1. Basics on Game Theory For Industrial Economics (According to Shy’s Plan)

2. The Reasons of Game Theory What –GT is the study of strategic interaction involving decisions among multiple actors Why –Economic and political world is made of rules, actors and strategies –GT is the right frame for studying competition What for –What are the alternatives of a game? –Can the behavior of other actors be predicted? –How to design the optimal strategies

3. What is a Game Normal Form Games –Players –Strategies –Payoffs Extensive Form Games

4. Concepts Actions Outcomes Payoffs

5. Concepts In general: –Playersi:1,...,n –A i : Set of actions of player i. –a i : One action of player‘s set A i –(a 1,..., a n ): Outcome –  i (a 1,..., a n ): Payoff for player i, according to the actions of other players Summarizing:,,,,

6. The Search for the Solution of a Game: Elimination of Strictly 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

7. Dominant Actions (and one remark on notation) The payoff  for player i, depends on his move and on the other’s moves (a 1,a 2,…, a i,…, a n ): We represent it in the form: An action a ~ is dominant for one player i, if:

8. Nash Equilibrium If the solution of the game is unique, it is a Nash Equilibrium. Example Definition of NE: P2 P1 LeftCenter Up 0,44,05,3 4,00,45,3 3,5 6,6 Right Middle Down

9. Games with Multiple Equilibria The Battle of the Sexes I He She OperaFootball Opera Football 2,10,0 1,2

10. Games without NE The Battle of the Sexes II He She OperaFootball Opera Football 2,00,2 0,11,0

11. The Best Response Function He She OperaFootball Opera Football 2,10,0 1,2

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

13. Some Questions What is a normal form game? What is a strictly dominated strategy? What is a NE in a normal form game? What are the advantages and the shortcomings of GT in the prediction of the strategic behavior?

14. Exercise In the next game in normal form, which strategies survive to the elimination of strictly dominated strategies? P2 P1 LeftCenter Up 2,01,14,2 3,41,22,3 1,30,23,0 Right Middle Down

15. Extensive Form Games 1 LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1122100 Characteristics: 1) Moves occur in sequence 2) All the previous moves are observed before choose the next one 3) Payoffs are common knowledge among the players

16. Backward Induction and NE 1 LR (2,0) 2 (1,1) L´R´ 1 L´´R´´ (3,0)(0,2) Induction: 1. Step 3. P1 chooses L´´ with u 1 = 3 instead of R´´ with u 2 = 0 2. Step 2. P2 anticipates that if the game reaches level 3, then P1 chooses L´´ therefore u 2 = 0. P2 chooses L´ with u 2 = 1. 3. Step 1. P1 anticipates that if the game reaches level 2, then P2 chooses L´ and therefore u 1 = 1. Then, P1 chooses L with u 1 = 2.

17. Strategies in Extensive Form Games One strategy is a complete plan of actions specifying a feasible action for each move in each contingency for which he can be called upon to act. 1LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1 1 2 2 1 0 0 P2 has 2 actions A{L,R} but 4 strategies. Strategy 1: If P1 plays L, then play L´, if P1 plays R, then play L´: (L´,L´). Strategy 2: If P1 plays L, then play L´, if P1 plays R, then play R´: (L´,R´). Strategy 3: If P1 plays L, then play R´, if P1 plays R, then play L´: (R´,L´). Strategy 4: If P1 plays L, then play R´, if P1 plays R, then play R´: (R´,R´). P1 has 2 actions A{L,R} S P1 coincides with A{L,R}

18. NE of Extensive Form Games Which strategy is the NE of the game? 1LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1 1 2 2 1 0 0 Strategy 1: (L´,L´) Strategy 2: (L´,R´) Strategy 3: (R´,L´) Strategy 4: (R´,R´)

19. Normal Form from Extensive Form1LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1 1 2 2 1 0 0 P1P2(L´,L´)(L´,R´)(R´,L´)(R´,R´) (L) (R) 3,1 1,2 2,10,02,10,0

20. Subgame Perfect Nash E. Definition: A NE is Subgame Perfect if the strategies of the players constitute a NE in each subgame. Algorithm for Identifying a 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.

21. Subgame Perfect Nash E. Example:1LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1 1 2 2 1 0 0 SPNE = (, ) Subgame 1 Subgame 2 R`L` Between 1 and 2, P1 prefers to play R.

22. NE and Subgame Perfect NE Subgame Perfect Nash Equilibrium vs. Simple NE SPNE is more powerful than NE, for solving Imperfect Information Games:1LR 22 L´R´L´R´ Payoffs P(1): Payoffs P(2): 3 1 1 2 2 1 0 0 SPNE = (R`,L`) Backward Induction = (R,L`)