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Basics on Game Theory Class 2 Microeconomics

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Introduction Why, What, What for Why Any human activity has some competition Human activities involve actors, rules, strategies Game theory formalizes the analysis of competition What GT is the study of strategic behavior of competing actors What GT for GT allows to analyze the alternatives set by the rules GT permits to prescribe the opponent’s behavior GT show how to design games

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Games Outline Normal Form Games: Players: Strategies: Payoffs: Concepts Actions: Outcomes: Payoffs: Objective: To find the solution of the game

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Eliminating Strictly Dominated Strategies Game’s Solution, Dominant-Dominated Strategies P2 P1 LeftCenter Up 1,01,20,1 0,30,12,0 Right Down 1,01,2 0,30,1 1,01,2 P1 Up Down P2 LeftCenter P1 Up P2 LeftCenter

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Nash Equilibrium NE Example Definition: P2 P1 LeftCenter Up 0,44,05,3 4,00,45,3 3,5 6,6 Right Middle Down

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Multiple Equilibriums The Battle of the Sexes He She OperaFootball Opera Football 2,10,0 1,2

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The Best Response Functions Multiple Equilibriums He She OperaFootball Opera Football 2,10,0 1,2

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Pareto Efficient Outcomes The Prisoners’ Dilemma P2 P1 DefectCooperate Defect Cooperate -1,-1-9,0 0,-9 -6,-6

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Exercise Find the Solution for the Game P2 P1 LeftCenter Up 2,01,14,2 3,41,22,3 1,30,23,0 Right Middle Down

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Exercise Find the Solution for the Game P2 P1 LeftCenter Up 2,01,14,2 3,41,22,3 1,30,23,0 Right Middle Down

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Extensive Form Games Games with Perfect and Complete Information Moves occur in sequence All previous moves are observed Payoffs are known by all the playersP1LR P2 P2 L´R´L´R´ Payoffs P1: Payoffs P2: 3 1 1 2 2 1 0 0

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Backward Induction (NE-1) Solutions for Extensive Form Games Step 3: Step 2: Step 1: P1 L R (2,0) (1,1) L´ R´ L´´ R´´ (3,0) (0,2) P2 P1

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Strategies The Concept of Strategy in Extensive Form Games Definition: P1: 2 actions, 2 strategies P2: 2 actions, 4 strategiesP1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0

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Nash Equilibrium 1 (Backward Induction) Strategies and Extensive Form GamesP1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0 Strategy 1: (L´,L´) Strategy 2: (L´,R´) Strategy 3: (R´,L´) Strategy 4: (R´,R´)

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Normal Form and Extensive Form Games Extensive Form Games as Normal Form Games P1P2(L´,L´)(L´,R´)(R´,L´) (R´,R´) L R P1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0 3,1 1,2 2,10,02,10,0

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Sub-Game Perfect Nash Equilibrium NE 2 Def: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. P1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0 Algorithm SPNE: Identify all the smaller subgames having terminal nodes in the original tree. Replace each subgame for the payoffs of one of the NE. The initial nodes of the subgame are now the terminal nodes of the new truncated tree.

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Sub-game Perfect Nash Equilibrium Extensive Form Games as Normal Form Games Def: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. P1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0 Algorithm SPNE: Identify all the smaller subgames having terminal nodes in the original tree. Replace each subgame for the payoffs of one of the NE. The initial nodes of the subgame are now the terminal nodes of the new truncated tree.

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SPNE and BI Extensive Form Games as Normal Form Games SPNE is more powerful than NE, for solving Imperfect Information Games: SPNE = (R`,L`) Backward Induction = (R,L`) P1LR P2 P2 L´R´L´R´ 3 1 1 2 2 1 0 0

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