Steps in Mixed Strategy Games

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Unit No.3.
DECISION SCIENCE
Presented By:
Dr. V. M. Tidake
Ph. D (Financial Management), MBA(FM), MBA(HRM) BE(Chem)
Dean, EDP & Associate Professor MBA
1
Sanjivani College of Engineering, Kopargaon
Department of MBA
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302-DECISION SCIENCE
Unit No.3. Games Theory
3.2.4 Steps in Mixed
Strategy Game
Presented By:
Dr. V. M. Tidake
Ph. D (Financial Management), MBA(FM), MBA(HRM) BE(Chem)
Dean EDP & Associate Professor MBA
2
Sanjivani College of Engineering, Kopargaon
Department of MBA
www.sanjivanimba.org.in

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GAMES THEORY
 At the End of the Session Student will be able to
understand-
A. Steps in Mixed Strategy Game.

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Games Theory
For the Following Mixed Strategy Game:
Firm B
Firm A
Where,
p - Probability that Player A uses Strategy A1
1-p - Probability that Player A uses Strategy A2
q - Probability that Player B uses Strategy B1
1-q - Probability that Player B uses Strategy B2
B1 (q) B2 (1-q)
A1 (p) a11 a12
A2 (1-p) a21 a22

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Games Theory
We have,
𝒑 =
𝒂 𝟐𝟐 − 𝒂 𝟐𝟏
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟏𝟐
𝒒 =
𝒂 𝟐𝟐 − 𝒂 𝟏𝟐
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟏𝟐
𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑮𝒂𝒎𝒆 =
𝒂 𝟏𝟏 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 𝒂 𝟏𝟐
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟏𝟐

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Games Theory
Solve the Following Mixed Strategy Game:
Player B
Player A
. Player B
. Player A
As the Game has no Saddle Point its a Mixed Strategy Game
B1 B2
A1 3 5
A2 4 1
B1 B2 Min.
A1 3 5 3 Maximin
A2 4 1 1
Max. 4 Minimax 5

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Games Theory
We have,
𝒑 =
𝒂 𝟐𝟐 − 𝒂 𝟐𝟏
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟏𝟐
𝒑 =
𝟏 − 𝟒
𝟑 + 𝟏 − 𝟒 − 𝟓
𝒑 =
𝟑
𝟓

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Games Theory
We have,
𝒒 =
𝒂 𝟐𝟐 − 𝒂 𝟏𝟐
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟏𝟐
𝒒 =
𝟏 − 𝟓
𝟑 + 𝟏 − 𝟒 − 𝟓
𝒒 =
𝟒
𝟓

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Games Theory
We have,
𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑮𝒂𝒎𝒆 =
𝒂 𝟏𝟏 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 𝒂 𝟏𝟐
𝒂 𝟏𝟏 + 𝒂 𝟐𝟐 − 𝒂 𝟐𝟏 − 𝒂 𝟐𝟐
𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑮𝒂𝒎𝒆 =
𝟑(𝟏) − 𝟒(𝟓)
𝟑 + 𝟏 − 𝟒 − 𝟓
𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑮𝒂𝒎𝒆 =
𝟏𝟕
𝟓

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EXERCISE
Hence,
The Optimal Strategy for player A is (p, 1-p) i.e. (3/5, 2/5)
or (0.6, 0.4) or (60%, 40%)
The Optimal Strategy for player B is (q, 1-q) i.e. (4/5, 1/5)
or (0.8, 0.2) or (80%, 20%)
The Value of the Game is 17/5 i.e. Player A will get Rs.
17/5 from Player B.

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EXERCISE
 State and Explain the steps in Mixed Strategy Game.

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For More Details Contact
Dr. V M Tidake
tidkevishal@gmail.com
tidkevishalmba@sanjivani.org.in

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Thank You