My God, has it really been over two years since my last post?!! The past two years flew by faster than a top hot getting back-to-back “Advance to Go” cards.

For anyone reading this post, thanks for stopping by. While this topic may not be as “exciting” as figuring out the shortest possible game, it hits upon some facts and statistics that may be counter-intuitive for many of you (as they were for me until today).

As anyone who has played a lot of Monopoly knows best, the world’s best-selling board game is ultimately a game of luck. But perhaps the most commonly overlooked piece of luck in any game of Monopoly is established before a single token is ever touched: move order.

That’s right! Going first matters…especially in a heads up (aka two-player) match. The idea behind this is very simple: the first player tends to land on properties a step earlier than his/her counterpart, which increases the chance of purchasing a new property. This, in turn, substantially increases player 1’s chances of creating a monopoly “naturally” on any property group that he/she first lands on.

But how much of an advantage are we talking about? Here are some numbers a simplified Monopoly simulator, which I wrote in Java, came up with (measuring percentage of all monopolies first established naturally):

- 2 Players -> 53.5% / 46.5%
- 3 Players -> 37.2% / 33.1% / 29.6%
- 4 Players -> 29.0% / 26.2% / 23.4% / 21.3%
- 5 Players -> 24.1% / 21.8% / 19.6% / 18.0% / 16.5%
- 6 Players -> 20.7% / 18.8% / 17.2% / 15.8% / 14.3% / 13.2%

To help clarify things, let’s look at the first line (2 players) and break down what the numbers mean. Simply put, of all 2-player Monopoly games in which at least one player obtained a full color group WITHOUT the need to trade, 53.5% of the time the first player is first to establish such a monopoly. While this may not seem significant, it certainly is statistically relevant and in fact might even be considered a landslide if it involved votes for an election.

So why did I say that this is most important for a 2-player game?? Well, this is simply because trading can help to offset the advantage of being first to naturally obtain a Monopoly in a multi-player instance (more than two) of the game. In other words, trading seldom occurs in heads-up Monopoly games because all trades require all players, and the player with a monopoly benefits little by trading with his opponent.

But even in games with as many as six players, the above table helps to drive home an important point (both in Monopoly and in business): being first is everything!

“Wait a second, Mr. Monopoly Nerd!” you may be saying to yourself. “How often does at least one player get a monopoly on his/her own (aka via luck alone)?” As always, I’m happy to break down the numbers:

- 6 Players: 27.2% of games
- 5 Players: 35.4% of games
- 4 Players: 48.1% of games
- 3 Players: 67.7% of games
- 2 Players: 92.5% of games

This last number astonished me at first! Is it really fewer than 1 out of 10 heads-up games that require trading to create monopolies?? First let’s do some simple probability to help verify that the simulator’s findings are reasonable:

- P(either player obtaining both Med. and Baltic) ~= 50%
- P(either player obtaining both Park Place and Boardwalk) ~= 50%

Logically, either player can grab the first of either Med. and Baltic or Park Place and Boardwalk, which then makes the long-term probability of getting both equal to the probability of simply getting the second. Since there are 2 players, the P(getting any property) ~= 1/2 or 50%.

P(either player obtaining both dark purples OR either obtaining both dark blues) = P(either obtaining dark purples) + P(either obtaining dark blues) – P(either player obtaining both dark purples AND either obtaining both dark blues) ~= 75%

Now, is it reasonable to assume that the remaining six color groups can make up for the approximately 20% remaining? I believe so, yes.

Still unable to shake the feeling that the number was too large, I started to think of reasons why my observations seemed to conflict with the simulator’s findings. I quickly realized that the dark purples are usually all but completely disregarded as a significant monopoly. Therefore, my opponent and I would normally trade for monopolies regardless of whether either of us owned the lowest color group.

When compensating for this by having my simulator ignore the dark purples, the revised estimated percentage of naturally-occurring monopolies was lowered to approximately 86%. Still higher than I remember, but that’s probably where human biases come into play. After all, a game that requires wheeling and dealing is a hell of lot more memorable than the slew of “lucky wins and losses”.

What can probably be said is that player 1 throwing 7 and getting advance to Mayfair/Boardwalk would be the best start and player 2 throwing a 4 (or 7 and go back 3 spaces).