Monopoly: The Mathematical Secret Behind Winning

Рет қаралды 13,356

Күн бұрын

This is Monopoly: The Mathematical Secret Behind Winning. Hope you enjoy!
Hannah's book: www.amazon.com/gp/product/146...
A paper I used to check my results and as assistance when I was stuck: www.jstor.org/stable/2688377?...
Software I used to calculate the probabilities: RStudio
Today, I delved into the search for a (somewhat) mathematically guaranteed way of winning Monopoly, or at least a mathematical tip on the properties you should aim for, and those you should avoid at all costs. I am currently a sophomore in college, so I apologize if an appalling mistake appears along the way; I proofread the script a few times, but it might have just slipped my mind. I'm completely open to any feedback! Always looking for ways to better myself and further understand the beauty of math.
The animations in the video were made using Grant Sanderson's (3blue1brown) wonderful tool, Manim. Here's the depo: github.com/3b1b/manim
Also, credit where credit's due: 3blue1brown is a huge inspiration to me. His videos never fail to make me fall in love more with math, and that's what I hope to achieve in my videos too as I continue to make them.
For queries of any kind, you can send me an email: mathchat9@gmail.com
This video deals with: Math, Monopoly, Games, Probability, Markov Chains

Пікірлер: 42

@_notch 2 жыл бұрын

You're making me want to play around with markov chains! Nice video :)

@peretternavn8287 2 жыл бұрын

What I thought. I aways go for orange, and usually win with it as well. Last time I kicked ass with the light blue ones. I guess a quick reasoning for orange and red is that 6-8 are most likely to happen and a red and orange occurs almost certainly within the third or fourth roll with those numbers.

@koukoumusicfu3768 Жыл бұрын

Hello, when you add epsilon to p_i, should it be 40* epsilon in total when you calculate the row sum? (6:00 in the video)

@lexxusoriginaldj 2 жыл бұрын

Love it, there is potential in these videos :)

@IQ.oo. 3 жыл бұрын

Thanks so much!

@justmeldin6062 Жыл бұрын

I thought this channel has millions of subs. When I saw 109 I sad I will be 110. Good work!

@nicolt6983 2 жыл бұрын

Idk man, I'm trying to figure out the best strategies for monopoly and what I've been seeing with a bunch of computer sims is that the most important color to buy is indigo giving it's owner a 31% more probability to not lose (0.6% margin error)

@in4theride75 2 жыл бұрын

Yeah, if you take money into account Boardwalk is far and away the most advantageous property to hold with a monopoly. This is simply due to the devastating impact of the renter to be able to afford infrastructure since you are statistically more likely to win the less money you keep on hand.

@tbn4lyfe 2 жыл бұрын

You assume that the Chance and Community Chest cards are reshuffled (uniformly distributed) after each visit to their respective squares. This is not the case in a conventional game of Monopoly as per the official rules, the drawn card is sent to the bottom of the deck. This enables "card-counting" of sorts and the time homogeneous-ness of your Markov Chain premise becomes unclear. Very good video nonetheless!

@nemenavarro9240 6 ай бұрын

Question, but here you are not taking into account the full rules of double roll. For instance it is possible to move up to 35 moves from every square, however you are only taking up to 12 but punishing for triple doubles. Is this correct, or I am missing something? Great video tho

@addymant Жыл бұрын

One thing that forever annoys me about Monopoly which will mean Markov chains will probably never capture the full picture is that the community chest & chance cards are never re-shuffled after setup, and used cards go to the bottom of the deck. Which means if a game goes on long enough, the probabilities change slightly as you gain more information about what the next card is, until you know for certain what it is when the full deck has been cycled through. Also, unfortunately, unlike houses, hotels, properties, and get out of jail free cards, there is no cap on the amount of money in play. The rules tell you to add extra money, which unfortunately means there are infinite states a game can be in, though you can only get to a finite subset of those states in a single turn

@reda1829 Жыл бұрын

Hello, can you please share the code that you used to solve the system to get the invariant distribution?

@BigBlackChicks 11 ай бұрын

@jessikacardozo217 3 жыл бұрын

Hi, I am writing a paper for school due very soon lol and wondering how you calcultated the marcov chains? what program do you use?

@mathium17 3 жыл бұрын

Hey! As stated in the description, I used RStudio to calculate the limiting distribution. But basically any software that can solve for the eigenvectors of a matrix will do the trick; bear in mind, automation is key here, since we're talking about a 40x40 matrix. That and the fact that R has a nice library that deals with Markov chains are the reasons I used RStudio.

@jessikacardozo217 3 жыл бұрын

@@mathium17 I saw the description right after I sent that. I appreciate your help! Thank you for the great video and putting everything into perspective for me.

@erwanbayiha4378 3 жыл бұрын

Are you writing the ib ?

@danishharoon 3 жыл бұрын

@@erwanbayiha4378 lol i am

@devranbakir1276 2 жыл бұрын

@@danishharoon Lmfao I'm doing my IA on this lol

@rhysdenno5993 2 жыл бұрын

Beautiful graphics

@QuetalsoyDiego 3 жыл бұрын

Great!

@samiulhaquerounok5787 Жыл бұрын

Please make a video about Aviator and it's best strategy

@tobysmith5857 3 жыл бұрын

I don't understand when you derive the probability of doubles why you divide by 3

@PaulKnutsonSther 2 жыл бұрын

I think it's because you can't get 3 double rolls in a single double roll: It can maximally happen every 3 rolls.

@gabrieltepin 2 жыл бұрын

@@PaulKnutsonSther Could you elaborate some more? I didn't understand this division by 3 neither :/ What do you really mean by "you can't get 3 double rolls in a single double roll"?

@PaulKnutsonSther 2 жыл бұрын

@@gabrieltepin Notes: 1. Sorry for a long and messy reply. This is as much me wring while exploring, as a direct reply to your question. 2. My (probabily incorrect) terminology: * When I say "roll", I usually mean throwing two dice. * "Double-roll" means throwing two dices, which then show the same face. E.g. 2+2, not 3+4. * "Round" is when it's a players turn. This is 1-3 rolls long. Notably, the only way for it to be 3 rolls long, is for the first two rolls to be double-rolls. ** ANSWER TO YOUR COMMENT ** My thought process (which I suspect is wrong) was that since we're measuring average likelihood per roll, and you need three rolls in a row to land in prison, you'd only be able to one of those every 3 rolls. So if the likelihood of getting P=3 number of double-rolls in a row is N, then the probability should be N/P = N/3 per roll (on average). ** ME DISAGREEING WITH MYSELF ** That being said, I've grown unsure if this really makes sense, for two reasons: 1. When calculating the likelihood per roll, you can't assume you'll always get three or more rolls. Not all sets of three rolls are valid. For example: * 2+2, 4+4, 6+3 would be valid, but not result in prison. * 2+2, 4+4, 5+5 would be valid, and results in prison. * 2+3, 4+4, 6+6 would not be valid, as your round would end after rolling 2+3. Thus not all rounds are 3 rolls long. The expected amount of rolls per turn is - as shown at the bottom - approximately 1.2, not 3. Therefore, maybe we should shich 3 with 1.2? The probability would then be 1/6^3/1.2. 2. However, it may also make more sense to calculate per round, not per run. After all, the only way you'll actually get three rolls in a row is that at least the first 2 rolls are equal/double-rolls. ** ME RAMBLING ABOUT MATHS AND SIMULATION ** I made a script to do science, and the results were: * 1 000 000 rounds * 1 194 651 rolls (the number of times two dice were thrown) * 4 582 prisons (triple-double-rolls, or whatever) In the script, the round is ended when dice 1 is not equal to dice 2. I.e. if the first roll is 2+3, then the round ends. Thus, the expected amount of rolls per round is approximately 1.2: 1 194 651 / 1 000 000 ~= 1.2 Using maths: * Pr[round length = 1] = 30/36 = 0.833 * Pr[round length = 2] = 6/36 * 30/36 = 0.139 * Pr[round length = 3] = 6/36 * 6/36 * 36/36 = 0.028 Pr[1] + Pr[2] + Pr[3] == 1 Expected rolls per round: P = 1*Pr[1] + 2*Pr[2] + 3*Pr[3] = 1.1944... ~= 1.2. Yay maths! Final note: I'm no mathematician, so I would easily be wrong about this. Feel free to double check or correct me. :)

@justa9560 8 ай бұрын

@@PaulKnutsonSther This is 2 years later, but I'd also like to say that out of the 11 possible rolls with two dice (2,3,4,5,6,7,8,9,10,11,12), only 6/11 of them are doubles. comparing this to the 5/11 rolls that cannot be doubles, you get 6/11 / 5/11, or 6/5 = 1.2. Don't know if that's valid or not, but either way, a funny coincidence

@gatoquantico3925 3 жыл бұрын

The streets following prison are the most frequented. I recommend buying those.

@odedsayar4345 2 жыл бұрын

Cool model, but buying as much as you can as early as you can is still the best strategy (and when you run out of money start mortgaging) - in this way you decrease your chances of paying the other players while increasing the chances they pay you. In thevlong run this increases your chance of getting to buy a whole city and build houses to make real bank

@Marshall.R 2 жыл бұрын

Well yes, but that's a very very general rule that doesn't really help for trading or other situations where there's a lot more at play than "spend money on something or not", so this knowledge of where players will visit the most often can still be very helpful

@billowen3285 2 жыл бұрын

I knew I recognised Manim :)

@katherinesullivan837 3 жыл бұрын

Can you please model my life?

@katherinesullivan837 3 жыл бұрын

For real what an interesting video and how well put it was. Enjoyed your enthusiasm. Good luck with your channel and cheers mate!

@le_etian 8 ай бұрын

Faites le percer!!

@aurelianspodarec2629 6 ай бұрын

In reality America can be a good way to win becaus of the high rent one needs to pay, which cn put a player out of game, I won and lost to many games landing on New York, because youc ana lso get there by a card - if you play enough games you notice America can also be one of the better ones to buy