Basics on Game Theory

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Basics on Game Theory

• Class 2
• Microeconomics

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Introduction

• Why
• Any human activity has some competition
• Human activities involve actors, rules,
  strategies
• Game theory formalizes the analysis of
  competition
• What
• GT is the study of strategic behavior of
  competing actors
• What GT for
• GT allows to analyze the alternatives set by the
  rules
• GT permits to prescribe the opponents behavior
• GT show how to design games

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Games

• Normal Form Games
• Players
• Strategies
• Payoffs
• Concepts
• Actions
• Outcomes
• Payoffs
• Objective
• To find the solution of the game

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Eliminating Strictly Dominated Strategies
P2
Left
Center
1,0 1,2
P1
Up
P2
Left
Center
1,0 1,2
0,3 0,1
Up
P1
Down
P2
Left
Center
Right
1,0 1,2 0,1
0,3 0,1 2,0
Up
P1
Down
Games Solution, Dominant-Dominated Strategies
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Nash Equilibrium

• Example
• Definition

P2
Left
Center
Right
0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
Up
P1
Middle
Down
NE
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Multiple Equilibriums
He
Opera
Football
2,1 0,0
0,0 1,2
Opera
She
Football
The Battle of the Sexes
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The Best Response Functions
He
Opera
Football
2,1 0,0
0,0 1,2
Opera
She
Football
Multiple Equilibriums
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Pareto Efficient Outcomes
P2
Defect
Cooperate


-1,-1
-9,0
Defect
P1
0,-9
-6,-6
Cooperate
The Prisoners Dilemma
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Exercise
P2
Left
Center
Right
2,0 1,1 4,2
3,4 1,2 2,3
1,3 0,2 3,0
Up
P1
Middle
Down
Find the Solution for the Game
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Exercise
P2
Left
Center
Right
2,0 1,1 4,2
3,4 1,2 2,3
1,3 0,2 3,0
Up
P1
Middle
Down
Find the Solution for the Game
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Extensive Form Games

• Moves occur in sequence
• All previous moves are observed
• Payoffs are known by all the players

Games with Perfect and Complete Information
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Backward Induction (NE-1)
P1

• Step 3
• Step 2
• Step 1

R
L
P2
(2,0)
L
R
P1
(1,1)
L
R
(3,0)
(0,2)
Solutions for Extensive Form Games
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Strategies

• Definition
• P1 2 actions,
• 2 strategies
• P2 2 actions,
• 4 strategies

The Concept of Strategy in Extensive Form Games
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Nash Equilibrium 1 (Backward
Induction)
Strategy 1 (L,L)
Strategy 2 (L,R)
Strategy 3 (R,L)
Strategy 4 (R,R)
Strategies and Extensive Form Games
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Normal Form and Extensive Form
Games
3,1 3,1 1,2 1,2
2,1 0,0 2,1 0,0
Extensive Form Games as Normal Form Games
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Sub-Game Perfect Nash Equilibrium
Algorithm SPNE
Def A NE is Subgame Perfect if the strategies of
the players constitute a NE in each subgame.
Identify all the smaller subgames having terminal
nodes in the original tree.
Replace each subgame for the payoffs of one of
the NE.
The initial nodes of the subgame are now the
terminal nodes of the new truncated tree.
NE 2
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Sub-game Perfect Nash Equilibrium
Algorithm SPNE
Def A NE is Subgame Perfect if the strategies of
the players constitute a NE in each subgame.
Identify all the smaller subgames having terminal
nodes in the original tree.
Replace each subgame for the payoffs of one of
the NE.
The initial nodes of the subgame are now the
terminal nodes of the new truncated tree.
Extensive Form Games as Normal Form Games
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SPNE and BI
SPNE is more powerful than NE, for solving
Imperfect Information Games
SPNE (R,L)
Backward Induction (R,L)
Extensive Form Games as Normal Form Games