monopoly grad talk
Post on 13-Apr-2017
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Introduction to
● Gameboard:
– 40-spaces
– 2 six-sided die● Gameplay:
1) Buy property
2) Trade
3) Wait
Introduction to
● Monopoly is boring● Only one time when you make actual choices:
trading
therefore, we need to know
● which monopolies are best
therefore, we need to know
● the probability of properties being landed on in the endgame
It's nontrivial because of spaces that send you to other spaces!
Linear algebra formulation
P0
P1
P2
.
.
.
P39
<0|0> <1|0>
<2|0> . . . <39|0>
<0|1> <1|1>
<2|1>
<0|2> <1|2>
<2|2>
.
.
.
.
.
.
<0|39>
<39|39>
Probability vector:P
i = probability to be on space I
Σj P
i = 1
Transformation matrix:< i | j > = P
i → j where
Σj < i | j > = 1
All matrix and vector elements are real and positive
A simplified example● Consider a game even more boring than Monopoly:
– 40-space blank board (Monopoly with blank spaces)
– 2 six-sided die
<0|0> <1|0> . . .<0|1> <1|1> . . .<0|2> <1|2> . . .<0|3> <1|3> . . .<0|4> <1|4> . . .<0|5> <1|5> . . .<0|6> <1|6> . . .<0|7> <1|7> . . .<0|8> <1|8> . . .<0|9> <1|9> . . .<0|10> <1|10> . . . <0|11> <1|11> . . .<0|12> <1|12> . . .. . .
0 0 . . .0 0 . . .1/36 0 . . .2/36 1/36 . . .3/36 2/36 . . .4/36 3/36 . . .5/36 4/36 . . .6/36 5/36 . . .5/36 6/36 . . .4/36 5/36 . . .3/36 4/36 . . .2/36 3/36 . . .1/36 2/36 . . .. . .
A simplified exampleApply matrix to to get1
00...
Indeed,it's the probability distribution for two 6-sided dice.
A simplified exampleApply matrix again...
first roll
second rollIt got shorter and wider. Yes, quite.
A simplified exampleApply matrix again...
first roll
second roll
third roll
By George, it looks like a particle traveling in a dispersive medium!
A simplified example
● Want to solve for endgame probability distribution (EPD)
● Apply matrix over and over: we expect to converge on EPD
therefore, we expect
● Applying matrix to EPD yields EPD
therefore, we expect
● EPD is an eigenvector of matrix with eigenvalue 1
It's also the only eigenvector that's entirely real and positive!
A simplified example
● We can get that eigenvalue numerically by applying matrix over and over to any starting vector
● We converge on the target eigenvector
Kids these days and their newfangled moving pictures...
Building the Transformation MatrixAssumptions:
● If a space A would send you to another space B, your final landing position is space B
● Players only attempt to roll doubles out of jail
● Rolling multiple doubles in a row counts as separate applications of the matrix
● Rolling from space 10 (jail/visiting jail) is one application of the matrix
Building the Transformation Matrix● Step 1: distribute dice roll probability distribution
Rolling from here (for example)
Building the Transformation Matrix● Step 2: “Go to Jail” sends you to jail
Rolling from here (for example)
Building the Transformation Matrix● Step 3: Rolling triple doubles sends you to jail
Rolling from here (for example)
Building the Transformation Matrix● Step 4: chance and community chest cards can send you places
Rolling from here (for example)
Property comparison
Avg. monetary gain per opponent's move for each property group
Avg. number of opponent's moves needed to offset development costs for each property group
property group property group
● Assume all properties are part of fully-developed monopolies
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