Basics on Game Theory

1. Basics on Game Theory Class 2 Microeconomics

2. Why, What, What for Introduction • 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

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

4. Eliminating Strictly DominatedStrategies P2 Left Center P1 Up P2 Left Center Up P1 Down P2 Left Center Right Up P1 Down Game’s Solution, Dominant-Dominated Strategies

5. Nash Equilibrium • Example • Definition: P2 Left Center Right Up P1 Middle Down NE

6. Multiple Equilibriums He Opera Football Opera She Football The Battle of the Sexes

7. The Best Response Functions He Opera Football Opera She Football Multiple Equilibriums

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

9. Exercise P2 Left Center Right Up P1 Middle Down Find the Solution for the Game

10. Exercise P2 Left Center Right Up P1 Middle Down Find the Solution for the Game

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

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

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

14. P1 L R P2 P2 L´ R´ L´ R´ 1 2 0 3 0 1 2 1 Nash Equilibrium 1 (Backward Induction) Strategy 1: (L´,L´) Strategy 2: (L´,R´) Strategy 3:(R´,L´) Strategy 4: (R´,R´) Strategies and Extensive Form Games

15. P1 L R P2 P2 L´ R´ L´ R´ 1 2 0 3 0 1 2 1 P2 (R´,R´) (L´,L´) (L´,R´) (R´,L´) L P1 R Normal Form and Extensive Form Games Extensive Form Games as Normal Form Games

16. P1 L R P2 P2 L´ R´ L´ R´ 1 2 0 3 0 1 2 1 Sub-Game Perfect Nash Equilibrium Algorithm SPNE: Def: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. 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. NE 2

17. P1 L R P2 P2 L´ R´ L´ R´ 1 2 0 3 0 1 2 1 Sub-game Perfect Nash Equilibrium Algorithm SPNE: Def: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. 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. Extensive Form Games as Normal Form Games

18. P1 L R P2 P2 L´ R´ L´ R´ 1 2 0 3 0 1 2 1 SPNE and BI SPNE is more powerful than NE, for solving Imperfect Information Games: SPNE = (R`,L`) Backward Induction = (R,L`) Extensive Form Games as Normal Form Games