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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

- 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 to play
- Choices
- Chance
- Information
- Utility

Representing Games

- Game tree
- Visual depiction
- Extensive form game
- Rules
- Payoffs

Game Types

- 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

Strategy

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

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

- 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

In normal form

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 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.

Game of imperfect information

In normal form

Another example: 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.

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

imperfect Info game

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

- 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 Game

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

The 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.

Prisoners’ Dilemma

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

Prisoners’ Dilemma

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

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

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

- At equilibrium: players - no change

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

Table 11.12

A game with no equilibria in pure strategies

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

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 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

- 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

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

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