Background



      The main mathematical concept that was used in this project is a part of Matrix Algebra called a Markov Chain. A Markov Chain consists of a sequence of observations or predictions each of which depends on the preceding one. The initial observations must be put into a matrix called a Transition Matrix. As the sequence of the Chain continues after some time the numbers in the sequence remain the same and will always remain the same as long as the initial conditions (those in the Transition Matrix) remain the same.
      The good example of a Markov Chain an example of the weather. Letís say, hypothetically, that every sunny day has an 80% chance of being followed by another sunny day, a 10% of being followed by a cloudy day and a 10% chance of being followed by a rainy day. A cloudy day has a 40% chance of being followed by a rainy day and 60% chance of being followed another cloudy day. A rainy day will always be followed by a sunny day, 100% chance. This information would go into a Transition Matrix like this:
			        SD   CD   RD
			SD[   .8     0      1   ]
			CDI   .1    .4      0   I
 			RD[   .1    .6      0   ]
      Now lets say today is Monday and it is a sunny day. A starting Matrix would be made with one in the first spot and zeros in the others because there is a 100% chance that today is a sunny day. The starting matrix would look like this.
			SD [ 1 ]
			CD I 0 I
			RD [ 0 ]
                          Monday
      Then the Transition and starting matrices would be multiplied together to find out what Tuesday could be like.
               SD    CD     RD          
	SD[   .8     0      1   ]       SD [ 1 ]            SD [.8 ]
	CDI   .1    .4      0   l    x  CD I 0 I    =       CD I.1 I
	RD[   .1    .6      0   ]       RD [ 0 ]            RD [.1 ]
               Transition                Monday             Tuesday
      To find the what the weather could be like Wednesday based on the fact that today is sunny day, the Transition Matrix was multiplied by the second matrix.
	      SD     CD     RD          
	SD[   .8     0      1   ]         SD [ .8 ]        SD [.74 ]
	CDI   .1    .4      0   l      x  CD I .1 I    =   CD I.12 I
	RD[   .1    .6      0   ]         RD [ .1 ]        RD [.14 ]
                 Transition                Tuesday         Wednesday
	
	And so on...

	      SD     CD     RD          
	SD[   .8     0      1   ]         SD [.74]         SD [.732]
	CDI   .1    .4      0   I      x  CD I.12I    =    CD I.122I
	RD[   .1    .6      0   ]         RD [.14]         RD [.146]
                 Transition               Wednesday        Thursday

	And so on... 

	        SD   CD   RD          
	SD[   .8     0      1   ]         SD [.732]         SD [.732]
	CDI   .1    .4      0   l      x  CD I.122I    =    CD I.122I
	RD[   .1    .6      0   ]         RD [.146]         RD [.146]
                    Transition              Thursday             Friday
	
	The probabilities have stabilized after four sequences.

                           SD   CD   RD          
	SD[   .8     0      1   ]          SD [.732]         SD [.732]
	CDI   .1    .4      0   l      x   CD I.122I    =    CD I.122I
	RD[   .1    .6      0   ]          RD [.146]         RD [.146]
                Transition                Friday             Saturday
      So, if one started with a sunny day it could be figured out what the weather could be like on Sunday or even some time next year. By doing that it has been figured out from these calculations that over all, every day has a 73.2% chance of being sunny, a 12.2% chance of clouds, and a 14.6% chance of rain. That is a Markov Chain in action.
      In Monopoly the same idea can be used. If figure you the chances of occupying any spot from starting at any spot, for example if a player starts at GO what are his chances of landing on any of the 40 spots on the board or starting from States Ave. what are his chances of landing on any of the 40 spots. Just like before when it was known that there was a 60% chance rain if we started from a cloudy day. Also, just like it was known there was 100% chance of starting with a sunny day, there is a 100% chance of starting a game from the GO spot. So, using this concept of the Markov Chain the objective can be achieved.