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Game Theoretic Analysis of Oligopoly.

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5 -20 -5 y n Y N 0000 Y N -20 5 1 22 The unique dominant strategy Nash Equilibrium is (y,Y) A game of imperfect Information The Prisoners’ Dilemma Y y stand for compete N n stand for collude

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5 -20 -5 y n Y N 0000 Y N -20 5 1 22 The Prisoners’ Dilemma A game of Perfect Information The only play at a Nash Equilibrium is (y, Y)

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10 3 0 10 -3 -4 T M B L RC -2 11 -5 2 L R C 12 -2 -3 -4 LRC 10 3434 1 2 2 2

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A:1 plays T 2 plays R if T, R if M, R if B B: 1 plays B 2 plays L if T, R if M, C 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

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

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Finitely Repeated Games Prisoners’ Dilemma

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5 -20 -5 y n Y N 0000 Y N -20 5 1 22 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 Y y stand for compete N n stand for collude Game 2

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5 -20 -5 y n Y N 0000 Y N -20 5 1 22 The Prisoners’ Dilemma A game of Perfect Information Y y stand for compete N n stand for collude Game 200 Player 1 plays y and player 2 plays Y if y and Y if n at the only Nash Equilibrium

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Finite Sequence of Entry Games

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

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

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Collusive Behaviour Reputation Building And Predatory Behaviour

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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) Analysis of the Infinitely Repeated Game Prisoners’ Dilemma

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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

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= 5-5/r PDV Y = 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)}] Present Discounted Value of playing collude forever (PDV N ) is 0 Present Discounted Value of playing Compete now (PDV Y ) is

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

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

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

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

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

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

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…………… S SSSS SS S S S G SS GGG G GG G GG 1 11 1 1 1 2 2 2 22 0000 -10 1 -10 -9 2 -9 -8 3 90 101 90 91 102 103 92 103 102 91 2 Rosenthal’s Centipede Game

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

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