# Is this true : «In a 2-player game of Monopoly, there is a 12% chance that the game will go on indefinitely.»

This supposedly «fun fact» was posted on a Facebook game page.

One commentator declared a 2-player game of Monopoly a zero-sum game;

I stated that the bank acts as a 3rd player, injecting and withdrawing cash.

Is there any mathematical validity to the statement that a 2-player game of Monopoly could go on indefinitely?

Edit: Concerning «indefinitely». Since the OP was making a distinct case of a 2-player game, and 3- or more -player games always end, for this question, I think we can assume that s/he meant that the 2-player game would never end.

• It is unclear what is meant by "indefinite" in this context. It can either mean an unlimited amount of time, or an undefined amount of time. If you think of a simplified sample game where we each have \$100, and we repeatedly flip a coin. If I win, you give me \$1. If you win, I give you \$1. Will this game ever end? As the number of flips tends towards infinity, so does the chances of the game ending. Eventually the game would end; but it could take an infinite amount of time to do so. So really it comes down to what the original post meant by "indefinite". – GendoIkari Jun 4 '20 at 17:36
• I can't see how Monopoly is a zero-sum game. Players get money from the bank, players give money to the bank (in terms of fines, house repair chance cards etc). – Philip Kendall Jun 4 '20 at 17:49
• @Gendolkari, Philip Kendall : You both make valid points... – Robert Miller Jun 4 '20 at 18:01
• There are only a few ways the bank can inject money into the game and that is by passing go and a few chance/community chest cards. outside of that it is just removing money from the game from the purchase of property, housing, and various fees from spaces, chance and community space cards. Unless both players are losing less than 200 dollars each turn around the board on average they will run out of money eventually. – Joe W Jun 4 '20 at 18:03
• Is the poster of the fact playing by the actual rules are using things like 'free parking' variants which lengthen the game? '12% seems such a weirdly accurate figure I suspect its just a made up 'fact'. Players getting cards like General Repairs will also keep removing cash from the game. – StartPlayer Jun 4 '20 at 19:51

The short answer is "Yes, but...".

The longer answer is, as per the paper in question, that a team of researchers did some calculations on what would happen in a 2-player game of Monopoly where both players follow very simple strategies (and a couple of things that aren't 100% by the rules), notably:

1. Always try to keep a small reserve of cash on hand to pay rent or other costs.
2. Always buy properties you land on where possible.
3. Never bid on properties that are up for auction.
4. Build houses according to a simple pattern.
5. Never pay to get out of jail (even on the third roll).
6. Always sell your Get out of Jail card to the bank for \$50 (which I'm pretty sure isn't a thing).

At the very least, #2, #3 and #4 are generally considered poor strategy - careful use of auctions can get you key properties on the cheap, and clever building of houses can deprive your opponent of their opportunity to build. Obviously the key here was removing most of the major decision points to keep their model manageable.

With those simplifications to the game, they then created a big state model of the game - all the possible things you could potentially see if you took a snapshot of the game at different points in terms of who owned what properties, how much money they have, what spaces they're on, etc. And then they modeled all the different paths the game could take between those states, to find the probability of going from one state to the next (e.g. if the current state includes "I have rolled doubles twice in a row", there's a 1 in 6 chance the next state will transition my position to "I am in Jail").

Then, with that bit transition model, they do some fancy maths to show how often the game winds to a close. You're right in saying that the game is not zero-sum, but the "banker" role can both add and remove money so it can be as much to blame for making the game go on forever as it can be the reason it finally ends.

They actually do this modelling in a few different ways, but all of their different methods all agree that if you run the game for an arbitrarily long time then there's about an 88% chance that one player or the other will win, meaning that there's a 12% chance that you'll never actually see the game end because both players wind up having enough money on hand to handle the ups and downs of the dice.

So, in a 2-player game of Monopoly, with a few rule changes, and where neither player makes any real decisions, there is a 12% chance that it will never end.

• The phrase "and where neither player makes any real decisions" seems to bear the semantics of "where neither player actually plays with intent to win". Viewed in that light, it's surprising that in 88% of games a winner does emerge. – Forget I was ever here Jun 5 '20 at 23:21
• Properties are never up for auction, because of the previous point. In two player monopoly, trading is a bad idea for one party. In the steady state, "sell your Get out of Jail card to the bank for \$50" is a simplification of "hold onto GooJ card until you exit Jail failing the third roll" – Caleth Jun 8 '20 at 12:42

Someone one the FB page where this question was originally posted found this answer from the

School of Operations Research and Information Engineering Cornell University Ithaca NY 14853, USA

ESTIMATING THE PROBABILITY THAT THE GAME OF MONOPOLY NEVER ENDS

At the end of the 10-page report, the following is stated :

All four of our estimators yield confidence intervals that suggest that the probability that the game goes on forever is close to 12%.

The answer to the question would therefore be : True

but I will have to read it to confirm this.