**Hey Everybody! I’m very sorry for not posting this on Saturday – last week was ridiculously busy. However, I promise that this week I will post a brand new design article on Saturday as usual. Until then, I have decided to reprint a pair of related articles about everybody’s favorite friendship-ending board game, Monopoly. See you on Saturday!**

# What is the Best Square in Monopoly?

For the last couple of weeks I have been on a winter break from college, which for me means traveling all over the Midwest trying to see all the family members that I rarely get the chance to see throughout the year. Not only did I get to spend time with my family that I haven’t seen in a while, but I also got the chance to play some games with them. And when you are looking for a game that the whole family can sit around the table and play, few games are more reliable than Monopoly. Sure, it might end up permanently tearing the family apart, but at least everyone knows how to play!

Monopoly has been around for over 80 years, and is one of the rare games that has not only survived all that time but thrived. Very few games have the same kind of brand recognition as Monopoly, and over the years it has become a household name. Heck, it even has an annual promotion with McDonalds! Because of this level of recognition, there are very few people who haven’t played this game (or one of it’s thousands of variations and knock-offs) at least once.

When playing with my family, one of the most interesting things that I noticed was that everybody had their own unique strategies and preferences.One of the most interesting things about these different strategies is how much value each player puts on the different board spaces. Certain spaces, such as the utilities, are often considered worthless while others, such as Boardwalk and Park Place, are highly coveted. But how accurate are these judgements? And, more importantly, what are the actual best spaces and monopolies on the board?

**To Be The Very Best, Like No One Ever Was**

Before we go any further, we should try to figure our what being the best square in monopoly actually means. For the most part, Monopoly has two goals. First goal – have enough money to survive landing on your opponent’s squares. Second goal – build up your properties enough so that your opponent’s cannot survive landing on your squares.

While these two goals may seem like they always fit together, this is not actually the case. Often in games of Monopoly players will end up spending far too much money trying to build up their own properties, without saving enough to survive their opponent’s squares and simply hoping that they will get lucky. On the other hand, there are other players who try to conserve their money too much, and end up crippling their income capabilities.

As far as simple survival, the corner squares are by far the best at simply keeping players alive. These four squares – Go, Jail, Free Parking, and Go To Jail, never have any negative consequences for the player and in some cases actually provide the player with free income. However, these squares are public spaces, and for the purposes of this article I am going to focus on properties that can be purchased.

In terms of conserving money, the most straightforward approach is to simply not buy any properties, but this is a recipe for disaster. However, another approach could be to try and get as much board coverage as possible without necessarily improving your properties. By getting a lot of coverage over the board you can minimize your chances of landing on an enemy square, and give yourself a chance to survive as long as possible. Unfortunately, this strategy has very low income potential and without some way to generate passive income you will eventually run out of money. You also have a very low chance of actually knocking out any other players from the game.

If you are simply looking for the square with the highest income potential, Boardwalk is the obvious choice. With a hotel it has the highest rent in the game, costing a back-breaking 2000 dollars to land on. That amount is enough to take out almost any player in one foul swoop. It’s partner Park Place is no slouch either, with a very respectable 1500 dollar maximum rent.

These properties have the potential to absolutely devastate any opponent who lands on them, and this potential can be very tempting for players. Unfortunately, it requires a lot of money to actually reach this potential. In order to build a hotel on Boardwalk you must first purchase the entire Monopoly -a $750 dollar face value, although it will likely cost you a lot more if you aren’t lucky enough to actually land on both spaces on your own. Then, you must build at least 4 houses on Park Place, as well as the actual hotel. Because each house costs you 200 dollars, this is a minimum of $2550 to achieve that maximum income potential.

This leads to a question of costs and benefits – yes, somebody landing on Boardwalk will devestate them, but it also leaves you $550 short of breaking even on your investment, and you will probably have to have some other source of income to finance your improvements on these squares. If it is landed on twice, however, you are now $1450 in the black. In order to decide if this is a good investment, we need to figure out how often somebody is actually going to land on your Monopoly, and how quickly you can make your money back.

**What Are the Odds? **

This brings us to another very important point to keep in mind when trying to find the “best” square in Monopoly – the odds of actually landing on that square. If somebody lands on your property nearly every turn, it is a far better investment than if nobody ever lands on it in the entire game.

At first this seems like a pretty simple question – the Monopoly board has 40 different squares, so the odds of landing on any square should be 1 in 40, or 2.5%. Unfortunately, it is not nearly this simple. There are several ways to move around the board without rolling the dice – Chance and Community Chest cards, the “Go To Jail” square, or rolling doubles 3 times in a row can all send you around the board to somewhere completely unexpected, and makes figuring out the odds much more difficult.

The easiest way to figure out the odds would be to run a computer simulation that models pieces moving around the board, and where they end up after various die rolls. After running this simulation several millions or billions of times, it would give us some pretty accurate numbers for how likely you are to end up on any particular space. Fortunately, somebody has already done this! In this article by Truman Collins, the author lays out the probabilities that they found for each square, as well as several other useful pieces of data that we will be using to reach our conclusions.

From this information, we can see some very interesting results. The first thing that jumps out to me is that Jail is by far the most frequently landed on space. This makes sense as there are so many ways to get to jail – you can land on it, you can land on the “Go To Jail” space, you can get a Chance or a Community Chest card, or you can roll 3 doubles in a row. All of this adds up to between a 6.2 and 11.6% chance of ending your turn on the Jail space. The reason for this huge discrepancy is based on your current strategy – do you pay to get out of jail immediately or try to stay in as long as possible – but either way, it is clearly the most visited space on the board.

The space that you are least likely to end your turn on (besides “Go To Jail, or Chance and Community Chest spaces which send you somewhere else on the board) is Mediteranean Avenue. This space is directly after the Go space, which means that it is impossible to land on from Go (as you cannot roll a 1). Ironically, St. Charles Place (the space directly after Jail) does not have this problem, probably due to the existence of a Chance card that sends players directly to St. Charles Place.

Using this information, we can finally figure out whether Boardwalk really is that good of an investment. To begin with we can see that the odds of finishing a turn on Boardwalk is somewhere between 2.62 and 2.48%, which puts it in the middle of the pack as far as probability of landing on it goes. Boardwalk actually has a rather unfortunate position (being directly before GO), but the existence of a “Go to Boardwalk” Chance card helps improve its… well… chances.

Based on the numbers, Boardwalk is actually a relatively good investment on it’s own. It takes about 25 rolls to earn back your investment with a hotel (which is 9th out of all properties with hotels) , and after that it has the highest average per-roll earnings of any property in the game. However, what brings it down is its partner, Park Place. In order to get the full value out of Boardwalk you must also heavily invest in Park Place, which is not nearly as good of an investment. Park Place takes about 45 rolls to pay off it’s hotel – the longest of any property in the game – and once you do, there are 5 other spaces with a higher per-roll income. So even though Boardwalk may be one of the best investments in the game, the Monopoly as a whole is not.

**To Buy or Not to Buy**

If Boardwalk isn’t the greatest investment in the game, then what is? First, let’s just look at the colored properties. Besides Boardwalk, the two properties with the best per-turn income are Illinois and Pennsylvania Avenues, which both provide nearly the same amount of income per turn. Of these two, Illinois Avenue is the better choice because it requires much fewer rolls to recoup your initial investment, and it is also much cheaper to fully upgrade than Pennsylvania (it costs at least $2630 to put a hotel on Illinois Ave., as opposed to $3520 for Pennsylvania Ave). This means that Illinois Avenue provides a much better bang for your buck than Pennsylvania Ave.

But even this is not the best investment in the entire game. In order to determine that, I am going to look at how much income you can expect from the fully upgraded properties, and compare them to the the total cost of upgrading. I will also be ranking entire Monopolies, not just individual properties. After making these calculations, the results for best investment on a property are as follows.

(Note: Scores are relative to the Orange Monopoly )

**New York Avenue, Tennessee Avenue, and St. James Place – 100****Connecticut, Vermont, and Oriental Avenues – 87.92****Illinois, Indiana and Kentucky Avenues – 77.05****Virginia and States Avenues, St. Charles Place – 74.88****Boardwalk and Park Place – 74.88****Railroads – 74.40****Marvin Gardens, Ventnor and Atlantic Avenues – 73.67****Pennsylvania, North Carolina and Pacific Avenues – 63.29****Mediterranean and Baltic Avenues – 58.45****Utilities – 30.68**

As you can see, the Orange monopoly is by far the most efficient investment of in the game. It’s excellent position makes the orange properties the most common spaces to land on after getting out of jail, and being at the end of the row they are relatively efficient to upgrade while still providing steady income potential.

There are also a number of other interesting things that can be gathered from these rankings. Firstly, each side of the board has two monopolies, and each monopolies at the beginning of the side tended to be less efficient than monopolies at the end of the side (with the exception of the red monopoly, which actually scored slightly better than the yellow one). The primary reason for this is that the cost of building houses remains the same for these properties as the later ones, but you tend to get less of a pay increase for building them.

Finally, as we all suspected, the utilities ranked dead last. They are by far the worst investments in the game, and have extremely low earning potentials even for their relatively low prices.

# What is the Value of a Monopoly?

Hey everybody! A few months ago I wrote an article that compared the different colored monopolies on the monopoly board (you can find it here). In that article I used math and statistics to examine several different aspects of the Monopoly board game, but there is one aspect of these games that I didn’t really deal with – trading.

Trading is a huge part of Monopoly strategy, and making good trades can easily be the difference between victory and defeat. And of all the different trades that can be made in a game of Monopoly, the most important trades are when you are trying to complete a Monopoly.

Perhaps it’s just the people that I play with, but I have found that when I am trying to trade with somebody to complete a Monopoly I end up paying through the nose. On the one hand, this makes sense – Monopolies are key to winning the game, because without them you cannot build houses, hotels, or maintain a reasonable income.

On the other hand, it can be difficult to know if you are getting a good trade or not. If you complete the Monopoly but don’t have any money left, you can end up in a far more dangerous position than you were previously. And without at least a few houses, it can be difficult to earn enough income to make up for your initial purchase.

In a game where paying too much can spell death for a player, it is important to make sure that you are not overextending yourself. Today, I am going to look at various business valuation techniques to try and answer the question – how much should you actually pay for that last property in your Monopoly?

**Warning: Economics Ahead**

**Let us Count the Ways **

According to Investopedia, there are a number of different ways of measuring the value of a business, including Market Capitalization, Times Revenue Model, Earnings Multiplier, Discounted Cash Flow, Book Value, and Liquidation Value. In this section, I want to go through these different methods and determine if any of them would be applicable to the game of Monopoly.

Market Capitalization is a way of measuring the value of a business through its share value. Basically, if your business has 100 shares, and each is valued at $100, then this would give you a market capitalization of 100 * 100 = $10,000 dollars. While this method is relatively simple to calculate, it doesn’t really apply to Monopoly.

The Times Revenue Method is a way of analyzing the value of a business by looking at how much it can earn over a given period of time, and multiplying it by some constant factor which is dependent upon the type of business. These multipliers usually lie somewhere between 0.5x and 2x, depending on how profitable the business is. This method seems like a promising area to look into for this question.

The Earnings Multiplier method is a way of determining the value of a business based on it’s profit and earnings, rather than just based on revenue. Generally this method compares the annual returns on a given stock to the purchase price of the stock, and uses this to compute a ratio known as the P/E ratio. While it may be possible to modify this method to make it apply to Monopoly, as is it doesn’t really seem like a reasonable metric for determining the value of a Monopoly.

Discounted Cash Flow estimates are made by estimating the future earnings of a company, adjusting them for predicted inflation rates, and using these adjusted earnings to estimate the present value of the company. I don’t think that this method would be ideal for Monopoly because there is no inflation, and the income potential of every property is always known.

Finally, there are the Book Value and Liquidation values of the company, which are both calculated by taking the company’s total assets and subtracting their total liabilities. The difference between these two is that Liquidation value assumes that you are liquidating all assets due to bankrupcy, which means that you are generally going to be getting lower than market value for your assets. These methods are extremely easy to calculate for Monopoly pieces, but are not great at evaluating the value of a Monopoly because they don’t take intangible assets into account – having all the pieces of a Monopoly is worth far more than the pieces are worth separately.

**Examining Our Options **

Based on the above, it seems like the most appropriate method for determining the value of a Monopoly would be the Times Revenue method. This method is usually based upon the annual income of a business but, while games of Monopoly can last a very long time, I don’t think actually figuring out how much a property can earn during a year is going to be the best way to value it’s income potential. Instead we have to determine what the equivalent of a year would be within a game of Monopoly.

We actually have a number of different options to choose from – a year could be represented as a single turn, a round (each player gets a turn), a single rotation around the board, or even an entire game. For this article, I am going to be using the metric of 1 complete circuit around the board = 1 year. This metaphor seems to be implied by the board itself – you collect a Salary every time you complete the circuit, and have to pay annual taxes.

The board is 40 squares around, and the average roll for two dice is a 7. This means that it would take a little under 6 rounds for a player to make a circuit around the board, on average. However, because you get to roll again if you roll doubles, and the odds of rolling doubles are 1 in 6, this brings the number of rounds down to around 5. This number is further affected by things such as the “Go to Jail” square, as well as Chance and Community Chest cards, but for this article I will be using the metaphor of 5 rounds = 1 circuit = 1 year.

The next thing we have to look at is the actual earning potential of these different Monopolies. However, calculating this is actually quite tricky, because it relies on a number of factors such as the number of players at the table and how much you can afford to improve the Monopoly.

First off, lets look at how the number of players affects the price of a Monopoly. First of all, the more other people are on the board, the more chances there are for people to land on your property every “year”. It also means that there is more total cash to go around, since each person gets the same starting amounts. Finally, more players simply means more competition for the same amount of resources, which will drive the prices up. Therefore, I propose that the estimated prices should have a “Opponent multiplier” (N) , which multiplies the income potential of a Monopoly by the number of other players on the board.

Secondly, the total value of a Monopoly is also related to how much you can do with it. The expected income for a Monopoly increases based on how many houses or Hotels you have on it, and if you are unable to improve the property then it’s value to you is actually much lower.

Finally, we need to figure out what our actual multiplier should be. In general, the multiplier used in these calculations is based on expected rates of growth and recurring revenue. In general, when buying a Monopoly you would expect the potential for growth to be relatively high. However, the recurring business is relatively low. At most, each player will only be landing on your property once every “year”, and usually not even that. Based on this, I am going to be choosing a multiplier of 2.

**The Price is Right **

Based on the above, we can finally complete the Times Revenue value of a Monopoly. Using calculations from this article by Truman Collins, we can determine the average earnings of each property per opponent dice roll. Then, we will multiply this number by 5 due to our calculation of 5 rolls per complete navigation of the board, and then multiply it by 7/6 due to the additional 1/6 change of rolling doubles each turn. We will then add up the earning potential of all properties that are part of a single monopoly, and multiply by our Times Revenue multiple of two. Finally, we will perform this calculation for each possible number of houses.

When performing these calculations, we also need to keep track of the up-front costs of actually constructing the houses and hotels. The cost of building the houses will significantly reduce the expected income of the first year, but will have little to no effect on the expected income of following years. Therefore, I propose to depreciate the construction costs of those buildings over the expected lifespan of the monopoly, which I would estimate to be around 10 years.

**Example Calculation**

Suppose we were looking at completing the Orange Monopoly and putting 3 houses on it, and there are 4 people at the table. The expected income per roll for each of these spaces with three houses on them is 50.015. We then multiply this number based on our multiples of 5 * 7/6 * 2 = 583.5. Because there are 3 opponents, this gets further multiplied by 3 and becomes $1750.53 (which I will round to $1751, because there are no coins in Monopoly). Because houses cost $100 each, and you must buy a total of 9 houses for each property to have 3, (depreciated over 10 years), this gets reduced to $1661 for the whole Monopoly.

Suppose that I had already purchased Tennessee Avenue and St. James Place, and am thinking about trading for New York Avenue. I have already paid $360, out of the total value of $1661, which means that the maximum I am willing to pay to complete this Monopoly is $1301 – quite a bit higher than the face value of $200! Keep in mind that if you are willing to pay this price, you must also have enough cash left over after purchasing to actually buy the houses as well (and probably even a little more above that).

**The Results!**

After performing this calculation on all the different Monopolies, and every number of houses, I came up with the following table:

(The table only lists one member of each Monopoly per row, which represents the whole Monopoly)

0 Houses | 1 House | 2 Houses | 3 Houses | 4 Houses | Hotel | |

New York Avenue |
$91 | $197 | $578 | $1661 | $2247 | $2834 |

Connecticut Avenue |
$32 | $65 | $195 | $630 | $944 | $1290 |

Illinois Avenue |
$115 | $242 | $732 | $2066 | $2558 | $3050 |

Virginia Avenue |
$56 | $111 | $362 | $1140 | $1594 | $1958 |

Boardwalk |
$146 | $278 | $854 | $2008 | $2397 | $2786 |

Marvin Gardens |
$126 | $271 | $858 | $2142 | $2689 | $3029 |

Pennsylvania |
$145 | $554 | $998 | $2366 | $2852 | $3292 |

Mediterranean |
$9 | $13 | $48 | $173 | $322 | $477 |

As you can see, the value generally increases significantly the more renovations the player can afford to place on it. In addition, most Monopolies don’t actually produce enough income to offset their costs until you are able to build the second or third houses.

**Until Next Time! **

**I hope you enjoyed this in-depth look at Monopoly mathematics, economics and strategy. As I mentioned at the beginning, I am planning on releasing a new article this Saturday. The topic is “Tiers in Competitive Games”, so if that sounds exciting to you make sure you subscribe to the blog on Twitter or here on WordPress so you be notified whenever a new post goes up. See you then!
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