Like I said in the last post, I spent some time working on the actual probabilities for each square in Monopoly. First thing first: Chance/Chest cards suck.
About half of the Chance/Chest cards can be classified as “move cards”, as they instruct the player to move to a square. Lets say you draw the “Go to jail” card from Chance and move to jail. That card now goes to the bottom of the stack, so the probability of anyone getting that card is now 0 for the next 16 Chance draws(there are 17 Chance cards), and 1 on the 17th draw. Therefore the probability of the squares is affected by the initial order of the C/C cards, since its common for not all the cards to be drawn during shorter games.
The number of players also affects things; obviously more players will go through the C/C cards faster, but it also complicates the “get out of jail” cards. I never pay the $50 to leave jail, therefore I either need a “get of jail” card or a doubles roll. If two other players already have the freedom cards, that means when I leave the jail it’ll be on a doubles roll. This means that I can’t land on squares 3,5,7,9, or 11 on a jail exit move. Though the effects are probably negligible in the grand scheme of things, they’re definitely present.
The only other factor, aside from standard two-dice probabilities, is that rolling three doubles in a row immediately moves the player to jail.
With all these factors considered, solving things gets pretty complex. Example: You’re on Water Works, and roll double 4s. You’re now on Chance. This Chance turns out to be a “Go Back 3 Spaces” card, putting you on Community Chest. The Community Chest card is an “Advance to Go” card. Since you had double 4s, you get to roll again; doing so gives you get double 1s, putting you on Community Chest. Go To Jail! At the end of it, you’ve visited four different squares on a single turn. On the other hand the same outcome could have occurred if double 1s were rolled from Water Works, as this would have put the player on the “Go To Jail” square.
Such complex move sequences are very rare, but are certainly possible. Therefore instead of trying to come up with an equation, I decided to calculate the potential pathways to each square and the probability of said pathways. I also took some liberties with C/C cards and simply treated them as a random draw. I’ve attached the final Excel sheet as usual:monopoly_final_paths
And, a screenshot of the final ordering and game board composition:
There are definitely some differences between the theory and the experimental results, but the general trends are fairly similar. Railways and orange still appear as the most popular properties. There’s a lot more “smoothness” in the theory though, especially in the middle range properties. The one last thing to do with this data would be to factor in costs of buying and rent, to see which are the best price wise. I’m sick of this game and data though, so maybe another day. 🙂