Short Course on Game Theory

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Short Course on Game Theory
Hamed Ghoddusi
Von Neumann Aumann
Selton Harsanyi
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• In war the will is directed at an animate object
  that reacts.

- Karl Von Clausewitz, On War
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Course Plan

• Part 1 Definition of games, Typical Examples of
  Games, Common Knowledge, Expected Utility, States
  and Strategies
• Part 2 Extensive Forms with Perfect
  Information, Backward Induction, Imperfect
  Information, Incomplete Information, Normal forms
• Part 3 Solution Concepts Rationalizibility,
  Strong and Weak Dominance, Nash equilibrium,
  Bayesian-Nash equilibrium, Correlated equilibrium
  , Sub-game perfection, signalling games
• Part 4 (If time permits) Trembling hand,
  repeated games, supermodular games

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Game Theory

• Study of strategic behavior

• Strategic behavior Taking into account the
  behavior of other
• players

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Examples of Strategic Behavior

• Industrial Competition
• Firm / Capital market relationship
• Voting decisions
• Auctions/Biddings

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Game to be discussed in the class

• Shareholders voting game
• Search models
• Durable goods monopolist
• RD
• Bargaining
• Job market signaling

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Games We Study

• Played once / more than once
• Finite number of players
• Finite/infinite strategy space

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Prisoners Dilemma
Silent
Cooperate
1 , 1
10 , 0
Silent
0 , 10
8 , 8
Cooperate
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Chicken Game
Drive
Stop
-100 , -100
1 , 0
Drive
0 , 1
Stop
0 , 0
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Hawk and Dove
Hawk
Dove
(V-C)/2
V , 0
Hawk
0 , V
Dove
V/2
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Battle of Sexes
Theater
Restaurant
Theater
4 , 4
10 , 5
Restaurant
5 , 10
4 , 4
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Game Theory
Decision Theory
Representation Theory
Solution Theory
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?Extensive Form

• Players
• Rules of Game
• Strategies

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Normal Form

• Players
• Space of Pure Strategies
• Payoffs

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Differences between Extensive and Normal Form?
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Games

• Competitive zero-sum games , non zero-sum
  games
• Non Competitive supermodular (complementary)
  games
• search games for instance

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Games

• One shot games
• Repeated games

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Information

• Perfect Information Chess
• Imperfect Information Cards
• Incomplete Information

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Characteristics of Players

• Self-interested, utility maximizer
• Rational Friedmans as if paradigm
• Remark One person and several identity
  situation

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Tree of Ulyssess problem
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Decision Under Uncertainty

• Decision over lotteries
• Von Neumann Morgenstern expected utility
• Risk Aversion

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States

• Mutually exclusive
• Collectively exhaustive
• Independent of players decision

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Information Sets
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Bayesian Decision Making

• Updating of prior beliefs
• P(EE) P(EnE) / P(E)

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Strategy
Detailed plan of actions, contingent to any
possible occurrence in the game a plan that
you could leave to somebody else playing for
you. Strategy as a function which maps states
of the world to the actions
A
W
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Strategy

• Strategy in the simultaneous move games
• Strategy in the Bayesian games
• Finite and infinite strategy space

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Example 1 How many strategies?
L
2
L
H
1
L
2
H
M
H
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Example 2 Equivalent Normal Form?
1
L
H
2
2
L
L
H
H
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What do you think if strategies always assign
the same value to some elements of the domain?
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Strategy in Extensive Form
A strategy for player X is a sub-tree of a game
tree which satisfies the following conditions

• It is rooted at the root of the game tree
• whenever it is player X's turn at a node that
  belongs to the subtree, exactly
• one of the available moves belongs to the
  subtree
• whenever it is not player X's turn at a node
  that belongs to the subtree, all
• of the available moves belong to the subtree

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Bargaining Games

• Gains from trade, the problem of distribution of
  benefits
• In the absence of market
• Disagreement value
• Rubenstein smart solution

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Rubensteins Bargaining Game
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Real Life Example

• Company-Union Negotiations

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Solution to Bargaining Game
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Extensive Form Games

• Set Theoretic Definition
• Graph Theoretic Definition

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Set Theory Representation

• (W,N)
• W set of Plays
• N collection of non-empty sub-sets of W (Nodes)
• w ? N
• Predecessor function

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Set Theory Representation

• Plays States
• Nodes Events
• Moves
• Terminal Moves

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Example
1
2
2
2
w5
w6
w3
w4
w2
w1
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Example Modeling of Bargaining

• W ?
• N ?

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Graph Theoretic Representation

• Graph (V,E)
• Nodes (States)
• Branches (Actions)
• History
• Immediate Predecessor

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Example
1
H
A
N
2
2
2
H
A
A
N
H
N
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Games with Perfect Information

• An Extensive Form
• Assignment of Decision Points
• Pay off function

Simultaneous move is ruled out.
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Games with imperfect Information

• An Extensive Form
• Collection of Information set
• Pay off function

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Example
1
2
2
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Sub Games

• Sub-tree
• Contains the whole information set

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Example Noisy Stackelberg
L
2
1
H
QL
2
1
QH
2
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Strategy Space
The product structure of strategies of all
players S S1 S2 S3 Sk ? Si S(-i)
Strategies of all player but player (i)
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Pure Strategy vs Mixes Strategy
The subset of pure strategies entering the mix
with a strictly positive weight is the support of
the mixed strategy. Point A pure strategy may
be strictly dominated by a mixed strategy even if
it does not strictly dominated by any pure
strategy
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Behavioral Strategy vs Mixes Strategy
When a player implements a mixed strategy, she
spins the roulette wheel a single time the
outcome of this spin determines which pure
strategy (set of deterministic choices at each
information set) she will play. When she
implements a behavior strategy, she independently
spins the roulette wheel every time she reaches a
new information set.
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Behavioral Strategies
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Kuhns Theorem
Every game of perfect information with finite
number of Nodes has a solution of backward
induction.
Comment If pay-offs to players at all terminal
nodes are Unequal then the solution is unique.
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Backward Induction
The concept of backwards induction corresponds to
the assumption that it is common knowledge that
each player will act rationally at each node
where he moves even if his rationality would
imply that such a node will not be reached
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Group Discussion Christine and Lion
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The centipede game
Go on
Go on
Go on
Jack
Jill
Jill
Jack
Go on
stop
stop
stop
stop
(5, 3)
(4, 7)
(2, 0)
(1, 4)
Go on
Go on
Jill
Jill
Jack
(99, 99)
stop
stop
(98, 96)
(97, 100)
(94, 97)
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Games With Incomplete Information

• Harsanyi Transformation Model the Game as a
  game
• with imperfect information

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Harsanyi Transformation

• Nature plays first
• Common knowledge about the original distribution
  of
• types (how nature plays)
• The realization of types are private information.

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Bayesian Games

• Bayesian updating rule

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Example Art Auction

• Private second price sealed auction
• Each player maintains belief u
• Prior and posterior beliefs

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Example Falklands War
(Cost of War 1 for both players)
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Equivalent Normal Form?
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Asymmetric Information

• Adverse selection
• Moral hazard
• Signaling / Screening

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Example Cournot Competition with Asymmetric
Information

• Know cost structure of Firm1 C1(q)cq
• Unknown cost structure of firm2 C2(q)cHq ,
  C2(q)cLq

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Example Insurance Signaling

• High Risk / Low Risk Customer
• Two policies
• Signaling through buying the insurance

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Example Job Market Signaling

• Qualified / Unqualified Worker
• Signaling through education degree