Game Theory

1
Game Theory Fundamentals for Industrial
Organization (Student Notes)
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
2
Game Theory Taxonomies
A Simplified Taxonomy
General Nature of Play Cooperative
Non-cooperative
Frequency of Play One-time Repeated
Timing of Play Simultaneous Sequential
Bierman and Fernandez (1998) Game Theory with
Economic Applications
N a t u r e o f I n f o r m a t i o n
Complete
Incomplete
N a t u r e o f P l a y

• Normal (Matrix) Form
• Nash Equilibrium
• Dominance Reasoning
• Pure / Mixed Strategies

• Bayesian N.E.
• Action Plans
• Revelation Principles

Static

• Extensive Form
• Backward Induction
• Subgame Perfection
• Credible Threats

• Perfect Bayesian N.E.
• Signaling
• Beliefs
• Bargaining
• Reputation

Dynamic
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
3
Formalizing a Game Matrix Form

Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
4
Formalizing a Game Extensive Form
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
5
Solution Methods / Equilibrium Concepts
Dominant strategy equilibrium Nash equilibrium
(NE) Focal point equilibrium Subgame perfect
NE (Sequential games) Intertemporally
supportable NE (Simultaneous repeated
games)
q q q q q
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
6
Dominance Reasoning / Dominant Strategy
Equilibrium
q q q
Elimination of dominated strategies A player has
a dominated strategy if there is one
action/strategy which always provides a lower
payoff than another strategy, no matter what
the other player does. If you cross off all
dominated strategies, sometimes you are left
with a clear equilibrium outcome.
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
7
Dominance Reasoning / Dominant Strategy
Equilibrium
This approach only works for strict dominance
B
b1
b2
b3
a1
3,2
5,4
4,3
A
4,2
3,6
a2
2,5
a3
6,3
1,3
5,4
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
8
Dominance Reasoning / Dominant Strategy
Equilibrium
Sometimes, dominance reasoning does not work
-- one or all of the players may not have a
dominant strategy
B
b1
b2
a1
2,1
0,0
A
1,2
0,0
a2
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
9
Nash Equilibrium (Nash, 1951)
In many games, a players optimal strategy
is contingent upon the other players choice
Player A If b1, then a1 If b2, then a2
Player B If a1, then b1 If a2, then b2
B
b1
b2
a1
2,1
0,0
A
1,2
0,0
a2
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
10
Nash Equilibrium (Nash, 1951)
In an n-player game, a strategy profile,
, is a Nash equilibrium if for each
player, i,
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
11
Nash Equilibrium (Nash, 1951)
Note that the equilibrium is defined in terms of
strategies, not payoffs Note also that all
dominant strategy equilibria are also NE.
NE a1, b1
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
12
Prisoners Dilemma / An Intriguing Case of Nash
Equilibrium
NE a1, b1
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
13
Focal Point Equilibrium (Schelling, 1960)
In games with multiple NE, one of the NE might
stand out because of some asymmetry that is
common knowledge to the players.
NE a1, b1 and a2,b2
F
If fairness is a criteria for both players, then
a1, b1 may emerge as a focal point
equilibrium.
B
b1
b2
a1
6,8
2,1
A
3,11
1,3
a2
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
14
Sequential Games / Backward Induction Subgame
Perfect NE
2,2
b1
a1
5,0
b2
A
B
0,5
b1
a2
3,3
b2
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
15
Sequential Games / Backward Induction Subgame
Perfect NE
This example is taken from Luis Cabral,
Introduction to Industrial Organization
(2000), pages 55 58.
2
1
1
2
2
-10,-10
10,-20
0,50
-10,-10
10,20
0,50
Rose-Hulman Institute of Technology / K.
ChristVA353, Industrial Organization
16
Sequential Games / Backward Induction Subgame
Perfect NE
b1 Price War
1,1
a1 Enter
b2 Status Quo
2,2
B (Incumbant Firm)
A (Entrant)
b1 Price War
0,-1
a2 Stay Out
b2 Status Quo
0,3
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
17
Repeated Play / Reciprocity and Supportable
Equilibrium
Suppose that player A adopts the following
strategy
Play a2. If Player B plays b2, continue to play
a2. If Player B plays b1, then play a1 until
Player B changes.
Such a strategy will mean that both players
face two possible streams of payoffs
Time, t
Strategy 1 Defection
2
2
2
5
Strategy 2 Cooperation
4
4
4
4
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization
18
Repeated Play / Reciprocity and Supportable
Equilibrium
Given a prisoners dilemma
B
Payoff from cooperation Payoff from
defection Payoff from being defected
upon Payoff from NE (Cournot competition)
b1
b2
a1
A
a2
In such a game, cooperation may be sustained
if the following condition is met
Rose-Hulman Institute of Technology / K.
Christ VA353, Industrial Organization