POM 335: Operations Research II Gaming Models

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POM 335 Operations Research IIGaming Models

• Yanni Papadakis

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Outline

• Game Representations
• Table Form
• Decision Tree Form
• Zero Sum Games
• Mixed Strategies
• Solve Using LP
• Solve Graphically

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Formulation Example
Supermarket Chain Expansion
SIZE35 of Total
10 Miles
15 Miles
SIZE20 of Total
20 Miles
SIZE45 of Total
Two Players Dominant Stores Only goes to big
towns will never go to C DPdiscounts Co. Could
open store in all three towns
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Supermarket Chain Expansion

• Question If each company has money for only one
  store, which town should they locate to?
• Market Realities
• If both chains locate in same town or at equal
  distance, then Dominant gains 65 of market
  share.
• If Dominant is closer, then it gains 90.
• If Dominant is farther, then it gains 40.
• Construct the Payoff Matrix.

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Supermarket Chain ExpansionPAYOFF TABLE
Option B DOMINATESAt least better in all cases
for Player II
Option B DOMINATESAt least better in all cases
for Player I
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Supermarket Chain ExpansionDECISION TREE
A
Player II
65
A
B
Player I
62.5
C
80
A
67.5
B
B
65
C
CHOICE RULE MAX for Player I Decision MIN for
Player II Decision
80
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US Senate Race Example

• Final 2 days before election
• Two politicians running
• Each has only three plans choices
• 1 day each city
• Both days Bigtown
• Both days Megalopolis
• Cannot adapt to first day plans, Visits
  committed to beforehand

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US Senate RacePAYOFF TABLE
MINIMUM 1 0 -1
MAXIMIN
MAXIMUM 1 2 5
MINIMAX
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General Rules

• Where Maximin Minimax we have a Nash
  equilibrium
• Maximin Minimax is the value of the game
• A fair game has a value of 0 (previous game was
  not fare)
• If a Nash equilibrium exists we call the game
  stable
• When game is not stable need to use mixed
  strategies

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Penalty Kick in SoccerPAYOFF TABLE
MINIMUM -1 -1 -1
MAXIMUM 0 0 0
MAXIMIN NOT EQUAL MINIMAX
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Penalty Kick in SoccerMIXED STRATEGY EQUILIBRIUM

• Play a mix of options randomly according to some
  probability loads xj (e.g., probability player I
  uses strategy j)
• Used very often in sports
• Baseball Dont pitch always fastballs
• Football Dont always pass or rush, but play mix
• Find equilibrium using LP

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MIXED STRATEGY EQUILIBRIUM
Player II Chooses Option 1 Player II Chooses
Option 2 Player II Chooses Option 3 Probabilities
sum to 1
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MIXED STRATEGY EQUILIBRIUM for Player II
Player I Chooses Option 1 Player I Chooses Option
2 Player I Chooses Option 3 Probabilities sum to 1
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Penalty Kick in SoccerMIXED STRATEGY EQUILIBRIUM
SOLUTION V-2/3 x1x2x31/3 Why?
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Mixed Strategies Example
Player I Probability x 1-x
Expected Payoffs 5-5x 4-6x -35x
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Mixed Strategies ExampleGraphical Solution
6 5 4 3 2 1 0 -1 -2 -3 -4
A 5-5x
B 4-6x
4y-3(1-y)2/11 ? y5/11
Expected Payoff
0 ¼ ½ ¼ 1 x
C -35x
MAXIMIN
4-6x-35x ? x7/11 and payoff 2/11
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Mixed Strategies Example LP Solution
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LP Solution with Excel
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LP Solution with Excel