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Chapter 5

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Chapter 5

Game Theory and the Tools

of Strategic Business Analysis

- Game theory applied to economics by John Von Neuman and Oskar Morgenstern
- Game theory allows us to analyze different social and economic situations

- 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

- 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.

- Numbers represent years in jail
- Each has a dominant strategy to confess
- Silent is a dominated strategy
- Nash equilibrium: Confess Confess

- Similarly for GM
- The Nash equilibrium is Price low, Price low

- Game of perfect information
- Player – knows prior choices
- All other players

- Player – knows prior choices
- Game of imperfect information
- Player – doesn’t know prior choices

- 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

Game of imperfect information

Represented in bi-matrix form

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.

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

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

- 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 perfect information

In normal form

- 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

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.

In normal form

- 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

imperfect Info game

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

- 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

- Similarly for GM
- The Nash equilibrium is Price low, Price low

- Numbers represent years in jail
- Each has a dominant strategy to confess
- Silent is a dominated strategy
- Nash equilibrium: Confess Confess

- Each player has a dominant strategy
- Equilibrium is Pareto dominated

- 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

- Coordination game
- Players - common interest: equilibrium
- For multiple equilibria
- Preferences - differ

- At equilibrium: players - no change

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

A game with no equilibria in pure strategies

- 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

IBM

DOS

UNIX

Toshiba

Toshiba

1

2

DOS

UNIX

DOS

UNIX

3

100

100

600

200

100

100

200

600

- 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

- 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

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

- 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

- 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.

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

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