Game theory in Operation Research ~ Pure Strategy
Pure Strategy : Game Theory
The simplest type of game
is one where the best strategies for both players are pure strategies. This is the case if and only if, the pay-off
matrix contains a saddle point.
Example: Pure Strategy in Game Theory
|
Player
B |
|||||
Player
A |
|
I |
II |
III |
IV |
V |
I |
-3 |
1 |
1 |
6 |
4 |
|
II |
5 |
3 |
2 |
4 |
3 |
|
III |
-5 |
-4 |
1 |
-3 |
7 |
|
IV |
6 |
4 |
-5 |
3 |
-7 |
What is the optimal plan for both
the players?
Solution.
We use the maximin (minimax) principle
to analyze the game.
|
Player B |
||||||
Player A |
|
I |
II |
III |
IV |
V |
Minimum |
I |
-3 |
1 |
1 |
6 |
4 |
-3 |
|
II |
5 |
3 |
2 |
4 |
3 |
2 |
|
III |
-5 |
-4 |
1 |
-3 |
7 |
-5 |
|
IV |
6 |
4 |
-5 |
3 |
-7 |
-7 |
|
Maximum |
|
6 |
4 |
2 |
6 |
7 |
|
Select minimum from the maximum of columns.
Minimax = 2, Player A will choose II strategy, which yields the maximum payoff
of 2.
Select maximum from the minimum of rows.
Maximin = 2, similarly, player B will choose III strategy.
Since the value of maximin coincides
with the value of the minimax, therefore, saddle point (equilibrium point) = 2.
The optimal strategies for both players are:
Player
A must select II strategy and player B must select III strategy.
The
value of game is 2, which indicates that player A will gain 2 unit and player B
will sacrifice 2 unit.
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