Page 1
Math 166-copyright Joe Kahlig, 10A
Section G.2: Mixed Strategies
In a strictly determined game, the optimal strategy is a pure strategy of selecting the row/column
for the saddle point. When the payoff matrix does not have a saddle point, i.e. is not strictly determined, a mixed strategy is used by the players.
Example: Here is the payoff matrix (in terms of the row player) for a game. Assume that the row
player will chose strategy R-1 40% of the time and use R-2 the rest of the time. Assume that the
column player will chose strategy C-1 15% of the time and C-2 the rest. If this game is repeated many
times using these mixed strategies, find the expected value of the game.

C-1

A = R-1  3
R-2
−3

C-2

−2 
1
Expected value of a Game: Let P and Q be the matrices representing the mixed strategies of the
row player, P, and the column player, Q, and let A be a payoff Matrix.
P =
h
p1
p2
pm
···
i



A=


a11
a21
..
.
a12
a22
..
.
am1
am2
···
···
···
···
a1n
a2n
..
.
amn






Then the expected value is given by
Example: Find the expected value of the game for these mixed strategies.

C-1

A = R-1  3
R-2
−3
A) P =
h
B) P =
h
0.75
0.5

C-2

−2 
1
0.25
0.5
i
i
Q=
Q=
"
"
0.3
0.7
0.6
0.4
#
#



Q=


q1
q2
..
.
qn






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Math 166-copyright Joe Kahlig, 10A
Definition: For every choice of strategy for the row player there is a best counterstrategy–that is
a strategy for the column player that results in the least expected value for the game. An optimal
mixed strategy for the row player is one for which the expected value against the column player’s
best counterstrategy is as large as possible.
Definition: For every choice of strategy for the column player there is a best counterstrategy–that
is a strategy for the row player that results in the largest expected value for the game. An optimal
mixed strategy for the column player is one for which the expected value against the column
player’s best counterstrategy is as small as possible.
Theorem: For a game with the payoff matrix given as
A=
"
a
c
b
d
#
Then the optimal mixed strategy for the row player is given by
P =
h
p1
p2
i
, where p1 =
d−c
and p2 = 1 − p1 .
a+d−b−c
The optimal mixed strategy for the column player is given by
Q=
"
q1
q2
#
, where q1 =
d−b
and q2 = 1 − q1 .
a+d−b−c
The value of the game(expected value) is
E = P AQ =
ad − bc
a+d−b−c
Example: Find the value of the game and the optimal strategy for the row and column players.
A=
"
3
2
−1
4
#
Example: Find the value of the game and the optimal strategy for the row and column players.
A=
"
15
−5
10
20
#
Page 3
Math 166-copyright Joe Kahlig, 10A
Example: Which of these strategies should never be used by the row player?
A=
R-1
R-2
R-3





C-1
4
6
5
C-2
5
7
6
C-3
2
3
8





Example: Which of these strategies should never be used by the column player?
A=
R-1
R-2
R-3





C-1
4
6
5
C-2
5
7
6
C-3
2
3
8





Definition: Row i is said to dominate row j if every entry of row i is
sponding entry of row j
Column i is said to dominate column j if every entry of column i is
ing entry of column j
to the corre-
to the correspond-
Rows/columns that are dominated may be removed from the payoff matrix without affecting the
analysis.
Example: Find the optimal strategy for the row and the column player.
A=
R-1
R-2
R-3





C-1
4
6
5
C-2
5
7
6
C-3
2
3
8





Math 166-copyright Joe Kahlig, 10A
Page 4
Example: Reduce the matrix by deleting rows or columns that are dominated by other rows or columns.

6
2
7
−3

6
4
2
1



A) 



B) 
2
0
2
−8
2
1
4
3
−2
9
6
6
3
4
1
3





1
2
8
1





Example: Find the optional strategy for both players.





4
5
−3
4
0
2
1
2
−8
3
−2
7
−3
2
0
3




