It is impossible to write a paper about Monopoly strategy without mentioning that an early monopoly of the orange or red properties is killer. They are the most frequently landed-on colors and only cost $1460 and $2030, respectively, to develop to three houses. The problem, of course, is that everybody knows this. That is why you typically have to bend a few fingers back to get anybody to sell you the third orange or red property, at least early in the game. Later you might ruin somebody through other means and seize the property you need to complete an orange or red monopoly, but by then they are no longer dominant.

Another frequently touted strategy is to quickly buy up the light blue properties and immediately put hotels on them, which only costs $1070. As you bankrupt the weaker players, you can use the property you seize to assemble another monopoly with higher rent and use it to bankrupt the stronger players. Unfortunately, with everybody starting at Go and seven being the most frequent die roll, in a four-person game it is almost certain that at least two different players will own the light blue properties. To a lesser extent, the same goes for the purple properties around the corner.

The yellow and green properties are less frequently landed on than the orange and red properties and the greens are too expensive to develop to three houses until one has obtained some other source of income. A dark blue monopoly is less expensive than a green one because there are only two properties; getting an early monopoly there is actually quite fortuitous. Unfortunately, while Boardwalk is 18th, Park Place is only the 33rd most frequently landed-on square. It is entirely possible that everybody will circle the board a dozen times and Park Place will still have her virginity intact. Getting them both early in the game is a bit improbable and trading for them late in the game is not a good idea, for reasons explained below.

The dark purple properties just ahead of Go are the least landed-on colored properties and their rent, even with hotels, is more of an annoyance than a bankrupting experience. Unless they are practically thrown at your feet, they are not worth picking up. And, with that, we have summarized what is commonly known about the colored properties.

It is my belief that people buy too much unmatched property early in the game. Buying one of everything is not really a step towards getting a monopoly of any of those colors. You can recoup half the cost by mortgaging a property but, in most cases, you will be forced to do so (either to pay rent or to develop a monopoly elsewhere) before the rent has recouped the other half. Except for the fact that its purchase blocked somebody else’s monopoly, it does you no good and leaves you poorer than if you had not bought it at all.

My strategy hinges on getting the other players to tie up a lot of wealth in properties that bring them no income (if they are mortgaged) or very little income (if they are undeveloped) but which they must hold to prevent another player from getting a monopoly. If someone has two of a certain color and I land on the other property, I buy it. But I don’t just sit there thumbing my nose at the would-be monopolist; I approach a third party who has a lot of cash and offer it to him at an inflated price, darkly hinting that, if he does not accept my terms, I will sell it to the would-be monopolist. (What’s a little extortion among friends, eh?)

Following is a table of the limits for bidding on property. The 1st column is for when none of the other properties is owned or when both are owned by different people. The 2nd column is for when you already own one property and you have the opportunity to buy a second one; or when somebody else already owns one property and is trying to buy a second one.

Suppose, on your first go round the board, you land on St. Charles Place.’ States and Virginia Avenue are already owned by two other players.’ Maybe you will eventually wind up with a purple monopoly, but probably not; it is just as likely that, when the trade wars begin in earnest a dozen cycles hence, you will decide to try for a monopoly in one of the other colors "whatever looks feasible.’ If you do not need this property you can always mortgage it for half, so the question that fills your mind is, will rent on it recoup the other half before powerful monopolies are created, causing you to mortgage it either to pay their rent or to buy houses for your own monopoly?

For St. Charles Place, the answer is, in a word, no.’ Truman Collins has been kind enough to compile how many opponent turns are required, on average, to recoup the cost of a single property, assuming that people get out of jail on their first turn, as they typically do in the beginning of the game.’ (Everybody knows to stay in jail as long as possible in the end of the game; if you pay to get out, there is a 45.5% chance of landing on an orange property at least once in the next two turns; if you roll doubles to get out, there is a 43.5% chance.)’ Because property can be mortgaged, we are really interested in Collin’s numbers by half.’ For St. Charles Place, it takes 259.1 opponent turns to earn $70 in rent.

It takes about 5.92 turns to go around the board and about ten cycles for four players to have purchased all the property.  At this time, the trade wars begin in earnest.  And, if I know anything about trade wars, I know that you can get farther with ready cash and a silver tongue than you can with a silver tongue alone.  Somewhat arbitrarily, let us say that the trade wars begin after exactly 10.6 cycles, or 188.3 opponent turns.  $120.9 = $70 (1 + 188.3 / 259.1) is all that you can expect from renting out St. Charles Place and then mortgaging it when the trade wars begin, and that is all that you should pay for it.  If you have already got another purple, then pay up to $132.2 for it.

What actually happens when you follow the bid limits set forth above is that everybody else piles into the light blue, purple, orange and red properties, buying them at list or higher.’ Then they run out of cash and, while some of the yellow, green and dark blue properties will be bought at list, others will be put up for auction and you can get one or two while staying within the bid limits, which are about 93% of list for the adjacent properties.’ (Marvin Gardens and Pennsylvania Avenue are the black sheep of the yellows and greens.)’ When the trade wars begin, use the cash you have amassed from your railroads and utilities (see below) to buy a yellow or green monopoly.’ Or make a deal with whoever else is holding yellows and greens so you get the yellows and he gets the greens.’ Then use your cash hoard to immediately put three houses on your yellow monopoly, leaving your hapless co-monopolist stuck with three greens that he cannot afford to develop.’

If you and another player hold, between you, all of two colors, there is a question of which one you should trade for.’ In the previous paragraph I recommended the yellows over the greens because they are cheaper to develop.’ That is one consideration, but the most important one is the probability of people landing on it.’ In general, the first monopolist to have someone (anyone) land on his property will win because he can then survive landing on the other monopolist’s property without having to sell houses.’ If you have to sell houses to the bank, it is unlikely that you will be able to ever buy them back.’ This is why the dark blue properties are great if you happen to land on them both early in the game but, if you trade for them late in the game, they may not survive against other monopolies.’ Everybody goes around and around without landing on them and, when someone finally does, you charge $2000 but only collect a couple hundred and a handful of mortgaged properties from the ruined player.

The following chart gives the probabilities per opponent turn, assuming people stay in jail as long as possible.  Giving someone the railroads they need to complete the set of four in exchange for an orange or red monopoly is okay if you can afford to put at least one house on them and still have $200 left over for a railroad fare; doing so for a yellow or green monopoly is only okay if you can afford to put at least two houses on them and still have $200 left over.  But if you cannot immediately put three houses on a monopoly of the less probable colors and still have $200 left over, they will probably not hold their houses against a railroad monopoly.

Because selling houses back to the bank is such a bummer, it is imperative that you hold enough cash to survive a go around the board.  But, at the same time, you do not want to be too conservative because, while you are trying to develop your monopoly, somebody else may land on the third property of their color and immediately put hotels on it.  In general, if you own nothing else than your monopoly, you need to hold $95 to survive a go around the board without having to sell any houses back to the bank if nobody lands on your property, but this question is complicated enough that it deserves a page of its own.

Typically, these trade wars do not result in just one but in two people holding monopolies.’ Hopefully, you are one of them and, hopefully, yours has higher rent either because the property is more expensive or because it is better developed.’ But even if you have the weaker monopoly, or no monopolies at all but enough cash to survive landing on someone else’s monopoly one or twice, it is still possible to win.’

Suppose two players are blocking each other in some property, neither has a monopoly but one is teetering on the brink of bankruptcy from having spent all of his cash and mortgaged most of his property to pay a hotelier’s rent, while the other has several unmatched properties around the board blocking people and enough cash to survive visiting the small monopoly but not the big one.’

The former player has lost hope and, if he does not like the player who owns the big monopoly because he gloats when someone lands on it, the doomed player may act as a kingmaker by selling all his property to someone else just to avoid having to listen to any gloating at his own funeral.’ So be nice and, if a weak player owes you a small rent on undeveloped property that will not bankrupt him but will push his cash reserves so low that someone else’s undeveloped-property rent will bankrupt him or make him mortgage something, then just ignore the rent.’ Maybe he will anoint you.

The latter player is convinced that he may yet win if only he can assemble enough cash to survive visiting the big monopoly.’ Offer him some cash in exchange for the property blocking the doomed player.’ It is just one of many properties he owns which are not producing any rent but are blocking somebody and, if he has not given much thought to either the inevitability or the consequences of the doomed player going under, he may let you have it.’ If the doomed player bankrupts himself to you (even a railroad can push someone under when they are that close to the brink), you will have a monopoly; if he goes bankrupt over taxes, you may be able to buy the property from the bank; and if he goes bankrupt to the player with the big monopoly, you are at least blocking that person from his second monopoly, which would have made him a juggernaut.

Sometimes the trade wars do not result in any monopolies.’ No two players hold, between them, all of two colors, which would have resulted in them both acquiring monopolies, though developed to a greater or lesser degree, depending on how much cash they have squandered on other properties.’ With everybody blocking each other, you wind up in a Mexican standoff.’ This link goes to a page which proves that Mexican standoffs are interminable.’ They are also really boring and, if you persist in going around and around in one, people will lose interest and chose another game.’ If you find yourselves in a Mexican standoff, you could, by mutual agreement, decide that Go will only pay $100 until someone goes under.’ Or you could just quit and call whoever has the most assets the winner.

Now let us do the same analysis as above for the railroads and utilities.’ Comparing the following chart to the first one will quickly convince you that railroads are a total, total money tree.’ Trust me on this:’ When you look around the table and see one of the players sitting on a Sentry Safe instead of a dining room chair, that fellow is the railroad baron.’ Watch out for him.’ He’s the kind of guy who will buy, at list price, Boardwalk from one player and Park Place from another, both mortgaged, immediately un-mortgage them and then put two houses on Park Place and three on Boardwalk.’ Oops!’ Didn’t know he had that much cash, did you?

Since everybody knows that railroads are worth at least their list price, they are rarely put up for auction. The bid limits shown above are more typically used for comparing property values. For instance, suppose that you have Pacific Avenue and Reading Railroad while you opponent has North Carolina Avenue and B & O Railroad, so your positions are comparable. Should you trade Pacific Avenue for B & O Railroad? You are giving up property that lists for $300 in exchange for property that lists for $200, not to mention opening up the possibility of your opponent landing on Pennsylvania Avenue and completing the green monopoly. However, with four players in the game there is only a 25% chance that he will be the first to land on Pennsylvania Avenue while there is a 43.75% chance that you will be the first to land on at least one of the two remaining railroads. So cross your fingers and make the trade, but wait until he is just past the third property so everybody else gets a chance to land on it before he comes around the board again.

Buying a second railroad has about the same return on investment as putting your first house on an undeveloped monopoly. Buying a third railroad has about the same return on investment as putting a second house on a monopoly that currently has one house per property. Buying a fourth railroad has about the same return on investment as putting a third house on a monopoly that currently has two houses per property. If you already have three houses on a monopoly, getting a fourth railroad is more important than further development of that monopoly. Note that mortgaging a railroad or utility does not diminish the rent on the others.

Utilities are, inexplicably, shunned by Tim Darling, who actually cites Collin’s work, though he apparently does not understand it. Suppose you own the Electric Company and have just purchased Illinois Avenue at auction, completing the coveted red monopoly. Then you roll and land on Water Works. A house for Illinois and the title deed to Water Works both cost $150. Which should you buy? Answer:   Water Works has a 60% higher return on your $150 investment than the house. Yet Illinois is the most often visited square on the board!

It is possible that Darling is confusing average income with return on investment. The average income per opponent turn is what matters in a Mexican standoff, while the number of opponent turns needed to recoup an investment is what matters when you have only $150 and must decide what to spend it on. Utilities in Monopoly are like owning a hot dog cart; the profit on the cart (gross earnings minus grocery bills minus minimum wage for you to run it) is hundreds of percent a year, far greater than Microsoft stocks. But Microsoft is a big enough company that they can absorb the investments of millionaires while there are a limited number of hungry people walking past your spot on the sidewalk, so further investment in a fancier cart is not going to make you more money.

Owning utilities and railroads is like having a dead-end job while owning a monopoly that you cannot afford to put houses on is like having an under-capitalized business.  Neither will lift you out of poverty but, if you have both, then you can use your wages to buy equipment for your business and, once it is properly capitalized, the business will lift you into the middle class.  Similarly, if you trade away everything you own but your two or three railroads to obtain a monopoly, the railroads will pay for houses and then the houses will pay for hotels and then the hotels will win the game.

So, bottom line, don’t get caught up in auction fever and bid more than the limits listed in the chart above for colored properties. But, at the same time, don’t lose your nerve when bidding on your second railroad. If the chart says that it is worth over three times its list price, then it is. Do not, of course, let your opponents know how much you value it. Moan and groan like you have to pull every dollar bill out of your nose, but keep upping the price by five dollars until you win the auction.

Good luck! Roll dem bones and hope for the best!

Proposed Rule Change

Even in the standard four-player game, the last player in the queue starting from Go buys significantly less property than the first two. In a five- or six-player game, he is basically a bystander. It is frustrating for the last in line to have all kinds of knowledge about bid limits and then just sit and watch while the other players buy all the properties from the bank at list price. Then, the first three players who have been buying everything they landed on invariably get into a Mexican standoff and the game becomes boring for everybody, propertied or not.

I propose that, on the first trip around the board, players not be allowed to buy at list price from the bank. When they land on property, it immediately goes up for auction and everybody gets a chance to bid on it. Unlike most home rules (like the ill-conceived one about putting tax money under Free Parking), this would speed up the game. Why? Because so many players have an almost religious faith in the orange and red properties that, if they are put up for auction while everybody still has their initial $1500, the bidding war will invariably escalate to four or five hundred dollars. This takes money out of private hands, which hastens bankruptcies and creates unstable situations that push people to the negotiating table.

Also, almost everybody has some pet theory about which color property is the best. My proposed rule change gives them a chance to test their theories by bidding on the color they like. As it is, people go for years touting a theory and, whenever they get bankrupted, claiming that their theory was not debunked because, in this particular case, they were frustrated by someone else landing on their favorite property and buying it at list price from the bank. Of course, strong players know that there is no “best” color; opening up bidding to everybody on the first go around the board just gives the weak players enough rope to hang themselves with.