Chapter 5
description
Transcript of Chapter 5
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Chapter 5
Game Theory and the Tools of Strategic Business Analysis
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Game Theory
0 Game theory applied to economics by John Von Neuman and Oskar Morgenstern
0 Game theory allows us to analyze different social and economic situations
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Games of Strategy Defined
0 Interaction between agents can be represented by a game, when the rewards to each depends on his actions as well as those of the other player
0 A game is comprised of 0 Number of players0 Order of play0 strategies0 Chance 0 Information 0 Payoffs
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Example 1:Prisoners’ Dilemma
0 Two people committed a crime and are being interrogated separately.
0 The are offered the following terms:0 If both confessed, each spends 8 years in jail.0 If both remained silent, each spends 1 year in jail.0 If only one confessed, he will be set free while the other spends
20 years in jail.
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Example 1: Prisoners’ Dilemma
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Prisoner 2
confess silent
Prisoner 1Confess 8, 8 0, 20
Silent 20, 0 1, 1
0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess
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Example 2: Matching Pennies
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Player 2
Heads Tails
Player 1Heads - 1, +1 +1 - 1
Tails +1 - 1 - 1, +1
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Example 3: Oligopoly Game
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General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Similarly for GM0 The Nash equilibrium is Price low, Price low
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Game Types
0 Game of perfect information0 Player – knows prior choices
0All other players
0 Game of imperfect information0 Player – doesn’t know prior choices
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Representing Games
0 The previous examples are of0 Simultaneous games0 Games of imperfect information
Games can be represented visually in0 Bi- matrix form
0 Table0 Dimensions depend on the number of strategies
0 Game tree0 Extensive form game
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Matching Pennies
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Player 2
Heads Tails
Player 1Heads - 1, +1 +1 - 1
Tails +1 - 1 - 1, +1
Game of imperfect informationRepresented in bi-matrix form
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Extensive form of the game of matching pennies
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Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.
Child 1
Child 2Child 2
Tails Heads
TailsHeads TailsHeads
- 1+1
+1- 1
+1- 1
-1+1
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Strategy
0 A player’s strategy is a plan of action for each of the other player’s possible actions
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Game of perfect information
13Player 2 (Toshiba) knows whether player 1 (IBM) moved to the left or to the right. Therefore, player 2 knows at which of two nodes it is located
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
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Strategies
0 IBM: 0 DOS or UNIX
0 Toshiba0 DOS if DOS and UNIX if UNIX0 UNIX if DOS and DOS if UNIX0 DOS if DOS and DOS if UNIX0 UNIX if DOS and UNIX if UNIX
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Toshiba
(DOS | DOS,DOS | UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Game of perfect informationIn normal form
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Game of imperfect information
0 Assume instead Toshiba doesn’t know what IBM chooses0 The two firms move at the same time
0 Imperfect information0 Need to modify the game accordingly
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Game of imperfect information
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Toshiba does not know whether IBM moved to the left or to the right, i.e., whether it is located at node 2 or node 3.
1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
In extensive form
Information set
Toshiba’s strategies:• DOS• UNIX
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Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Game of imperfect informationIn normal form
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Equilibrium for GamesNash Equilibrium
0 Equilibrium 0 state/ outcome0 Set of strategies0 Players – don’t want to change behavior 0 Given - behavior of other players
0 Noncooperative games0 No possibility of communication or binding
commitments
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Nash Equilibria
chosen is *s when i player to payoff i player of choicestrategy
choicesstrategy ofarray -
i
),...,(
),...,(*
**1
**1
n
*i
n
ss
s
sss
20
ii
nini
n
Ss
ssssss
sss
in all for If
mequilibriu Nash a is - ii
ˆ
),...,ˆ,...,(),...,,...,(
),...,(***
1***
1
**1
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Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Nash Equilibrium: Toshiba-IBM
imperfect Info game
The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?
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Dominant Strategy Equilibria
0 Strategy A dominates strategy B if0 A gives a higher payoff than B 0 No matter what opposing players do
0 Dominant strategy0 Best for a player0 No matter what opposing players do
0 Dominant-strategy equilibrium0 All players - dominant strategies
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Oligopoly Game
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General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Ford has a dominant strategy to price low 0 If GM prices high, Ford is better of pricing low0 If GM prices low, Ford is better of pricing low
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Oligopoly Game
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General Motors
High price Low price
FordHigh price 500, 500 100, 700
Low price 700, 100 300, 300
0 Similarly for GM0 The Nash equilibrium is Price low, Price low
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Prisoners’ Dilemma
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Prisoner 2
confess silent
Prisoner 1Confess 8, 8 0, 20
Silent 20, 0 1, 1
0 Numbers represent years in jail0 Each has a dominant strategy to confess0 Silent is a dominated strategy0 Nash equilibrium: Confess Confess
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Prisoners’ Dilemma
0 Each player has a dominant strategy0 Equilibrium is Pareto dominated
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Elimination of Dominated Strategies
0 Dominated strategy0 Strategy dominated by another strategy
0 We can solve games by eliminating dominated strategies
0 If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable
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(a) Eliminating dominated strategies
Player 2
1 2 3
Player 11 2, 0 2, 4 0, 2
2 0, 6 0, 2 4, 0
(b) One step of elimination
Player 2
1 2
Player 11 2, 0 2, 4
2 0, 6 0, 2
(c ) Two steps of elimination
Player 2
1 2
Player 1 1 2, 0 2, 4
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(a) Eliminated dominated strategies
Player 2
1 2 3
Player 11 20, 0 10, 1 4, -4
2 20, 2 10, 0 2, -2
(b) Reduced game eliminating column 3 first
Player 2
1 2
Player 11 20, 0 10, 1
2 20, 2 10, 0
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Games with Many Equilibria
0 Coordination game0 Players - common interest: equilibrium0 For multiple equilibria
0Preferences - differ 0 At equilibrium: players - no change
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Toshiba
DOS UNIX
IBMDOS 600, 200 100, 100
UNIX 100, 100 200, 600
Games with Many Equilibria
The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX
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Normal Form of Matching Numbers: coordination game with ten Nash equilibria
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Player 2
1 2 3 4 5 6 7 8 9 10
Player 1
1 1, 1 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
2 0, 0 2, 2 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
3 0, 0 0, 0 3, 3 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
4 0, 0 0, 0 0, 0 4, 4 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0
5 0, 0 0, 0 0, 0 0, 0 5, 5 0, 0 0, 0 0, 0 0, 0 0, 0
6 0, 0 0, 0 0, 0 0, 0 0, 0 6, 6 0, 0 0, 0 0, 0 0, 0
7 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 7, 7 0, 0 0, 0 0, 0
8 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 8, 8 0, 0 0, 0
9 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 9, 9 0, 0
10 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 10, 10
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Table 11.12
A game with no equilibria in pure strategies
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General 2
Retreat Attack
General 1 Retreat 5, 8 6, 6
Attack 8, 0 2, 3
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The “I Want to Be Like Mike” Game
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Dave
Wear red Wear blue
Michael Wear red (-1, 2) (2, -2)
Wear blue (1, -1) (-2, 1)
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Credible Threats
0 An equilibrium refinement:0 Analyzing games in normal form may result in equilibria
that are less satisfactory0 These equilibria are supported by a non credible threat0 They can be eliminated by solving the game in extensive
form using backward induction0 This approach gives us an equilibrium that involve a
credible threat0 We refer to this equilibrium as a sub-game perfect Nash
equilibrium.
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Toshiba
(DOS if DOS,DOS if UNIX)
(DOS | DOS,UNIX | UNIX)
(UNIX | DOS,UNIX | UNIX)
(UNIX | DOS,DOS | UNIX)
IBMDOS 600, 200 600, 200 100, 100 100, 100
UNIX 100, 100 200, 600 200, 600 100, 100
Non credible threats: IBM-ToshibaIn normal form
0 Three Nash equilibria0 Some involve non credible threats.0 Example IBM playing UNIX and Toshiba playing UNIX
regardless:0 Toshiba’s threat is non credible
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Backward induction
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1
2 3
IBM
ToshibaToshiba
UNIXDOS
UNIXDOS UNIXDOS
600200
100100
100100
200600
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Subgame perfect Nash Equilibrium
0 Subgame perfect Nash equilibrium is0 IBM: DOS0 Toshiba: if DOS play DOS and if UNIX play UNIX
0 Toshiba’s threat is credible0 In the interest of Toshiba to execute its threat
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Rotten kid game
0 The kid either goes to Aunt Sophie’s house or refuses to go
0 If the kid refuses, the parent has to decide whether to punish him or relent
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Player 2 (a parent)
(punish if the kid refuses)
(relent if the kid refuses)
Player 1(a difficult
child)
Left(go to Aunt Sophie’s House)
1, 1 1, 1
Right(refuse to go to Aunt Sophie’s House)
-1, -1 2, 0
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Rotten kid game in extensive form
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• The sub game perfect Nash equilibrium is: Refuse and Relent if refuse• The other Nash equilibrium, Go and Punish if refuse, relies on a non
credible threat by the parent
Kid
Parent
RefuseGo to Aunt Sophie’s House
Relent if refuse
Punish if refuse
-1-1
20
11
1
2
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Application 1: Collusive Duopoly
0Example: The European voluntary agreement for washing machines in 1998
0The agreement requires firms to eliminate from the market inefficient models
0Ahmed and Segerson (2011) show that the agreement can raise firm profit, however, it is not stable Firm 2
eliminate Keep
Firm 1eliminate $1,000 $1,000 $200 $1,200
keep $1,200 $200 $500 $500
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Application 2: Wal-Mart and CFL bulbs market
0 In 2006 Wal-Mart committed itself to selling 1 million CFL bulbs every year
0 This was part of Wal-Mart’s plan to become more socially responsible
0 Ahmed(2012) shows that this commitment can be an attempt to raise profit.
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1
2 3
Wal-Mart
Small firmSmall firm
Do not commitCommit to output target
Do notCommit Do notCommit
9045
50040
8060
10050
When the target is small
The outcome is similar to a prisoners dilemma
Application 2: Wal-Mart and CFL bulbs market
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1
2 3
Wal-Mart
Small firmSmall firm
Do not commitCommit to output target
Do notCommit Do notCommit
8030
50035
90100
10050
When the target is large
When the target is large enough, we have a game of chicken
Application 2: Wal-Mart and CFL bulbs market