years after Nash got the Noble Prize for this, Roger Meyerson wrote a very nice
paper in general economic literature where he argues very convincingly that this theorem lies at
the basis of modern economic thought.
In the following sense: Also because it inspired the price equilibrium theorem, but also because it
attacked it as a prototype of what results can be assumed as a corollary of agent rationality. And
so meanwhile, so for all this half century, there is the one question had stayed open. How can we
compute a Nash equilibrium?
And it became clear that the phenomenon, sort of min/max price can be computed efficiently by
linear programming. But Nash equilibrium seemed a little more difficult. It always seemed a little
more difficult than that. There were many attempts. Many people have tried to find algorithms for
that.
Some algorithms were found like Lemkey Houser [phonetic] algorithm, for example. People also
tried to find algorithms for price equilibria, which was a harder problem. A lot of work in
economics on this. And it's an interesting question why are we interested in algorithms for Nash
equilibrium, since this is just a statement about games and economies that they have an
equilibrium. Why do we need to find it? Let me know. And one thing to think about it is as
follows: That why is Nash equilibrium such an influential concept. Why does Meyerson tells us it
lies at the basis of modern economic thought?
And the reason is that it's a prediction. It sort of says that when you have a bunch of agents, a
bunch of people and they're faced with situations that involves a conflict and they are both
strategic and rational, then they're going to behave this way. And the point is that the prediction,
if it's not sort of supported by an algorithm, is a poor prediction. So that as Kamel has put it very
graphically if your laptop can't find it, then neither can the market.
So if you tell me that your market is going to, is the source of stability, then I want to see your
market. Okay?
All right. So this, I think, is a very valid question. And it hadn't sort of seriously occurred to
people that there is a fundamental difficulty here. But there is. And there's a theorem, I guess
2005 now, by Costilakis [phonetic], Paul Berger and myself that says that finding a Nash
equilibrium is intractable and the technical term is PPA decomplete.
I know what the people who are hearing this for the first time are thinking believe it has nothing to
do with my last name, even though I defined this class. And it had not occurred to me that it's -and I know, I know, and until a reviewer of the journal paper pointed it out.
And the argument I gave to the editor was that if authors were not allowed to use initials of their
names, that's discrimination against me. [laughter].
And that's about it. Okay. So let me explain to you what this means so you know. Imagine the
following. Imagine that you are given a directed graph. And somebody points out to you that one
of the nodes is unbalanced. Okay? Then it's sort of very simple mathematics to realize there
must be some way, there must be another unbalanced node. Unbalanced, I mean the integer is
not equal to the degree. Make sense?
Of course, if you have enough time and the graph is not big, so you can look after and find it,
that's not a big problem. So let me make it a little harder for you. Imagine I don't give you this
graph, but this graph is a huge graph, exponential graph. And basically what I give you is a
computer program that you give it sort of like the names of two nodes and tells you if there's an
edge from one to the other.
And then I convinced you that node 0, let's say, is unbalanced. And then I tell you, well, then it's
a finite graph. There must be another unbalanced node, find it. Another problem seems a little
harder, right? In fact, it looks so hard that you might be tempted to take pencil and paper and
start trying to prove it's impossible, that we need exponential, we need the following to search all
possible nodes.
And this temptation lasts until you realize that it's an NP, because you can always guess
assuming that the degree of the nodes are very small. So you see my point? That it's an NP, so
it cannot be. But still it's -- that's what PPDA means. It's polynomial part of the argument for
directed graphs. That's the unfortunate name that led to this.
So this means -- this means that in some sense for Nash equilibria, it's a weaker notion than NP
completeness, we don't know exactly how much weaker. But in some sense we face -- we find
ourselves with a Nash equilibrium with the same position that we were in the early '70s vis-à-vis
the stability [indiscernible]. There's some evidence of intractability. What do we do?
And people have done the obvious things for -- they have looked at special cases. They have
looked at approximation algorithms and there have been some successes. There have been
some failures. There have been some negative results. So there's now the last five years this
has been an active field of research. And what I want to tell you about is not about these works,
about approximation, about special cases and all that. But I want to tell you about something
else, which is basically twists of the problem. Let's twist the problem. And look at other problems
that are relatives of the Nash equilibrium. In fact, other problems that for a long time had thought
to be easier.
And see what happens. And I'm going to talk about three things. Very briefly I'm going to talk
about repeated games. And then I'm going to talk about two other things, slightly more detail
about equilibrium selection and about risk. Okay. And the title of my talk gives away -- the plot
thickens. It means that if you look at these which sort of traditional game theoretic thought was
implying that these are sort of easy to tread on, you get more catastrophic results. So that's the
idea.
So let me very briefly tell you about repeated games. So I have defined a game, but I'll show you
sort of a way of representing games. So that's pay of player one. This is the pay of player two.
And imagine that you have these diamonds, these four points. That's a game. I'm not going to -anybody going to guess what's the game? That's prisoner's dilemma.
Because this -- if we both defect, we get this. If we both collaborate we get this. If one
collaborates and defects, the two players, everybody understands the geometry of the thing?
Now, suppose that we play -- so now I'm talking about repeated games now. Suppose that we
decide to play prisoner's dilemma a million times. Okay. So it's two players who decide to play
this. Okay. So here is fact one that there is this -- first of all we cannot hope to go out of this
diamond. This is sort of our -- but the point is that within this diamond there is this individually
rational division which says that anything here is good. Of course I like this better than this. So
we know. But anything here is sort of reasonable, in the sense that it's above this point.
This point is something called a threat point, which is sort of the lowest values they can guarantee
for myself. The bottom value. You can say listen. Worst comes to worst I'll play defect and I can
guarantee for myself. The other guy says the same thing. So anything above that is gravy. So
it's fine. That's what individual rational means.
And then in 1970 came the famous game here is, came up with Aman and Shannon came up
with something called the -- they never published it and they chose to call it the Folk Theorem.
And the Folk Theorem says is following. That suppose -- forget now the prisoner's dilemma.
Imagine we play a complicated game with many players. Right?
Then again you have something called the threat point, which is sort of you can find. You can
define easily. And then what you do is you select any point here. And the idea is that by
repeated play, there is a Nash equilibrium that analyzes this point. Okay. And the proof of the
folk theorem goes as follows: You see what I'm saying? There is a behavior, there is a way that
we can agree to play and it's an equilibrium play. In other words, it's rational for all of us to play
that realizes this kind of payoff. In fact, can be as close to the collaborative pay off as you want.
It's a great result. You see from prisoner's dilemma if you play repeatedly you can really do very
well. But here is how the general idea goes, all right? How the proof goes.
Suppose that this is rational. If it's not rational, you know, there is a rational point next to it. So
find the rational point. And suppose that this is the rational point is something like ->>: Rational meaning.
>> Christos Papadimitriou: Sorry. Rational in the ancient Greek sense [laughter]. Right? So the
point is this is a rational point. You can express it as a complex combination of these corners.
So it's a complex combination. All right. So suppose it is half this, one-third this, one-sixth this
and zero this.
It's something like that. Then what we say, what the people agree is the following: We're going
to go through this procedure. We're going to play three times this, twice this, once this and then
three times this, twice this, once this, and so on. So the prescribed amount of time. And we do
this forever. And so I'm going to realize this point, okay, which is -- okay. And the point is, okay,
that we are going to -- this is going to be a Nash equilibrium. And it's going to be a Nash
equilibrium because eventually the threat -- if anybody does anything different, okay, even to my
favor, even to anybody's, I don't care, if anybody sort of defects from this agreement, then
everybody else are going to gang up to him and lead him to the threat point. You see what I'm
saying?
So are going to play against you min/max. All right. So this is sort of, this is the proof of the -- so
the conventional wisdom, because of the Folk Theorem was, okay, we don't quite know how to
answer this. But the point is in repeated games, which in some sense is a variable situation,
there is a huge wealth of equilibria, that are in some sense as good as you want.
>>: Iman and Shopper [phonetic]?
>> Christos Papadimitriou: Iman and Shopper, right. So then sort of there is a bunch of us -and, you know, Jennifer and Christian were part of it, but also Nicole here. So we're part of the
troop that actually proved this, we proved the theorem and Adam Kalai and [indiscernible]. Am I
forgetting somebody? Nicole is not here. All right.
So the theorem is the following: That that's okay for two players. So if you have just two players,
it's fine. And the Folk Theorem works. Well, for at least three players it breaks down sort of
annoyingly, sort of catastrophically. You can't realizing any point in the individual rational region
is PPD complete.
So what is the goal of the Folk Theorem, in other words to realize at some point in the IR region is
sort of as hard as playing -- you know, what this says is that if you have three or more players,
and you want to find the protocol to play arbitrary games so that you are always in the individual
rational region, listen, why don't you solve the one short game and play repeatedly, play always
the same. It's not easier.
And the particular method that the Folk Theorem suggests is even worse, because finding the
threat point is NP hard, okay, for three or more players. But for two players, and people have
observed that, you can actually implement the -- which is an interesting -- an interesting ->>: These are exponentially sized games? Are you dealing with ->> Christos Papadimitriou: No, no, we're dealing with sort of there are three-person games with
N strategies, let's say, who are sort of ->>: Are there points of the strategies or triple strategies for each of them?
>> Christos Papadimitriou: Basically -- I guess what it's saying is that -- no, no. I mean, it's a
three-player game.
>>: So possible ->> Christos Papadimitriou: Yes, of course.
>>: Otherwise, it would be hard to find NP is polynomial.
>> Christos Papadimitriou: To find the threat point. The threat point is going to be a mixed
strategy. Right. So basically you can do it for two, because for two the threat point is min/max
theorem. It's a two-player game. But for three, if you tried to solve it for three or more. So you
get into trouble. Okay. Good. That was helpful.
So what I'm saying is that this is sort of, I have three results that I'm going to tell you about sort of
in the same, around the same, in the same vein.
So that even if you change the proofs, if you change them in directions that were thought to be
easier, you get sort of into a complex territory. So let me briefly tell you about equilibrium
selection. So as you know, so there's this new area called algorithmic game theory. Sort of it's
the synthesis of two areas. The synthesis of two areas is not always easy, because areas have
their own mindset and so on.
So if you ask me what is the difficulties synthesizing computer science theory and game theory,
advanced two things, that, first of all, the Nash equilibrium, the basic concept in game theory is
something declarative, not algorithmic. In fact, in a true way, in a real way.
The second one is that it's not even deterministic. So the second piece that can't find equilibrium
and there are too many of them. So both of these point to a way that playing games is not an
algorithmic -- is not an algorithmic -- its nature is not a task of algorithmic nature, because it's not
deterministic and in fact it's not easy.
And so this, the multiplicity of equilibria, has been a huge problem of game theory. And people
have been working on it, so have been thinking about it. And by far the most influential proposal
is due to Harsanyi and Selten, who, in 1992, they published a monograph called Harsanyi Selten
[indiscernible] shared with Nash the Noble Prize in 1994.
They published the monograph called General Theory of Equilibrium Selection. And basically it
contained a few ideas about how to overcome this problem, because by that time it was, of
course, broadly understood to be a mine field for game theory.
And in that book sort of perhaps the most sophisticated part of the idea is the following: Suppose
that we want to sort a game. Let's call it G-1. Okay. Now, Harsanyi and Selten are economists.
They assume that people have bayesian priors about the world.
And so basically you see this game and each one of these players, there are many players
playing this game, they assume that they know what everybody else will play. So they sort of
know that player 1, he's just going to equally sort of modernize between the first two strategies.
Player 2 is going to play this. So they have a prior that sort of tells them how these people are
going to play. They have a distribution, prior distribution.
Once you have a prior about what everybody else will play, okay, sort of -- then it's not a problem
at all. You know exactly what to do. You just -- I'm not going to solve the problem for you, but it's
real easy. You calculate what is the expected outcome of each of your strategies and sort of then
you pick this one.
So the point is that there is another game. G-0, which, in the beginning, when you know nothing
about what people, how people are going to play, you are convinced that this is the game you're
playing. And this game is a very easy game. In general it's going to have a unique Nash
equilibrium. And with high probability, with probability 1 in the space of the game it's going to
have a unique Nash equilibrium. But sort of life is going to betray you. People are not going to
play the way you expect.
So slowly you are going to learn. And in any event, you want to know how to play this game.
You don't care about that game. And Harsanyi Selten say the following: Imagine now that you
have -- okay, that you have -- what you do is you sort of at time -- you play some kind of
interpolation of the two games. That's an easy game. That's a real game. Okay. That's the
game you play at time T.
Okay. And what they prove is the following: So it's a sort of very interesting result sort of
probability. That the space of all games, sort of there is a path. This has a unique equilibrium.
This has a unique equilibrium, and there is a path of unique equilibria you follow. And with high
probability this path is going to end up with a unique -- it's going to eventually suggest a unique
equilibrium in the resulting game.
So this game could have 1700 equilibria. But only one of them will be selected if you sort of
follow this reason.
>>: Continuous, not a repeated game, just imagining the sequence?
>> Christos Papadimitriou: Yes, you imagine the sequence of games. Right. Right. And then in
the longest chapter of this book, sort of they explain why this is a reasonable thing to think that
the equilibria are selected this way. In fact, so we know that there is something called Browder's
theorem, which Browder's fixed point theorem. If you are lucky enough to prove nice fixed point
theorem, it's really terrible luck to have your name be Browder, so close to Brower. So the six
point theorem says that if you have sort of two functions from domain to itself, okay, to continue
the construction of the domain itself, then there is sort of, there is a whole continuum of functions,
so that each one of them is the fixed point of the interval and interpolated function. So that's
Browders six point theorem. So basically there's a lot of work, there's been something called the
homeotopic method, which many of you have probably heard and many of you have used.
The homeotopic method is very much used for fix point problems, and also used for control
problems. It's the following, you want to solve the problem, looks hard but you know how to solve
this problem. So why don't you solve this problem and then continuously so until you transform it
here.
So the theorem we prove that is joint work with Goldberg, with David Goldberg of the University of
Liverpool, is that finding the Harsanyi Selten Nash equilibrium. For example, Nash equilibrium
was going to be selected by the equilibrium selection procedure of Harsanyi-Selten is prepare for
a shock. Is base complete. Base complete means unimaginably. That means harder than
complete.
>>: Finding G-0, that's the first step, easy?
>> Christos Papadimitriou: Finding the first step is easy, yes. But the point is that you have to go
through an exponential number of [indiscernible], not only that, but exponentially times you may
have to come back and find other fixed points in the same game.
And so I can -- this is a complicated proof. But I can give you -- it's very -- it's the very sort of
basic essence of it. And it's the following: You remember the original problem, right, the Papad
problem; that said that I'm giving you an unbalanced node. I want you to find another unbalanced
node somewhere here, okay? All right?
And that's Papad complete.
>>: Doesn't have to be reachable, your ->> Christos Papadimitriou: Exactly. That's the point I'm making, okay. So imagine that so that's
Papad complete. That's actually the basic problem in this class Papad. But now imagine that I
change it. That I change slightly the problem. And I'm telling you, all right, find me another
unbalanced node but please find me another unbalanced node in the corrected component.
And this is always possible. Okay? So it's equally possible. So the problem becomes space
complete now. That's sort of -- that's a very strange phenomenon and complex theory that we
discovered sort of while working on this problem, that if you ask a similar, very benign, if you ask
a similar very benign -- and intuitively the reason is that connectivity in the succeeding graph is
space complete.
Okay. So you are combining -- so the second ingredient is nasty. So we know the connectivity.
Okay. So sort of -- and we know it's not very hard to prove it. Except you have to do some
old-fashioned automaton theory that I almost forgot about. You have to make touring machines
that are invincible, which is a trip.
>>: I hear there's a book that tells you how to do that.
>> Christos Papadimitriou: Yes, you're right. Yes, of course. If I forgot, you guys -- yeah, yeah.
You're well versed in this. Yes, of course.
So that's sort of a -- now let's come to this. All right. So let me give you another game. So this
is -- that's a famous game, called matching pennies. You've seen it before.
Okay. It sort of forces you to randomize between, okay, basically we flip a coin. Sorry. Both flip
coins -- sorry. We both choose heads or tails. Sorry. And if we both choose the same thing, I
win. If we choose different things you win. So it's sort of matching pennies.
And this is one of the original great games. So there is nothing you can do except randomize.
So you have to flip -- instead of choosing heads or tails, you have to pick another way. There's
no other way. That's Nash equilibrium. Let me make it a little harder for you. I'm adding another
game. And this other game is minus 10 to the 6 plus 10 to the 6.
Sorry. Plus, I guess minus. Minus 10 to the 6, plus 10 to the 6. Actually, you know what I mean.
And I'm doing it -- so I'm giving the role player two more options. So to play the same game more
huge stakes. Remember, my subject is risk. All right? In fact, let me make it a little better.
Imagine that in fact I'm cheating. So I give you an X every time he wins. Now game theory says
everybody will play, who plays has to play this. Because this is stupid now, right? Because now
you're expected to win half a dollar. And you shouldn't do anything else. Then you stop thinking,
what if I'm the role player and this is the price of my house? Okay? Maybe I'll play this. And the
question is how do you explain this.
So this brings -- this is what I mean by risk. So this is an example you can decrypt. But risk is
something that people have thought in economics have thought tremendously about. For
example, the whole first chapter of the 61st chapter of [indiscernible] book was utility theory. And
in that chapter, in fact Bernoulli -- in fact, back in the 17th, 18th century, two of the Bernoullis had
proposed the question that Saint Peters Paradox and provided the answer. Logarithmic utility.
And [indiscernible] said basically it's not money that matters but it's the utility of money. And if
you look at it this way, then the mystery of this game goes away, because what you do is you
transform -- you transform utility sort of money and utility and you play the game on the utilities.
If you play the game on the utilities, sort of you played up the law of the game with minuscule
probabilities, if you deal anywhere close to where most people's utilities are.
But what I'm saying is this defines -- so this sort of asks for an interesting sort of, how do you call
it? -- formulation. So this is something that people have thought for a long time. In fact, what I'm
going to show you is a variant of the original formulation due to [indiscernible] formula. So what
you say, imagine you have a general -- what is a person's attitude towards risk? And one answer
is that you might have a general function that maps probability distributions of the [indiscernible],
lots of economies like to call them, probability distributions of the [indiscernible]. Let's say a finite
support to the two areas.
So that's a good definition. So basically no restriction, whatsoever. But what I'm telling you is if I
give you, if I present to you a lottery, there should be something -- there should be an amount of
money to offer to you. Is everybody aware of that issue?
And let's see some examples. Expectation is one. Okay? You know everybody -- everybody's
favorite way of mapping distributions to numbers, expectation. But maybe you are risk averse.
So maybe you want something like that. Expectation minus variants. Or expectation minus
squared variants, I don't care. It behaves sort of the same for my purposes. Or maybe you are
risk-seeking, in which case that's a good one. Okay. Why not?
We are very general here. So maybe what another -- what's another one? Let's see. Imagine
that somebody is facing a life-saving operation. So all they care about is the outcome more than
100K or not. So maybe what you're going to maximize your evaluation is the probability that X is
greater than 100,000. That's a possibility.
Or maybe what you want to maximize is X such that probability -- sorry. A says that probability X
bigger than A is at least .95. This says sort of, okay, accidents happen. Sort of I don't want to
risk absolute guarantee. But I want the bottom line. So something -- I want to maximize what I'm
going to almost certainly win. Okay. Here is another one which may find some crazy but there is
another prize supporting it.
So that's a good one. So imagine that sort of I want to -- I want to -- I'm interested in maximizing
the average between the min and the max that are supported by non-zero probabilities. Okay.
Okay. So this sort of covers the whole space of violations I'm interested in. Okay?
So what's my point? My point is the following, that suddenly there is a new concept of Nash
equilibrium. You get it? What is Nash equilibrium? So there's something called V Nash
equilibrium. V1, V2 Nash equilibrium. What is Nash equilibrium after all? We are all choosing
strategies, mixed strategies, probability distribution. So that the resulting distribution cannot be
improved in expectations for us by changing the distribution, by changing the strategy. Is this
clear?
So imagine now that we don't use -- we say does not improve our valuation, whatever it is. And
for other players, perhaps they have their own valuation. Okay? And so for each one of these,
you can ask yourself what is the -- can you find the Nash equilibrium? And, of course, we know
that here you can. Okay, that's what Nash's theorem says. That's Nash's theorem. If you're
interested in the expectation, you get Nash's theorem.
>>: You can find ->> Christos Papadimitriou: You can find modular this. Yes. So, in fact, sorry, there is a Nash
equilibrium and it's Papad complete to find, exactly. So let's call this a very good case. You will
see what I mean. Okay. Anybody else? Anybody knows what another one has a green
checkmark?
This one has a green checkmark. Okay. Here is why. Because basically I have a game. I'm
interested only in probability, in the cases where I get more than 100,000. So I transform the
payoff so I put 0 if it's not 100,000, one if it's bigger than 100,000, and then I play the game. Is
this clear? So this is the magic of utility.
In here I have a utility. My utility is 0 if it's less than 100,000, 1 if it's bigger than 100,000. If I
have a utility and basically here's what [indiscernible] formula says, there's a utility function if and
only if there's many probabilities in the audience. If and only if my valuation is linear in the
probabilities, is a linear function of the probabilities. Not the payoffs, the probabilities. That's
what we're interested in.
Okay.
>>: This applies if all players are using evaluation?
>> Christos Papadimitriou: Sorry. Yes. So, right. So basically you can mix these two. Okay?
And you get -- okay. So in some sense you can mix any green checkmarks. Anybody know?
There's another one. Another that's positive. You know which one it is? It's this one.
Okay. So basically it says that, yes, there's always Nash equilibrium, Nash's theorem holds.
And, in fact, you can find it exactly as easily as you can find the Nash equilibrium. Okay. And to
make a long story short, everything else is read with my ->>: [indiscernible].
>> Christos Papadimitriou: Sorry?
>>: Experience, inexperience.
>> Christos Papadimitriou: Mine this. This doesn't, this doesn't.
>>: Did you forget about the variance here?
>> Christos Papadimitriou: [chuckling] so both hold in both cases, if it's square root or not. If it's
standard deviation of variance, or basically any reasonable. So I'll tell you what is the
mathematical -- so basically here's the mathematical sort of probability that guarantees this,
okay?
It's because variance is, I know this is going to be, many of you are going to disagree until you
listen to me at the end. Variance is a concave function of the probabilities. If you consider it as a
function of probabilities, it's concave.
Okay. So basically Nash's theorem does not only hold for linear functions of the probabilities,
also holds for concave functions of probabilities. And the proof is not very novel. It mimics
Nash's proof. So it just goes through. Okay. And so the bad news is that if it's not concave, then
we don't know the good case. And this means two things. Nash equilibrium may not exist. And
it's NP hard to tell.
>>: The zero sum case?
>> Christos Papadimitriou: Zero sum can become very bad. So if what Ron is asking, what if it's
V zero sum. Even if you start from a zero sum game, if your valuations are nasty.
So let me tell you, let me give you sort of -- this sort of -- so soon after the phenomenon
Morgenstern book appeared, there were a lot of, coming from economics, from finance, there
were a lot of criticism of the utility function, the expected utility hypothesis, so to speak. And
Markovski and Allai and later Carmen Adversky, Carmen Adversky defined something, that this is
the sort of stylized of what Carmen Adversky proposed.
Sort of Markovski and later Allai proposed this. And this is sort of what I proposed. So I just
wanted six examples. And so the bottom line is that for many of the most interesting Nobel Prize
winning sort of proposals for understanding risk Nash's theorem evaporates.
And I think I'll stop here and ask for questions.
[applause]
>> Yuval Peres: Questions.
>>: Given how to do Nash equilibrium, why do games like Sutherlands [indiscernible] converge
so fast?
>> Christos Papadimitriou: Sutherlands?
>>: Sutherland had a paper ages ago on how to divide up PP1 at MIT. [indiscernible] on its own.
He claimed it works at one place, I put the paper on the POSIX desk and the next week the
market was stable. And so why do some of these market convert so incredibly fast, given those
theorems?
>> Christos Papadimitriou: This is a fair question. Listen, okay, you don't expect a negative
result in complexity sort of to stop the world. So listen, we have seen this before. Okay? In
the '70s people started proving NP complete problems, NP complete. And everybody said, yeah,
but I can solve the traveling salesman problem in no time at all.
So then there is the way I like to explain to game theorists, a negative result is really the opening
move in interesting game. It's not the end of the world. It's your move now. What are you going
to do? What are you going to say? But the point is the following: That in game theory, you
know, so let me remind you, Meyerson said that -- the universality theorem is the basic of modern
economics -- basically there's something fundamental about the assurance that every single
game has an equilibrium. It's not like sort of don't worry, your game will have an equilibrium. So
it's a fundamental result so we know that serves us all. And so that assures you that the world is
benign and every game has an equilibrium.
And what I'm saying sort of is that this is extremely fragile. So if you take like the reality of risk
into account. So it goes away immediately.
>>: So regarding this comment about basically like a checkmark, I know about the E minus
variance. So one way to resolve this situation, this impossibility is the following. So you have
these various strategies and you have probability of the strategies, and if you can tell that your
dash paper may not exit. But draw another game where my strategies are what property vector I
choose. And then apply the Nash equilibrium there. It may not have any meaning.
>> Christos Papadimitriou: But it does -- the point is that Nash's theorem does not hold for
infinite, for ->>: Modular ->> Christos Papadimitriou: Right.
>>: Let's say you can discretize this space, we'll always talk about probability within one percent.
Then in this new game, which is like very huge metrics.
>> Christos Papadimitriou: You're going to have a mixed Nash equilibrium. What do you get out
of that? So I haven't had -- I haven't done this. What I known is in practice what this means is
that basically both players, what happens, there are very simple games. That's where our
negatives all start.
Very simple players who both players sort of know they start playing the first strategy, then
together they switch to 0 strategy. So they do other strategy.
So at the same time. So they never cross. They never -- it's either -- you know, concavity means
that the extremes win. I'm sorry, convexity means that -- if you maximize convex functions this
means that extremes are going to win. And so basically you jump from ->>: But if you do the utility function as the probability which will be output of the game, if you -- if
that's your strategy input what probability said I'm using, then Nash will take you randomized over
the probability ->> Christos Papadimitriou: So, you know, I doubt it works. Let's work it out. So I haven't gone
through this. I've gone through this exercise in order for a positive, in order to get algorithms for,
approximate algorithms for algorithms, but I don't know what happens here. Yeah, I'll tell you.
>>: Maybe you could say a little more about the dynamics. This is an area I know very little
about. On one area you have Nash equilibrium, normative things and the market should go this
way.
>> Christos Papadimitriou: If the market goes somewhere, it's going to be there.
>>: It should be there.
>> Christos Papadimitriou: Or one of these.
>>: But on the other hand you've got a dynamic system where the players are sort of
[indiscernible] you've got the equilibrium. So the whole connection between dynamic market
evolution and these notions.
>> Christos Papadimitriou: It's fascinating, yes, of course. So there is -- I mean in some sense a
good learning -- a good learning algorithm is an alternative equilibrium concept. And people have
been -- there are results now that say that some powerful algorithms, sort of like boosting, like
exponentials, multiplicative updates, they solve zero sum games but they don't solve it in 3 by 3,
two-player games. So that's of course a very interesting -- very interesting -- the proofs are hard.
So this means that there might be something there. There is probably something there.
>>: There are certainly cases where you have Nash equilibria, examples where you have
dynamic systems corresponding to them simulate, they don't settle on ->> Christos Papadimitriou: Yeah, yeah.
>>: Between ->> Christos Papadimitriou: Yeah, yeah. So that ->>: Example of species of lizard that's been found between three types and in the period of three
years they go between three types of behaviors.
>> Christos Papadimitriou: I see. You know, so with my students, we recently proved the result
along the same lines. So it's a little subtle. That you don't just to prove that the dynamical
system has a certain behavior, because maybe that's what -- maybe you are going through -that's a winning -- that's an equilibrium mixed strategy. You want to be sure that it has behavior
that is [indiscernible] and exponential [indiscernible] so you cannot have -- that it goes very close
to one and then -- so it cannot, the average cannot -- the average is not good. So basically you
are sort of -- it's a dynamical system on the average of the running average program.
>> Yuval Peres: Any further questions? Thanks.
[applause]