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Subtitles generated by robot

00:08

right let's jump straight into it so the
title for my talk to
it is time for a change except not
really there was something missing from
the title and that's a comma what I
really wanted to talk about is not time
for a change but time for a change
because I think that we have left out
time from our considerations of risk

00:40

especially risk management the way we
think about risks investment ups and
downs in in general so I think that time
and it's irreversibility are really key
to developing a better understanding now
on this slide here you will notice a
grasshopper and the grasshopper
I met Gresham College I'm very very
aware of the fact that I crash from
college and a great honor to be here the

01:10

grasshoppers of course aggressions
symbol in this coat of arms and I
thought I put the grasshopper on all the
slides of something some relevance to
the gresham context so when you see a
grasshopper in one of the slides tonight
and I don't mention what that is about
then please please do ask me now
continuing with the playful theme
because mathematics and everything else

01:41

in life really should be playful I
thought I'd start tonight's lecture with
my favorite game and the game has the
following rules you're given a hundred
dollars and you invest them in in this
game but you're tossing a coin once the
minute and if the coin shows heads you
win 50 percent of your wager if it shows
tails you lose 40 percent it's just to
explain to you how this how this works
over time people be repeating this game
here's one one sequence we're playing

02:13

for five minutes it's very simple I'm
tossing a coin now on the first two
tosses here
I lost 50% each of the wager at that
time so on the first sorry 40% I lost
40%
on the first toss I lost 40% I went down
to 60th and I lost 40% of the 60 I went
down further and so on then I gained I
lost I gained so you get these
trajectories of wealth developing over
time in in this game I want to
understand what happens in this game so

02:43

I thought I play I played a little a
little longer now I'm playing this for
an hour so I I have 60 coin tosses in
this sequence here I'm playing it for a
little bit longer because I'm thinking
that perhaps I'll get some idea of the
sort of general dynamics of this thing
if I just watch it for a little while so
this is the 60 minutes but I don't
really see much in this trajectory I
just see noise it just seems sort of
random what we usually do in that
situation is we'll try again and see
what sort of spread we have right so I

03:16

played this 10 times for for 60 minutes
and I developed these 60 these 10
trajectories and yeah we develop some
idea for for how far they're spreading
how much I could gain or could lose and
so on how likely things may be maybe I
want to play 20 sequences I could
continue doing this if the screen will
just fill up with colorful lines and
eventually I'm action are learning very
much so the next step in the analysis is
to say let's let's average somehow let's

03:48

get rid of this this noise I'm really
just interested in the fundamentals here
so I'm now averaging at every minute at
every moment in time I'm averaging over
the 20 trajectories that that have
generated and I'll get a line that looks
like this you can see that it's less
noisy but it's still it's still quite
noisy so maybe 20 sequences is not
enough I should order a thousand
sequences it's still somewhat wobbly up
there so how about a million sequences
well that looks pretty good if I

04:19

averaged over a million sequences I can
clearly see that this is a favor of the
game and this is probably what our
intuition would tell us right from the
beginning we were tossing a coin there
was a 50/50 chance of either losing 40
percent or gaining 50 percent now 40
percent is less than 50 percent so on
average we should be gaining in this
in this game and this is exactly what
happens we regain on on average but what
does it actually mean we looked at an

04:50

average and in this case the type of
average is called an ensemble average
and that's because we looked at this
ensemble of a million players if you
like who are playing this game or
million trajectories that we that we
generated that the immediate question is
is this relevant to me because this is
just an average and I'm I don't have
access to that average I will just
generate one trajectory and that will be
me now the reason why that can be
problematic is that I can't I can't go
back in time if I don't like the result
if I end up with a particularly unlucky

05:20

sequence I'll end up destitute but I
can't go back and try again because I've
already lost everything
alternatively I could say I can't access
parallel worlds where my luck was better
or I can't force other people who played
the game to share their phenomenal
winnings with me so this may not be
relevant for me I'm really interested in
one trajectory so I want to see if I can
get rid of the noise in this system in a
different way and that's really the
theme for for this lecture get rid of
the noise through time rather than

05:53

considering many many parallel systems I
want to consider what happens if I play
this for a very very long time so let's
do that this is the original sequence
that I showed you right at the beginning
when I'm playing this game tossing a
coin once a minute for 60 minutes and
now I'm saying I want to get rid of the
randomness by considering longer and
longer time sequences so I'll play it
instead of for an hour for day here's
what happens over day you can see that
the fluctuations become smaller if you
if you look at this it really just
bounces up and down you don't know what

06:25

is happening here the fluctuations are a
little smaller and this inset up here is
just the original trajectory and these
these lines you indicate that I'm
blowing up the initial 60 minutes - to
get to that inset so this is really just
continuing the same game that I played
in the beginning
still to me this is pretty noisy so
maybe I want to play for a week instead
of just a day so now I'm playing for a
week noise diminishes I'll play for a

06:59

year
okay now I've played for a year for 12
months and the perhaps surprising thing
is that I've lost the noise has
completely disappeared so there's really
no reason to believe that there's any
uncertainty in this in this result but
I'm losing so what on earth is going on
I have two perspectives the first is the
sort of intuitively right perspective
we're averaging we have a favorable game
we should be winning on average the

07:30

second perspective is a time perspective
where and sort of averaging over a large
ensemble I've averaged over time so
essentially I've picked maybe I've
picked one individual I just followed
that for a very very long time and I
could pick any single individual if I
just follow it for long enough then the
noise will average out over time but the
effect of time is a different effect
than the effect of averaging out over
the ensemble now this ensemble
perspective here over emphasizes the

08:03

exceptional successes so it contains
phenomenally unlikely individuals that
just made a killing in this game and
I'll illustrate it in a second and they
they pull up this average but that
average is not reflective of what
typically happens so if I took any one
of them and I just play for a long time
I will see that every single one of them
will lose in the long run nonetheless
this perspective may be relevant to a
collective so if we really are a million
people and we really aren't sharing our

08:34

resources this may be this may have some
relevance on the other hand the time
perspective clearly immediately has
relevance to the individual to myself
because I will only be one of those
trajectories in this situation where
averaging in the ensemble yields a
different result from averaging over
time is called non agar DISA tea so this
is an on a gothic game
now what does every DISA t/o I have to
apologize for the resolution of the

09:06

screen here this would be red boxes
Agnew DISA tee very unfortunately is
used in different ways in the physics
community from how it is used in the
economics community in physics we've
just seen that there are these two
perspectives the ensemble perspective
where you remove randomness from the
system by letting an ensemble size go to
infinity
get very large to remove randomness the
other perspective is to let time go to
infinity become very large to remove
randomness in physics when we say a
system is ergodic we mean that both

09:40

perspectives are equivalent and
therefore this last game was non
negative in economics a Gurda City means
that the laws of a process don't change
at least this is what I've been able to
extract from the economics literature so
far
now the logical relationship between
these two uses of the word is that
physics like audacity implies economics
ethnicity but not the other way around
so you can have many systems whose laws
don't change over time meaning they are

10:11

a Ghatak in the way that economists use
the word but they are not agaric in the
sense that physicists use the word
meaning that time and ensemble
perspectives are different now it's said
that this has something to do in this
case in this game with with the
distribution I said that there are very
unlikely enormous enormous lis which
individuals in the population that
basically make up for all those poor
souls who are who are losing overtime no

10:43

one has asked me yet about the the
grasshopper now what I want to show you
is a histogram so how many people in the
population I'm now calling as a
population is ensemble how many people
in the population have a certain amount
of of money so here's here's the wealth
and on this axis here I'm drawing how
many people there are with with a given
wealth I'm starting every one according
to the initial condition of the game and
starting everyone at $100 and then I'll
let time run and in
minute of time I'm playing this for 60

11:16

minutes I will draw you this the
cystogram okay I'll tell you why I put
the graph suppose anyone what that yes
ray you look like you might want to yes
so that's good yeah no it has to do with
the histogram the history of the word
histogram was first used at Gresham
College by one of your predecessors
buying a Pearson yeah right so let's
construct these 60 histograms over time
there we go there they are constructed

11:54

so after 60 minutes this is what the
distribution looks like in this system
oh sorry I haven't told you what this
green line means the green line tracks
the wealthiest individual in the
population so what happened after 60
minutes after we played this game for 60
60 times in a row a million people have
played this one third roughly one third
of the population is essentially
bankrupt so they have less than $1 I
just said that as an arbitrary limit for
bankruptcy
one of them has close to a hundred
million dollars and that's what gives

12:27

you this average that has nothing to do
with the typical experience the average
is actually sitting somewhere somewhere
around here at $2,000 but that's really
just because of these very few
exceptional outliers out there so this
is what is happening in this game right
that's my initial remarks and now we'll
come to the actual talk this is the
basic concept and I'll start I'll start
applying that to a few classic problems

12:57

in economics the first will be the
leverage problem so the question is how
much should I invest in a favorable
investment in a favorable game so let's
look at these two perspectives again the
time perspective and the ensemble
perspective they both have problems
the ensemble perspective looks very nice
but the problem is that it's
inaccessible we would have to be able to
access all these parallel universes so
if that's not really available to us
the time perspective it doesn't look so

13:30

nice it's too risky but maybe maybe we
can find a compromise maybe we can trade
them off and somehow get the best of
both worlds and the idea there is to
keep some of your money but what
happened here was that I was investing
the hundred dollars that someone gave me
in the beginning and then I kept
investing whatever I had at every moment
in time but I don't have to do that I
could say well I'm not going to invest
$100 I'll invest less so I'm investing
less in a favorable game and maybe
that's a good idea

14:02

because not investing something and
sitting on it in this case is like
freezing your money in time so you do
have excess it's almost like you could
go back in time because you can access
that stuff that you haven't spent in the
beginning so this is this is the the
central ideas to keep some money safe in
this game
meaning reduce your leverage what we had
here in the beginning was a leverage of
100% meaning we bet everything we had so
I'll now sequentially reduce this

14:32

leverage and show you what what happens
in the time perspective and in the
ensemble perspective in the ensemble
perspective you can already guess what
happens it's a favorable game and now
I'm investing less in it so my money
will grow more slowly
what's interesting watch what happens to
the time perspective there as I'm
investing less in every round in this
favorable game
I'm also losing less and I'm not just
losing this I can actually go to the

15:01

level where I'm gaining so this idea of
freezing money in time love not
investing it keeping some of it safe
actually works so I can turn this this
game that and the time perspective was
really horrendous and ruinous for us it
can turn that into a favorable work a
little thing by keeping some of my money
safe and every time step so this is what
what I did here if I continue reducing
the leverage of course at some point you
see the time perspective the time
performance going down again because
well you're investing less interesting

15:36

nothing of course I'm not gaining
anything I can also short this game so
can take the position of the house I can
offer this game to people this is like
short-selling a pretty good asset
actually and that would be really stupid
because it is a favorable game after all
so if I'm shorting it that's that's not
a good idea so you can see that selling
short not not so good here what happens
if we do the thing that is usually
associated with the word leverage namely
investing more than we own in every
round I can do that by borrowing money

16:07

well let's do that we started with 100%
let's go 250 well at 150 well you just
saw what happened to until the time
trajectory that it really collapses
let's go to 200 oh my god but compare
that to the ensemble perspective now the
travel is that much of the mathematics
that is used in the analyses investment
analyses risk management analyses
focuses on expectation values
expectation values is another word for
most horrible averages so we're really

16:38

often focused on this perspective and
it's also that is the intuitive
perspective for some reason that's how
our intuition works I'll give you a
favorable game you think on average it's
fine that's true it just depends on what
you mean by average so the ensemble
average perspective just becomes better
that and now the time has completely
disappeared because at this leverage
again t-to go bankrupt so you're just
wiped out immediately I can keep going
it just looks phenomenal and the in the
ensemble perspective is the message here
well what would be the message is from
from these two perspectives if you're

17:11

really thinking about Possible's then
the the recommendation of this game
would be just leverage as much as you
can the trouble is that you will
immediately go bankrupt as we've seen in
the time perspective the time
perspective says there's an optimal
leverage it says this is something that
doesn't exist in the Azam ensemble
perspective so it's not a disagreement a
quantitative disagreement it's a
qualitative disagreement the time
perspective says an optimal leverage
exists if we exceeded it's bad for you
so in the in the case of our game the

17:43

optimal leverage was 1/4 invest no more
than a quarter two gamblers these
results
unknown as the Kelly criterion and in
gambling this is usually derived or
losses I've only seen it derived from
information theory we have used a
different argument here so it's not so
much that we've invented a new
mathematics but we've invented a
different interpretation maybe a ritual
interpretation or a deeper
interpretation than what is known in in

18:16

gambling we've used really the argument
of time irreversibility that you can't
go back in time the same techniques that
we've developed you can apply to other
to other processes for example geometric
Brownian motion which is the most
important mathematical model in in
finance and I've written that up in a
paper and I think it's dishonest to give
a lecture and not show all the technical
details of what I'm doing so I thought

18:50

I'd just give you all the technical
details and you can read them for
yourselves there they are
it should be enough this is another way
of illustrating this perspective so here
you have you have many many worlds right
so the ideas you start in one state of
the world up there and time progresses
down on this slide so this world has
potential this world splits into
potential futures as you go down the
number of possible worlds increases

19:21

exponentially what reality does or time
what time does is it picks out one
trajectory through all these
possibilities so that's reality one of
those trajectories when we're thinking
about the time perspective we are
averaging along one of those
trajectories when we're thinking about
the ensemble perspective we are
averaging across all the parallel worlds
all the possibilities that could have
that could have materialized so I come

19:52

to the second part of my talk there's an
Petersburg paradox and again I won't go
into too much technical detail but I'll
give you a bit of the history of this
problem which is really quite
fascinating
I believe the st. Petersburg paradox in
essence I'm just giving you the essence
of the problem it was noticed in the
early 18th century that if you offer a
ticket for a lottery that is associated
with an infinite expectation value of

20:24

the payout depending on how you
structure that lottery you can end up
with a case where players are not
willing to pay much for it so they have
an infinite expected gain from this
lottery but they're not willing to pay
much for it and people who are baffled
by that in the early 18th century of the
reason for that of course instead I got
a theory had not been developed and
these ideas of Agra disa T are 115 years
younger than that and the resolution

20:56

that I proposed that Michael was
referring to earlier on is that the
expectation value is really irrelevant
to the player so he has no reason to
behave according to it instead what
really matters is the time average
growth rate the way the player will fare
over time in this case under
multiplicative dynamics which is roughly
which is not a bad approximation for for
many wealth processes oh Michael is

21:28

pointing to a a grasshopper well what is
that doing there now right so the
solution is expectation really not
really relevant to the player time
average is on the other hand when you
compute them and you compare them to
human behavior they actually match so
this perspective describes what people
do very very well this paper that I
wrote about that was published in the
philosophical transactions by the Royal
Society and I put the grasshopper on top

22:02

of the Royal Society and now someone can
guess well so I just wanted to point out
that Gresham College was really the the
meeting point or the the origin of the
Royal Society this is this is where that
all
started to another very nice connection
will see more Royal Society later so I
told you that I will talk a little bit
about the history of this problem

22:32

and here it is but I would go a lot
further back then then Michael had
prepared you for so we'll jump right
into the Middle Ages although I know
very little about this I heard that
John Duns Scotus a medieval philosopher
was one of the first people to formally
talk about the idea of parallel worlds
of somehow consistent states of affairs

23:03

that could in principle occur so he had
I'm sure that people had that know so
much earlier on that there are these
other possibilities out there we tend to
think like that but but he made that
somewhat somewhat formal in his
philosophy then in the the 60 in the
1500s in the sixteenth century
k'tano appears and I could probably give
three lectures just about Gaddafi is
really a fascinating man but in in the
context of this lecture I'm mentioning

23:35

him because he wrote a book about games
of dice roughly in the 1550s no one
really knows when it was written because
it could be published in his lifetime
because of course it was sinful to
gamble and this book was all about
gambling it's actually a wonderful book
and it's full of full of good advice
because life's a gamble I guess in that
book apart from lots of good advice and

24:06

deep insights he he begins to count
favorable instances in in games of
chance so he he gets very close to
developing the idea of an ensemble
average and of probabilities and so on
so this is this is perhaps where the for
most development starts the next two
people I have on here
Driggs that grasshopper should be pretty
obvious to you
Briggs of course was the first professor

24:40

of geometry at Gresham College and
Napier invented the logarithm and Briggs
then co-developed with Napier the
logarithm and the date 1615 up here is
the date of their meeting in in
Edinburgh where Briggs suggested some
changes to logarithms and tricks that
computed a lot of them published big
tables for shipping and so on navigation
and now the reason I put this up here is
that in this whole story when you get to

25:11

the actual technical details the
logarithm is the key to almost
everything this is absolutely wonderful
function and I I recommend that you
spend some time thinking about it but
the real formal story starts in 1654
with an exchange of letters between
Pascal and farmer and in that exchange
of letters Pascal of imago essentially
discussing the question how a bet the

25:43

pot in a bet should be split if the bet
never never comes in so you're playing a
game and you have to abandon the game
before it's finished and everyone has
bet something on the outcome of the game
well but then you have to run away so
you're maybe you're playing some game of
dice and suddenly the police bursts in
and you all have to jump out the windows
because the gambling is illegal and you
all run away and then later on you you
assemble somewhere someone took the pot

26:14

of money with him and now you sit down
and you say well but I had four points
in this game starting at three and how
do we split this pot and in a fair
manner and first caliph Omar came up or
mostly family came up with it with the
idea of giving to everyone who was
involved in this gamble the expectation
value of his winnings at the moment the
game was interrupted
so this is where expectation values come
from note that the idea of the expert

26:44

value is a moral idea the question was
how do we split this plot in a fair way
there's not really something about
correct or right or wrong rose it was
about fairness and the early debaters
has a lot to do with moral concepts
because color of course was famous for
for using probability theory to justify
his his religious beliefs now in this
exposition I'm jumping straight to
Nicolas Bernoulli I could talk about
Huygens and Hallie and I have enough

27:15

time so I will mention something about
those two Huygens has a interesting
paper only three years after the
exchange of letters between between
Pascal and firma hogan's writes in one
of his papers that it is really the same
thing if if someone gives you either
three or seven shillings with equal
chance it is the same thing as if he
should give you five feelings and this
really summarizes the belief about
randomness in the in the seventeenth

27:47

century the idea was that you can
completely disregard fluctuations of
course this idea that you're only
focusing on the on the expectation value
means you don't care about miss you
don't care about fluctuations and it
gets you somewhere but not very far in
the in the 1690s Halle wrote a paper
this is maybe the first paper about
pricing financial products in these
papers about pricing life annuity so

28:18

it's a particular kind of financial
product the idea behind life annuities
is that some king wants to go to war or
build a castle to do fancy things that
Kings do and he doesn't even have money
but he has lots of subjects so he
promises those subjects a pension for
the rest of their lives and says well if
you give me amount X now I will pay you
amount y every month until until you die
for the rest of your life the good thing

28:51

there for the King is that he gets a lot
of money right now and sometimes that's
that's needed like you want to go to war
or you want to
build a new palace and the trouble for
the king is that he may charge the wrong
price and then end up paying dearly for
for this quick money that he needed so
some of you have to balance these two
and Hallie suggested that the price the
king should charge as the expected
payout and that he has to give the

29:22

people who buy this this insurance and
that makes a lot of sense so think about
what that means so expected payoff that
means we are talking about ensemble
averages that means we're talking about
the ensemble perspective but what's the
system here well the system really is an
ensemble the king is interacting with
many many individuals that are
essentially independent systems so there
this perspective works but that doesn't
mean it works for everything it really
is an ensemble internet works now in
nineteen eighteen nineteen in 1713
Nicolas Bernoulli invented the listen

29:53

Petersburg paradox this idea of the very
favorable lottery that no one wants to
pay anything for and what he really
wanted to do there Nicolas Bernoulli was
smash this religious belief in
expectation values he wanted to say no
no no there's more to randomness than
expectation values this must be clear to
you
Nicolas padule-- is also one of the
first people to think about extreme
values and in extremes you know that the
expectation value is not very
interesting if you're building a dike

30:25

you're not interested in the expected
height of the next flood you're
interested in the height of an extreme
flood and the expectation value is
really is really irrelevant there so he
was arguing this point for ever and ever
Nicolas was very very creative in his
writing he was very playful with
different ideas different games and
randomness that he invented and this
particular game the sim Petersburg
paradox comes from an exchange of
correspondence with with more more in
1713 so Nicolas Bernoulli invents this

30:56

game Daniel Bernoulli then proposes a
solution and in that solution this is
only for those of you who know what I'm
talking about in that solution he
proposes utility
essentially proposes that people are not
interested in the money
they receive but they are interested in
what that money means to them the
usefulness of that money and now you can
pick any function you like that
translates an amount of money into a
usefulness and you can pick that
function in such a way that your

31:28

infinity in the expectation value
disappears but that's a very arbitrary
thing to do so it's a very dangerous
thing to do because this function this
utility function was not further
specified yes you can get rid of the the
the infinity but you introduce a lot of
arbitrariness that's the problem in in
the newly's solution in 1812 Laplace
wrote a book he talks about probability
theory and he mentions Daniel
Bernoulli's solution to st. Petersburg

31:58

paradox and in doing so he corrects a
little error that Daniel Bernoulli made
now Laplace was a gentleman of course
and he he did not mention that he was
correcting the newly he just said oh
this is how Bernoulli solved the problem
but he didn't say that there was
anything wrong with it he just said that
he solved it in this way which wasn't
quite true he just implicitly corrected
something and then after Laplace
everyone who talked about this in
Petersburg paradox just quoted Laplace

32:29

you can see it in the notation so
there's a famous textbook from 1865 by
Todd hunter who talks about again the
history of probability theory and he
uses the same notation as as Laplace
does when he talks about Bernoulli's
problem and Bernoulli's solution so
Laplace really solved the problem
corrected everything and and so on still
following the same idea of this utility
of Bernoulli now something very major
happened to the way we think about

32:59

randomness in the 1870s 1860s with the
development of statistical mechanics so
this is where everything changes this is
where the idea of a greedy City begins
everything that was before was only
about expectation values was only about
counting favorable instances in a set of
possibilities essentially
but in the 1860s 1870s Maxwell and
Boltzmann were working on statistical
mechanics so now they had a physical
system and in this physical system you

33:33

could you can do experiments very
controlled precise experiments so this
is no longer this vague field of human
behavior and how we feel what our
characters are and so on we're now
talking about molecules this is where
this is where Maxwell and Boltzmann were
coming from so they didn't have the
excuse to introduce functions to get rid
of strange behaviors of expectation
values and that meant that they had to
really dig deeper and find out what what
the issues were and so maximum Boltzmann

34:06

said well it's it's we always use these
expectation values but really the reason
we're doing that is that in many systems
or in some systems and the ones that we
are dealing with right now the
expectation values are identical to the
time averages so there's there's the big
insight and they said well of course
that doesn't have to be that way
sometimes they are very different so
let's just call the systems method that
works at gothic and that's where where
the electricity notion comes from and
web starts so from the 1870s we really

34:37

have different concepts in dealing with
randomness than than before 100 years
later so I will lie to you now in the
1950s
Kandee published a paper where he used
these ideas when he used the idea that
what was really important to you is how
you fare over time not in an ensemble of
parallel worlds and hypothetical
possibilities so he wrote this paper in

35:06

1954 it was immediately accepted and
celebrated by the economics community
and he received his Nobel Prize for
economics only a few years later as did
Ed Thorpe who then used Kelly's methods
extended them a little and solved the
problem of blackjack famously went to
use could see him here in Las Vegas went
to Las Vegas and actually demonstrated
in a real-life experiment that you can

35:39

use these ideas to beat blackjack then
in 1991 thomas cover wrote optimal
portfolios and of course also received
his Nobel Prize soon after for economics
except that was all a lie these people
never refused received Nobel prizes they
were shunned by by the economics
establishment instead something else
happened in 1934 karl menger wrote a
paper that proves that everything I've

36:11

just said in this whole lecture is wrong
this was mathematically proven in 1934
and what happened after that was that in
1950 one can arrow looked at megohms
paper and noticed that Vegas paper was
quite hard to read so he tried to fish
out the essential argument and the
essential results and he wrote that down
in a very clear in a very clear way
in 1968 then Samuelson

36:43

I think introduced this new notion of
ergodicity in economics and he said that
well for something to be for us to be
able to treat something scientifically
that something has to be a gothic and
you can see if he's using the notion of
ergodicity that's prevalent in economics
and that makes sense because he says the
laws that govern the system mustn't
change over time if they change over
time what can we do with it we can't
predict anything then what
so that's right but that's not actually
what people meant by agar DISA tea and

37:15

he didn't quite notice that so he said
something is ergodic you can do science
with it if it's not a ghatak all you can
do is history you can just tell what
happened in the past but it has no
bearing on the future so that's that
1976 Markowitz mentioned that he accepts
mengas argument from 1934 in 1977
Samuelson wrote another paper about
Menger and he said that manga is a
modern classic that stands above all

37:44

criticism in 2009 can arrow
said that a deeper understanding
referring to the newly a deeper
understanding of this Sam Petersburg
paradox was only achieved with manga in
2011 Peters that's me
so that manga was wrong there's actually
an error in in manga and the error goes
as follows Megha needed to show and this
is now where all the equations come in
right I think this is the only slide

38:18

with equations so Megha needed to show
that a quantity a plus B is infinite so
he came up with the following argument
he said well I can show that a is
infinite positively infinite so I don't
really have to worry about what B is
because infinity plus anything is still
infinity it's not bad but the problem is
that B actually diverges negatively so
it becomes minus infinity and it becomes
minus infinity faster than a becomes
plus infinity and the consequence of

38:49

that is that a plus B is not plus
infinity but minus infinity so you can
see that this is a somewhat subtle error
and if that's in a paper that's not the
clearest paper you've ever read then
it's easy to read over it and not not
see it still it's surprising that this
stayed in the literature for 77 years
without anyone pointing it out now what
do you do when you see something like
this you see a classic that stands above
all criticism it's wrong now I should

39:21

mention that okay I lied about these
people in their Nobel Prizes
but these people all have Nobel Prizes
so arrows Samuelson Markowitz are all
economics Nobel laureates and I'm saying
that the endorsements are endorsements
of something that is false so I
contacted can arrow and it took some
time and help of good friends at the
Santa Fe Institute Murray gell-mann was
very helpful Jeff West was very helpful

39:52

and eventually the summer in August Ken
was kind enough to meet up with me and
sit down in a room we locked ourselves
up for for three hours
actually with Murray gell-mann and we
went through every little detail in this
problem in this manga problem and after
our meeting Ken agreed that manga was
indeed wrong so for me that's a big step
forward and I'm moving Ken now Albert on
this penultimate slide I will explain to

40:25

you why some people here on the right
and some are on the left up there I have
put ensemble perspective and up here I
have put time perspective so the people
who are closer to the ensemble
perspective with me thinking I've put on
the right and those who are closer to
the time perspective when they're
thinking I put on the left and following
arrows statement that Megan was wrong I
have moved him to the middle which is a
good place to be so that's the end of
the two specific problems I wanted to

40:55

talk about and I'll come to a few
conclusions that I can draw out to any
length but let's see a quick summary so
what happened here we've really in this
work we've reframed we reframed risk
management how you think about risk
instead of thinking about risk
preferences and such things I want to
think about the effects of time so the
key concept really for me is time what
you need to do to be able to do that and

41:26

this is this is really crucial I've seen
this in comments on blogs that people
have written so on that this is
something that's difficult to get is you
have to extend the probability spaces
that you're used to thinking about to
dynamical systems so the ideas it's not
enough to think about just probabilities
probabilities don't get you far enough
in addition to the probabilities you
need a dynamic you need to know how
things move over time and that's
actually not in classic measure
theoretic probability theory so there's

41:58

a little addition but we know how to do
that that's called dynamical Systems
Theory we don't need any utility
functions that's very nice and then I've
just mentioned we found this 77 year old
error in a paper that had been endorsed
by several Nobel laureates we found a
slight error and
Bernoulli's work now the fact that we're
finding these old arrows is just the
reason for that is that we have a
clearer conceptual framework and that
makes it possible to spot these arrows
it's not no one no one on any of these

42:30

slides is stupid
they're all phenomenally intelligent
people they're probably much smarter
than I am but they had a different angle
and therefore they couldn't they
couldn't find these these arrows yes and
the Kelly criterion is is out there it
has a slightly different perspective the
mathematics looked completely identical
as long as you have multiplicative
dynamics that's what the Kelly criterion
is developed for our arguments of time
averages extend beyond multiplicative

42:59

dynamics so what next what can we do
with this that's all very nice we solved
some classic problems that other people
have solved before using different
methods and we suggest a new one to us
this is a proof of principle
so we're saying this perspective that
we've developed is very powerful because
we can solve these old problems with new
concepts and new concepts open up new
possible questions other questions that
you can address one well Michael
actually protogynous in his introduction

43:31

this really makes you think about time
versus parallel worlds in your in your
daily life I think that there's
something in our culture where we we
undervalue time and we behave as if we
had access to it two parallel worlds an
example that I always think about is is
mass production we have all these
products that look exactly identical and
if one breaks we can we could just
replace it it's as if we could go back
in time but as if we could switch it and
I wonder if that does something to our

44:00

to our psychology to the way we approach
life and and time well I don't think I
have to tell anyone here that it's easy
to underwear you time that that's really
the only thing that is ever lost that
has ever really truly destroyed us is
time so I think there's there's a lot of
deep thought in these concepts on a more
practical level we can we can start
thinking about market stability in in it
in a completely different way and I
have started doing that with my with my

44:32

colleague at the London mathematical
laboratory Alex animal in a paper that
we've what we've been unable to publish
so far because we call a new concept
stochastic market efficiency where we're
saying that markets are efficient in a
way different from how we usually think
about them they are efficient in the
sense that you can't beat the market by
leveraging an investment in it and this
is a really curious concept it makes a
lot of sense because you can imagine if

45:03

if you could just do that everyone would
do it
but what if everyone did that well it's
inconsistent it's unstable so there must
be something more than just price
adjustments there must be something like
adjustments of fluctuations of
correlations they're constrained by this
new concept of market efficiency that
you cannot beat the market by leveraging
it another big area that opens up from
this new perspective is inequality and

45:34

welfare economics and I just want to
take you back to this to this very
central slide to this talk where we have
the two perspectives
you know Samba perspective and the time
perspective and the population average
increases but the typical individual is
is losing that says something you
already know that it says something
about the distribution here and well
inequality in economic terms has
something to do with the distribution so

46:07

we can learn something about the
evolution of of these distributions and
well really what does what this
illustrates is a fundamental conflict
between between a collective and an
individual so you can have this sort of
disenfranchisement where people feel
they're not participating it's very nice
that we see GDP for instance increase
but we may not be participating and so
it's it's even worse than that
the typical individual main

46:41

be participating in economic growth that
is that is evident in GDP because GDP is
one of these ensemble averages so
they're they're deep messages that goal
that go far that they have big practical
consequences finally now conclusions
something I've learned in this project
is that we really need to rethink the
way we communicate in in economics and I
mean this I mean the field the

47:13

scientific the academic field of
economics no economics journal has
published anything of what I've just
mentioned here some rejection letters
from economics journals know economics
conference has let us speak we applied
to conferences we said we'd like to come
and tell you about something we found we
have not been invited of course this has
been published but it's been published

47:43

by other by other entities the Royal
Society has published it the Santa Fe
Institute has published it there's a
TEDx talk on the Internet
quantitative finance has published it
there are various blogs that you can
find and there's a lot of interest now
from from financial firms another
message that I'm taking home from from
from this from this whole project is
that I think we've given up too easily
when it comes to economics
scientifically we've given up too easily

48:15

and here's an argument that I've often
heard made by people who deal with
economic systems they say well economic
economic systems cannot be predicted
because of reflexivity
it goes like this you make a prediction
about a system then the system responds
to your prediction and that invalidates
your prediction so your prediction is
useless
but if you can't make predictions about
something you also can't use scientific
method because scientific method relies
on predictions predictions are what you

48:46

use to test your hypotheses and if that
doesn't work then just the whole
framework disappears and this is
actually a claim made by many who deal
with economic systems they say this
is a different animal you cannot you
cannot treat that with scientific method
I don't believe that I truly disagree
and I think I've seen it in my work that
it's it's right to disagree with this I
believe that this is wrong first of all
because not all predictions elicit a
response what do I mean by that I mean

49:17

that I can make predictions about a
system that are completely useless and
I'm really into making useless
predictions because I think that making
useful predictions of focusing on them
is is an anthropocentric so nature is
much richer than that nature has much
more structure then what is useful to
humans and if we only focus on what is
useful to humans and we miss a lot of
that structure so research science also
economic science should be driven by

49:48

curiosity not by fixing practical
problems because furiosa tea gets you to
those structures in nature that you can
understand it's irrelevant whether that
is useful or not in the end this may
become useful but don't worry about it
just let yourself be guided by by your
curiosity and I think that we need to do
that so let's make predictions that
don't elicit responses meaning make
Lieut useless predictions of course
other predictions that are useful may
still be valid actually if if they if

50:21

they recursively take the effects into
account and the last line on the slide
is this hints at a utilitarian
understanding of Science in economics I
mean I mean that in economics we seem to
focus on the useful things and I say
let's not do anything useful we're
scientists we're not businessmen finally
mathematics I really learned a lot about
mathematics in this I was trained as a

50:52

physicist so for me the world started
around 1900 there wasn't really anything
before then and now I've looked at
mathematics a little bit and it's just
wonderful this goes back there were
centuries before the 20th century and it
actually goes back millennia and
and you really begin to feel this that
you're part of a part of a society of a
community of people who've thought hard
about difficult problems for thousands
of years that's a wonderful wonderful

51:23

thing so mathematics is much richer than
a tool that we use to I don't know what
you use it for mathematics is much more
subtle than then appreciated in in
economics and I think we've seen that in
these two notions of a Gerda City for
example the the notion that the
mathematicians have come up with the
physicist mathematical physicists have
come up with is is more refined it's
more precise mathematics is also more
vague strangely and that's because it's

51:54

really concept dependent so when you go
through these through these old papers
you're reading you're reading Bernoulli
and you're asking yourself did he did he
make a mistake here well you suddenly
realize that depends it depends on what
he was trying to say maybe he's talking
about something completely different he
had a different idea and this thing that
seems wrong to me is not wrong it's
wrong from my perspective or not from
his perspective and mathematics is full
of that things can be once you've
established the concepts things can be

52:26

right or wrong correct or not in
equation the two sides of the equation
are the same or not but to get to the
concepts to what's relevant that's
really where where mathematics is a lot
of fun
mathematics is also more advanced I
think that economics mostly uses
mathematical techniques from from the
19th century at best of the early 19th
century and finally mathematics is also
more correct as we've seen in the case
with manga and with that I thank you

52:58

very much for your attention

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