1. 4
Game Theory
To accompany Quantitative Analysis for Management, Twelfth Edition,
by Render, Stair, Hanna and Hale
Power Point slides created by Jeff Heyl
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2. LEARNING OBJECTIVES
After completing this module, students will be able to:
Understand the principles of zero-sum, two-person games.
Analyze pure strategy games and use dominance to reduce the size of a game.
Solve mixed strategy games.
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3. MODULE OUTLINE M4.1 Introduction M4.2 Language of Games
M4.4 Pure Strategy Games
M4.5 Mixed Strategy Games
M4.6 Dominance
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4. Introduction
Game theory is one way to consider the impact of the strategies of others on our strategies and outcomes
A game is a contest involving two or more decision makers, each of whom wants to win
Game theory is the study of how optimal strategies are formulated in conflict
Game models classified by number of players, sum of all payoffs, and number of strategies employed
Two-person and zero-sum games
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5. GAME PLAYER Y’s STRATEGIES
Language of Games
Two lighting fixture stores
A duopoly
Advertising strategies change
Table M4.1 – Store X’s Payoff Matrix
GAME PLAYER Y’s STRATEGIES Y1
(Use radio) Y2
(Use newspaper)
GAME PLAYER X’s STRATEGIES X1 3 5 X2 1 –2
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6. Language of Games Game outcomes STORE X’s STRATEGY STORE Y’s
OUTCOME (% CHANGE
IN MARKET SHARE)
X1 (use radio)
Y1 (use radio)
X wins 3 and Y loses 3
Y2 (use newspaper)
X wins 5 and Y loses 5
X2 (use newspaper)
X wins 1 and Y loses 1
X loses 2 and Y wins 2
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7. Dominance
The principle of dominance can be used to reduce the size of games by eliminating strategies that would never be played
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8. Dominance Reduce the size of this game Can be reduced to Y1 Y2 X1 4 320 X3 1
Can be reduced to Y1 Y2 X1 4 3 X2 2 20
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9. Dominance Reduce the size of this game Can be reduced to Y1 Y2 Y3 Y4X1 –5 4 6 –3 X2 –2 2
–20
Can be reduced to Y1 Y4 X1 –5 –3 X2 –2
–20
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10. Mixed Strategy Games
Players will play each strategy for a certain percentage of the time
Called a mixed strategy game
Commonly solved using the expected gain or loss approach
Each player plays a strategy a particular percentage of the time so that the expected value of the game does not depend upon what the opponent does
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11. Mixed Strategy Games For Player Y solve 4P + 2(1 – P) = 1P + 10(1 – P)
Table M4.4 – Game Table for Mixed Strategy Game
PLAYER Y’s STRATEGIES Y1 Y2
PLAYER X’s STRATEGIES X1 4 2 X2 1 10
For Player Y solve
4P + 2(1 – P) = 1P + 10(1 – P)
P = 8/11
1 – P = 1 – 8/11 = 3/11
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12. Mixed Strategy Games For Player X solve 4Q + 1(1 – Q) = 2Q + 10(1 – Q)
Table M4.4 – Game Table for Mixed Strategy Game
PLAYER Y’s STRATEGIES Y1 Y2
PLAYER X’s STRATEGIES X1 4 2 X2 1 10
For Player X solve
4Q + 1(1 – Q) = 2Q + 10(1 – Q)
Q = 9/11
1 – Q = 2/11
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13. Mixed Strategy Games Y1 Y2 P 1 – P Expected gain X1 Q 4 2
Table M4.5 – Game Table for Mixed Strategy Game with Percentages (P, Q) Shown Y1 Y2 P
1 – P
Expected gain X1 Q 4 2
4P + 2(1 – P) X2
1 – Q 1 10
1P + 10(1 – P)
4Q + 1(1 – Q)
2Q + 10(1 – Q)
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