MATH 166 Spring 2016
c
Wen
Liu
G.2
G.2 Mixed Strategy Games
Expected Value: Suppose a game has a payoff matrix

a11 a12 · · ·

a22 · · ·
 a
A =  21
···

am1 am2 · · ·

a1n

a2n 


amn
and the row player uses the strategy p = [p1 p2 · · · pm ], where each pi is the probability of choosing
the ith strategy, and the column player uses the strategy q T = [q1 q2 · · · qn ], where each qj is the
probability of choosing the jth strategy, then the expected value for those two strategies is
E(p, q) = pAq

= [p1 p2
a11 a12

a22
 a
· · · pm ]  21

am1 am2
···
···
···
···

a1n

a2n  


amn
q1
q2
..
.





qn
Example: Consider the game with the payoff matrix
−1 1
A=
2 4
Find the expected values for the strategies below and determine which the row player should use.
(a) p = [0.2 0.8] and q T = [1 0].
(b) p = [0.7 0.3] and q T = [1 0].
Remark: From the example above we see that if a player uses different strategies that the payoffs
will change. There exists an optimal strategy for each player that we shall call p for the row player
and q for the column player. If both players use their optimal strategies, the expected outcome is
v = E (p, q), the value of the game. The goal of game theory is to find the optimal strategy for each
player.
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MATH 166 Spring 2016
c
Wen
Liu
G.2
Example: Find the optimal strategy for the row player and the value of the game given in previous
example.
Generalized 2 × 2 Game: If the payoff matrix A is given by
a11 a12
A=
a21 a22
Then the optimal row strategy is
p = [p1 p2 ]
a22 − a21
=
a11 + a22 − a12 − a21
1 − p1
and the optimal column strategy is
q = [q1 q2 ]
a22 − a12
=
a11 + a22 − a12 − a21
1 − q1
The optimal value of the game is
v = E (p, q)
= pAq
a11 a22 − a12 a21
=
a11 + a22 − a12 − a21
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MATH 166 Spring 2016
c
Wen
Liu
G.2
Example: Josh and Sandy play a card game where each player has two cards to show. Josh has the
three and four of hearts and Sandy has the two and five of diamonds. Both show a card at the same
time. If the sum is even, Josh wins the sum of the two cards in dollars. If the sum is odd, Sandy wins
the sum of the two cards in dollars. What is Josh’s optimal strategy and the value of this game?
Definition: Row i is said to dominate row j if every element in row i is greater than or equal to
the corresponding element in row j. Also column i is said to dominate column j if every element in
column i is less than or equal to the corresponding element in column j.
Example: Find the value and the optimal strategy for the row player for the game with the following
payoff matrix


−2 −1 0
B =  −1 0
1 
2
3
−4
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