Markov Chain

Here is a snake and ladder game and this was the markov chain matrix 

4.1snakesAndLadders

A = [

%1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %1

0 W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %2

0 0 W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %3

0 0 0 W W 0 0 0 0 0 0 0 0 0 W 0 0 0 0 0 %4

0 0 0 0 W 0 W 0 0 0 0 0 0 0 W 0 0 0 0 0 %5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %6

0 0 W 0 0 0 W 0 0 0 0 0 0 0 0 0 W 0 0 0 %7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %9

0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 0 0 %10

0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 0 %11

0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 %12

0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 %13

0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 %14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 %15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 %16

0 0 0 0 0 0 0 0 0 0 W 0 0 0 0 0 W 0 W 0 %17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W T %19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 %20

];

The complete Matlab program is here

% Author: Tim Chartier

% Title: snakes.m

%% Probabilities for different outcomes of rolling a die

D1 = 1/3; % stay

D2 = 1/3; % move ahead 1

D3 = 1/3; % move ahead 2

%% Transition matrix for small game board

B = [

D1 D2 D3 0 0 0 0 0 0 % 1

0 D1+D3 D2 0 0 0 0 0 0 % 2

0 D2 D1 0 D3 0 0 0 0 % 3

0 1 0 0 0 0 0 0 0 % 4

0 0 0 0 D1 0 D3 D2 0 % 5

0 0 0 0 0 0 0 1 0 % 6

0 0 0 0 0 0 D1 D2 D3 % 7

0 0 0 0 0 0 0 D1 D2+D3 %8

0 0 0 0 0 0 0 0 1 % 9

];

W = 1/3; % 1 Step

T = 2/3 % 2 Step

%% Transition matrix for Large game board

A = [

%1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0

W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %1

0 W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %2

0 0 W W W 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %3

0 0 0 W W 0 0 0 0 0 0 0 0 0 W 0 0 0 0 0 %4

0 0 0 0 W 0 W 0 0 0 0 0 0 0 W 0 0 0 0 0 %5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %6

0 0 W 0 0 0 W 0 0 0 0 0 0 0 0 0 W 0 0 0 %7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %9

0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 0 0 %10

0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 0 %11

0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 0 %12

0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 0 %13

0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 0 %14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 0 %15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W W W 0 0 %16

0 0 0 0 0 0 0 0 0 0 W 0 0 0 0 0 W 0 W 0 %17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W T %19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 %20

];

%% Create initial vector

v = zeros(1,20);

v(1) = 1;

%% CREATE INITIAL VECTOR

%%=zeros(1,20);

%%p(1)=1;

%% Iterate through turns

G = v’; % this vector stores the probability vector for each turn

for i = 1:30

v = v*A;

G = [G,v’];

end

%% ITERATE THROUGH TURNS

K = p’;

%%for i = 1:

%% This graphs the probabilities as a visual way to analyze the game

Ginvert = -G + 1;

image(256*Ginvert), colormap(gray(256)), axis off equal tight