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?
First of all, the red is always worse than yellow. You don’t have to pay the property, so only the cost of development counts. And red development costs the same as yellow, but with worse rents.
With $1100, you can buy 2 hotels on the magenta and 3 on the light blue. You can buy 7 houses on the red or yellow. Overall, in this quiz there will never be more than 12+7+7=26 houses on the board, so no housing shortage.
Light blue is definitely the best.
Magenta saves some money, but they won’t help if you land on the 3-house spots your opponents own…. it’s also relatively rare to end up on them (wrong side with respect to jail, only 1 step from to GO!, etc.). So I’d pick yellow.
Hey Paolo! Thanks for your comment; clearly it was well constructed and thought out, which I always appreciate. That said, please double check your math/reasoning behind your answer for number of houses/hotels on the magenta group: houses there cost $100 each. Remember that the differential in houses between any pair of properties of the same color group may not exceed one, as per official Monopoly rules.
Of course, I don’t think that the above correction would change your answer much, so I will proceed with addressing another point that you made. You have correctly noted that the cost of development is equivalent on the red and yellow color groups but that the revenue collected per hit is higher on the yellows. However, this added revenue is actually almost exactly offset by the greater probability of landing on a red space.
See the below one-on-one analysis that I’ve done (using my simulator) between the reds and yellows (assuming the same amount of starting liquidity [x axis] and a “smart” building strategy by each player): the win percentage [y axis] differential is as close to a coin flip as it gets!
Also, could you go into some explanation as to why you feel that light blue is best? I’d love to get your insights into the problem.
Hi, I was confusing magenta with dark purple. Magenta is interesting because it lets you build 3 houses, which is where rents have a big jump. I would have guessed it to outperform yellow, but I would have been wrong (though by a small amount). As expected, however, orange outperforms magenta: even considering “go to St. Charles Place”, it costs the same to build, gives higher rent, and it’s easily reached from the jail.
Even with this mistake corrected, I still prefer the light blue, and the simulator agrees.
You spend really a lot less money on building, and you have a pretty good chance that the opponent land on it on the first throw. Depending on the exact rules you use for the timing of construction activities (beginning of your turn vs beginning of everybody’s turn), this may impact their ability to build on the red or yellow.
Does your simulator compute probabilities only for “Player 1 goes first, then 2, then 3″? Could you make it randomize that part as well?
Ah ok…that makes sense about the color mixup.
The simulator actually gives magenta the edge over red and yellow, but it is not as big as the edge afforded to light blue. Perhaps there are some slight differences in how you setup the position, which actually highlights one of the major points behind the simulator: there are many seemingly subtle factors in determining win percentages.
Since my example has everyone starting on GO, the short-term probability of the two players without the light blues landing on the light blue squares before they have a chance to spend their 1100 dollars is quite high. Note that this is still the case when substituting the light blues with the magentas…but to a lesser extent. As you astutely mentioned, the magentas will still surpass the critical 3-house level with $1100, which really leaves short-term probability as the determining factor behind the difference in win percentage.
I encourage you to play around with the starting squares, and you’ll notice a significant change in the win percentages (ie. put them all on Free Parking). Perhaps more than any other color group on the board, the win percentages of the owner of the light blues are particularly dependent on the opponents’ starting squares.
And to address your other questions/points:
- the official rules of Monopoly state that you may build at any time.
- I plan on making the position input functionality of my simulator a lot more robust, but for now the move order is determined by the player numbers (1 then 2 then 3 …)
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Hey, just came across this blog, don’t know if you’re still updating, but I’m a fan of Monopoly too (and a fellow board-game blogger) and I thought I’d chime in.
Mathematically I’m inclined to lean towards the yellows. They’ll give a better payout per house than reds, and the probability of landing isn’t so much lower that it would swing me. They’re an end-game monopoly, one that could deliver that final blow.
But Monopoly is also about the given situation. At $1100 you can get to the 3rd house level (barely) for red or yellow, but be completely strapped for cash (7 houses * $150/house = 1050), with two monopolies ready to make you tear down. For the magentas you get your 3rd house on all of them 9 houses * $100/house = $900), greatly increasing the odds of tagging your opponents, with the added benefit that yours is the first monopoly they’ll pass. Either opponent takes that, and their monopoly will be crippled for the first go-around, easing your chances of getting counter-hit.
I’d say take the magentas, build to 3 houses each, refrain from building the fourth to deal with a potential double rent hit from two houses ($200 + $450 rent – ($300 + $330 combined red and yellow rent) = $20), and hope to get a hit in.
Alternatively I would consider taking the reds, hoping to dodge the magentas my first one or two rolls. I’m not a huge number cruncher, so probabilities vs. rent payout and likely scenarios I wouldn’t know (and I’m not factoring in turn order, though I think that’s not as relevant since houses can go up between turns), but I feel like it’s a big gamble.
As for the light-blues, I’d avoid them. The situation is decent, but the probabilities of landing are not as great as I’d like, and if the game lasts much longer than these next few critical rolls you don’t have a great game-ending Monopoly. In that scenario I’d take my chances hopping over the blues and take the reds, hoping to get a hit before I get to yellow.
Thanks for your reply! Obviously you’re no amateur when it comes to board games.
I’m not sure whether you entered my proposed “Monopoly problem” into my simulator (http://pandora.dyndns.biz/monopoly), but you may be surprised to see that the light blues are the way to go.
In this exaggerated scenario (all players starting at Go), the short-term probability of landing on the light blues overwhelms the long-term considerations that you have mentioned. While the blues’ knock-out potential pales in comparison as the amount of cash in the game increases (as you mentioned), in this case it’s all about getting that first hit in. If they all miss, then the situation changes of course, but remember that you have three players each with a 14/36 chance of landing on either Oriental, Vermont, or Connecticut before you even touch the dice!
P(at least one hit) = 1 – P(zero hits) = 1 – (11/18)^3 ~= .7718
That’s just too high a percentage to pass up.
I took a look at the simulator (cool stuff BTW) for magentas but not light blues; I wanted to hand out all the deeds and it took a while to click on all of it, and I’m at work, so I didn’t check. But it was good to know the magentas is the right decision.
The math is good, but not as good as that last equation. It’s 3 players, so 2 opponents (you landing there is irrelevant). So:
P(at least one hit) = 1 – P(zero hits) = 1 – (11/18)^2 ~= .6265
Good, better than 50/50, but not as good as 77%.
I look forward to any future blog entries you have!
Jeez…that shows how long it’s been! I could have sworn that the scenario was for four players.
Instead it was a two-part question, each with three players. And in the part you addressed, the magentas of course are best. I encourage you to play around with positioning the tokens in various scenarios…it’s pretty remarkable what a difference board position can make.