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Pure Strategy - Math Problem Example

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Pure Strategy

Player A would prefer to have the largest value possible from those highlighted. In the diagram this can be seen to occur when the line V = 2 + 7X intersects with V= 6 - 3X which occurs at the point as determined below.








Value of the game is as follows.




The minimax criteria is to maximise the minimum values. To find the solution for B's strategy we need to identify and solve the 2 by 2 pay-off matrix associated with A's optimal strategy. In this case that will be the columns associated with the lines V= 2 + 7X and V= 6 - 3X. Hence
3
9
6
2
Q3.

a) Solution:

The transformation is as follows.

Total Number of audience = 10000000
Total audience size as a pair in each matrix element = 100000
Total value of a pair in each matrix element = 10000000/100000 = 100
Subtracting half the total value of a pair as computed above from each element of the payoff matrix defined in the question.
Half the total value = 50

Payoff Matrix



SkyCast





Historical
Drama
Comedy
Contemporary
Drama
Variety

Mini Series
60 - 50
58 - 50
49 - 50
52 - 50
RayTV
Detective
55 - 50
52 - 50
54 - 50
59 - 50

Film
55 - 50
54 - 50
47 - 50
55 - 50

Variety
47 - 50
54 - 50
56 - 50
61 - 50


Transformed Matrix:



SkyCast





Historical
Drama
Comedy
Contemporary
Drama
Variety

Mini Series
10
8
-1
2
RayTV
Detective
5
2
4
9

Film
5
4
-3
5

Variety
-3
4
6
11


b) Solution:




SkyCast





Historical
Drama
Comedy
Contemporary
Drama
Variety

Mini Series
10
8
-1
2
RayTV
Detective
5
2
4
9

Film
5
4
-3
5

Variety
-3
4
6
11

Comparing fourth column with the third column 2 > -1, 9 > 4, 5 > -3...
In the diagram this can be seen to occur when the line V = 2 + 7X intersects with V= 6 - 3X which occurs at the point as determined below.
The minimax criteria is to maximise the minimum values. To find the solution for B's strategy we need to identify and solve the 2 by 2 pay-off matrix associated with A's optimal strategy. In this case that will be the columns associated with the lines V= 2 + 7X and V= 6 - 3X. Hence
The value decides the type of the show to be put out in order to maximise the ratings over the coming TV season. The value against Detective has more gain (approximately 3.33) and hence it has to be put out in order to maximise the ratings. ...Show more

Summary

Pure Strategy occurs when the best strategy for each player is to choose the same option for all plays of the game. If there is a pure strategy, A plays i and B plays j (say), then the ijth element (the payoff to A per play) is called a saddle point.
Player A would prefer to have the largest value possible from those highlighted…
Author : craigtromp
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