Monopoly Quiz of the Week

Here’s the scenario:  

You are playing against two opponents and have all properties split up among the three of you, but none of you have a complete color group / monopoly and must therefore trade.  You each have about 1100 (give or take a few bucks) in cash and are all coincidentally sitting on “GO”.

After a lot of arguing, negotiating, and compromising, a most interesting choice has been reached.  Each player will end up with one of the following color groups:  magenta, red, or yellow.  It just so happens that all other properties are mortgaged, but these are not.

Assuming that you are able to convince your opponents to settle for any of the three aforementioned monopolies, which should you choose and why?  What factors are the most sensitive to this decision?

I look forward to your answers, and good luck!

** Now imagine the same scenario, except that the magenta properties are substituted with the light blues. How does this impact your decision and why?

Monopoly Simulator Updated!

** Update – As of 11/3/12, the simulator will now validate all entered position information, in an effort to avoid invalid or corrupted sample games / win percentages.

Well, it’s taken a bit longer to post an update than I had originally planned, but I guess that’s what happens when the storm of the century turns everything upside down for a week!

Head on over to http://pandora.dyndns.biz/monopoly/simulator.html to check out some of the radical and exciting changes made. My personal favorite is the added functionality for getting a sample game and literally watching it play out in “Auto Action” mode.

An brand new tutorial / sample usage video can be found here:

As always, please hit me up with any bugs, limitations, or shortcomings that you would like to see fixed. Here are some of the next steps I’m planning on taking:

1. I will be making the entire position setup reversible, which is certainly a lot less frustrating than having to start from scratch after a mouse slip.
2. I’ll be implementing a server-side method of storing previously-tried positions, which will help go back-and-forth when analyzing slightly different starting positions.
3. I will continue to perfect the timing of actions taken on the board, as well as expanding the simulator’s building strategy to handle multiple monopolies.

Without further ado, enjoy!! Keep an eye out for a new video tutorial / trailer.

Monopoly Simulator Online!

After several weeks of Java, Javascript, and PHP coding, I am proud to present the beta release of the Monopoly Nerd’s online simulator!

Here is a brief video tutorial on how to use the tool. Keep in mind that it is still far from a finished product, but the numbers seem quite reliable.

As always, I welcome your comments, questions, and suggestions.

Do You Know the REAL Rules of Monopoly?

I thought that I would make today’s post a little more interactive than usual, so I encourage all of you to make your comments below.  They may not show up right away, but I will do my best to approve them as quickly as possible.  Once I get enough comments and replies, I will re-post the questions along with the official answers.

Now…onto today’s challenge!  So you think you know how to play Monopoly, huh?  We’ll see. 

Below are 10 questions that hit upon specific (or general) rules of the game.  Feel free to comment with your answers to any or all of them.  Good luck:

1. How much money does a player receive when landing on GO?
2. How much money does a player receive when landing on Free Parking?
3. How much rent is paid to the owner when landing on Boardwalk, assuming it is unmortgaged but the owner’s Park Place is mortgaged?
4. Situation:  You roll a seven, land on the chance space near Free Parking, and pick up the “Advance to Nearest Utility” card.  How much money do you owe the owner of Water Works, assuming that it is unmortgaged and he/she also owns Electric Company?
5. How much does it cost to unmortgage New York Avenue?
6. What is the highest payment possible for the owner of three railroads?
7. Which came first…the Monopoly square spaces or the street names in Atlantic City?
8. Is it possible to receive money from another player via trade if you do not possess any properties?
9. If you are able to spend — via mortgage and cash — up to 1100 to build houses on the yellow properties (currently without houses or hotels and assuming no housing shortage), what is the maximum number of houses that can stand on Marvin Gardens after spending your money?
10. You are grinning ear to ear after building hotels on the orange properties at what seems to have been the perfect time.  After doing so, the remaining twelve houses in the bank were quickly scrounged up by your desperate opponents, who could each only build one or two houses around the rest of the board.  You roll the dice, land on Luxury Tax, and owe the bank $75 (or $100 if you play the new Monopoly).  In an effort to build your hotels, you mortgaged every other property you own and have only $20 cash in hand.  Assuming that trading with your jealous opponents is off the table, how do you pay the bank?

Good luck to all who participate! 

It’s NOT Just a Game of Luck!

This photo highlights the importance of board position when making a critical trade. Using my recently-completed Monopoly simulator, I’ve been able to precisely quantify the jump in win% when owning the light blues (versus oranges) and building when your opponent is on Mediterranean (the MOST DANGEROUS place for him/her to roll from) versus Connecticut (the SAFEST place).

The numbers along the bottom represent how much cash each player has before building, and the assumption is that both players build as aggressively as possible at the end of each of their turns…until one of them goes bankrupt.

All other property is mortgaged, to eliminate unwanted noise or variance.

My favorite is when starting with $1000 each. As the person ending up with the light blues, your probability of winning jumps from a miserable 32.51% to a favorable 57.7% just by timing it when your opponent is in solid striking distance!

The Railroad Shuffle

**Note: This post assumes that the reader has a decent amount of Monopoly experience and fully understands the ideas behind mortgaging and unmortgaging.

For those of us who have been there before, it’s one of the trickiest late-game decisions to make: should you mortgage your valuable set of four railroads to help build an additional four houses on the oranges (or two houses on the dark blues, etc), or is it not worth the loss of significant income you are already getting ($200 per railroad hit)?!

Fortunately, there is a highly effective and easy solution to help obtain the best of both worlds. I call it the railroad shuffle.

The key behind this tactic is that the rent owed by someone landing on your railroad is based SOLELY on the mortgage-status of that particular railroad (NOT all four) and the number of railroads you OWN (regardless of their status). In other words, if someone lands on an unmortgaged Reading Railroad (one of four that you own), you are still owed $200 even if the other three railroads are mortgaged. Pretty sweet indeed!

So how do we use this gem of a rule to our advantage? Well, it’s actually quite simple. We mortgage three out of the four railroads we own (or two out of three), use the extra $300 to improve our houses/hotels situation, and continue to benefit from owning the full set of railroads.

But how to know which railroad to keep “alive”?? Well, since mortgaging/unmortgaging are allowed at any time, we can simply apply the following procedure before each roll:

1. Assess the current roller’s board position. Which railroad is within striking distance (note that there will ALWAYS be at least one railroad within 12 spaces or a single roll’s total)?
2. Is the closest railroad currently unmortgaged? If yes, then we are done.
3. If no, then we SWAP the currently-unmortgaged railroad for this one. The price for doing this…$10!!!

That’s right! It only costs $10 to mortgage one railroad and unmortgage another (this is because each railroad has the same value and there is a 10% unmortgaging fee on the mortgage value of any property).

So in short, what I like to call the railroad shuffle costs a mere $10 max (often it costs nothing if more than one opponent are near each other) per opponent roll to always have a chance at getting $200 on that roll. From a long-term expected value perspective, let’s look at the math to see when it makes sense to apply this strategy:

• Let C represent the cost of swapping an unmortgaged railroad (aka $10).
• Let p represent the probability of your opponent landing on the newly-unmortgaged railroad.
• Let R represent the revenue made if your opponent lands on the railroad.

For the cost to be justified, we setup the following break-even analysis/formula:

p*R >= C

To solve for p, let’s look at the aforementioned scenario of owning four railroads:

p*200 >= 10
p >= 1/20

So, it looks like this strategy makes statistical sense for all but two rolls: 2 and 12 (see footnote for caviat).

Now when owning three railroads:

p*100 >= 10
p >= 1/10

In this case, the railroad shuffle makes sense for the following space differentials/rolls:

5,6,7,8,9

So even from a strict numbers perspective, the railroad shuffle makes a lot of sense! And when we consider that the $100 or $200 gained from one railroad hit is often the difference between three houses on New York (for example) or two houses, the importance of the railroad shuffle becomes even more apparent.

Of course if you have plenty of cash on hand AND houses or hotels on a nice color group, then this strategy is not only unnecessary; it is counter-productive (remember that there are ways for your opponents to land on your railroads via Chance cards too).

** A quick note on the special case of an opponent being two spaces away from the nearest railroad:

Whenever this occurs, then your opponent is actually within striking distance of TWO railroads (2 and 12 spaces away). This, of course, doubles the probability of his/her landing on one of your railroads but requires having two (rather than one) railroads unmortgaged. So unless you’re ok with leaving two railroads unmortgaged, I would just stay away from doing anything for these rolls.

The Importance of Going First in Monopoly

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:

1. P(either player obtaining both Med. and Baltic) ~= 50%
2. 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”.