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 let’s compare right cause I’ve read a lot of your book 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 2×2 grid I’ve just made up You don’t have very many comments in your code I have very few comments 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 excellent good kid few and less another
one what about this is the most this is 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

100 thoughts on “The Mathematics of Winning Monopoly

  1. So what happens if I write a book of my simulation? Would I then get all the credits and all the money? 🙂

  2. From what I can tell, the stations are the most profitable cards, please explain why you said they were rubbish. When I play a game of monopoly the stations always give me the most and gives me a higher chance of winning.

  3. I’d love to put 3 houses. But you can’t. You have to balance them within 1 so it makes sense to do 3 + 3 + 3 = 9 or 6 depending on set size. Sure you can do 2,2,3 etc but return on 3,3,3 is better assuming you have the cash.

    Best house rules: Free parking chance money.
    Prison means prison. Seriously who takes rent in prison?!
    Insurance. Sell insurance in your negotiating phase. Or to skip a rent payment you can’t make. Say you want a green for a red, offer 1 time insurance if they land on a green.

  4. There's a BIG problem with the "don't buy the dark blue" squares, which is when a player chooses not to buy a property, the other players get to buy them in an auction. Which means it's not good enough to say "don't bother with Park Lane", because it's obviously worth bothering to get in an auction, the real question being "What is the MOST I should pay for it?".

  5. Eating a peanut butter and onion sandwich to calm the mind whilst watching this.

  6. I don’t mean to take the wind out of your sails, but this analysis (with Markov chains) has been done before. I wish I still had the article handy, but I remember reading it back in 2013 (I think).

  7. Yes, finite houses, but not finite money. This is explicitly covered in the rules; 'The Bank never "goes broke." If the bank runs out of money, the banker may issue as much more as may be needed by merely writing on any ordinary paper.'

  8. Buy everything you land on, mortgage what you can't get monopoly on and deny monopolies to everyone else.

  9. I always land on the railroads and utilities, while opponents continuously skipped right over my yellow and green properties.

  10. The real key to winning is owning one property of each colour, thus preventing other players from being able to get a monopoly, and therefore meaning that you quite literally control monopoly.

  11. Its been my experience the best way to win is to buy every property you land on and make the best of what you get. Occasionally you may not buy a property but only if you are cash poor and already have one of that colour. Put only three houses on each set. Build more only after you have more cash then needed to survive the worse case scenarios

  12. Are the Mathematical equations and working in the book? I am thinking of doing a school paper on Winning monopoly and would like to know

  13. What is your luck of winning if after turn 1 you have a shitty train station and 1 of all colors have been taken by the 3 other players?

  14. There is more to it: Calculating your own and your opponent's rolls for deciding if you want to buy houses at a certain moment or save the cash. You can also negotiate prices of properties, so even selling the very best properties can be a good deal if the price is right and you can get good value for your cash terms of other properties or real estate.

  15. Why did they change it to Park Lane? There's a reason for the named in Monopoly — it's based on Atlantic City? Is this a weird Mandela effect thing? Also, the green board looks wrong. No no no, I don't trust them.

  16. The trick of it is not "which to invest in?", but rather, "should I just buy everything as I go (so I have stuff to trade later); OR should I only buy the properties that are deemed most valuable given the number of opponents?"

    But that's a question of strategy, not probabilities and rate of return.

  17. BTW, some properties give a better rate of return with only 3 (green) or 4 (dark blue) houses, whereas others maximize their return on investment with hotels (orange and red).

  18. Has anyone ever noticed that £200 every time you pass Go is basically a system of universal basic income?
    Do you think you could introduce new fiscal policies to ensure no-one ever goes bankrupt in a game of monopoly?

  19. Nice to to know this. But in reality you just buy every street you land on and try to bargain for the best possible once all streets are taken. It's not that you really have the choice to not buy any specific street.

  20. 3:57 You're actually meant to place used chance/community chest cards at the bottom of their piles. The order stays the same according to the official rules.

  21. monopoly is fun when you play it right, it's even better when digitalised cause then you don't need to do the maths and it goes much much faster.

  22. 8:30 good point that u did count the probability of landing on each square from every square, but did you exclude the one right next to it? Cuz the minimum is 2 steps

  23. How do they account for people not buying (so auctions) trades, and so forth where people make other choices?

  24. The orange properties are the most valuable when they have 3 or more houses on them. If short on cash 3 houses is the best. If flush with cash 3 hotels is best.

  25. I actually only go for the light blue and brown places (the cheapest on the board), no matter the number of opponents, and I’ve lost only one game in the past 4 years (a game I didn’t feel like playing to begin with). May be luck, but I think that these probabilities only work of everyone is playing according to the same probabilities. But, thats not the case. Some like buying certain places, and some don’t. Therefore, bidding changes the game. Also, the mere luck of others getting to places before you get the chance. Lastly, at least here in PR, we make trades and deals after a certain point in the game, so that changes everything. My point being that its not as simple as buying certain colors based on the number of opponents.

  26. Keep the railroads&utilities, give away(preferably exchage with railroads&utilities) others to your opponents with the condition of never paying in that color.(For example give a yellow place for a railroad to an opponent but make a deal about never paying in a case where you end up on a yellow owned by said opponent. That player is probably trying to gather all the yellows so paying to other opponents on a yellow is out of question). Do that in every chance you get and try to make every player stronger than they are to others but have no strength over you. In fact you will get their money if they land on railroads/utilities. In the begining noone will ever see you coming and even if they do, they will not care as they are getting a property they want for almost nothing. Towards the middle of the game weaker players are more likely to make this kind of deal as they need something to compete with others. Later start to buy properities from the losing players as you need to control the board by either owning properties or the deals you have made and if a player loses, your deals with them wont be enough to control the board. Plus, it will be easier to buy from them as they will be needing money. In the endgame where you are against the strongest player(except you of course), you can start to enjoy your victory because you will not be paying a dime in almost any of the squares on the board whereas they will start crying and use their 1 dollar bills as a napkin but you will be taking even those dollars out of their hands.

  27. "Finite money, finite houses! We're examining how bad capitalism is!"

    ummm…..I don't think that word (capitalism) means what you think it means….

    There are no finites in capitalism.

  28. The version with Park Place and Illinois Avenue is the true version, and the version with UK names is a UK-localized version; one of hundreds of custom versions.

  29. That's all interesting but if your opponent has the optimum sets already, or a single from such sets, your only chance if winning is to get any set you can and hope, despite the slight disadvantage, that your yellow or purple properties will win you the game.

  30. Guys, I got you want to do something else, but where the hell am I supposed to put my houses and hotels!?!?
    Can you get to the God damned point for Heaven's sake!?!?

  31. Ouch, that Millennial renting joke hurts.

    "Permanent ownership of something. The trick is to have done that in the past."

  32. all fine and good. you now know which properties are best to have. but as almost all of them make back their money over the course of a game, i feel like generally buying even sub-prime spots is a better thing to do than to leave them for your opponents to buy (as that will cost you money in the long run, instead of turning profit).
    basically, you should keep a certain amount of money in cash and invest the rest when ever you can.
    and then of course with the houses: the rent difference between house 2 and house 3 is simply far bigger than the other steps, which makes it reasonable to make sure you get them all to 3 first. and of course, within the same colour it does matter which one you put more houses on. the 3rd ones always bring more money per hit but the probability of a hit can obviously be lower.

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