EKONOMSKA ANALIZA PRAVAEKONOMSKA ANALIZA PRAVA

Game Theory

Outline of the lecture:I. What is game theory?

II. Elements of a game

III. Normal (matrix) and Extensive (tree) form

IV. Equilibrium Concepts and examples

V. Repeated Games

VI. Sequential Games

I. What is Game Theory?

•Unilateral vs. Bilateral/Multilateral situations• Strategic behavior

‘GT is a set of tools and a language for describing and predicting strategic behavior’ (Picker)

• Expected utility• Rationality• Cooperative vs. non-cooperative games• Simultaneous vs. sequential games

The law frequently confronts situations in which there are few decision-makers and in which the optimal action for one person to take depends on what another actor chooses. These situations are like games in that people must decide upon a strategy. A strategy is a plan for acting that responds to the reactions of others. Game theory deals with any situation in which strategy is important. Game theory will, consequently, enhance our understanding of some legal rules and institutions. To characterize a game, we must specify three things:◦ the players◦ the strategies of each player, and◦ the payoffs to each player for each strategy.

II. Elements of a game

• Players: individuals taking decisions. Goal: maximize expected utility

• Action/moves: all actions the player can choose. 2x2!• Strategy: rule telling which action to take at any point

(in 2x2 games, actions and strategies are identical)• Payoffs: amount of money, utility etc. the player

receives when game is played• Equilibrium: combination of strategies the players will

choose• Equilibrium concept: rule that defines the equilibrium

III. Normal and Extensive form

Normal form (matrix)

Player A

Player B

Up

Down

Left Right

0, 10 20, 5

5, 20 10, 0

III. Normal and Extensive form

Extensive form (tree): nodes and branches

‘Initial node’ or ‘starting node’

III. Normal and Extensive form

Extensive form (tree): nodes and branches

A

‘Initial node’ or ‘starting node’

III. Normal and Extensive form

Extensive form (tree)

AUp

Down

‘Branches’

III. Normal and Extensive form

Extensive form (tree)

A

B1

B2

Up

Down

‘Decision nodes’

III. Normal and Extensive form

Extensive form (tree)

A

B1

B2

Up

Down

‘Information set’

III. Normal and Extensive form

Extensive form (tree)

A

B1

B2

Up

Down

Left

Right

Left

Right

‘Branches’

III. Normal and Extensive form

Extensive form (tree)

A

B1

B2

Up

Down

Left

Right

Left

Right

‘Terminal nodes’

A, B

0, 10

20, 5

5, 20

10, 0

IV. Equilibrium Concepts

Definition: a rule that defines an equilibrium

1. Dominant strategies

2. Iterated dominance

3. Nash equilibrium

4. Maximin strategies

1. Dominant strategies

Definition: a players strictly best response to any strategies the other players might choose.Example: Prisoners’ Dilemma

Player A

Player B

Betray

Silence

Betray Silence

-8, -8 0, -10

-10, 0 -1, -1

IV. Equilibrium Concepts (dominant strategy)

Prisoners’ Dilemma

Player A

Player B

Betray

Silence

Betray Silence

-8, -8 0, -10

-10, 0 -1, -1

Prisoners’ Dilemma

IV. Equilibrium Concepts (dominant strategy)

IV. Equilibrium Concepts2. Iterated dominance

Supermarket game:• 2 supermarkets (Albert Heijn and Bas v.d Heijden)• Charge high, medium or low price• Profits per customer per week: €12 with high, €10 with

medium and €5 with low price• Both stores have fixed clientele of 3,000 people• Floating clientele of 4,000 people shops at cheapest

store (if identical price: both 2,000)

Idea: (repeatedly) wipe out dominated strategies until you can solve the game

Bas van der Heijden

high medium low

Albert

Heijn

high 60, 60 36, 70 36, 35

medium 70, 36 50, 50 30, 35

low 35, 36 35, 30 25, 25

IV. Equilibrium Concepts (Iterated dominance)

(wipe out dominated strategies)

Bas van der Heijden

high medium low

Albert

Heijn

high 60, 60 36, 70 36, 35

medium 70, 36 50, 50 30, 35

low 35, 36 35, 30 25, 25

IV. Equilibrium Concepts (Iterated dominance)

(remaining game: prisoners’ dilemma)

Competition law!

Bas van der Heijden

high medium low

Albert

Heijn

high 60, 60 36, 70 36, 35

medium 70, 36 50, 50 30, 35

low 35, 36 35, 30 25, 25

IV. Equilibrium Concepts (Iterated dominance)

IV. Equilibrium Concepts3. Nash-Equilibrium

Definition: a strategy combination is a Nash equilibrium if no player wants to deviate from his strategy given that no other player does (iterated) dominant equilibrium is also a Nash equilibrium!

This equilibrium occurs when each player’s strategy is optimal, knowing the strategy’s of the other players.

A player’s best strategy is that strategy that maximizes that player’s payoff (utility), knowing the strategy’s of the other players.

So when each player within a game follows their best strategy, a Nash equilibrium will occur.

IV. Equilibrium Concepts (Nash)

Coordination game

Which equilibrium will occur? No communication!

Possible solutions:• Size of payoffs• Repeated games• Focal points

IV. Equilibrium Concepts

4. Maximin

Idea: maximize your minimal payoff

Nash relies on rationality of both players. When in doubt, maximin is safer

Player A

Player B

Up

Down

Left Right

1, 0 1, 1

-1000, 1 2, 1

IV. Equilibrium Concepts (Maximin)

Nash: {down, right}

Maximin: {up, right}

V. Repeated games

Repeating a prisoners’ dilemma

Relevant issues

• Infinite number of times cooperation• Finite number of times no cooperation

• Short term gain by cheating• Long term gain when cooperating• Discount factor for future gains• Assessment of probability of next period

Tit-for-Tat: start with cooperation, and imitate in subsequent rounds

VI. Sequential Games

Players move in turn instead of simultaneously

rollback

• First Mover Advantage (setting the stage)• First Mover Disadvantage (revealing information)