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EKONOMSKA ANALIZA PRAVA

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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

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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

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Game theory 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.

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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

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III. Normal and Extensive form Normal form (matrix) Player A Player B Up Down LeftRight 0, 1020, 5 5, 20 10, 0

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III. Normal and Extensive form Extensive form (tree): nodes and branches ‘Initial node’ or ‘starting node’

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III. Normal and Extensive form Extensive form (tree): nodes and branches A ‘Initial node’ or ‘starting node’

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III. Normal and Extensive form Extensive form (tree) A Up Down ‘Branches’

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III. Normal and Extensive form Extensive form (tree) A B1B1 B2B2 Up Down ‘Decision nodes’

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III. Normal and Extensive form Extensive form (tree) A B1B1 B2B2 Up Down ‘Information set’

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III. Normal and Extensive form Extensive form (tree) A B1B1 B2B2 Up Down Left Right Left Right ‘Branches’

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III. Normal and Extensive form Extensive form (tree) A B1B1 B2B2 Up Down Left Right Left Right ‘Terminal nodes’ A, B 0, 10 20, 5 5, 20 10, 0

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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

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Player A Player B Betray Silence BetraySilence -8, -80, , 0-1, -1 IV. Equilibrium Concepts (dominant strategy) Prisoners’ Dilemma

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Player A Player B Betray Silence BetraySilence -8, -8 0, , 0-1, -1 Prisoners’ Dilemma IV. Equilibrium Concepts (dominant strategy)

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IV. Equilibrium Concepts 2. 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

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Bas van der Heijden highmediumlow Albert Heijn high60, 6036, 7036, 35 medium70, 3650, 5030, 35 low35, 3635, 3025, 25 IV. Equilibrium Concepts (Iterated dominance)

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(wipe out dominated strategies) Bas van der Heijden highmediumlow Albert Heijn high60, 6036, 7036, 35 medium70, 3650, 5030, 35 low35, 3635, 3025, 25 IV. Equilibrium Concepts (Iterated dominance)

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(remaining game: prisoners’ dilemma) Competition law! Bas van der Heijden highmediumlow Albert Heijn high60, 6036, 7036, 35 medium70, 3650, 5030, 35 low35, 3635, 3025, 25 IV. Equilibrium Concepts (Iterated dominance)

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IV. Equilibrium Concepts 3. 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.

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IV. Equilibrium Concepts (Nash) Coordination game Which equilibrium will occur? No communication! Possible solutions: Size of payoffs Repeated games Focal points

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IV. Equilibrium Concepts 4. Maximin Idea: maximize your minimal payoff Nash relies on rationality of both players. When in doubt, maximin is safer

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Player A Player B Up Down Left Right 1, 01, , 12, 1 IV. Equilibrium Concepts (Maximin) Nash: {down, right} Maximin: {up, right}

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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

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VI. Sequential Games Players move in turn instead of simultaneously rollback First Mover Advantage (setting the stage) First Mover Disadvantage (revealing information)

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