## Monopoly Chances

While the whole world surely heard about it before, I just today ran across a web page purporting to give the probabilities and expected incomes for the various squares on a Monopoly board. There are many similar versions of this table around — the Monopoly app for iPad even offers the probability that your opponents will land on any given square in the next turn, which is superlatively useful if you want to micromanage your building — and I wouldn’t be surprised if there are little variations and differences between tables.

What’s interesting to me is that the author, Truman Collins, works out the answers by two different models, and considers the results to probably be fairly close to correct because the different models of the game agree fairly well. There are some important programming differences between Collins’s two models (both of which are shown, in code written in C, so it won’t compile on your system without a lot of irritating extra work), but the one that’s most obvious is that in one model the effect of being tossed into jail after rolling three doubles in a row is modelled, while in the other it’s ignored.

Does this matter? Well, it matters a bit, since one is closer to the true game than the other, but at the cost of making a more complicated simulation, which is the normal sort of trade-off someone building a model has to make. Any simulation simplifies the thing being modelled, and a rule like the jail-on-three-doubles might be too much bother for the improvement in accuracy it offers.

Here’s another thing to decide in building the model: when you land in jail, you can either pay a \$50 fine and get out immediately, or can try to roll doubles. If there are a lot of properties bought by your opponents, sitting in jail (as the rolling-doubles method implies) can be better, as it reduces the chance you have to pay rent to someone else. That’s likely the state in the later part of the game. If there are a lot of unclaimed properties, you want to get out and buy stuff. Collins simulates this by supposing that in the early game one buys one’s way out, and in the late game one rolls for doubles. But even that’s a simplification: suppose you owned much of the sides of the board after jail. (You’re likely crushing me, in that case.) Why not get out and get closer to Go the sooner, as long as it’s not likely to cost you?

That Collins tries different models and gets similar results suggest that these estimates are tolerably close to right, and often, that’s the best one can really know about how well a model of a complicated thing represents the reality.