Friday, May 13, 2016

Monopoly PR6: Rent on Yellow Properties




Question:   A person starts on Kentucky Avenue and throws the dice once.   (We will consider ramifications of doubles in a future post but for now let’s assume the person does NOT go again if he throws doubles.) 

The player’s opponent owns all three yellow properties --  Atlantic, Ventnor, and Marvin Gardens.    The rents at these three properties for two houses and for three houses are presented below.  


Rent at Atlantic, Ventor, and Marvin Gardens
Two Houses
Three Houses
Atlantic
330
800
Ventor
330
800
Marvin Gardens
360
850


What is the likelihood that the person will not land on any of these three properties after one throw of the two dice?

What is the most likely outcome after one throw of the two dice?

What are the expected value, standard deviation, and skewness of rents given two house and given three houses?

Analysis:   The landing probabilities for one throws of the two dice from the starting point of Go were calculated in a previous post. 


 The table below updates these landing probabilities for the situation where the starting point is Kentucky Avenue.



Outcomes from One Roll of Two Dice Starting From Kentucky
Sum of Dice Rolls
Property Landed On
Probability
2
Indiana
0.027777778
3
Illinois
0.055555556
4
B&O Railroad
0.083333333
5
Atlantic Ave
0.111111111
6
Ventnor Ave
0.138888889
7
Water Works
0.166666667
8
Marvin Garden
0.138888889
9
Go to Jail
0.111111111
10
Pacific Ave
0.083333333
11
North Carolina Ave
0.055555556
12
Community Chest
0.027777778


The probability of not landing on any of the three yellow properties is 0.611111.   You can get this by summing up probabilities for all non-yellow properties or you can sum the probabilities for the yellows and subtract from one.   Remember the probability of the complement of an event is 1- probability of the event.  

The probability of landing on Atlantic from Kentucky is 4/36.  The probabilities for Ventnor and Marvin Gardens are 5/36.   The probabilities for all non-yellows are 1-14/36, which is 0.611111.

The square the person is most likely to land on is Water Works because the probability of getting a 7 on the sum of the two dice is higher than any other probability. 


The table below outlines calculations for expected value


Outcomes from One Roll of Two Dice Starting From Kentucky
Sum of Dice Rolls
Property Landed On
Probability
Rent Two Houses
Rent Three Houses
2
Indiana
0.027777778
0
0
3
Illinois
0.055555556
0
0
4
B&O Railroad
0.083333333
0
0
5
Atlantic Ave
0.111111111
330
800
6
Ventor Ave
0.138888889
330
800
7
Water Works
0.166666667
0
0
8
Marvin Garden
0.138888889
380
850
9
Go to Jail
0.111111111
0
0
10
Pacific Ave
0.083333333
0
0
11
North Carolina Ave
0.055555556
0
0
12
Community Chest
0.027777778
0
0
Expected Rent Two Houses
135.28
Expected Rent Three Houses
318.06


The formula for the standard deviation is   SD= (E(Rent-E(Rent)2)1/2


The formula for the skew is E(Rent-E(Rent))3/SD3


Results for the case with two houses are presented below.


Expected Rent, STD Rent, Skewness Rent With Two Houses
Property Landed  On
Probability
Rent Two Houses
Expected Rent
(rent-E(rent)2
(Rent-E(rent))3
Indiana
0.028
0
135.3
18300
-2475594
Illinois
0.056
0
135.3
18300
-2475594
B&O Railroad
0.083
0
135.3
18300
-2475594
Atlantic Ave
0.111
330
135.3
37917
7383233
Ventor Ave
0.139
330
135.3
37917
7383233
Water Works
0.167
0
135.3
18300
-2475594
Marvin Garden
0.139
380
135.3
59889
14656161
Go to Jail
0.111
0
135.3
18300
-2475594
Pacific Ave
0.083
0
135.3
18300
-2475594
North Carolina Ave
0.056
0
135.3
18300
-2475594
Community Chest
0.028
0
135.3
18300
-2475594
Expected Rent
135.3
STD (Rent)
170.2
Skewness (Rent)
0.5




Switch the rent numbers in the spreadsheet and get the results for three houses.



Results with Thee Houses
Property Landed  On
Probability
Rent Two Houses
Expected Rent
(rent-E(rent)2
(Rent-E(rent))3
Indiana
0.028
0
135.3
18300
-2475594
Illinois
0.056
0
135.3
18300
-2475594
B&O Railroad
0.083
0
135.3
18300
-2475594
Atlantic Ave
0.111
800
135.3
441856
293711258
Ventor Ave
0.139
800
135.3
441856
293711258
Water Works
0.167
0
135.3
18300
-2475594
Marvin Garden
0.139
850
135.3
510828
365100020
Go to Jail
0.111
0
135.3
18300
-2475594
Pacific Ave
0.083
0
135.3
18300
-2475594
North Carolina Ave
0.056
0
135.3
18300
-2475594
Community Chest
0.028
0
135.3
18300
-2475594
Expected Rent
318.1
STD (Rent)
438.9
Skewness (Rent)
1.5



The increase in rent due to the addition of the third house increases expected rent, variability of rent and the skewness.   The increase in skewness occurs because the financial differential of falling or not falling on the yellow properties increase.

In life you might want to even out cash flows by spreading houses on more property.   In monopoly you want to bankrupt the opponent, which is often done by concentrating your buildings on a narrower set of properties.


Additional Work:  The monopoly problems can be extended in a number of directions.   Often in the game a person has two monopolies and is faced with a choice of concentrating building on one monopoly or spreading building on many monopolies.   This issue can be studied.

Also, in previous posts I looked at the probability of landing on different squares after throwing doubles and going a second time.


This post looks at the probability of moving seven squares including outcomes when the player throws a double on the first throw.

The post looks at the probability of going 15 squares on one turn.   (Hint: this requires the player throw doubles on the first turn.)


The current problem of landing on yellows can be modified to consider additional paths to the yellows once the player throws a double.


Next week I hope to create a post with 5 or 10 monopoly math problems with links to the answers.

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