Game Theory

Algebraic Method

Consider the zero sum two person game given below:

  Player B
Player A    I II
I a b
II c d

Formulas:

The solution of the game is:

A play’s (p, 1 - p)

where:

p  =
d - c
--------------------
(a + d) - (b + c)

B play’s (q, 1 - q)

where:

q  =
d - b
-------------------
(a + d) - (b + c)

Value of the game, V  =
ad - bc
--------------------
(a + d) - (b + c)


Example 1

Consider the game of matching coins. Two players, A & B, put down a coin. If coins match (i.e., both are heads or both are tails) A gets rewarded, otherwise B. However, matching on heads gives a double premium. Obtain the best strategies for both players and the value of the game.

 
Player B
Player A     I II
I 2 -1
II -1 1

Solution.

This game has no saddle point.

p  = 1 - (-1)
-----------------------
(2 + 1) - (-1 - 1)		 =
2
----
5

1 – p = 3/5

q  = 1 - (-1)
-----------------------
(2 + 1) - (-1 - 1)		 =
2
----
5

1 – q = 3/5

V  = 2 X 1 - (-1) X (-1)
--------------------------
(2 + 1) - (-1 - 1)		 =
1
----
5


Example 2

Solve the game whose payoff matrix is given below:

 
Player B
Player A     I II
I 1 7
II 6 2

Solution.

This game has no saddle point.

p  = 2 - 6
-----------------------
(1 + 2) - (7 + 6)		 =
2
----
5

1 – p = 3/5

q  = 2 - 7
-----------------------
(1 + 2) - (7 + 6)		 =
1
----
2

1 – q = 1/2

V  = 1 X 2 - (7 X 6)
--------------------------
(1 + 2) - (7 + 6)		 =
4

 

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