Modeling

with Linear Algebra

Maxwell Gregoire

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)

Endgame probability distribution

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

Questions?

Space 10 is very important