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>> Nikhil Sevanur Rangarajan: Hi, All. It's my pleasure to welcome Leonard Schulman.
Leonard's a faculty at Cal Tech. He's also the director of the Cal Tech Center for
Mathematics Information. Leonard is a well known in computer science. His interests
span almost all of the theory from quantum computing to error tracking codes to
approximation algorithms and you name it.
And game theory, too. He's going to talk about solvency games today. Leonard has
been here almost two months and this is kind of his farewell talk, I guess, his last week.
We hope he keeps coming back. So over to Leonard.
>> Leonard Schulman: Thanks. And just thanks for inviting me to visit Microsoft. And
it's been great to visit. And there's two more days in which you can chase me to criticize
errors in the talk. So I'm not out of here until Friday.
So we're going to talk about what I did a few years ago with Noam Berger and Nevin
Kapur and both at Cal Tech and Vijay Vazirani at Georgia Tech.
But the first is a prelude. There's a saying in Hebrew that says: All roads lead to Rome.
And I've since learned there's a new Hebrew saying that says: All random walks lead to
Redmond.
So this talk is sort of about finance but also sort of about random walks. And of course
the connection between -- it's not on my computer. Good. In the Wizard of Oz there's
somebody standing behind the screen manipulating sudden things happening. That's
what this feels like, because my computer is stable.
>>: Just take it ->> Leonard Schulman: Right. Right. As I was saying, this talk is sort of about finance
and sort of about random walks and the connections between these two fields or types
of questions is long, long established, and this is from the '70s I think, this book, that sort
of told you how to successfully make a lot of money on Wall Street by just trusting to
Brownian motion or something.
Strangely, we were sort of talking about this problem at the beginnings of it. We
discovered that as far as we could tell, and I would love to have my eyes opened to
literature we couldn't find, all this literature on how to make a lot of money by gambling
speculatively, so all this kind of trying to increase the rate of increase toward infinite
wealth doesn't seem to answer some really basic questions that we asked about just
avoiding ruin.
Now, investing for speculative game can be -- as I say -- very alluring. So for a while
some of you may remember when these predictions came out. There was, 2000, the
book was published called "Dow 36,000 - The New Strategy for Profiting for the Coming
Rise in the Stock Market." The Dow today is 11,000.
There were other books, one said 100,000. One said 40,000. Apparently you could sell
books just by just picking a big number.
But we learned our lessons, everybody, right? Some of us learned our lessons. This
one was published in 2006. It seems like these books are very good predictors of the
future about two or three years after their publication.
So we're kind of, in this talk, interested not in this highly speculative end of the question,
but actually for us, you know, it was more like what's the investment strategy of a typical
person. Maybe middle class, doesn't have to be middle class. A person who actually
isn't so interested in speculation, but really what you want to do is you have a standard
of living.
You want to keep your house and food on the table and educate your kids, and you just
want to invest to avoid catastrophic disruption.
So that's a very different investment goal than Dow 36,000 or 100,000 apparently they
couldn't think of really big numbers. This same kind of question you might also think
should come into the thinking of an insurance company. An insurance company actually
makes money by convincing you that they will stay solvent then they can sell you
insurance premiums, right?
As a matter of fact, the prices in an insurance policy, companies themselves have
ratings, and the premiums they can charge depend upon those ratings. Those ratings
we've discovered recently can be very unreliable. But in case that's how that market
works.
The same thing happens in the banking sector. If you don't have insurance deposit,
which is a sort of nice innovation in the United States, banks have the same issue of
having to convince potential depositors that they in fact will stay solvent. In the U.S. it's
true once you go above a certain amount. So there are these two investment goals.
One is maximizing profit. The other is minimizing the probability of ruin. And they're
related but they're not the same.
And this is actually the source of why we regulate these industries. Or if you've read the
papers in the last year or so, maybe we don't really regulate these industries, but we
think we do.
So I want to start to be a little more specific about the type of phrasing of the question I
ask. And again I'll tell it through a little story. Suppose you're a wonderful card player.
You know the odds of whatever this card game is perfectly and you can calculate on the
fly. And you show up at a party and it's a bunch of amateurs.
There's two kinds of games you can play and you unfortunately didn't arrive -- you really
want to clean them out, but the worst thing would be the alcohol isn't very good. The
worst thing would be to run out of your $50 stake and be stuck at this party.
So there's two card tables you could play at. And they're the following odds every time
you play a round. You can choose what table to play at every time. Depending on how
much money you have in your pocket. But you arrive with $50.
Table A looks like this: Table A, these are your personal odds, you, the expert. So the
expert has positive bias. Positive drift. The expert can on average earn money. Here
are his odds. He loses $3 50 percent and he has 50 percent chance of winning $4. So
it's nice. He has an expected gain of 50 cents every round of play. Table B looks a little
different.
It's the same game but scaled up by a factor of 10. So 50 percent chance you'll lose $30
and 50 percent chance you'll win $40. So expected gain per hand is much more, it's $5.
So just psychologically, if you're interested, if you're a speculator, you can see all the
kind of type A jocks heading toward table B. But if what you're really worried about
running out of money, you want to go to table A because you'll prolong by a factor of ten
time to ruin if you do get to ruin.
So that was the easy comparison to make. We can all just ->>: [inaudible].
>> Leonard Schulman: Sorry. Actually my last sentence was incorrect. But anyway,
what I said was just wrong. But this is the comparison. Playing table B is the same as
playing table A but if you all had arrived with $5. That's the way to make that
comparison. Sorry about that.
There are more subtle comparisons you'll have to make, though. That one was
particularly trivial. Table A, let's say, is just like you had before. You might lose $3 or
win $15 even stakes.
But maybe it's like this. Maybe you either win or lose $3 but you have these sort of more
nuance probabilities 0.42 versus 0.58. Positive drift. But this one's maybe a little harder
to figure out which table you should play at.
So the type of questions that we'll want to address is trying to resolve that kind of
decision. If the best strategy is not to maximize the expected wealth, which would be
sort of the expected drift, let's call it, which would be sort of the trivial thing to do, then
what is is it? You could say you should be doing the lob but that's not really the right
thing either.
So what should your best strategy be? And apart from the kind of playing games
version of this, sort of sociologically we were curious at the time about this kind of
question. You watch people living life and making very different kinds of investment
decisions depending on it seems depending on whether they're wealthy or poor.
And you don't want to say that poor people are making bad decisions and that explains
why they're poor. Sometimes that might be appropriate. But I think the question we're
more interested in is the rationalization of this issue.
That if you're rich, you can afford to take risks that by and large are profitable. But if
you're poor, the probability of ruin prevents you from kind of taking these risks.
So again let's try and be a little more specific. An investment or I'll start calling it an
action, is going to be for us a finitely supported probability distribution on the integers.
And just to show you tables A and C from our game, [inaudible] if you start from some
origin zero which would be like your current wealth there's 50 percent chance you go to
that wealth minus 3 and 50 percent chance that you go to that plus four and here is table
C.
And in our setup, you have a choice between finitely many actions. So this is the
framework. So that every value of the wealth which would be some number W you
choose an action, sample from the distribution. The action is a probability distribution.
And that's your new wealth.
And this game continues until you are broke. And then the game is over and you would
like to play simply to minimize the probability of going broke.
A strategy is a mapping from the positive integers to the space of actions. And it just
says ->>: Wait a minute. Can I -- when I analyze having gone broke you have to play a game
each time?
>> Leonard Schulman: You have to play the game each time.
>>: Probably going broke when?
>> Leonard Schulman: Ever.
>>: Ever. Okay. So ever.
>>: [inaudible].
>> Leonard Schulman: Yes. So in fact I don't know, I say it on some slides but let me
say it now. In fact we're going to say we're going to assume that every one of these
actions has positive drift. So okay.
So when you fix a strategy, a strategy means which action you play at every value of
wealth. We'll let P of W be this what I'll call the ruin function or the -- given that you start
with wealth W and you play this strategy, what's the probability you ever go broke.
And you like to minimize P of W. So here are two very easy facts. So this audience, it's
totally obvious. There exists a strategy for which P of W is strictly less than 1 for all
positive W. If and only if there exists some investment in this collection that has positive
drift and also you can play Jacobean strategy.
I should say with respect to bullet one here, again I'm going to assume that actually all of
them have positive drift for some technical reasons later on.
And now coming back sort of -- well, one of the things --
>>: Let me ask about, so you lack deception?
>> Leonard Schulman: Right now it wouldn't make any difference. But it actually ->>: You might want to play ->> Leonard Schulman: You might want to play something with negative drift. And
there's some point in the results where that makes things complicated and we don't
actually know how to deal with it.
Okay. So I'm going to try describe this as sort of this additive. Everything will be
phrased additively. Since we're not talking about expectation of your wealth or anything,
it doesn't really matter whether we're working on the [inaudible] logarithmic scale, in
terms of kind of, if we're talking about investment policies motivation, it makes more
sense mentally to think of the multiplicative version where this additive walk is happening
on the logarithms which just means you invest in some thing and it goes up by two
percent or down by two percent sort of thing.
And then going broke is not actually hitting zero dollars in that scenario but hitting
whatever your broker charges you for minimum transaction fee. So everything
translates perfectly.
And now back to the motivation about why do you have to play. The motivation why you
have to play, again if you're thinking about this not as a probablist but an economist,
there's no such thing as a risk-free investment. There's always inflation. You sort of
have to make some choices.
So this is the weirdest part of the talk. The statement that as far as I can tell there's no
literature on precisely this problem, because as you can see it's a simply stated problem.
There's, of course, lots of work on gamblers ruin and probability. But that doesn't have
to do with -- here we're talking about sort of not a single walk but this strategy business.
Let me talk about what I know about the literature. And I warn you pretty much what you
see on the slide is my knowledge of finance. So just -- if you've ever looked at how
mutual fund managers get rated and things like this, there are these ratios that people
study, the sharp ratio is maybe the best known of them, where you look basically at
empirical data on something, mutual fund or a single walk but this strategy business.
Let me talk about what I know about the literature. And I warn you pretty much what you
see on the slide is my knowledge of finance. So just -- people -- if you've ever looked at
how mutual fund managers get rated and things like this, there are these ratios people
study, the sharp ratio is maybe the best known of them where you look basically at
empirical data on something, mutual fund or a stock or anything, and you look at the
expected profit relative to some benchmark, relative to the S&P 500 or whatever your
favorite benchmark is or whatever is appropriate for that type of investment vehicle and
you divide by the standard deviation. Again, just gathered empirically.
The higher this is the better the risk/reward ratio. And the guideline people offer is sort
of that whatever your comfort level of risk you should invest in something with a high
sharp ratio.
So these are some kind of heuristics that nothing at all relevant immediately to our
question. So I'll just leave it there. Now in probability theory, people have definitely
looked at sort of this kind of decision making with respect to avoiding ruin.
And I'm basically aware of these three papers. The motivation is extremely similar. In
fact, everything I've said up to now except the post-2000 news and market stuff is very -could be taken equally from the introduction from some of these.
But the model is different. They only study where there's only one investment available.
And your decision is how much to invest.
So you're trying to actually make a profit and there isn't this business that I said a
second ago about inflation, so in that sense it's very, very different. So there is sort of -there is sort of a safe choice and you're trying to escape from zero. So in that sense it's
different.
And operations research, [inaudible] is a special case of some much more general
model called Markov decision processes, where you have some space and you can
make a decision. You're trying to minimize some penalty function or maximizing some
reward function and it definitely belongs to that extremely general framework.
It does not belong to the kind of really well understood special case as far as I can tell it
does not belong to the extremely special -- this special case that's very well understood
called multi-armed bandits. Again it doesn't seem to be treated by that literature.
All right. So here's some questions we're going to try and answer and give at least
partial answers for. First of all, is it true that in our -- does our model reflect kind of our
understanding of reality that actually investment strategies do depend upon your current
wealth. That certainly seems very likely. But the question is does this model actually
capture that phenomenon?
Now, there's a related phenomenon that I think seems natural which is that somehow it's
really only this business of being poor that makes you choose something other than -there's some optimal strategy that kind of drives you off to infinity as fast as possible or
something like that, and that once you're wealthy enough you can just do that and all the
finer points of avoiding ruin happen in some finite region or near zero that seems like a
reasonable suggestion.
And if so, we would like to answer which is is that investment, which of your actions?
And if it's not so, of course we would like to know what the tail of this sequence looks
like. And finally we would like to know can you actually compute how to behave. So
these are the kinds of things we would be interested in.
So I'm going to spend maybe two slides or three slides on something that's purely
probability theory and for that reason especially for this audience I'll go through it very
fast, which is just what's the theory of no action, just random walk with drift. And partly
just to set up my notation.
I was thinking of taking out these slides. But I'll leave them in. Suppose you have this
one action A we're talking about. It's supported between the interval minus L to K. You
can lose at most L dollars and you can gain at most K dollars.
It's specified by sequence of probabilities, indexed to Q minus L through QK. These are
probabilities. They sum to one. They're ruin for PW remember is the ruin function. P of
W is in the kernel of the following map. So it's a -- you map the vector P of W to minus P
of W plus the expected, plus the average of all the things you would go to. So this
second term here is with probability QR.
I go and I get the ruin probability of whichever random step I took. And the true ruin
function should be harmonic with respect to this iteration. So it should be in the kernel of
this linear mapping.
And, of course, you have to match the boundary conditions to the problem which are that
P of W is equal to 1 for W less than or equal to 0.
So again for this audience, I'm sure you're all happy with this. But for those who are not
it's like a Laplacian and there's this asymmetry and it's not like a Laplacian.
So the kernel of this map is the dimension K plus L. And it's spanned by these
exponential eigenfunctions. So this is an eigenfunction. It's a function of W where these
lambda sub I, there's K plus L sub roots. And they're the roots of this rational function.
The rational function has a minus 1 for the fact that you left from position zero and it has
a plus Q sub R times X to the R, R ranging from minus L to K. This function is in the
kernel for obvious reasons and they do span the kernel.
And, of course, we want to pick a particular [inaudible] to the kernel matching the
boundary conditions. These are the boundary conditions for W between 0 to minus L
plus 1, the P of W is 1.
There's something suspicious about this. It seems like you should need K plus L
boundary conditions. And we only got L boundary conditions. So we have the missing
conditions. They're not missing for two reasons. One is sort of abstract argument.
Notice that the ruin function has to tend to 0. These are positive drift walks. So no
matter what you do if you start wealthy enough you're probably not going to be ruined.
So that means that the eigenvalues greater than 1 or the roots, I should say, greater
than 1 can't actually show up with non-0 weight in this function.
And then you have to prove this little theorem that says that this kind of function has
exactly L roots in the open unit disk. And there's a nice little perturbation argument that
shows that.
There's another argument that's more constructive. Here I'll assume that the GCD of the
positive 2 sub Ks is 1. Otherwise you get some degeneracy. Everything is still true but
you get some degeneracy. And there's something called -- I think it's called ladder
random variables. Probably somebody in the room, probably many in the room can -there's a nice way of looking at this which is for I between 1 and L let alpha sub I be the
probability that if you start with wealth W, then the first time you go below W, you hit W
minus I.
Okay. So these alpha sub Is probabilities they don't sum to 1 because of course you
may not go below W. But they're positive and they sum to at most 1.
Then there's -- if you write things like this you can get a nice kind of iteration on P of W,
that sort of only works forward. The hard part about writing the other equation is you
depend upon larger and smaller values, here you can just work your way up on the P of
Ws. You get this equation. The probability of ruin is that the sum with these alpha sub
Is the probability of ruin given that you went down to W minus I.
And this gives you a certain transform matrix, gives you recurrence for Ps. If you look at
the eigenvalues of this matrix they're precisely the roots of that polynomial that way in
the open unit disk. This formulation sort of gets rid of the other eigenvalues.
There's a unique positive eigenvalue. So this is going to be important for us later on. To
point out when I have these L roots in the unit open unit disk, there's exactly one of them
on the positive real axis. That's going to be real important.
And this is the [inaudible] Frobenius root of this matrix. Now, I'm done with kind of the
one action case. Let's now talk about strategies. And sketch out what the answers to
those questions look like.
So, again, we'll get a fixed some finite collection of some actions script A. We need to
define a partial order on strategies. So remember strategies are mappings from the
natural numbers into the actions.
P sigma W is going to the probability of going broke at W following sigma. So this is the
ruin function of sigma. And there's a partial order on strategies which says sigma is less
than or equal to pi.
The smaller one is the good one here, right? If for every W the ruin function of sigma W
is less than or equal to ruin function of -- that should be a pi in that second superscript
there.
And we say that sigma is optimal if sigma is less than or equal to pi for all pi. That's an
obvious definition. And the interesting thing is that it makes sense. In other words,
we're going to show there exists optimal strategies.
So here's a theorem. Given any finite collection of actions script A there exists an
optimal strategy. And furthermore sigma is optimal if and only if it's locally optimal.
Locally optimal means that sigma just has to have this inequality has to be less than
strategies that differ from it in exactly one position. So they do the same thing for all
values of wealth except they disagree, and there's one wealth where they choose a
different action.
Okay. So this theorem says sort of two types of interesting things. One it says there
actually is sort of a globally optimal strategy, and the other is that sufficient witness for
that is that you're locally optimal.
So I'm going to sketch out what this looks like. None of it should be I think too -- it's all
pretty straightforward. But so here's the sketch of how you do these things. We call a
sequence of ruin probability so our ruin function P, we'll say it's subharmonic with
respect to some strategy sigma. If for all values of wealth P of W is less than or equal to
the expectation of P of W prime, where W prime is chosen according to this random walk
one step of the random walk starting from W. So that means where W prime is equal to
W plus J, where you sample J from the distribution according to sigma.
Okay. So this is -- so that's the statement that sigma, I'm sorry, that P is subharmonic
with respect to sigma. And this is the expression, where Qs these transition probabilities
are those of sigma.
And for this purpose we want subharmonic to also include the condition that is true of all
the functions we care about which is P tends to 0 in the limit tends to infinity.
And there are the obvious corresponding definitions for super harmonic and harmonic
where you just change this sign for super harmonic and put equality for harmonic.
Here's the lemma we need. Fix a strategy sigma. And any boundary conditions. So, for
example, it could be the boundary conditions of all 1s which is sort of what we carry
about overall but for purpose of lemma it's useful to know that this is true for any
boundary conditions, and the statement goes if V is subharmonic, so V now is our
function. If V is subharmonic with respect to sigma and P is the unique harmonic
solution corresponding to sigma, so for any strategy sigma there's some ruin function,
that's the one that's harmonic. Then V of W is less than or equal to P of W for all W.
I won't supply the proof. A lot of this, like various other proofs in the paper, mimic what
you know from sort of complex analysis from the maximum principle and say if there's a
violation, there has to be a violation, has to be a violation of inequality on the boundary
of the region. Here the regions are sort of finite prefixes of the integers. But same proof
technique.
So now we want to go back and prove that theorem about that there's an optimal and
there locally optimal is sufficient witness for optimality. We'll start with the furthermore,
the fact that if you're locally optimal then that shows you're optimal.
Let's let S be the ruin probabilities of the ruin function of some locally optimal function
sigma. Let pi be any other strategy. By this lemma up here, S is actually subharmonic
with respect to pi. So that means that S is less than or equal to the ruin probabilities of
pi. Because your local optimality just -- it's enough just to look at this one location W.
So S is less than or equal to the ruin probabilities of pi. So that's sort of surprisingly
short but it works. So now we actually need to show that there is something that's locally
optimal.
So, well, first of all, we should note something really simple which is the -- I'm going to
take kind of a trivial topology, which is on the space of strategies, which is just add to
discrete topology on all actions in any position and I take the product topology. It's a silly
space but it's nice to say things this way. That's compact. Just by taking it off.
And now, okay, that sounds like a stupid definition. But now I'm going to look at the
following quantity for a particular strategy pi I want to look at the sum of all of the ruin
probabilities. Okay? Now, because of the positive drifts of the strategies, I know that
this sequence is dropping off exponentially.
If I go to a sufficiently high wealth probably not going to see the origin. So this always
converges. In fact, by really just using the same sentence I just said, this function of pi is
actually continuous in the product topology. If I fix a long enough prefix, I can make the
sum as long as I want. That's all I would say. There we go. Some strategy sigma
minimizes this some. And then sigma is actually locally optimal, because if it differed
from some other strategy pi at some location W, its ruin probabilities would probably be
subharmonic with respect to that other strategy pi.
So there is an optimal strategy and we can sort of have a short witness for it, if omega
counts as a short witness.
All right. So now we're going to come back. Remember we had the other questions like
is it true then once you get far enough from the origin you can just make your life simple
and just always play this strategy that drifts the fastest or something like that. So is that
really true? So here's the theorem.
Let's look at the leading root. So the [inaudible] Frobenius root. So for each of these
different actions A, B and C, remember there was some number of roots in the open
number of disk. The number depends on how much you could lose maximum with that
particular action.
But each one of them has one Frobenius root. Meaning one root that's positive. And
between 0 and 1. So those were what I'm going to call these lambda 1s. Lambda 1 of
action A. And lambda 1 of action B. And so forth. The statement is if lambda 1 of some
action A is strictly less than lambda 1 of all the other actions, then, yes, then there's
some tail in which you only play action A. So let's go back to an example I showed you
at the beginning of the talk. This is where we had these two tables. One you could lose
$3 or win $4, with even odds. And then there's this other table where you would lose
$30 or win $40, with even/odds.
And this again was the really easy example to look at. In that example lambda 1
happens to be 0.291 for A and table B you just take the tenth root of that. Of course, it's
bigger and this is worst. So that was obvious by inspection even back then. Now we
have a more complicated table or complicated alternative between A and C to remind
you you could then lose $3 or win $3 but the odds were in favor of winning.
So you can calculate lambda 1 of C happens to be 0.898, which is less than and
therefore better than 0.921. So table C is is in fact the rich man's strategy in that game.
Even though table A has a faster drift toward infinity.
And I didn't ask for a show of hands early in the talk about whether everybody thought
the positive drift would be the rich man's strategy, if it existed. Because I'm nice.
Anyway, so it's an odd phenomenon. Once you think about these roots enough, you can
convince yourself that this is really the right intuition.
>>: This is true even for A and B. B has a bigger drift than A?
>> Leonard Schulman: It's true even for A and B. Even that disproves this. Yes, that's
absolutely right. But -- good point.
>>: Good A and C maximum loss ->> Leonard Schulman: It's the same, right. Okay. Yes. Now, there's also an
algorithmic version of the theorem I just showed you which is under the same conditions,
namely that there's some action whose leading root dominates, is less than the leading
roots of all the others.
If you pull out some of the conventional algorithms, actually I think any of the
conventional algorithms used for Markov decision processes you will actually converge
to the optimal strategy. Actually, the only one I know for sure is value iteration. Value
iteration is a simple thing you -- I'll do the bullets in order. So the ingredients in showing
this to be correct is first of all we need some explicit bound on where the pure tail starts
because we actually want to say something about how long it takes to compute this
thing.
So we can make these previous things quantitative and get some bound and then at the
tail where you've cut off the tail you sort of need to change the recurrence and actually
put in those boundary conditions, the fact that you know it's going to recur. So you can
do that.
Then you run value iteration. Value iteration is you start all the probabilities of ruin or
your current guess of them to be 0. And then you update making a greedy update. So
you sort of said, well, if I'm going to choose at every W the action that would minimize
my chance of ruin based upon my current estimate of what the chances of ruin are, and
you just update it every time. And this updates in particular the ruin function. What you
really update is the ruin function, not so much the strategy, if you will.
>>: Leonard, you can give exactly the interpretation, the real probability in Jth steps in
the iteration.
>> Leonard Schulman: Good. That's right. That's right. That's right. So why does this
actually converge? Actually, there's a reason why we converge from there. So this
didn't occur to me.
But this -- the way you bound the convergence rate is by pretending that the updates
were done with respect to the unknown optimal strategy. And which is just a linear
update ruin. Therefore you can just look at its spectrum and show it has operating
normalistic. That operating norm is less than 1 but how much better than 1 it is depends
on how finely this separation space.
On the other hand, I did a big setup job on this so-called rich man's strategy. The
reason for the big set up job is it's false in general.
So here's explicitly how you can falsify it. If you have two investments whose -- again,
there's Frobenius, these are two strategies, both of which lose at most $2. And if you
cook them up, so that the leading root is identical. So they fail the conditions of the
previous theorem, but they have different, they have one more root that you care about
and those roots are different, then in fact the optimal strategy has an alternating tail,
eventually.
And I'll just make a very quick sketch of slightly weaker claim that it's just not a pure tail,
which is shocking enough. It was to me when we did this. I'm not saying you need to
be.
So suppose you're pure and you're playing A from capital W and on and we'll let S be
the ruin function of sigma of this supposedly optimal sigma, then you can write down,
once you get a little bit past W, you can write down exactly what the ruin function looks
like. It's a combination of these two exponentials where here lambda is this common
prone Frobenius root and lambda 2 is the other root of A. The sum of two exponentials.
I want to point out now that lambda 2 of A of 2 of B are negative, remember that I
emphasized there's only one positive root.
And since this was a real polynomial and you've only got one other root, it's negative.
Then if you look at the value of playing B at W plus 3 minus this ruin function at W plus
3, you get an expression which, yes, so this is some -- sorry, there's a mistake in this
expression. There's a QB -- QB was the rational function of -- mistake -- main thing to
pay attention is this term, sorry for the error in this second term there where you have
lambda two of A to the power of W plus three times some constant. We don't care what
that constant is.
But if you look at W plus four you have the same kind of expression with W plus four
remembering lambda two of A is negative, these guys of opposite sign, and this was this
quantity this difference we're looking at here is the kind of sub or super harmonicity. So
one of these is not as good as it could be and you can make an improvement.
Simple corollary of that is even when there's a pure trail it might start arbitrarily. Not
bounded by K and L it's bounded actually by the separation of the roots.
>>: Can you probably the answer is yes, I assume, but can you -- so the optimal strategy
may be in the tail is not just to pick the choice which would be best if you picked that as
a pure [inaudible].
>> Leonard Schulman: What?
>>: So given the wealth, you can say what would be a probability of if you always picked
option A, option I and that gives you a number.
>> Leonard Schulman: Yeah.
>>: And so presumably it's not true that the optimal strategy RFW is to pick the optimal
[inaudible].
>> Leonard Schulman: I'm sorry, I'm not sure why I'm not understanding this.
>>: So if for each investment HI and HW is a probability ->> Leonard Schulman: If you would just ->>: If you always played that.
>> Leonard Schulman: The word function of this pure [phonetic].
>>: Right. So ->> Leonard Schulman: Oh, I see. You're just saying ->>: It can't be but the optimal strategy is not just to pick the one that's best ->> Leonard Schulman: There's no reason for it to be that. I don't have ->>: That probability is basically the lambda to the power W. So it's not ->> Leonard Schulman: But the setup conditions before the pure tail kicks in ma either,
too. . For example, there might have been a third strategy that's not even playing a role.
>>: Right. Relatively shifts the calculations.
>> Leonard Schulman: But the question you're asking is sensitive to -- well, it's sensitive
to what is ->>: Right. I imagine [inaudible].
>> Leonard Schulman: So I don't know that that wouldn't be, but I would be really
surprised. I don't know. So there's a bunch of open problems. I think our results don't
really go all that far. We don't, for example, know if the tail of the optimal strategy has to
be periodic. We don't have an algorithm for the case where the rich man strategy
theorem fails.
Probably you want to answer the first part first, the first question first because if you at
least knew there would be periodic it would be clear that there would be a short witness
for what happens in the tail and you could just write that down. But if it turns out that the
answer to the first question is no then the second one becomes a much more interesting
question because some more elaborate characterization.
Finally, there's really no reason except technical reasons to stick with the case of integer
increments. Really. Last time I checked my broker statement, I wasn't losing or making
exactly an integer number of percentage on each of my investments.
It just doesn't really make sense, except it's a lot easier to do. So I think that would be, if
something to get a handle on.
>>: Existence of an optimal strategy might somehow use some kind of compactness of
something but not necessarily ->> Leonard Schulman: Everything becomes an operator -- it just becomes technically ->>: But you can evaluate a situation where there's no optimal strategy simply because
there's always something a little bit better than ->> Leonard Schulman: No.
>>: If you only have finite actions ->>: Yes, but you might ->>: Extend it to the [inaudible].
>>: Yeah.
>> Leonard Schulman: So still finitely many actions. Still finitely many actions but the, if
you will, the group that these guys generate is dense. So then I sort of don't know what
to ->>: Okay.
>> Leonard Schulman: Okay. So as Nikhil said I'm out of here soon. I had a wonderful
time. Thanks for having me here. I hope I see you again soon. That's it.
>>: Thanks for all the pictures you've shown, I Love Sushi.
>> Leonard Schulman: I went to I Love Sushi. It was wonderful but taking my family to
the aquarium was actually even more wonderful.
[applause]