# The Mathematics of Winning Monopoly

I'm here with Hannah Fry
hello
and we're going to be looking at the optimal
mathematical way to play Monopoly
yes, indeed
I think
So you were looking into this 'cause you were writing a book
I've got it here - "The Indisputable Existence of Santa Claus"
A lovely looking book \{mumble\}
It's a - look at it - it's shiny and everything
It's so lovely
Oh and it's been signed by the authors, look at that
And so, you were writing a book all about
the mathematics of Christmas, and in the UK
As it is a serious Christmas tradition
It is tradition to argue over Monopoly
Alright so in here, there's your book, what did you find?
Well, the key to winning at Monopoly, beating everyone at Monopoly
It is realizing that not all sets are created equal
so there are some squares that you are much more likely to land on than the others..
..and therefore making more
-Yes, and this is the point
cause a lot of people think
well the best ones are the ones with the high rent..
..and they just get better and better as you get arouend the board..
..but you need to land on it before they pay off
-Exactly right, exactly right
So if dice rolls were the only thing
If it was just, you know, what you throw out of twelve
Ah, Then every square would be equal - It wouldn't, you know
Yeah there would be a slight clumping when you start but after a couple of laps, evenly distributed
Exactly right
The thing is though
There are a few curve balls in this game which change the probabilities
Right
So some of them come in the form of these chance and community chest cards
ahh because there are cards in here that will send you to other places around
Right
So its not as straight forward as going around the board
Its a bit more complicated
Cards send you to different places
you might go to jail
end up over here
Exactly
You end up here a lot
You do
and in fact jail ends up being the most visited square on the board
Ahh yes that, that
Okay so I think our results match, cause I found, I found
From memory, Just over six percent of all goes end in being...
I don't remember what my numbers were
Check it, check it, check your number
Wait, Wait, Wait
Oh, interesting
Oh really what did you get
Not six percent
Not six percent?
No
Uh oh
I've got just over four percent
Really?!
Yeah
I'm off
I'm way off
Ok so shall we,
We used different methods
But I've deliberately not read the monopoly chapter...
...Cause I didn't want to pollute my own research, so I'll show you what I got
Yeah, ok
I've actually, I haven't got like a, you know, a full slide show presentation but I have my spread sheet of results
I'm disappointed in you Matt
So what I did was, was I just simulated a lot of games
So let me fire this up
So did you effectively have one imaginary player going around the board
Yeah I thought I wanted to have multiplayer but then I couldn't be bothered to program, I thought of programming
They don't even make money, It's the most boring, they just go around the board
I've, I've written it, that's my bar chart, don't look ahead
Ok, ok I'm sorry
Where's my code, Where's my code, here's my python code, ok right so
It's one person playing by themself, and all they do is they write down which square they
They still do all the community chests and everything
But they just keep track of where they were and after one hundred goes they play again
And do they keep track of how many double they roll in a row
Yes!
So, so what I did was, for some strange reason, I listed all thirty-six results you get when you roll two dice...
... and then I get a random number from zero to thirty-five and that just the index of what they rolled
but then down here double rolls...
if its one of these specific indexes, those are the doubles...
I see
That's like a 2x2 grid I've just made up
These are the dice indexes I use for double rolls
I've seen neater code in my life
Haha shut up
It's all right!
So once you go through
insulin the person rolls this tracks doubles and if they go past three
doubles they go to jail which is position tenth I've just numbered so the
board starts go is position zero and it goes all around position 39 and then it
just tracks those and then the only thing I've done to make it realistic is
at the beginning of each game you get them that the Community Chest and chance
are in order and then you shuffle them up and then as you go and if you round
the cards if the number left in chance is 0 you refill it and reshuffle it so
you're realistically shuffle and then work through shuffle work for us quite
please so I think that how many how many goes
to your player have your 2 knows best so I ran this for 1 million games each of
100 goes so I think that in the limit as the number of games that your simulated
player plays yes the whole person gave and run the board yeah it should match
what I did it slightly different way you did it theoretically I did do it right
so my most as though it's like a road atlas if you
like of the probability of getting from one square to every square right like a
big table because I didn't look at all any of the probabilities I just
hardwired here's where it sends you and my and you can see I'll show you my six
percent which is really sort of working out the province's going along right yep
yep there it is right Oh our blessed series one on them there's an Excel faux
pas and then it's a face get rid of that okay so I have taken out all the non
financial properties so if I bring up financial properties where I put those
where they it'll heat my taking any chances so here we go here we go so go
to jail is zero because no one ever ends they go this is where you end up and
then jail Joe Joe Joe Joe there I got six point three percent of all goes end
up on there okay what did you get for Parklane I pop claim to me as the safe
is to start line or Park Place yes in the US version 2.1 interesting that's
exactly what I got really yeah I'll hold on I know what's happened
does your jail include visiting jail and go to jail yes uh hi split them up
different ah right you split them out that's why I've got just over six what
you're led to yeah just a sick sick I I didn't mess it up
more fun than playing Monopoly what are the probabilities which one so I have
too far the square o Illinois's Avenue yeah Illinois revenue that can most
visitors ground board oh wait three point two I think that's my most visited
what I'm so sorry Oh three point eight do after J yes it is my most visited
after Joe three-pointer before the same answers all right actually I feel
surprisingly vindicated well you surprised I didn't get it wrong no it's
right between us wow wow that's that's mass for you you
know two different methods same result so good right so there's my final buy
try that I got and this is where I stopped
so I'd written the code in such a way that I could now tweak the way the game
runs and I was going to start factoring in other things like buying properties
having money all that so that I drew a line under it there
is in the end of the story right just the probabilities it's at the end of the
story because ultimately monopoly is about winning money and crushing your
relatives yeah and so for instance Park Lane is an interesting one right so both
of us have Park Lane is the least visited Square why it's just to recap
you did Markov chains for your pure system right so if you want to have a
look at this using Markov chains was just genius it's basically you did a
network yeah of every single P like the probably going from anyone to any other
one yeah if you start off it go it's quite easy to work out right yeah I know
what the probability of throwing any number yeah you're like a description
here like that you can then tweak that according to the chance to come into
chess cards oh yeah which is again very easy to do and you
can work out in fact that you have a one point two percent chance of visiting
jail on your very first guy really converting all other dice exactly one
point two yeah tell the time you play Monopoly once a year that's about once
every eight you three years that you can expect to get your first which would be
an even more wonderful beginning so about once a lifetime you should roll
boom jail anyway as you carry on going round though you need to take into
account the probability of going from any square to square the first time
since you're big you just take into account doubles right three doubles in
or I guess I'm you to do um so but it is possible yeah calculate the probability
of going to any square from any Square takes a long time yes a long time but
eventually you end up with a table right so it's kind of from every square to
risk a massive grid yeah then what you do is you use something called a Markov
chain so effectively if you know you get one player one the boot Moonves piece um
and every time that you move the boot onwards you split it up so it's like you
have a fraction of the boot according nice you distribute the boot around for
ability yes exactly and then repeat and repeat and for all those squares they
distribute and edit and eventually you end up with the boot distributed
proportional to the odds of being exactly on any one square carry on going
no come a point where those fractions of boot they're moving and then splitting
off and then moving and spitting off won't end up changing because it matches
what's coming at the same time and that would give you the kind of
long-term probabilities of every square wave okay but you then you get the same
result as me spending an hour in Python a person cost but you can be more
confident in your results right though from first principles right it's true
uh-oh it's just nice to get you hands over the matrix multiplication that's
true I mean I'm little envious having it to do that so right it's once you got
that now this is where I stopped that's not the end of the story because money
comes into account so Park Lane for instance is a good kind of Square to
what was Park Lane in Park Place there we go fine press the one with the
pocket hello in the park in the park position yeah so that's the least
visited square that you can build on on the board but it's also earns absolutely
tons right so yeah hotel on Park Lane will get you in the British version 1500
pounds and barely get your hotel room yeah yeah in the actual Park Lane which
is like the streets that way right now yeah so actually maybe it's not quite so
bad right so what you have to do is you have to take into account the not just
the probabilities but the amount of money that you stand to earn and that
changes things yeah cuz it yet so what did you find all that yeah so what you
can do you can put those two things together you can work out how much you
expect each square to earn for every role that you're compared to your
expected return per opposition role for each that's nice exactly so here you
have it so you've got a plot each of these properties start on the x-axis
depending on how much you have to invest that you investment you're a negative
money exactly and then every time that one of your competitor rolls the dice
doesn't expected amount that you can yep and they you gradually build back up
again and then suddenly portly doesn't look so bad anymore okay Mayfair is the
clear standout win on the border oh my goodness yeah so you're in you're in
profit about 25 opposition rolls later and your well in front of us we can see
and is this like a normal number of rolls you'd expect in a game cuz I just
guessed 100 so you actually have about 30 rolls if you play properly
if you don't play any house I'll know this money on and the thing is though
this also isn't the end of the story it's socializer
because you can't just buy Park Lane on its own or Mayfair on its own it's very
elderly you have to yeah so when you take that into account
when you when you build in all the sex together this is the the real graph that
you want since this is the expected return for each set exactly right yeah
that's nice and now suddenly actually Park Lane is so rubbish that it
completely drags down that Mayfair effect really yeah basically navy-blue
said Obama but well I'm not worth it too expensive too little chances than being
landed on because of partly why not worth it okay so what is worth it okay
so what is whether it changes depending on how many opponents you have because
there's different numbers of expected opponent rolls exactly so you all right
and the amount that you you expect to earn per control changes depending on
which set you go for right so essentially if you have just one
opponent yeah you split the game to go on for about 30 rolls of your opponent
then the light blue set all the oranges that are the best ones they're also the
best ones early on in the game right how many of you if you've got two or three
opponents then it's orange and red that you want to go for what and any more
than that anyone three runners three green that's straight forward so two
opponents blue and orange three opponents orange and red and more than
that Green yeah sees you two remember we think we can do that don't bother be
neighborly don't bother me yellow train station try stations utility please
please no juice yeah right it doesn't make a difference
how late in the game you are like if you have a chance to get the greens quite
late on and you go well there aren't that many opponent rolls left
oh I don't know I guess it depends what set you get when the game ends yeah the
thing is only once you get later in the game the amount of money changes as well
so the point amount that it's sort of relative to what you have the amount you
have to invest is sort of changing as well so I mean once you have the money
to invest in it because I was my end goal for my program was to simulate the
amount of money and have different players with different ratios of money
invested to cash reserves to see if there's a it maybe there's a nice
threshold like always have 80% of your net worth in property and 20% in
reserves and I was going to play loads like millions of people with different
strategies against each other and see which was the
best which is what the simulation methods perfect yeah yeah because you
can just do that over and over I'm sure you know what I haven't even looked it
up online yet so maybe someone else has done that and has already come back the
results there is one final thing actually I'd like yeah actually probably
my number one tip for playing Monopoly right which is how quickly you expect to
make back your money because sometimes you've got sort of new you're choosing
whether to build an extra house or change to a hotel on a property yeah and
the question is how quickly should you do that
so the other thing that the numbers say is that you will get your money back on
three houses quite quickly right houses takes a long time five no hotel -
because all the extra money are spending on the property developer exactly Oh for
any young people any Millennials watching the notion of owning a house
macing so sometimes instead of renting you can exchange a lot more money for
permanent ownership of something the trick is to have done that in the past
yeah that's where young people going wrong with these days yeah yeah exactly
right so this is a real fantasy game now more than anything else right it's true
the one advantage though also to only having three houses yeah been investing
in hotels early on because there are a limited number of houses in the game you
can block your father's sorry why here you get am literally a monopoly on the
houses they can't develop past a pass which is another hospital that stupids
you know I've had to be like people like oh you could get a paper you you know
like there's finite houses finite money we're learning about how bad capitalism
is that's group your family so we can go have some fun playing another game
that's more violent yeah that's brilliant okay so this is all the graphs
and plots and working out and Markov chains and everything are in here as
well as other great chapters on wrapping pre-mixed I mean you're you're hugely
biased so there it is the indisputable evidence existence the indisputable
existence you know when you're promoting a book you should at least read all of
the title I feel my Santa Claus and so you can get this all your favorite
online bookstores have it and we have it on mass gear in fact if you buy this
book from mass Kia it will be signed by both the authors yourself and
Thomas yep sign on the sheet and this specific copy that we've been waving
around in the video I'm going to sign it as well down the bottom the first person
to buy on mass gear and say can I have the copy from the video you will get
this specific copy so ask nicely this one could be yours actually one quick
thing before we finish I ever I warned you I've got one complaint about your
book one major glaring mistake and that's
that all your snowflakes have eight points they've all got fourfold symmetry
what is this embarrassed and ashamed I think you should be like unbelievable
way through and I checked it it's not even like an explanation about why
snowflakes graphic designer you're gonna blame a graphic designer
somewhere now I take those messes are you by and by the time the book was
finished yep and they sent from the final proof okay whatever
it's good publish the thing it's nearly Christmas guys will be here to leave a
precise fire case and it's a really important thing if you can get over the
snowflakes this is available online and no longer are nasty you know not really
no maybe we should just start by agreeing this is a terrible terrible
game right but what do we we were backstage at a show somewhere and when
we start to talk about the fact we were both analyzing the maths of Monopoly and
then we both complained about what a terrible terrible game is you have done
it different ways though that's very true which is what complaining about the
game or analyzing them probably both anyway yeah yeah cuz you hate the house
rules I bought yeah I think hospitals are terrible so um you know this idea of
all the money goes on free parking worst idea ever
what it means is that you you're essentially putting a big cash injection
into the game exactly moment where yeah the quantitive
easing yeah yeah yes exactly why why draw out their suffering
okay I can tell there's a house rule we my family could face because it's so
many good board games and we do play the board games that are fun the same reason
about once a decade we're drawn back to play that and we
have so many arguments actually the main problem is I try and make them more fun
by introducing more dynamic elements of the game you know well you know why just
don't think that right I have leasehold you have a shared investment between two
players and I offer like a loyalty card yeah yeah I tried to payday loan system
one time did not work out a little bit of low-level loan sharking really helps
the game I'll be honest yeah how extortionist but I was able to
communicate it in a mathematically complicated way but the rest my family
saw right through that they know how to use attack better
so probably the is that the sole object of this game is force your friends or
family in politics never gonna make for us to just party our community other
games you meant to obliterate them from the face of the earth don't get me wrong
right all steal their supply of rock right I think you're right you're right
it's not it's not it's not a fun game so you should be able to win quickly
efficiently and get on to a better game if you're gonna do it you want to do it
properly