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Chapter 5. Game Theory and the Tools of Strategic Business Analysis. Game Theory. Game theory applied to economics by John Von Neuman and Oskar Morgenstern Game theory allows us to analyze different social and economic situations. Games of Strategy Defined.

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Chapter 5

Chapter 5

Game Theory and the Tools

of Strategic Business Analysis


Game theory
Game Theory

  • Game theory applied to economics by John Von Neuman and Oskar Morgenstern

  • Game theory allows us to analyze different social and economic situations


Games of strategy defined
Games of Strategy Defined

  • 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

  • A game is comprised of

    • Number of players

    • Order of play

    • strategies

    • Chance

    • Information

    • Payoffs


Example 1 prisoners dilemma
Example 1:Prisoners’ Dilemma

  • Two people committed a crime and are being interrogated separately.

  • The are offered the following terms:

    • If both confessed, each spends 8 years in jail.

    • If both remained silent, each spends 1 year in jail.

    • If only one confessed, he will be set free while the other spends 20 years in jail.


Example 1 prisoners dilemma1
Example 1: Prisoners’ Dilemma

  • Numbers represent years in jail

  • Each has a dominant strategy to confess

  • Silent is a dominated strategy

  • Nash equilibrium: Confess Confess



Example 3 oligopoly game
Example 3: Oligopoly Game

  • Similarly for GM

  • The Nash equilibrium is Price low, Price low


Game types
Game Types

  • Game of perfect information

    • Player – knows prior choices

      • All other players

  • Game of imperfect information

    • Player – doesn’t know prior choices


Representing games
Representing Games

  • The previous examples are of

    • Simultaneous games

    • Games of imperfect information

      Games can be represented visually in

  • Bi- matrix form

    • Table

    • Dimensions depend on the number of strategies

  • Game tree

    • Extensive form game


Matching pennies
Matching Pennies

Game of imperfect information

Represented in bi-matrix form


Extensive form of the game of matching pennies
Extensive form of the game of matching pennies

Child 1

Heads

Tails

Child 2

Child 2

Heads

Tails

Heads

Tails

+1

- 1

+1

- 1

- 1

+1

  • 1

    +1

Child 2 does not know whether child 1 chose heads or tails. Therefore, child 2’s information set contains two nodes.


Strategy
Strategy

  • A player’s strategy is a plan of action for each of the other player’s possible actions


Game of perfect information
Game of perfect information

In extensive form

IBM

DOS

UNIX

Toshiba

Toshiba

1

2

DOS

UNIX

DOS

UNIX

3

200

600

600

200

100

100

100

100

Player 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


Strategies
Strategies

  • IBM:

    • DOS or UNIX

  • Toshiba

    • DOS if DOS and UNIX if UNIX

    • UNIX if DOS and DOS if UNIX

    • DOS if DOS and DOS if UNIX

    • UNIX if DOS and UNIX if UNIX



Game of imperfect information
Game of imperfect information

  • Assume instead Toshiba doesn’t know what IBM chooses

    • The two firms move at the same time

  • Imperfect information

    • Need to modify the game accordingly


Game of imperfect information1
Game of imperfect information

In extensive form

IBM

Information set

DOS

UNIX

  • Toshiba’s strategies:

  • DOS

  • UNIX

Toshiba

Toshiba

1

2

DOS

UNIX

DOS

UNIX

3

600

200

100

100

100

100

200

600

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.



Equilibrium for games nash equilibrium
Equilibrium for GamesNash Equilibrium

  • Equilibrium

    • state/ outcome

    • Set of strategies

    • Players – don’t want to change behavior

    • Given - behavior of other players

  • Noncooperative games

    • No possibility of communication or binding commitments



Nash equilibrium toshiba ibm
Nash Equilibrium: Toshiba-IBM

imperfect Info game

The strategy pair DOS DOS is a Nash equilibrium. Are there any other equilibria?


Dominant strategy equilibria
Dominant Strategy Equilibria

  • Strategy A dominates strategy B if

    • A gives a higher payoff than B

    • No matter what opposing players do

  • Dominant strategy

    • Best for a player

    • No matter what opposing players do

  • Dominant-strategy equilibrium

    • All players - dominant strategies


Oligopoly game
Oligopoly Game

  • Ford has a dominant strategy to price low

    • If GM prices high, Ford is better of pricing low

    • If GM prices low, Ford is better of pricing low


Oligopoly game1
Oligopoly Game

  • Similarly for GM

  • The Nash equilibrium is Price low, Price low


Prisoners dilemma
Prisoners’ Dilemma

  • Numbers represent years in jail

  • Each has a dominant strategy to confess

  • Silent is a dominated strategy

  • Nash equilibrium: Confess Confess


Prisoners dilemma1
Prisoners’ Dilemma

  • Each player has a dominant strategy

  • Equilibrium is Pareto dominated


Elimination of dominated strategies
Elimination of Dominated Strategies

  • Dominated strategy

    • Strategy dominated by another strategy

  • We can solve games by eliminating dominated strategies

  • If elimination of dominated strategies results in a unique outcome, the game is said to be dominance solvable


Games with many equilibria
Games with Many Equilibria

  • Coordination game

    • Players - common interest: equilibrium

    • For multiple equilibria

      • Preferences - differ

    • At equilibrium: players - no change


Games with many equilibria1
Games with Many Equilibria

The strategy pair DOS DOS is a Nash equilibrium as well as UNIX, UNIX


Normal form of matching numbers coordination game with ten nash equilibria
Normal Form of Matching Numbers: coordination game with ten Nash equilibria


Table 11 12
Table 11.12

A game with no equilibria in pure strategies



Credible threats
Credible Threats

  • An equilibrium refinement:

    • Analyzing games in normal form may result in equilibria that are less satisfactory

    • These equilibria are supported by a non credible threat

    • They can be eliminated by solving the game in extensive form using backward induction

    • This approach gives us an equilibrium that involve a credible threat

    • We refer to this equilibrium as a sub-game perfect Nash equilibrium.


Non credible threats: IBM-Toshiba

In normal form

  • Three Nash equilibria

  • Some involve non credible threats.

  • Example IBM playing UNIX and Toshiba playing UNIX regardless:

    • Toshiba’s threat is non credible


Backward induction
Backward induction

IBM

DOS

UNIX

Toshiba

Toshiba

1

2

DOS

UNIX

DOS

UNIX

3

100

100

600

200

100

100

200

600


Subgame perfect nash equilibrium
Subgame perfect Nash Equilibrium

  • Subgame perfect Nash equilibrium is

    • IBM: DOS

    • Toshiba: if DOS play DOS and if UNIX play UNIX

  • Toshiba’s threat is credible

    • In the interest of Toshiba to execute its threat


Rotten kid game
Rotten kid game

  • The kid either goes to Aunt Sophie’s house or refuses to go

  • If the kid refuses, the parent has to decide whether to punish him or relent


Rotten kid game in extensive form
Rotten kid game in extensive form

Kid

Go to Aunt Sophie’s House

Refuse

Parent

1

2

Punish if refuse

Relent if refuse

-1

-1

2

0

1

1

  • 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


Application 1 collusive duopoly
Application 1: Collusive Duopoly

  • Example: The European voluntary agreement for washing machinesin 1998

  • The agreement requires firms to eliminate from the market inefficient models

  • Ahmed and Segerson (2011) show that the agreement can raise firm profit, however, it is not stable


Application 2 wal mart and cfl bulbs market
Application 2: Wal-Mart and CFL bulbs market

  • In 2006 Wal-Mart committed itself to selling 1 million CFL bulbs every year

  • This was part of Wal-Mart’s plan to become more socially responsible

  • Ahmed(2012) shows that this commitment can be an attempt to raise profit.


Application 2 wal mart and cfl bulbs market1
Application 2: Wal-Mart and CFL bulbs market

Wal-Mart

When the target is small

Commit to output target

Do not commit

Small firm

Small firm

1

2

Commit

Do not

Commit

Do not

3

90

45

500

40

80

60

100

50

The outcome is similar to a prisoners dilemma


Application 2 wal mart and cfl bulbs market2
Application 2: Wal-Mart and CFL bulbs market

Wal-Mart

When the target is large

Commit to output target

Do not commit

Small firm

Small firm

1

2

Commit

Do not

Commit

Do not

3

80

30

500

35

100

50

90

100

When the target is large enough, we have a game of chicken


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