rjnoe@riccb.UUCP (Roger J. Noe) (08/29/85)
I really underestimated net interest this time. I'm accustomed to sending out queries and making comments and hardly ever getting any response, which I naturally interpret as apathy. So it's very surprising to hit upon a subject which prompts not just a few but several (I count 14 so far) netters to actually respond by mail. I'm stunned! The Monopoly data will be posted in a separate article very shortly. This article will explain my methods and interpretation of some of the results. Those not interested in all the gory details should just read this article. Those interested in just the data but not what it means or how it was arrived at should just read the other article. The rest of you read both, obviously. Where to begin? Oh, yes, the Monopoly board. It has 40 squares, right? Well, not really. The In Jail/Just Visiting corner should really count as two squares. So the location of one's token on the board completely deter- mines the "state" the player is in at the time, regarding movement around the board, right? No, because of the "roll three doubles consecutively and you are in jail" rule. So we need three states for each of the 41 locations for having rolled no doubles to get there, one double, and two doubles. On the third roll of doubles from the latter state one goes to jail. The three states for in jail are just arrived, failed one roll (no doubles), and failed two rolls. On the third roll the token leaves jail by that count, whether or not it is doubles. There, we have 3*41=123 states that completely describe movement over the board. Construct a vector of 123 real numbers and call it P. P[n] denotes the overall probability of being in state n. Of course, each P[n] is nonnegative and the sum of all the P[n] is 1. Now construct a 123-by-123 matrix of real numbers T. T[i][j] denotes the probability one gets to state i given that one began in state j. Then if we sum the products T[i][j]*P[j] over all 123 values of j, we should end up with P[i]. This is just the matrix equation TP=P, which is equivalent to (T-I)P = 0 for the identity matrix I the same size as T. So if we can compute T we can do any Gaussian elimination on the matrix T - I and the result will be P. The only problem is that obvious ways of coming up with the matrix T for Monopoly will make T-I have rank 122. But if we add a row to T-I that defines that the P[n] all sum to 1 and change the right-hand- side from the 123-vector of all zeroes to a 124-vector with the last row changed to 1, everything will work. The only problem that remains is to construct T. Most of the transitions are easy, since they are just rolls of the die. Probabilities of rolling numbers on the die are easily computable. If it's doubles (also easily computable), we advance not just to another square but to the next "level". If we're already at the highest (i.e. third) level, rolling doubles moves us to jail (at the lowest level). This is the only way jail is reachable by roll in one move. (Landing on the Go To Jail square and then going to jail counts as two moves.) The complications arise from the Chance and Community Chest cards. Of the 16 Community Chest cards, only two move the token other than by roll, to Go or to Jail. After figuring these moves away from Chest squares, we compute the probabilities of rolling away and scale them down by a factor of 14/16. Chance is harder because ten of the 16 cards move the token, and there are some things like advance to the NEAREST railroad or utility. But there is only one possible destination for each of these cards for each of the three Chance squares and this can be coded ahead of time. By the way, my Chance deck has two "Advance to the Nearest Railroad" cards, perfectly identical. Does anyone else have different? Is this a mistake with my set or is this correct? I have assumed in my derivation that it is. The rest is easy. Go To Jail lands one In Jail 100% of the time. If one is In Jail and it's rather late in the game, the best strategy is to take your time getting out. This is also assumed in my derivation. This is actually easier to compute than rolling away from other squares. That's all there is. This completely constructs the rather sparse matrix T. It's a very quick Gauss elimination of the 124 rows and 123 columns to get the 123 state probabilities P. Adding the probabilities of being in each state corresponding to a single square gives the overall probability of being on each of the 41 spaces. This leads to In Jail being the most popular space (at 8.7%) due entirely to our chosen strategy. The other ones are, in descending order: 2.7774% Illinois Ave. 2.7058% Go 2.6822% B. & O. Railroad 2.6191% Free Parking 2.6144% Tennessee Ave. 2.6068% New York Ave. 2.6007% Reading Railroad 2.5836% Chance (between Kentucky and Indiana Avenues) 2.4839% St. James Place 2.4615% Water Works 2.4446% Pennsylvania Railroad 2.4320% Community Chest (between St. James and Tennessee) 2.4250% Electric Company 2.4239% Kentucky Ave. 2.3804% Indiana Ave. 2.3732% St. Charles Place 2.3621% Community Chest (between N. Carolina and Pennsylvania) 2.3551% Atlantic Ave. 2.3411% Pacific Ave. 2.3358% Ventnor Ave. 2.3069% Go To Jail 2.3051% Boardwalk 2.2966% North Carolina Ave. 2.2612% Marvin Gardens 2.2487% Virginia Ave. 2.1860% Pennsylvania Ave. 2.1256% Short Line 2.0367% Income Tax 2.0309% Vermont Ave. 2.0211% Chance (between Short Line and Park Place) 2.0185% Chance (between Oriental and Vermont Avenues) 2.0176% States Ave. 2.0131% Connecticut Ave. 1.9862% Just Visiting 1.9792% Oriental Ave. 1.9115% Park Place 1.9062% Luxury Tax 1.8912% Baltic Ave. 1.8835% Community Chest (between Mediterranean and Baltic) 1.8637% Mediterranean Ave. That's really more of a range than I first anticipated for the 40 regular squares. But it doesn't tell the whole picture, since the name of the game is money, not probabilities. We can compute expected revenues of the properties in color groups easily, merely by multiplying the probability of being on the property's space by the rent. This will give different figures for the property outside of a monopoly, undeveloped in a monopoly, and with one through four houses or a hotel. These results are pretty much as expected. The highest is Boardwalk with a hotel for $46.10 per token move. Mediterranean all by itself generates less than 4 cents per token move. Expected revenues for utilities and railroads are harder. If one reaches a railroad from a Chance card, the rent doubles. One need merely increment the probabilities of being on each of the three railroads by the amount already contributed from having reached there from the respective Chance locations and cards. For utilities, the rent is four times the roll on the die if one is owned, ten times the roll if both utilities are owned. But if reached by a Chance card, the dice are rolled again and multiplied by ten to get the rent. So the probability of reaching a utility has to be broken down into its constituent parts and the expected roll upon reaching the utility has to be computed. The most telling statistic for each property is to take the expected revenue and divide it by the cost (purchase price of that single property plus the cost of developments on that property), giving a return on investment rate. (Actually, one should go through each permitted combination of houses in each group and compute return on investment on the basis of expected revenue and total cost of the whole property group.) There are actually some surprises here. For example, in the yellow color group the best return on investment is with Atlantic Avenue and the worst is Marvin Gardens, unless they all have 1, 2, or 3 houses. The rent structure for Atlantic and Ventnor is much better than that for Marvin Gardens at the 4 house/hotel levels. The highest peak return on investment rates for individual properties are the three orange color group properties, followed by Boardwalk and then the three light blue color group properties. But not all properties reach their individual peaks when they have hotels on them. The exceptions are the green and dark blue properties. Pacific and North Carolina Avenues peak individually at either three or four houses (equal) while Pennsylvania Avenue peaks at three houses. Park Place also peaks at three houses while it's best to construct a hotel on Boardwalk to achieve the highest return on investment there. This leads to some unexpected combinations of buildings to get the highest return on investment for certain groups. The best development of the dark blue properties is to have three houses on Park Place and four on Boardwalk. For the green group, the optimal point is four houses on Pacific Avenue and three houses on the other two lots. This leads to a ranking of relative desirability of property groups based on the optimal return on investment rates: 3.617% Orange 3.190% Light Blue 2.801% Dark Blue 2.764% Red 2.741% Red Violet (or Maroon) 2.670% Railroads (all four) 2.658% Yellow 2.338% Green 2.124% Blue Violet (or purple) 1.113% Utilities (both) Keep in mind that this is based on optimal return on investment, which requires a lot of investment! You can graph expected revenue versus required investment by property group from which one can see at a glance the best investment given how much money is available. It's handy for making fine distinctions and dispelling mistaken notions. I know quite a few people who think the greens are just wonderful but it's obvious from this that they aren't, even if one does have the money to develop them. This kind of analysis is a handy tool when it comes to negotiating trades. So many people play Monopoly completely ignorant of the mathematics of the game that it's a snap making deals that appear fair or even in the other player's favor when it's really working for you alone. -- Roger Noe ihnp4!ihopa!riccb!rjnoe
rjnoe@riccb.UUCP (Roger J. Noe) (08/29/85)
There are form feeds separating sections of these results. See article <512@riccb.UUCP> for explanations. Overall Probabilities of Being on each Space 2.7058% Go 1.8637% Mediterranean Ave. 1.8835% Community Chest 1.8912% Baltic Ave. 2.0367% Income Tax 2.6007% Reading Railroad 1.9792% Oriental Ave. 2.0185% Chance 2.0309% Vermont Ave. 2.0131% Connecticut Ave. 1.9862% Just Visiting 2.3732% St. Charles Place 2.4250% Electric Company 2.0176% States Ave. 2.2487% Virginia Ave. 2.4446% Pennsylvania Railroad 2.4839% St. James Place 2.4320% Community Chest 2.6144% Tennessee Ave. 2.6068% New York Ave. 2.6191% Free Parking 2.4239% Kentucky Ave. 2.5836% Chance 2.3804% Indiana Ave. 2.7774% Illinois Ave. 2.6822% B. & O. Railroad 2.3551% Atlantic Ave. 2.3358% Ventnor Ave. 2.4615% Water Works 2.2612% Marvin Gardens 2.3069% Go To Jail 2.3411% Pacific Ave. 2.2966% North Carolina Ave. 2.3621% Community Chest 2.1860% Pennsylvania Ave. 2.1256% Short Line 2.0211% Chance 1.9115% Park Place 1.9062% Luxury Tax 2.3051% Boardwalk 8.7021% In Jail Expected Revenue per Move (dollars) Utilities Owned one both 0.746 1.600 Electric Company 0.762 1.736 Water Works Railroads Owned one two three four 0.713 1.427 2.853 5.707 Reading Railroad 0.674 1.348 2.697 5.394 Pennsylvania Railroad 0.751 1.503 3.005 6.010 B. & O. Railroad 0.531 1.063 2.126 4.251 Short Line Base Monopoly with Number of Houses Shown With rent 0 1 2 3 4 hotel 0.037 0.075 0.186 0.559 1.677 2.982 4.659 Mediterranean Ave. 0.076 0.151 0.378 1.135 3.404 6.052 8.511 Baltic Ave. 0.119 0.237 0.594 1.781 5.344 7.917 10.885 Oriental Ave. 0.122 0.244 0.609 1.828 5.483 8.123 11.170 Vermont Ave. 0.161 0.322 0.805 2.013 6.039 9.059 12.079 Connecticut Ave. 0.237 0.475 1.187 3.560 10.679 14.832 17.799 St. Charles Place 0.202 0.404 1.009 3.026 9.079 12.610 15.132 States Ave. 0.270 0.540 1.349 4.048 11.243 15.741 20.238 Virginia Ave. 0.348 0.695 1.739 4.968 13.661 18.629 23.597 St. James Place 0.366 0.732 1.830 5.229 14.379 19.608 24.837 Tennessee Ave. 0.417 0.834 2.085 5.735 15.641 20.855 26.068 New York Ave. 0.436 0.873 2.181 6.060 16.967 21.209 25.451 Kentucky Ave. 0.428 0.857 2.142 5.951 16.662 20.828 24.994 Indiana Ave. 0.555 1.111 2.777 8.332 20.830 25.691 30.551 Illinois Ave. 0.518 1.036 2.591 7.772 18.841 22.963 27.084 Atlantic Ave. 0.514 1.028 2.569 7.708 18.686 22.774 26.861 Ventnor Ave. 0.543 1.085 2.713 8.140 19.220 23.177 27.134 Marvin Gardens 0.609 1.217 3.043 9.130 21.070 25.752 29.849 Pacific Ave. 0.597 1.194 2.986 8.957 20.669 25.262 29.281 North Carolina Ave. 0.612 1.224 3.279 9.837 21.860 26.232 30.604 Pennsylvania Ave. 0.669 1.338 3.345 9.558 21.027 24.850 28.673 Park Place 1.153 2.305 4.610 13.831 32.271 39.187 46.102 Boardwalk Return on Investment per Move (per cent) Utilities Owned one both 0.497 1.067 Electric Company 0.508 1.158 Water Works Railroads Owned one two three four 0.357 0.713 1.427 2.853 Reading Railroad 0.337 0.674 1.348 2.697 Pennsylvania Railroad 0.376 0.751 1.503 3.005 B. & O. Railroad 0.266 0.531 1.063 2.126 Short Line Base Monopoly with Number of Houses Shown With rent 0 1 2 3 4 hotel 0.062 0.124 0.169 0.349 0.799 1.147 1.503 Mediterranean Ave. 0.126 0.252 0.344 0.709 1.621 2.328 2.745 Baltic Ave. 0.119 0.237 0.396 0.891 2.137 2.639 3.110 Oriental Ave. 0.122 0.244 0.406 0.914 2.193 2.708 3.191 Vermont Ave. 0.134 0.268 0.474 0.915 2.237 2.831 3.264 Connecticut Ave. 0.170 0.339 0.494 1.047 2.427 2.747 2.781 St. Charles Place 0.144 0.288 0.420 0.890 2.063 2.335 2.364 States Ave. 0.169 0.337 0.519 1.124 2.444 2.811 3.066 Virginia Ave. 0.193 0.386 0.621 1.307 2.846 3.212 3.470 St. James Place 0.203 0.407 0.654 1.376 2.996 3.381 3.652 Tennessee Ave. 0.209 0.417 0.695 1.434 3.128 3.476 3.724 New York Ave. 0.198 0.397 0.590 1.165 2.532 2.586 2.624 Kentucky Ave. 0.195 0.390 0.579 1.144 2.487 2.540 2.577 Indiana Ave. 0.231 0.463 0.712 1.543 3.019 3.058 3.086 Illinois Ave. 0.199 0.399 0.632 1.388 2.654 2.670 2.682 Atlantic Ave. 0.198 0.395 0.627 1.376 2.632 2.648 2.660 Ventnor Ave. 0.194 0.388 0.631 1.403 2.633 2.634 2.634 Marvin Gardens 0.203 0.406 0.609 1.304 2.341 2.341 2.296 Pacific Ave. 0.199 0.398 0.597 1.280 2.297 2.297 2.252 North Carolina Ave. 0.191 0.383 0.631 1.366 2.376 2.342 2.318 Pennsylvania Ave. 0.191 0.382 0.608 1.274 2.213 2.161 2.124 Park Place 0.288 0.576 0.768 1.729 3.227 3.266 3.293 Boardwalk