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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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PowerPoint Slides by Robert F. BrookerCopyright (c) 2001 by Harcourt, Inc. All rights reserved. Strategic Behavior Game Theory –Players –Strategies –Payoff.

PowerPoint Slides by Robert F. BrookerCopyright (c) 2001 by Harcourt, Inc. All rights reserved. Strategic Behavior Game Theory –Players –Strategies –Payoff.

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