Game Theoretic Analysis of Oligopoly

1. . Game Theoretic Analysis of Oligopoly

2. The Prisoners’ Dilemma 1 Y y stand for compete N n stand for collude n y 2 2 N N Y Y 5 -20 0 0 -5 -5 -20 5 A game of imperfect Information The unique dominant strategy Nash Equilibrium is (y,Y)

3. 1 n y 2 2 N N Y Y 5 -20 0 0 -5 -5 -20 5 The Prisoners’ Dilemma A game of Perfect Information The only play at a Nash Equilibrium is (y, Y)

4. 1 T B M 2 2 C R R L L C 0 10 -3 -4 10 3 -5 2 3 4 -2 11 2 L C R 10 -1 12 -2 -3 -4

5. B: 1 plays B 2 plays L if T, R if M, C if B A:1 plays T 2 plays R if T, R if M, R if B C:1 plays M 2 plays R if T, L if M, C if B Only C is a (Subgame) Perfect or ‘Credible’ Nash Equilibrium

6. 1- Entrant 3m 3m -1m 2m 2- Incumbent 0 7m Tough Soft 2 Stay Out Enter Credible Threat Equilibrium 1 The two Nash Equilibria are 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter

7. Finitely Repeated Games Prisoners’ Dilemma

8. 1 N n stand for collude Y y stand for compete n y 2 2 Game 2 N N Y Y 5 -20 0 0 -5 -5 -20 5 The Prisoners’ Dilemma A game of Perfect Information Player 1 plays y and player 2 plays Y if y and Y if n at the only Nash Equilibrium

9. 1 N n stand for collude Y y stand for compete n y 2 2 Game 200 N N Y Y 5 -20 0 0 -5 -5 -20 5 The Prisoners’ Dilemma A game of Perfect Information Player 1 plays y and player 2 plays Y if y and Y if n at the only Nash Equilibrium

10. Finite Sequence of Entry Games

11. 1- Entrant 3m 3m -1m 2m 2- Incumbent Game with two sequential entries 0 7m Tough Soft 2 Stay Out Enter 1 The two Nash Equilibria are 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter

12. 1- Entrant 3m 3m -1m 2m 2- Incumbent Game with two hundred sequential entries 0 7m Tough Soft 2 Stay Out Enter 1 The two Nash Equilibria are 1: Stay Out 2: Tough if Enter 1: Enter 2: Soft if Enter

13. Collusive Behaviour Reputation Building And Predatory Behaviour

14. Prisoners’ Dilemma Analysis of the Infinitely Repeated Game Both play the Tit-for-Tat Strategy Start with n or N (Collude) Stick with n or N (Collude) until the other player deviates and plays Y Play y (or Y) forever once the other player has played Y (or y)

15. Either player payoff structure is as follows Get 0 always if stick with n (or N) Get 5 one-off with play y (or Y) and then (-5) forever

16. Present Discounted Value of playing colludeforever (PDVN) is 0 Present Discounted Value of playing Compete now (PDVY) is PDVY = 5 - 5/(1+r) -5/(1+r)2 - 5/(1+r)3 – ….. = 5 – (5/(1+r) +5/(1+r)2 + 5/(1+r)3 - …..) = 5 – 5/(1+r) *[1/1-{1/(1+r)}] = 5-5/r

17. Entry Games Analysis of the case of an Infinite Chain of Sequential entry • Incumbent: always play tough if enter All Entrants : Play Stay out if the incumbent has no history of playing soft. Otherwise enter

18. Payoff structure for incumbent: Get 7m forever Payoff structure for each entrant: Get 0 forever

19. After any entry: Get 2m one-off with play tough and then 7m forever Is the threat ‘credible’?

20. Present Discounted Value of playing Threat strategy (PDVT) is PDVT = 2 + 7/(1+r) +7/(1+r)2 +7 /(1+r)3 – ….. = 2 +7 /(1+r) *[1/1-{1/(1+r)}] = 2 +7/r

21. Present Discounted Value of playing Soft strategy (PDVS) is PDVT = 3 + 3/(1+r) +3/(1+r)2 +3 /(1+r)3 – ….. = 3 *[1/1-{1/(1+r)}] = 3(1+r)/r 2+ 7/r > 3(1+r)/r If and only if r < 4

22. A Duopoly Game involving two firms A and B 2 (2, 0) qB 1 2 qA (4, 1) 2 1 (2, 2) qB 1 (3, 1) Show that Cournot (Stackelberg) ideas are similar to Nash (Subgame Perfect Nash)

23. Rosenthal’s Centipede Game G G G G G 1 2 1 2 1 2 S S S S S S 0 0 -10 1 1 -10 -9 2 -8 3 2 -9 …………… G G G G G 2 2 1 2 1 1 S S S S S S 90 101 101 90 91 102 102 91 92 103 103 92

24. A Game of Loss Infliction Y – Player 1 gives in to threat y – Player 2 executes threat 1 Perfect Nash Equilibrium 1 plays N 2 plays n if N But is 1 plays Y 2 plays y if N non-credible? Y N 2 -1 +1 y n Top Number is 1’s Payoff -2 -1 0 0