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From: rjnoe@riccb.UUCP (Roger J. Noe)
Newsgroups: net.games,net.math
Subject: The Mathematics of Monopoly (prolegomenon)
Message-ID: <512@riccb.UUCP>
Date: Thu, 29-Aug-85 12:54:37 EDT
Article-I.D.: riccb.512
Posted: Thu Aug 29 12:54:37 1985
Date-Received: Sat, 31-Aug-85 07:30:49 EDT
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Xref: watmath net.games:2160 net.math:2225

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