one name it's not on my slides and I'm
like suddenly okay
I know but just like suddenly God
the reception field
okay so maybe we'll start
hello hi everybody I would like to
welcome you all to this evening's
lecture my name is Elaine E&L and I am
the chair of the math department here at
Stanford this lecture is one in a series
of public lectures that it was organized
by the Stanford mathematics department
and is sponsored by the Stanford
mathematics Research Center and the
friends of Stanford mathematics and if
you would like to be notified of events
like this and you are not on a waiting
list on our mailing list please send me
an email at chair at math dot
stanford.edu
this evening were very fortunate to have
as our speaker professor Jordan Alan
Berg before we begin I would like to
tell you a couple of things about our
speaker this evening Professor Alan Berg
grew up in Potomac Maryland a son of two
statisticians he got interested in
mathematics at a young age he went on to
Harvard for college where he got a
degree in mathematics in 1993 then he
got a master's in fiction writing from
John Hopkins well it is and then he
returned to Harvard to get a PhD in
mathematics in 1998 under the guidance
of Professor Barry Mazur after spending
a few years at Princeton as a postdoc he
joined the faculty of University of
wisconsin-madison in 2004 where he is
now the John T MacArthur professor of
mathematics professor elenberg won
several awards and honors let me just
mention a few here in 2001 he won the
Young Investigator award from NSA in
2005 he got the Sloan fellowship and
also the Career Award from the National
Science Foundation in 2012 he was part
of the inaugural class of the fellows of
the American Mathematical Society and
just recently in 2015 he became a
Guggenheim Fellow professors elenberg
research is centered around numbers
theory in arithmetic algebraic geometry
and more broadly across several other
areas of mathematics on the more
algebraic side it also includes work on
a variety of counting problems in number
theory using symmetries and ideas from
topology and also part of a long-term
collaboration with two other colleagues
here at Stanford in Stanford math
departments professor Akshay batten
cottage and Tom Church just earlier this
year professor Allen bird made a
breakthrough in community oryx with his
solution to a pattern matching question
called the capsid problem which was
featured in May issues of quanta
magazine in addition to his research
professor Alan Berg has been writing
numerous expository articles on
mathematical topics for a variety of
publications including the New York
Times Slate The Wall Street Journal
Wired in The Washington Post he is also
the author of two books the first one
are novel the grasshopper King was
published in 2003 and more recently the
book how not to be wrong the power of
mathematical thinking that was published
in 2014 which was on the New York Times
and The Sunday Times bestseller and a
winner of the 2016 or allure book prize
the world so we are really delighted to
have Jordan tonight speak about how to
use math to get rich in the lottery I
should say there is a catch that most
likely this will not help you get rich
in the lottery but instead Jordan will
use this topic as an excuse to take us
on a tour some interesting mathematics
so please join me in welcoming professor
Alan Berg tonight
well thank you guys so much I'm just
checking everyone's still here after I
only revealed that I'm not gonna tell
you how to get rich okay it's always a
great pleasure to be here and that
amidst the incredible mathematical
atmosphere of Stanford I should say I
have been doing math of these people
non-stop since like nine o'clock this
morning so but I think I still have like
this much energy left I tell you all
this story because you know telling
stories about math is like really what I
do in this book and like what I do when
I'm writing articles in the newspaper on
mathematical topics so let me tell you a
story about lottery so what you're
looking at here it's a drawing for the
Massachusetts state lottery a game
called cash windfall actually for
reasons I don't fully understand you can
go to youtube and see a record of every
single drawing that was ever held in
this game what you are looking at
although to be honest it looks like all
the others is a picture of the last ever
drawing of cash windfall and the point
of this story is to explain why it was
the last ever drawing of cash windfall
so in order to explain that I've got to
start by telling you a little bit about
how lotteries work which maybe some of
you know but maybe it will be a review
so first of all you know in math if this
thing it's kind of complicated like a
lottery we like to make it simpler to
explain it so let's here discuss a
version of the lottery where well if so
I was thinking realistic tickets cost
two dollars which is about how much a
real lottery ticket usually costs but
here let's not have this kind of
complicated structure let's just say
there's one big prize the jackpot which
is three hundred dollars and one in
every two hundred tickets is gonna win
so when we think about playing the
lottery we usually think about it sort
of in the long term we say what happens
if you play the lottery a lot and people
who play the lottery by the way do play
the lottery a lot
so this is not an unrealistic assumption
we say what if you play this claw turi a
thousand times well if you win one if
one out of every 200 tickets is a winner
you'll probably win about five times and
so at all you're gonna come away with
$1,500 or a dollar fifty per ticket
which sounds I mean that sounds pretty
good like fifteen hundred dollars is not
a trivial amount of money until you
remember that you actually spent two
thousand dollars in order to get those
1000 tickets to win that $1,500 in some
sense now you already understand what
lotteries are about um let me let me
just sort of start by telling you about
the piece of mathematical terminology we
use to describe this I'm gonna introduce
it only in order to criticize it so
usually what we would say is that the
expected value of the ticket is a dollar
and fifty cents and you remember that
was like that was like how much per
ticket you were likely gonna win and
this is one of those mathematical terms
that you know everybody kind of wishes
we could go back in history and take it
back and change it and call it something
else because after all this expected
value one thing it is definitely not is
the value you expect the ticket to have
it's exactly the opposite it's like
literally a value that the ticket cannot
possibly have the ticket is either worth
nothing or it's worth $200 it's
definitely not worth a buck 50 and yeah
we call it dollar fifty the expected
value so maybe a better name if we could
go back in time and change it it would
be something more like the average value
because that's really what it represents
and that's somehow the first step in our
analysis the lottery is to compute
expected values or average values and
say if we play this game a long time how
much per ticket are we likely to win or
lose and on average in this game i just
showed you this sort of toy version of
the lottery you would expect to lose
about 50 cents per play so i said this
was sort of not a very realistic lottery
and actually one respect in which it was
it's not realistic is that it's actually
incredibly generous to the player
compared to a real lottery so what
you're looking at here is the payoffs
from an actual Massachusetts lottery
game from around 2005 I don't know the
exact date
here's how this looks I'm sure you
remember I'm sure you counted how many
balls there were in the cage and the
very first slide but in case you didn't
there was 48 balls when the cage spins
you pull out six of them and of course
you at home pick your six numbers and
try to match as many as you can of the
ones the come out of the cage so if you
match all six that's great you win the
jackpot and that's some incredibly large
number all the money goes in the jackpot
pool but the chance of that happening is
very low about one in nine point three
million so but of course that very
rarely happens and you want people to
want to play you want people to feel
that they have a chance to win something
and so a real jackpot has all these
lower tier prizes right if you match
five out of six that's still a very
handsome prize of $4,000 if you match
four out of six now that's starting to
be not so unlikely right one in eight
hundred times if you buy ten tickets a
day you're probably gonna see that
happen from time to time and you add 150
bucks that sort of feels pretty good
actually one clever thing about this
game I like is that if you match two out
of the six they don't give you two
dollars they just give you another
ticket so you can keep playing that's
very clever um anyway when you add all
this up you can figure out that for
example the one let's say the black pot
is 1 million dollars the 1 million
you're gonna win only about one every
ten million times so probably people
really don't play that long but if you
did it could mean in math you like to
sort of like extrapolate things out to
infinity if you played 10 million times
you would win 1 million dollars on
average which is about 10 cents of play
not very much and if you actually add up
the values of all these prizes
you get a truly measly number just under
80 cents of expected value for your $2
ticket so that's much worse than the
baby game I showed you and most state
lotteries look something like this so
here's the thing in the time period I'm
showing you Massachusetts had a problem
with its lottery and the problem was
that people were not playing it and it's
probably not because people like made a
complicated table like this it's more
psychological than that running a state
lottery is in some sense an exercise in
a both applied math and applied
psychology what happened was that nobody
won the jackpot for a long time and you
know the jackpot pool just kind of piled
up and up more and more money in it and
on one sense that makes the lottery more
enticing when the jackpot is very large
but people started to get a little
dispirited and people start to feel like
they're not gonna win and people stop
playing and when people when fewer
people are playing it's even less likely
that somebody will hit the jackpot and
you can see that there's sort of like a
downward death spiral of this of this
lottery so the people at the
Massachusetts lottery had an idea they
they borrowed an idea from from a game
in Michigan which is also called
windfall a recently discontinued game
called the roll down this is how the
roll down worked they said well if
nobody wins the jackpot we don't want it
to just keep on getting bigger and
bigger and bigger and nobody wins it
let's say if it goes over a certain
threshold it was 2 million that they
picked if it goes over 2 million if that
week nobody wins the jackpot we're not
just gonna let the money pile into the
pot we're gonna roll that money down
we're gonna take that money and put it
into all the lower prizes and make them
work and make them worth more so we're
not gonna have this thing where nobody
where nobody wins that money is gonna
get paid out so their goal was to have a
lottery system that was more enticing
that seemed to the players like a better
deal and they did a good job and they
did get a lot more people to play
in fact um they sort of did too good of
a job it's always the great
accomplishment of my life if I can get a
laugh with the table
that's awesome edition that's what you
really want so what you're looking at
here is the payoffs for cash windfall in
February 7th 2005 which was the first
roll down drawing on the new game so if
you compare this I mean it's quite
startling right this hitting four out of
six which before was worth four hundred
dollars on this day because nobody would
won the jackpot was worth almost $2,400
hitting three out of six I forgot how
much I was worth before which is a mere
$5.00 prize before was now a six dollar
prize that's something that is $60 prize
you know paying attention that's good
and this isn't it is really not at all
unlikely so in this case the expected
value of a $2.00 ticket even though
you're not going to win the jackpot it's
actually five dollars and 53 cents so
that is a pretty good deal by the way
you might ask why do I happen to know
the exact payoffs for February 7th 2005
of all days well the reason I know it is
because I read it in this document
you're not really supposed to be able to
read this from where you're sitting but
what you are looking at is a letter from
the Inspector General of the
Commonwealth of Massachusetts to the
state treasurer of Massachusetts this is
the first page of a 25 page PDF document
we just tried to explain what had
happened to the Massachusetts State
Lottery and it's a very interesting
story i well I in fact I would venture
to say that this is the only municipal
fiscal oversight document you will ever
read that makes you wonder if somebody
has the movie rights to it so the reason
I know that February is the exact
payoffs on February 7th 25 20 2005 the
first day of the the first roll down of
the new game was that on that day the
state lottery got a call
a cashier in a star market in Cambridge
Massachusetts sort of saying like so
some college kids just came in and
bought $5,000 of lottery tickets is that
strange like should we allow that
cashiers are supposed to call in if
there's sort of an unexpectedly large
lottery purchase so what had happened
let me back up a little bit so as it
happened in January of 2005 which is
sort of the inter semester period at MIT
they have um a tradition of having sort
of short-term three-week independent
studies in between the semester and a
student named Brian Harvey he was a
senior at MIT at the time he had what in
retrospect is now seen to be the
incredible good luck of doing an
independent study about the expected
value of state lotteries so we can
presume that he made a table pretty much
like the one I just showed you and as
you can imagine the next thing he did
was sort of like go around to everybody
in his dorm and say like if it's okay
you should really give me like all the
money you have right now so I can go to
the star market and buy lottery tickets
and that is exactly what you did and
this group of people did indeed about
triple their money on that first drawing
make they call themselves random
strategies that was the name of their
that was the name of their group
actually interestingly nothing you may
say this is something how very not
random what they chose to do but um but
random was the name of their dorm MIT
being MIT they all lived in a place
called a random hall actually and by the
way let me just tell a little more of
the story that was not the only high
volume purchase of lottery tickets on
that day so there was also a huge
purchase in a drugstore in Quincy
Massachusetts on the Southshore which
turned out to be from a large group of
biomedical scientists at North
who were known as the who were known as
the dr. John lottery Club I feel like
that's the best name that anyone had and
there was also a big purchase in
Williamsburg Massachusetts and that
purchase was made by jerry selbee who
was a retired engineer from michigan and
now for listening comprehension who
remembers why else I have mentioned
Michigan in this tie
exactly so Jerry Selby was a guy who for
many years had been making a lot of
money playing Michigan windfall until
Michigan closed down that game and he
absolutely could not believe his eyes
when he saw that a year later
Massachusetts opened the same game so if
you look on your map you will see that
Williamsburg which is in the northwest
corner of the state is in fact the
geographically closest point in
Massachusetts to the state of Michigan
so he immediately drove there to lists
of until he crossed the border and got
to the first convenience store he could
find and he started buying tickets so
the whole story it's like many amazing
twists and turns and let's just say I
mean because this kept on happening in
the lottery kept on rolling down and
this group of people kept on buying
tickets with the money they made they
sort of you know you roll your
investments back into your plan and you
buy more tickets until it came to the
point where on a given roll down day
approximately 80% of all tickets sold to
anyone in the state of Massachusetts in
this game were being sold to either
random strategies the dr. Jahn lottery
club or jerry selbee so this is it so
this is what happened and how does the
story end it's like a bit anticlimactic
but here what you're seeing is the front
page story of the Boston Globe from July
31st 2011 and this is when the globe ran
the story of like that something had
gone awry with Massachusetts cash
windfall that this game was essentially
being used as a piggy bank by a very
large by several groups
of high-volume betters and at this point
the game is up once this story comes out
because you as I said lotteries are
about math but they're also about
psychology so when people felt that the
game was rigged
then people stopped playing and then the
money dries up right then it doesn't
work anymore
so in some sense I've told you the whole
story but if you look at the story from
a mathematical lens I mean when you read
a story like this you start to sort of
have some puzzles and it's really those
puzzles that I want to talk about
tonight there's two one easy one hard
let's start with the easy one the first
puzzle is how could you actually get
away with this I mean may I remind you
that the state knows who wins the
lottery because I have to give you the
money right it's not a secret so the
state certainly knew that the winning
tickets were being sold in the measure
of 80% to the same three convenience
stores so you have to wonder like how
could no one catch on to what was
happening and and and this this one is
easy the answer is that the state
totally figured it out and totally knew
what was going on in fact now I'll sort
of pull aside the criminal but I told
you that the first thing that Brian
Harvey did when he realized the
structure of cash windfall and this
loophole that allowed it to be used as a
piggy bank I told you that the first
thing he did was go and round up all of
his friends take all their money and go
buy a lottery tickets but that was
actually the second thing he did um the
first thing he did being the kind of
like good honest young man who goes to
MIT was he took the tea out to Braintree
Massachusetts where the state lottery
headquarters was and if it sort of like
went in um and he showed them his table
like something like this and he said
like I'm planning to buy as many lottery
tickets think I can afford this week and
make a lot of money is that legal
and the inspector general's report does
not record what he was told but it must
have been something like sure go knock
yourself out because because we know
what happened we know that he bought
tickets and kept on buying tickets and
by the way I didn't emphasize this but
if you looked at the date of that news
story that was 2011 okay so six years
okay so that's in some sense the answer
to the puzzle but it instantly spawns a
sub puzzle which is why did the state
not care that this was happening okay I
shall explain so here's how here's my
this is now you're seeing the limit of
my powerpoint skills so here's a diagram
of what happens in this lottery game in
a lottery the state is gonna take two
dollars from every ticket okay and when
it takes that two dollars a dollar and
20 cents goes back into the prize pool
and 80 per day 2 cents has taken his
revenue right to like build roads and
pay schoolteachers and put up
streetlights and all the things that
state revenue does so for the sort of
the value proposition for the state is
actually incredibly simple 80 cents of
revenue per ticket that's it no more
computation needs to be done what that
means is that the state doesn't care who
wins that is absolutely irrelevant the
state cares about one thing and one
thing only and that is how many tickets
are sold and these guys were buying a
lot of tickets so I think this is
something that when the story was
initially reported it was being reported
as that these guys had somehow were
stealing money or maybe not quite
stealing but sort of somehow funneling
money from the state but the inspector
general reports that actually over the
life of this scheme the state actually
came out almost ten million dollars
ahead of where they would have been
without these high volume betters and it
ends
a state of confusion because it is sort
of hard to claim that you got scanned if
you made ten million dollars maybe Trump
could do it I don't know so I mean this
diagram should show you if the money
wasn't coming from the state where was
it coming from and the answer of course
is that it was coming from the other
people who were playing the lottery
because remember these guys were only
playing on the roll down day only when
the table looked like the good table I
showed you not what they would look like
the bad table I showed you but everybody
else in Massachusetts was playing every
day just like people do when they play
the lottery and so the way to think of
it is that this was basically a big cash
transfer from regular lottery players to
random strategies and the other
high-volume betters with Massachusetts
taking 80 cents each time it went by so
for this point of view you can see why
Massachusetts does not halt this
practice because Massachusetts was
making Bank from this practice in fact I
think the right way to think of it you
know a lot of people may be influenced
by this sort of similar story about MIT
kids who were like winning a blackjack
sort of said these guys figured out a
way to beat the house and what I want to
say is that that metaphor is completely
wrong because what random strategist
were doing was they were making a huge
number of bets each one of which has
positive expected value maybe sometimes
they were gonna win sometimes they were
gonna lose but because the bests were
weighted towards them if they made a lot
of bets in the long run they were gonna
come out of the house they were gonna
come out ahead and if that is your model
you are not beating the house you are
the house and that I think in the end is
the right quantitative way to see what
was going on here but random strategies
had in some sense created a kind of
virtual casino which was them at which
the other players would play and what
was the state well the state was the
state so it sort of looks a lot like
this right
I mean they are the casino people
happily line up to kind of sort of lose
money at a fairly steady rate
occasionally win but mostly lose and
each time that happens Nevada kind of
comes in and reaches in at an and takes
a little bit of the money oh no what
happened
I need a tech guy to come save me if I
get cancelled like go back to the way it
was
did that guy leave yeah okay I mean if I
had canceled and I couldn't just cancel
the entire talk I don't know what was
gonna happen okay okay so right so
Massachusetts had no more incentive to
shut down random strategies but Nevada
has to shut down all the casinos in Las
Vegas would make no sense for them to do
that um so that's one puzzle solved but
I want to kind of talk about a slightly
deeper one so I've sort of been
presenting this story with all three of
these groups the same but they were not
the same there was one difference the
dr. John lottery Club and jerry selbee
used what's called the Quick Pick
machine do you guys know what the Quick
Pick is how many of you got a lot of
liars here tonight okay I'm okay the
Quick Pick machine is that when you buy
a lot of lottery tickets you just have
the machine spit out random numbers
because you know it doesn't really
matter what numbers you have the machine
doesn't you know the lot the balls in
the case don't actually know like what
your birthday is or like what your cat's
birthday is or anything like that
their numbers are random and so it
doesn't really matter what numbers you
get so to save time you just have the
computer pick numbers for you
and if you're buying 200,000 tickets as
random strategies were doing at the high
point of their scheme it would
definitely save you a lot of time to use
the quick pick but they did not do this
they had their 200,000 tickets that they
liked they were filled out by hand and
those are the ones that they used this I
found extremely puzzling it's mentioned
in the expected general's report but
they don't sort of comment on how
strange it is maybe I'll just take I'm
doing well with time so let me just a
little excursus this is like for the
kids to give good life advice this may
sound like awesome like Wow
like you can just like play lottery and
like make money if you sort of happen to
be smart and see a loophole okay so
first of all there's a lot of extremely
boring work involved if you actually
sort of read about what these guys did
filling out 200,000 lottery tickets by
hand extremely boring even if you even
if you don't do that it turns out that
if your income for the year is is made
of lottery tickets you have to fill out
a tax form probably each win and loss to
like verify that you uh you know your
net income so this is like incredibly
time consuming and you have to save them
all in case you get audited and if your
main source of income is lottery tickets
you do get audited so so I so I talked
to Jerry Selby who was sort of that I
mean he had been doing and remember for
like six years in Michigan and then an
years in Massachusetts he actually had
to build a barn behind his house in
order to hold his hundreds of plastic
tubs have used losing lottery tickets
and altogether these guys in random
strategies they made three and a half
million dollars which sounds awesome
but that took six years and there was
ten people now ask yourself if you
graduated from MIT and you just like
wedding got a job so I'm just saying
kids like I don't recommend this as like
your life plan I think I mean I think
they probably could have made more money
and somehow maybe did something more
fulfilling but for some people the joy
of beating the system is its own reward
right so I think that's the way that's
the way to say okay that's like I had to
top step away from mathematics and give
like a life lessons interlude so anyway
the question is why would you fill out
the tickets by hand
so in math again we're trying to
understand a problem let's try to make
it simpler and hope that we don't and
hope that we don't lose the main
features of what we're trying to study
so what you're looking at here is a game
for some reason people often call it the
Transylvanian lottery and it's much
simpler than the real lottery because
instead of 48 numbers there's only seven
and the cage instead of giving you six
balls with numbers on them it only gives
you three and what I like about this is
it it's simple enough that I can
literally write down all of the possible
jackpots on a single slide all the way
of choosing three numbers out of seven
which for the combinatorics fan is a
number called seven choose three which
is 35 and and I encourage you to sort of
check whether you believe that I've got
them all so let's imagine that you are a
high-volume better in this game but in
this game there's only 35 tickets so
it's sort of I mean the high-volume
better is maybe not that high let's say
for us a high-volume better is somebody
you buy seven of these tickets okay
that's pretty much that's like one fifth
of all possible tickets and the question
is how should you do this well first of
all I got to tell you what the rules are
of this lottery so again let's make this
simpler we can't have like is that many
tiers of prizes when there's only three
numbers I'm just gonna sorry I'm just
gonna quickly when we have that little
glitch it changed my view and I want to
see it sorry let me just bring this back
to where you are I like this view better
okay so in this lottery there's only
three things that can happen if you get
the jackpot that's great you win six
dollars if you get two out of three so
let's it's a if the jot the jackpot was
137 and you have one two seven you get a
smaller prize called a deuce you only
win two dollars and if you only get one
out of the three or zero out of the
three you get nothing okay do this sort
of mimics the features of a real lottery
the more the more numbers you get right
the more money the more money you win so
what I want to look at is what would
happen if you played this game with a
quick pick so you're over again you're a
big better and you get oh I didn't
expect that to happen
okay
okay so what happens is something like
this as you can imagine I mean you might
get very lucky and win a lot of money
you might win very little money if
you're unlucky this kind of a spread a
nice sort of smooth curve and you can
check that your expected winnings in
this game is six dollars on average
you're gonna get two point four of these
deuces that's four dollars and 80 cents
and you're gonna get zero point two
jackpots as you could actually check by
the way right because you have one fifth
of the tickets so the chance that you
have the jackpot is one in five so about
every time every five times you're going
to hit the jackpot so let's do this cuz
I think I've got plenty of time we're
gonna kind of actually play this game a
little bit now if you um sometimes when
I have like a smaller room and people
are desks we do this as in an
interactive way I think we can actually
there's so many kids here that I sort of
want to do it is that okay okay let's do
it it's decide I heard some Rando yeah
guy yell yes and I've decided that's my
cue okay so what you're gonna do and I'm
gonna use this time to try to get my
view that I like back by the way um I
have picked seven tickets that I like
why don't you guys pick seven tickets
that you like but before you do I'm
gonna tell you the rules of the game
this is gonna be an elimination game the
rules of the game are though somebody
one of you guys whoever feels the most
random is gonna pick random numbers for
us if if you win six dollars or more you
get to stay in and if you in less than
six dollars you're out well I asked you
Sparrow because we solve it the expected
value of this game was six dollars so
probably some of you guys are gonna get
more some of you guys are gonna get less
well play ap rounds let's see who stays
in okay so if you are here especially if
you're a kid if you maybe if you can
memorize seven tickets that's great
but maybe if you have like a piece of
paper or something - JA something down
right down right down seven tickets
choose ones that you like
and I'll give you guys a minute to pick
out ones that you like
I mean as a thought by the way while
you're doing this you might want to kind
of think about a bad strategy would be
to pick seven copies of the same ticket
you could do that but that would be bad
for this game because then it's very
feast or famine right you might get
seven jackpots but very likely you're
gonna get nothing and be out so that's
something to think about in terms of
okay now while people are finishing up I
need somebody who the very important job
maybe Brian will do it Brian will you be
my ball cage and like sort of randomly
yell out numbers do you feel random
enough to do it I'm talking to you okay
okay so people got there uh yes Robbie
that was not the reason okay I have some
have several people got their tickets
all right let's do it
okay Brian Brian is gonna oh and let me
emphasize order doesn't count in case it
wasn't clear order does not count
doesn't matter what order the numbers
are in okay so now Brian Conrad
director of the American Institute of
mathematics I've given him a
mathematically menial task to do you're
gonna yell out three numbers between one
and seven okay three five four otherwise
known as three four or five all right
let's see how we do now open it so
everybody's counting I'll put up mine
and we'll see how I did okay so I first
of all check if I the jackpot but I
don't I don't have three four or five I
have one four five so I get two dollars
for that right I have um I have three
four six and I have and I have three
five seven I get two dollars for that so
I've got six I'm still alive okay
everybody's everybody three four five
our people confused what's the question
what oh it's so six dollars for a
jackpot
two dollars have you got two out of
three so I add I have three of those
juices I have three tickets than one two
out of three the one for five the three
four six and the three five seven are
you guys adding up your winnings
who's got a lot anybody get like ten or
twelve a lot okay I wanted to who got
the most anybody got 14 okay no I don't
see any so I got some 12s was that the
top score we got twelve okay a small set
of people like pretty good but I
definitely I'll get them out of tens and
twelve whose out who got totally hosed
on this one I got less than six
okay good number of people on but who's
still in okay good let's do it again
but remember if you're out you're out
okay this is elimination okay Brian
if one two five let's look okay I am
once again
Jack Paulus but but I have one two three
that's two when one four five that's so
- I have that again that's good let's uh
one two five and I have two five six so
I'm up to six so I'm in the clear
and then and then I'm out so I've got
six again okay um who's still in okay
that's a theater group but a lot of
people still it okay let's do it again
Brian two four seven oh I've got jackpot
okay so I already see it I've already
got six but nonetheless I'm gonna see
how much more I want let's see two four
seven
cube bye and that's it I got nothing
else okay now who's still in okay still
smoking let's do let's do one more on
what do you say what do one more and
then I'll be it
okay three five six okay I'm jackpot
list three five six I have two five six
oh look I'm saved at the end because I
have two five six three four six I've
got two of those and then three five
seven so I'm still in okay who else is
still in okay all right this is good
about Lake looks might even be like 15
people around there okay we could keep
doing this but then you wouldn't him
okay there's always one guy doesn't want
to hear the end of my talk I was like
this just got good let's just do this is
that okay I think for those who want to
keep playing I think I have explained
the rules well enough that you can play
at home okay so uh what did you just out
of curiosity what did you guys notice
about my awesome numbers
yes I see so it's not that I'm
guaranteed a jackpot right cuz las no we
didn't get a jackpot but somebody back
there said I get $6 each time you might
have noticed that I got $6 every time
you know one thing about games is
usually definitely it's good to decide
your strategy first and then set the
rules that's a basic principle of game
playing and that's exactly what I did so
my my set of cards has a very special
feature which is that its payoff matrix
as many of you observed looks very
different from what you get with a quick
pick so my set of cards has 2% of
tickets as the property but no matter
what the jackpot is I will win $6 never
more never less so this is that rare
case when the expected value is indeed
exactly the value that I expect I will
always get $6 with my cards every time
now you might ask why is this good why
would I want to do this besides that I
made up this cockamamie rule that was
exactly designed to privilege my set of
tickets well in life right if you're if
you think of yourself as an investor and
this game is a kind of investment for
the people playing it um usually when
you're investing if you can get the same
return with less risk you're happy right
if you can get the same on average
amount of profits but not risk mut
losing money um that's very good that's
especially true if you're in what's
called leverage right if you've borrowed
the money that you're playing with so
again step with me back to 2005 if you
have asked all the kids in your dorm for
a lot of money until you have $5,000 and
then you buy lottery tickets with it and
you lose and then you go back to all
your friends and say well according to
my calculations in the long run if you
keep on giving me more of your money
will eventually win even at MIT that is
a very tough sell so you do ideally you
want to build a strategy what's called
in finance a hedge that somehow keeps
all your return
without having any risk and I think the
reason that they were so careful about
their tickets was something very much
like this so let me say I want to say a
little bit about how did I come up with
this crazy set of tickets
let me draw show you a picture of it
this is a diagram of my tickets and well
you can sort of see that this diagram
has seven lines on it well one of the
lines looks like a circle but I'm going
to call it a line each of these lines
have has three points on it and the
three points on the line are my tickets
right you see one two three along the
bottom one six seven along the side that
circle is two five six which is one of
my tickets this is a beautiful object of
geometry called the Fano plane and the
thing I want to say to you tonight about
it I mean I think you may have the right
to object when I call that circle a line
why because the circle is not aligned
okay that's pretty solid objection
nonetheless I'm going to tell you that
it is a line and the reason is that it
because in mathematics at least
mathematics the way we do it in 2016 and
maybe even the way we've did it done it
you know since about the late 19th
century is that we don't really like to
say like what a line is we like to say
what is a line defined to be and the way
we approach it is we say well a line is
the kind of thing that behaves the way a
line is supposed to behave and how do we
know how lines are supposed to behave
well Euclid told us so right we know we
took geometry we know the rules for line
so let me sort of put a little slogan
underneath this picture
what's our sort of acid test for whether
these are lines
well should they they should obey
Euclid's axioms then every two lines
intersect in a single point and every
two points are contained in a single
line and it's kind of fun to do this you
can sort of look and see if that's true
I pick two points like one in five okay
there's the line one four five if I pick
two points four and six okay there's the
line like three four six I've drawn the
line through them two and five okay
the line between them is the circle and
it's sort of fun to like sort of let
your eyes roam over this diagram and see
that it um then actually Abbes eastward
by the way so if there are people in the
audience who are actually like taking
high school geometry right now or like
recently did yes okay so there's
something I sort of slightly lied
because I said okay the lines and the
points behave the way that Euclid told
you they're supposed to behave but
there's something actually different
about this that it's not the same as
Euclid's axioms anybody see it okay what
ah so that is a mere cosmetic artifact
of the fact that I've had to project
this onto a two-dimensional screen for
you but really the only points in this
diagram were where the numbers are so it
may look to you like there's an
intersecting I get to use my laser
okay I'm gonna use it it may look to you
like there's an intersection there but
that is only an illusion because that's
not a point okay so that's a good
objection but it's not the objection I
have in mind I'm saying that these rules
are actually not Euclid's the rules in
one minor respect that's my job man so
the difference is merely this that and
you put in geometry not every two lines
intersect in a single point two lines
can also be parallel and that I think
all right-thinking people see is a
defect in Euclidian geometry it's things
when rules have exceptions they
shouldn't have exceptions this is what's
called a projective geometry which is
much nicer where it's very simple like
any two distinct lines intersect in a
single point there's no exception for
parallel line so in this kind of
geometry no lines are parallel and if we
have like another hour together I would
tell you a whole beautiful long story
about the development to perspective
painting in 16th century Italy where
they sort of understood that when you
look at like a road from like far away
like the two edges of the road even
though they're parallel in the
three-dimensional world like seem to
meet at a single point called the
vanishing point and it's exactly because
they were sort of
instinctively learning how to do
projective geometry like on a flat
canvas another time I'll come back and
we'll talk about that the point I want
to make is that somehow this abstract
discussion we're having about the rules
of Euclidean geometry are exactly what
make this set of tickets have the
property that I wanted to have because
if you remember when my my menial slave
Brian Conway like yelled out the numbers
one to five how did I know I was gonna
get exactly six dollar as well because I
know there's a line through one and two
so I had a ticket that had those two
there's my two dollars I know there's a
line through two and five so I have a
ticket that has two and five and there's
two more dollars and there's a line that
has one in five and so I have a ticket
but as those two dollars of course the
other option is that when Brian picks
three points that are actually all
aligned then I win the jackpot and I win
nothing else but either way I went
exactly six so it's exactly the sort of
geometric feature of of these points
that make it have this magical property
now we've gotten a bit away from the
actual situation of cash windfall so I
have to say like once I understood sort
of what they were trying to do I became
obsessed with trying to figure out what
configuration of actual lottery tickets
and cash went ball they might have used
this is an example of a object in math
called a combinatorial design and so I
just went hunting through the literature
and combinatorial designs and eventually
after a lot of work and a lot of digging
through references I found the following
a paper of rhf Denniston from 1976 which
exactly gives you a configuration of
about 200,000 tickets which you can't
quite guarantee you're gonna win a lot
of those very valuable five out of six
prizes but you can almost guarantee it
you can guarantee that you have like a
98 percent chance of getting at least
five of those five out of six prizes if
you buy these particular 200,000 tickets
and if you bought a random 200,000
tickets you would take a substantially
larger risk of losing money and just in
to show them how far mathematics can
come I have to sort of show you
a very interesting comment the denistoun
writes his paper just just to show how
far the culture of mathematics has come
he's clearly quite embarrassed but he
used a computer in order to sort of like
he sort of season feel like a real
mathematician would have worked this all
out with a pencil and so he would soon
he'll he makes it clear but okay like
yes I checked it on a computer but I
certainly didn't use the computer to
figure it out that was on me we don't
really have that attitude anymore I
think nowadays in any kind of
large-scale operation like this the
computer is seen as our partner and
there's no shame in the using a computer
to help us discover so well okay I've
left one question unanswered I have now
I've shown you what they could have been
doing is that what they actually did I
wish I could tell you that I knew I
never was really able to get these folks
to talk to me directly about what about
what they did I think they're happy to
have that chapter of their life behind
them so I'll just say this I I was able
to get them say roughly this sort of
some version of this risk minimizing
strategy was what they had in mind but I
wasn't able to sort of get them to speak
up to the actual configuration they use
so we'll only say this I don't know if
they used this this paper I found but if
they didn't I think they should have um
so I'll stop there and take questions
thank you guys so much
[Applause]
if you have questions please line up in
front of these two microphones there is
one on this side and one on the other
side so that they can pick up any
questions
can you walk walk walk to one of the
microphones please no I have a quick
question which is you know about your
thing about that
what they actually did I mean do you
know the amounts of money they won and
how many times and so forth because
obviously knowing that they won a
relatively even amount of money each
time would tell you that maybe they were
trying to guarantee that so III know the
total amount but I don't know the sort
of week by week which is what I would
have to know for that and sometimes
noted knowing the total amount they won
over this over the six years it's kind
of like knowing the expected value and
it's whether or not they were sort of
like winning and losing big week to week
my expectation is they were not my
expectation was at least the random
strategies guy had like heads that risk
away but I don't have any direct way of
looking at their Ledger's and knowing
that here's whatever here I just read
and I can't remember where but there are
these people around the world that have
won the jackpot like three or four times
over the course of their did you read
that article I don't know if it was in
time or I don't know where I read it but
it's a group of people and it's five to
six of them that have won the big huge
lottery jackpots multiple times more
than twice any I don't know I mean is
that just completely random that these
single people have it could be I mean I
think what they I think with stuff like
this it's always sort of fun to think
about let me put it this way if somebody
like won Powerball like three times
you'd be sure to hear about it right so
most people are not going to but you
have to ask you
out of all the people in the world who
play and now in your store you're saying
they want lotteries in different places
so you sort of have to say out of all
the people in the world who play all the
lotteries would you expect someone or
some two people or some five people to
have won multiple times thinking that
just the way news spreads news of the
unusual spreads more rapidly than news
of the usual like you wouldn't divide a
story like Time magazine some schmuck
lost again like I mean so I mean I think
it's really a matter of like the
individual case you sort of have to sort
of say like how unlikely is that someone
somewhere in the world would have this
happen to them and then that I would
hear about so in some cases you still
can be quite amazed and in other cases
you realize that it's not so unlikely
after all yeah I'm yeah my question is
since the the table was published in
advance I assume and the three major
betters knew what that was going to be
one could imagine that if they bet
enough did the the state would lose
money on that date and then have to
recoup it later is that something that
happened regularly or do you know no one
it can't happen by design because the
state is always putting a certain amount
of money I mean I mean this the state is
just the only amount of money that goes
into the prize pool is that so okay so
let me explain and you may say like wait
if everybody you got those five numbers
got $40,000 so that's not the way it
works and any lot in any lottery if a
bunch of people will hit the same number
they have to share the winnings oh I see
so what happened is I mean I didn't
really go into it in the talk but it was
not always tripling your money that was
at the very beginning but what happened
is the more these guys bet the more
ticket splitting there was and the more
the profit went down so for most of the
time they were doing this their profit
was running about fifteen percent so
that also explains why were there no
more than three groups because they were
sort of in an equilibrium worth another
big group joining would have meant that
everybody would have lost money so they
sort of found an equilibrium and sort of
stayed there I see so so the the
expected
you actually was dependent upon how much
how many tickets were purchased for the
for the players yes yeah okay thank you
yeah you kind of just answered my
question but I was wondering how did it
go on for seven years like it seems like
eventually you know the the expected
value would be like two dollars and one
cent right because more people would
keep like did they have some agreement
so maintain this 15% premium or like I
think this is like sort of the in some
sense is like what the basic question of
economics if you're like a true believer
in classical economics you know you're
like oh is that a $20 bill on the ground
like no it can't be because someone
would have picked it up right right so
instead of and so now in theory profits
will eventually go to zero because if
you can buy more I mean but that's that
doesn't happen in real life I don't
think it's because there was an explicit
agreement but I think somehow that often
happens in real markets that I mean like
in real markets like companies do make
substantial profits even without
explicit collusion my sense is you know
I don't even have any way to verify with
me but Selby told me that the random
strategy is people had approached him
about sort of maybe taking turns and
alternating that he didn't want to do it
he felt like I mean with it's very lots
of interesting ethical questions on what
they were doing so he felt like now
that's not fair
I don't really I don't really know
exactly how the light is draw but that
he was clearly like but yes I it's an
interesting question I mean about
markets like sort of how they arrive at
that particularly equilibrium but that
was where they sat for awhile so I don't
understand why the state of
Massachusetts didn't take a further
advantage of tickets like why they
themselves didn't immediately
participate in the lottery or at least
raise that raised the raise the ticket
price and make a larger revenue because
if the winnings were $5 53 cents
couldn't they have raised the ticket
price to like $4 and still get people to
play
so that's a great question and again I
feel like this whole thing if I were
like an economist instead of
mathematician this whole thing would be
about like the failure of what they
teach you in equina want to actually be
true so here's what what happened I mean
people decide whether to buy things yes
in part based on whether they think the
value of the thing is worth more than
they paid for it as they'll tell you in
in classical economics but probably more
than that they decide based on whether
they think the price is fair so without
even being a professional economist I
can tell you what would happen if after
lottery tickets costing $2 for 20 years
the the math the state lottery said okay
from now on lottery tickets are $4
people would like burn down the lottery
office and and nobody would buy lottery
tickets because you look like that's not
fair a lottery tickets cost $2 I mean I
think I think that's what I think that's
actually the answer I think that's what
happened that day that in that business
that you sort of you can't do sudden
price hikes because people will get
angry and not if the state of
Massachusetts only cared about the
amount of tickets it's sold why didn't
it do like better scheme than Michigan
so that the they get even more than the
amount they get like a really really
good scheme that would get even more
tickets so your question actually speaks
to an interesting one was it I don't
even really know how much Massachusetts
knew about what had happened in Michigan
I mean I I have to say this is where you
see that I'm a math professor not a real
journalist because I I tried to like
penetrate the sort of public relations
wall of both the Massachusetts Lottery
and the Michigan Lottery and I got
nowhere
so the Michigan lottery would not
confirm that they knew that jerry selbee
had been making like millions of dollars
in their lottery even though obviously
they knew but they would not tell me
that the Massachusetts lottery would not
tell me whether they did this on purpose
or not and I and there I actually truly
don't know I don't know if they just
like read about it and thought it was a
cool idea and didn't sort of anticipate
what would happen and once it started
happening they knew for sure but I'm not
sure they planned it and I and I can't
figure out how to figure it out if
anything if there's any real journalists
who know how I should have pursued this
in the room
please tell me I have to okay the first
one how many tickets does a mission
Michigan guy really buy in that corner
store so I don't know the exact number I
think he was comparable to the random
strategies guys so I think he maybe he
and his he had family members too
we're doing like a sort of 200k ticket
buy and who actually won that lottery on
that day on that particular day I know
that the random strategies folks about
triple their money I don't because it's
not recording the inspector general's
report I don't know exactly how much the
other groups won I know that the brand
strategies guy I think took away about
fifteen thousand on there five thousand
of bats no kids are always first did
Massachusetts change to a different
lottery system at the end or yes so they
went back to a more traditional lottery
and of course what you should do right
after you hear this talk is you should
like run home and like look up the state
lotteries to see if anyone's running
this game I do this from time to time I
think so that the Michigan thing really
got no publicity I mean they shut down
their game but as far as I know it was
there was never any reporting about
jerry selbee in the context of michigan
this game I think I'm not saying this
was like a huge national story or
anything but I think like among people
who run state lotteries they probably
all read this story so I think it
probably got enough publicity that um
that nobody is that nobody's doing it
right there are other there are other
states that roll then run role down
games but there there's a they're a much
less generous role down you can close my
computer at the close my computer you
just close it
yeah about that hard okay this is what I
like to use their computer okay hi so I
have a question but first I want to say
I read your book while I was hiking
mountain Kilimanjaro I actually wrote
you an email about that and you you're
the guy yeah I'm the guy there's a
picture of this guy my Facebook page so
highest your book is ever been I think
that's graduated but my question is so
with fanos plane you that works with
seven points mm-hmm can you do a similar
trick like if you had more numbers then
exist in the Transylvanian lottery can
you do that so yes but it's only for
very specific numbers that it exists so
as I said this is very beautiful theory
of like finite projective geometries and
combinatorial designs and in some sense
this is this is actually a very active
research topic where there was a big
breakthrough not like maybe like two
years ago I don't know when was key
Bosch's theory okay there we go so it's
showing that in some sense there are
designs that have some of these features
of every size once the size is big
enough and that was quite I mean I think
it wasn't surprising it was true but we
didn't know it was so close to being
proved and I was just so mean it's an
active topic people are thinking about
even today okay let's take one more
question if there is or otherwise let's
think Jordan again last question okay
last question tonight okay so you're
talking about your draw diagram and how
it covers everything is there a faster
way to do it without drawing things out
because when you get into the bigger
scale as you'd probably have a lot more
lines and it would probably be a lot
harder to understand when you look at it
yes if you were good at doing like
modular arithmetic like addition to
multiplication modulo some number do
they keep that at the Palo Alto schools
I didn't read it so it's sort of nice to
do it in the new math back in the old
days so yes there are there are sort of
like there are ways for instance that I
could have taken those seven points and
identified when with the seven strings
of three bits other than zeroes
zero and there would be a sort of a very
clear relation telling me like which
sets of three I had and you could do a
similar thing for for larger
configurations and that would be the way
in practice if you needed to remember it
you could do it without drawing a
picture okay so let's think Jordan
[Applause]