I'm trying to understand a good way to calculate the probability of landing on each box of the Monopoly board. In Monopoly you can move both rolling dice and when extra events happen (e.g., chance/community chest cards).

Due to that heterogeneity of moving rules, I would like to simulate the game to calculate landing frequencies, but I have two issues about that:

  1. if I simulate a game with only one player it's the same of a simulation with $n$ players? (e.g., the usage of chance/community chest cards could depends on the number of players)
  2. Is the simulation a good way to tackle the problem? I mean: isn't there an approach better than mere computer simulation?
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  • $\begingroup$ Simulation certainly seems like a natural idea...if nothing else, all the cards you might draw mess the probabilities up considerably. Not sure if one person simulation is ok...been a while since I played. Can my opponents moves alter my location? If I can't pay rent or if I draw a specific card? $\endgroup$ – lulu Dec 2 '16 at 11:57
  • $\begingroup$ @lulu yes. If player $A$ picks the card "go to box $b$", player $A$ goes to box B and then the card go to the bottom of the deck of cards. Therefore next player will pick another card, but in terms of box landing frequencies, it seems the same. I'm looking for a general frequencies distribution, if any, regardless of the number of players... $\endgroup$ – floatingpurr Dec 2 '16 at 12:10
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    $\begingroup$ If you can simulate for one player, you most likely can simulate for N with minor adjustments. Simulation is certainly a good idea. So go for it. An then tells us about it :-) $\endgroup$ – Rolazaro Azeveires Dec 2 '16 at 13:33
  • $\begingroup$ @RolazaroAzeveires that is sure. : ) I am wondering if there are substantial differences on the calculation of landing frequency. $\endgroup$ – floatingpurr Dec 2 '16 at 13:43

If there were no Chance or Community Chest cards or triple doubles rules landing you into Jail then every property would have an equal 1/40 likelihood, but that's not the case and that's what makes the difference, and that is what needs to be analyzed.

Those two decks have you advancing to the nearest utility or railroad (which depends on where you are located), or specifies a railroad (see a railroad theme going on here), or specific places (St. Charles, Go or Jail, etc.).

Also, the likelihood of triple-doubles rolled on the dice $\frac{1}{6^3}$ lands you into Jail no matter where you are located.

Also, a typical game of Monopoly is limited to laps around the board measured in dozens not infinity, this should make the realistic odds to strongly favor squares that are 7, 5 & 6 moves ahead (and repeated several time) from the key properties where a play is "advanced to" (e.g. nearest railroad and Jail being the largest odds).

In short, if someone wants to design a brute force game analyzer, I think you would be best served to determine the odds of being sent to various locations (like the railroads and others) and from those locations determine the odds with a limited number of dice rolls to see which properties end up being best.

We all know it's going to end up being the railroads, then utilities, and then favoring some orange, red and yellow (at least based on what my experience gut tells me), but then it's proving this with math.


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