GAME THEORY

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GAME THEORY

• STRATEGIC DECISION MAKING

Assoc. Prof. Dr. Ramazan Sari
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OUTLINE

• INTRODUCTION
• NORMAL FORM GAMES
• EXTENSIVE FORM GAMES

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I. INTRODUCTIONOLIGOPLY

• Small number of sellers,
• Product may be either homogeneous (pure
  oligopoly) or differentiated (differentiated
  oligopoly)
• Entry is possible but difficult.

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OLIGOPLY

• Sellers must recognize their interdependence
• The action of one seller may affect another and,
  thus cause that seller to respond in ways that
  will affect the first seller.

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Responds???

• No action at all.
• Many actions
• Doing the same thing,
• Doing the opposite
• Starting a price war
• Increasing production
• Decreasing production
• ..................

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The difficulty of formulating models of oligoploy
stems from the many ways that firms interact.
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There is no general model of oligopoly
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MODELS OF OLIGOPOLY

• The Sweezy Model (The Kinked Demand Curve)
• The Cournot Model (A competition in quantities)
• The Bertrand Model (A competition in prices)
• The Stackelberg Model
• Price Leadership
• Game Theory

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GAME THEORY

• Combination of all!

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What is Game Theory?

• GT is an analytical tool that is used to evaluate
  situations where individuals and organizations
  can have conflicting objectives.

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Where can we use GT?

• Any situation that requires us to anticipate our
  rivals response to our action is a potential
  context for GT.
• Games Checkers, poker, chess, tennis, soccer
  etc.
• Economics Industrial Organization, Micro/Macro/
  International/Labor/Natural resource Economics,
  and Public Finance
• Political science war/peace (negotiations)
• Law Designing laws that work
• Biology animal behavior, evolution
• Information systems System competition/evolution

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Where can we use GT? (cont.)

• Business
• Games against rival firms
• Pricing, advertising, marketing, auctions, RD,
  joint ventures, investment, location, quality,
  take over etc.
• Games against other players
• Employee/employer, managers/stockholders
• Supplier/buyer, producer/distributor,
  firm/government

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Contributors

• John von Neuman and Oskar Morgenstern (1944) The
  Theory of Games and Economic Behavior
• John F. Nash (1950-53) Nash equilibrium
• Martin Shubik (1959) Strategy and Market
  Structure
• Reinhard Selten (1965) Sub-game perfect
  equilibrium
• John Harsanyi (1967-68) Games with incomplete
  information

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Why GT is Important?

• It helps us understand the strategic interaction
  situation
• It forces us to think strategically
• It provides a standard taxonomy for scientific
  approach to making strategic decisions
• Provides new insights.
• Both art AND science

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GT science or art?

• GT is the science of rational behavior in
  interactive situations.
• Good strategists mix the science of GT with their
  own experience.
• To be literate in the modern age, you need to
  have a general understanding of GT. P.Samuelson

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GT in the News

• lately game theorists have focused on real-world
  issueshow to raise auction proceeds by revealing
  the bids and how Wal-Mart can coexist with local
  retailers. Consultants are jumping in. McKinsey
  Co. has set up a game theory unit. Forbes, Nov.
  7, 1994
• GT is hot. The Wall Street Journal, Feb. 13,
  1995.

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GT in the News (cont.)

• FCC hired game theorists to construct rules of an
  auction for new wireless phone systems licenses.
  In response, communications companies hired game
  theorists first to negotiate with FCC and then
  help prepare optimal bids given the rules of the
  auction. Business Week, Mar. 14, 1994.
• GT, long an intellectual passtime, came into its
  own as a business tool. Forbes, July 3, 1995

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Some Terminology

• Strategy
• Payoffs
• Rationality
• Common knowledge of rules
• Equilibrium

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A Strategy

• A course of action taken by one of the
  participant in a game.
• Example
• In a response to a price change the strategies
  would be
• Change price
• Do not change price

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The Payoff

• The result or outcome of the strategy
• Examples
• Profits
• Revenues
• Utility
• ...

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The Rationality

• How good is each player at pursuing his/her aim?
• Much of game theory assumes that players are
  perfect calculators and flawless followers of
  their best strategies.
• Two essential ingredients of rationality
• Complete knowledge of ones own interest,
• Flawless calculation of what actions will best
  serve those interests.

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The Common Knowledge of Rules

• The players have a common understanding of the
  rules of the game.
• List of players,
• The strategies available to each player,
• The payoffs of each player for all possible
  combinations of strategies pursued by all
  players,
• The assumption that each player is a rational
  maximizer.

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An Equilibrium

• A strategy combination that consists of the best
  strategy for each player in the game.

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II. NORMAL FORM GAMES

• These are static (one-shot) games of complete and
  imperfect information
• We focus on games where
• There are at least two rational players
• Rationality is common knowledge
• Each player has more than one choices
• Players make their strategy choices
  simultaneously
• The outcome depends on the strategies chosen by
  all players there is strategic interaction

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3 Components of Normal Form Games

• A normal (strategic) form game consists of
• Players set of players
• Player 1, Player 2, ... Player n
• Strategies a strategy set for all players
• S1 S2 ... Sn
• Payoffs a payoff assigned to every contingency
  (every possible strategy profile as the outcome
  of the game)
• u1 u2 ... un where ui S1 S2 ... Sn?R.
• ui(s1, s2 ... sn), for all s1?S1, s2?S2, ...
  sn?Sn

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AN EXAMPLE

• Players Two competing firms.
• Firm 1
• Firm 2
• Strategies
• No price change
• Price increase
• Combination of Strategies
• both of firms increase their prices,
• neither firm increases its price,
• Firm 1 increases its price but Firm 2 does not,
• Firm 2 increases its price but Firm 1 does not.
• Payoffs Firms can make profit() or loss(-).

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Creating Payoff Matrix
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The Payoff Matrix
Firm 2
Price Increase
No Price Change
No Price Change
10, 10
100, -30
Firm 1
140, 20
Price Increase
-20, 30
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NASH EQUILIBRIUM

• A set of strategies such that none of the
  participants in the game can improve their
  payoffs given the strategies of the other
  participants.
• A game may have more than one N.Es.

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Rules of the Solution
Firm 2
Price Increase
No Price Change
No Price Change
10, 10
100, -30
Firm 1
140, 20
Price Increase
-20, 30
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The Payoff Matrix
Firm 2
Price Increase
No Price Change
No Price Change
10, 10
100, -30
Firm 1
140, 20
Price Increase
-20, 30
(No Price Change, No Price Change) (10,10)
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POP QUIZE??!!EXAMPLE (Price Competition)
Coca Cola
Pepsi Cola
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EXAMPLE (Price Competition)
Coca Cola
Pepsi Cola
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Prisoners Dilemma

• Two suspects are caught and put in different
  rooms (no communication). They are offered the
  following deal
• If both of you confess, you will both get 5 years
  in prison (-5 payoff)
• If one of you confesses whereas the other does
  not confess, you will get 0 (0 payoff) and 10
  (-10 payoff) years in prison respectively.
• If neither of you confess, you both will get 2
  years in prison (-2 payoff)

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Matrix Format of Prisoners Dilemma
Prisoner 2
Do Not Confess
Confess
Confess
-5, -5
0, -10
Prisoner 1
-2, -2
Do Not Confess
-10, 0
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DOMINANT STRATEGY

• Whenever a player has a strategy that is strictly
  better than any other strategy regardless of
  other players strategy choices, we say that the
  first player has a dominant strategy.

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Equilibrium in Dominant Strategies

• A strictly dominant strategy is the one that
  yields the highest payoff compared to the payoffs
  associated with all other strategies.
• Rational players will always play their strictly
  dominant strategies.
• Rational players know that other players will
  play their strictly dominant strategies.
• If all players have strictly dominant strategies,
  the equilibrium consists of those strategies.

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Matrix Format of Prisoners Dilemma
Prisoner 2
Do Not Confess
Confess
Confess
-5, -5
0, -10
Prisoner 1
-2, -2
Do Not Confess
-10, 0
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Dominant Strategy of P1Payoffs of P1
Prisoner 2
Do Not Confess
Confess
Confess
-5
0
Prisoner 1
-2
Do Not Confess
-10
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Dominant Strategy of P1

• CONFESS is a strickly dominant strategy of
  Prisoner 1.

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Dominant Strategy of P2Payoffs of P2
Prisoner 2
Do Not Confess
Confess
Confess
-5
-10
Prisoner 1
-2
Do Not Confess
0
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Dominant Strategy of P2

• CONFESS is a strickly dominant strategy of
  Prisoner 2.

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EQUILIBRIUM of Prisoners Dilemma
Prisoner 2
Do Not Confess
Confess
Confess
-5, -5
0, -10
Prisoner 1
-2, -2
Do Not Confess
-10, 0

• For both P1 and P2 the strictly dominant strategy
  is to confess
• Equilibrium in dominant strategies (confess,
  confess)

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DOMINATED STRATEGIES

• If there exists a strategy A that always yields
  higher payoffs than strategy B regardless of what
  other players do, then B is said to be dominated
  by A.
• Iterated elimination of strictly dominated
  strategies
• Even if there are no strictly dominant
  strategies, some strategies may still be
  dominated
• Rational players will never play their dominated
  strategies.
• Eliminating dominated strategies does not always
  lead to a unique solution
• If eliminating dominated strategies solves the
  game by pointing out a unique outcome then that
  outcome is also the Nash Equilibrium.

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EQUILIBRIUM of Prisoners Dilemma
Prisoner 2
Do Not Confess
Confess
Confess
-5, -5
0, -10
Prisoner 1
-2, -2
Do Not Confess
-10, 0
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EQUILIBRIUM of Prisoners Dilemma
Prisoner 2
Do Not Confess
Confess
Confess
-5, -5
0, -10
Prisoner 1
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EQUILIBRIUM of Prisoners Dilemma
Prisoner 2
Confess
Confess
-5, -5
Prisoner 1
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EXAMPLE 1
MOBIL
Wide
Narrow
Narrow
14, 14
-1, 16
SHELL
1, 1
Wide
16, -1
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EXAMPLE 1 (SOLUTION)

• MOBIL
• Dominant Strategy WIDE
• Dominated Strategy NARROW
• SHELL
• Dominant Strategy WIDE
• Dominated Strategy NARROW
• EQUILIBRIUM
• (Wide, Wide) (1,1)

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EXAMPLE 2
MOBIL
Wide
Narrow
Narrow
14, 14
2, 16
SHELL
1, 1
Wide
16, 2
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EXAMPLE 2 (SOLUTION)

• No Dominant Strategies
• There are Two N.E.s
• (Wide, Narrow) (16, 2)
• (Narrow, Wide) (2, 16)

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EXAMPLE 3
MOBIL
SHELL
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EXAMPLE 3
MOBIL
SHELL
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ZERO-SUM GAMES (Strictly Competitive Games)

• A game in which one players gain is always
  matched by another players loss.
• For example
• Tennis, soccer,....
• Land won/lost in a war
• .

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Location of Ice-Cream Track Game
Ahmets Location
Mehmets Location
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Matching Pennies Game

• Not a coin toss
• Players decide on which side of the penny to
  show.
• If both show the same side, P1 pays 1 to P2
  otherwise P2 pays P1 1

Player 2
Player 1

• There does not exist a Nash equilibrium in this
  strictly competitive game

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Coordination Games

• Games in which players face the problem of
  coordinating their strategies
• Failing to cooperate may or may not be costly
• RD coordination
• Destructing nuclear weapons
• Two types of coordination games
• Battle of the Sexes Game
• Assurance Game

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Battle of the Sexes
Güler

• Dominant strategies none
• Dominated strategies none
• Two Nash equilibria
• (Erener, Erener)
• (Aksu, Aksu)

Mehmet
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Battle of the Sexes Games

• Each player prefers a different equilibrium
• Both players prefer coordination
• If they fail to coordinate, they do not receive
  extremely bad payoffs
• How to achieve coordination? (If the game is
  played more than once)
• Rules
• History
• Focal point

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Battle of The Sexes After 30 Years of Marriage
Güler

• Two Nash equilibria
• (opera, opera)
• (movie, movie)
• Focal point equilibrium (opera, opera)

Mehmet
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Focal Point

• Players expectations must converge to reach a
  focal point.
• However, there is no theoretical backing behind
  the focal point concept.

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Assurance Games
Technology Adoption Game

• 2 Nash equilibria
• (new, new)
• (old, old)
• Focal point equilibrium
• (old, old)

Firm 2
Firm 1
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Technology Assurance Game

• Both firms prefer the (old, old) Nash
  equilibrium. However, they cannot decide for sure
  unless they see some kind of assurance from the
  other player that (old, old) equilibrium is what
  the rival also has in mind.
• How to give assurance
• Strategic move
• A player may make a preemptive move to signal the
  intentions
• Focal point
• If both players achieve significantly higher
  payoffs in the focal point, the high payoffs may
  be adequate to provide such assurance (as in the
  technology game)

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Pure Coordination Game

• Nash equilibriua
• (Up, Left)
• (Down, Right)
• Payoffs do not indicate any focal points

Sometimes convergence of expectations may be
beyond mathematical explanations
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MAXIMIN STRATEGIES

• It is not pure profit maximizing strategy
• Rather, it is designed to avoid highly
  unfavorable outcomes.

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• This strategy specifies that each player in the
  game will select the option that maximizes the
  minimum possible profit (or other desirable
  outcome).

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EXAMPLE
Firm 2
Firm 1
68
EXAMPLE
Firm 2
Firm 1
69
EXAMPLE
Firm 2
Firm 1
70
EXAMPLE
Firm 2
Firm 1
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III. DYNAMIC EXTENSIVE FORM GAMES

• Game Trees
• Games of complete and perfect information

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Assumptions in Dynamic Extensive Form Games

• All players are rational.
• Rationality is common knowledge
• Players move sequentially. (Therefore, also
  called sequential games)
• Players have complete and perfect information
• Players can see the full game tree including the
  payoffs
• Players can observe and recall all previous moves

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What is a Game Tree?
Player 1
Right
Left
Player 2
Player 2
A
C
B
D
P11 P21
P12 P22
P13 P23
P14 P24
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Game tree
x0

• A game tree has a set of nodes and a set of edges
  such that
• each edge connects two nodes (these two nodes are
  said to be adjacent)
• for any pair of nodes, there is a unique path
  that connects these two nodes

a path from x0 to x4
a node
x2
x1
x3
x4
x5
x6
x7
x8
an edge connecting nodes x1 and x5
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Strategy

• Recall A strategy for a player is a complete
  plan of actions.
• It specifies a feasible action for the player in
  every contingency in which the player might be
  called on to act.
• In a game tree, a strategy for a player is
  represented by a set of edges.
• A combination of strategies (sets of edges), one
  for each player, induce one path from the root to
  a terminal node, which determines the payoffs of
  all players

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Entry Game
Migros
Aggressive
Normal
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680 -50
730 0
700 400
800 0
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Finding Nash Equilibria in Dynamic Extensive Form
Games

• How to find the Nash equilibria in a dynamic game
  of complete information
• Construct the normal-form of the dynamic game of
  complete information
• Find the Nash equilibria in the normal-form
• This may yield equilibria with incredible
  threats, therefore we need a refinement.
• Subgame Perfect Equilibrium (SPE)
• A Nash equilibrium of a dynamic game is
  subgame-perfect if the strategies of the Nash
  equilibrium constitute a Nash equilibrium in
  every subgame of the game.
• Subgame-perfect Nash equilibrium is a Nash
  equilibrium.
• Every finite dynamic game of complete and perfect
  information has a subgame-perfect Nash
  equilibrium that can be found by backward
  induction.

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Entry Game
Migros
Aggressive
Normal
Wal-Mart
Wal-Mart
Enter
Enter
Stay out
Stay out
680 -50
730 0
700 400
800 0
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Entry Game
Migros
Aggressive
Normal
Wal-Mart
Wal-Mart
Enter
Stay out
Stay out
730 0
700 400
800 0
80
Entry Game
Migros
Aggressive
Normal
Wal-Mart
Wal-Mart
Enter
Stay out
730 0
700 400
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Entry Game
Migros
Aggressive
SPE is (aggressive, stay out)
Wal-Mart
Stay out
730 0
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A Generic Game Tree with 3 Players
P1
Right
Left
Middle
P3
2 1 1
P2
X
Y
F
G
P3
P3
3 3 3
-9 2 1
A
C
E
B
D
3 2 2
-2 1 5
4 4 6
-3 3 3
10 2 5
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A Generic Game Tree with 3 Players
P1
Right
Left
Middle
P3
2 1 1
P2
X
Y
F
P3
P3
3 3 3
C
B
-2 1 5
4 4 6
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A Generic Game Tree with 3 Players
P1
Right
Left
Middle
P3
2 1 1
P2
Y
F
P3
3 3 3
C
4 4 6
85
A Generic Game Tree with 3 Players
P1
Middle
SPE P1 plays middleP2 plays YP3 plays C
(Middle, Y, C) (4, 4, 6)
P2
Y
P3
C
4 4 6
86
Software Development Game

• Macrosoft and Microcorp
• Macro first player
• Marketing strategies Expensive and Cheap
• Micro second player
• Strategies Enter (E) and Stay out (S)

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Game Tree for S/W Game
Macro
Expensive
Cheap
Micro
Micro
Enter
Stay out
Enter
Stay out
800 0
380 -250
430 0
400 100
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Solution of the S/W Game

• SPE Macro plays Expensive and Micro Plays
  Stay out.
• Micros empty threat I will enter regardless of
  what Macro does
• A rational Macro would not believe in this threat
• However, Micro can come up with a commitment
  device such that carrying out the threat becomes
  its only best option.
• If Micro can create a credible commitment, the
  outcome of the game may change.

89
References

• Contents of this presentation are gathered from
• Game Theory with Economic Applications, Bierman
  and Fernandez, 1998, Addison Wesley Reading,
  Massachusetts.
• Game Theory Introduction and Applications,
  Graham Romp, 1997, Oxford University Press New
  York.
• A Primer in Game Theory by Robert Gibbons,
  1992, Prentice Hall New York.

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References

• Games of Strategy by A.K. Dixit and S. Skeath,
  1999, W.W. Norton and Company, New York.
• Thinking Strategically by A.K. Dixit and B.J.
  Nalebuff, 1991, W.W. Norton and Company, New York
  (Also available in Turkish via Sabanci University
  Press)
• X. Liu (Summer 2003) Lectures in Game Theory,
  Graduate School of Industrial Administration,
  Carnegie Mellon University
• http//www.andrew.cmu.edu/user/xinming/gametheory/
  
• U. Soytas (2001 to date) , Lecture notes in
  Strategic Games for Managers, Dep. of BA, METU.