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UNIT III: COMPETITIVE STRATEGY

UNIT III: COMPETITIVE STRATEGY. Monopoly Oligopoly Strategic Behavior. 11 /2. Strategic Behavior. Dominance Reasoning Best Response and Nash Equilibrium Mixed Strategies Repeated Games The Folk Theorem Cartel Enforcement. Dominance. Definition

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UNIT III: COMPETITIVE STRATEGY

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  1. UNIT III: COMPETITIVE STRATEGY • Monopoly • Oligopoly • Strategic Behavior 11/2

  2. Strategic Behavior • Dominance Reasoning • Best Response and Nash Equilibrium • Mixed Strategies • Repeated Games • The Folk Theorem • Cartel Enforcement

  3. Dominance Definition Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T1 T2 T3 T1 T2 T3 0,2 4,3 3,3 4,0 5,4 5,3 3,5 3,5 2,3 0,2 4,3 3,3 4,0 5,4 5,6 3,5 3,5 2,3 S1 S2 S3 S1 S2 S3 • (S2,T2) • (S2,T3) Sure Thing Principle: If you have a dominant strategy, use it!

  4. Nash Equilibrium Definitions Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to(s,t) for all s. T1 T2 T3 Nash Equilibrium: aset of best response strategies (one for each player), (s*, t*) such that s* is a best response to t* and t* is a b.r. to s*. (S3,T3) 0,44,0 5,3 4,0 0,45,3 3,5 3,5 6,6 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  5. Nash Equilibrium T1 T2 T3 Nash equilibrium need not be Efficient. 4,4 2,3 1,5 3,2 1,1 0,0 5,1 0,0 3,3 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  6. Nash Equilibrium T1 T2 T3 Nash equilibrium need not be unique. A COORDINATION PROBLEM 1,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  7. Nash Equilibrium T1 T2 T3 Multiple and Inefficient Nash Equilibria. 1,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 3,3 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  8. Nash Equilibrium T1 T2 T3 Multiple and Inefficient Nash Equilibria. Is it always advisable to play a NE strategy? What do we need to know about the other player? 1,1 0,0 0,-100 0,0 1,1 0,0 -100,0 0,0 3,3 -3 0 -10 -1 5 2 -2 -4 0 S1 S2 S3 S1 S2 S3

  9. Button-Button Player 1 Player 1 hides a button in his Left or Right hand. Player 2 observes Player 1’s choice and then picks either Left or Right. How should the game be played? Left Right L R L R (-2,2) (4,-4) (2,-2) (-1,1) Player 2 GAME 2.

  10. Button-Button Player 1 Player 1 should hide the button in his Right hand. Player 2 should picks Right. Left Right L R L R (-2,2) (4,-4) (2,-2) (-1,1) Player 2 GAME 2.

  11. Button-Button Player 1 What happens if Player 2 cannot observe Player 1’s choice? Left Right L R L R (-2,2) (4,-4) (2,-2) (-1,1) Player 2 GAME 2.

  12. Button-Button L R L R Player 1 -2, 24, -4 2, -2 -1, 1 Left Right L R L R (-2,2) (4,-4) (2,-2) (-1,1) Player 2 GAME 2.

  13. Mixed Strategies Definition Mixed Strategy: A mixed strategy is a probability distribution over all strategies available to a player. Let (p, 1-p) = prob. Player 1 chooses L, R. (q, 1-q) = prob. Player 2 chooses L, R. L R -2, 2 4, -4 2, -2 -1, 1 L R GAME 2.

  14. Mixed Strategies Then the expected payoff to Player 1: EP1(L)= -2(q) + 4(1-q) = 4 – 6q EP1(R) = 2(q) – 1(1-q) = -1 + 3q Then if q < 5/9, Player 1’s best response is to always play L (p = 1) L R -2, 2 4, -4 2, -2 -1, 1 L R (p) (1-p) (q) (1-q) GAME 2.

  15. Button-Button q LEFT 1 5/9 RIGHT 0 Player 1’s best response function. p*(q) 0 1 p GAME 2.

  16. Mixed Strategies Then the expected payoff to Player 1: EP1(L)= -2(q) + 4(1-q) EP1(R) = 2(q) – 1(1-q) (Equalizers) q* = 5/9 and for Player 2: p* = 1/3 EP2(L) = 2(p) - 2(1-p) EP2(R)= -4(p) + 1(1-p) L R -2, 2 4, -4 2, -2 -1, 1 L R (p) (1-p) (q) (1-q) NE = {(1/3), (5/9)} GAME 2.

  17. Button-Button q LEFT 1 5/9 RIGHT 0 q*(p) p*(q) 0 1/3 1 p GAME 2. NE = {(1/3), (5/9)}

  18. T1 T2 1.Prisoner’s Dilemma 2.Button – Button 3.Stag Hunt 4.Chicken 5.Battle of Sexes 2x2 Game S1 S2 x1,x2 w1, w2 z1,z2 y1, y2

  19. T1 T2 also Assurance Game NE = {(S1,T1), (S2,T2)} GAME 3. Stag Hunt S1 S2 5,5 0,3 3,0 1,1

  20. T1 T2 also Hawk/Dove NE = {(S1,T2), (S2,T1)} GAME 4. Chicken S1 S2 3,3 1,5 5,1 0,0

  21. T1 T2 NE = {(S1,T1), (S2,T2)} GAME 5. Battle of the Sexes S1 S2 5,3 0,0 0,0 3,5

  22. Opera Fight F Solving the Game NE = {(O,O), (F,F)} GAME 5. Opera Fight 5,3 0,0 0,0 3,5 Find the mixed strategy Nash Equilibrium

  23. Battle of the Sexes P2 5 3 0 (0,0) (1,1) (5/8,3/8) 0 3 5 P1 GAME 5. NE = {(O,O); (F,F); (5/8, 3/8)}

  24. Battle of the Sexes P2 5 3 0 (0,0) (1,1) (5/8,3/8) 0 3 5 P1 GAME 5. NE = {(1,1); (0,0); (5/8,3/8)} (p, q); (p, q); ( p , q )

  25. Battle of the Sexes q OPERA 1 3/8 FIGHT 0 q*(p) if p<5/8, then Player 2’s best response is q* = 0 (FIGHT) if p>5/8 q* = 1 (OPERA) 0 5/8 1 p GAME 5.

  26. Battle of the Sexes q OPERA 1 3/8 FIGHT 0 q*(p) p*(q) 0 5/8 1 p GAME 5. NE = {(1, 1); (0, 0); (5/8, 3/8)}

  27. Battle of the Sexes Opera Fight P2 Opera Fight equity (0,0) 5,3 0,0 0,0 3,5 bargaining power or skill efficiency (1,1) (5/8, 3/8) P1 GAME 5. NE = {(1, 1); (0, 0); (5/8, 3/8)}

  28. Existence of Nash Equilibrium There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w). Prisoner’s Dilemma Battle of the Sexes Button-Button GAME 1. GAME 5. (Also 3, 4) GAME 2. q 1 0 0 1 0 1 0 1 p

  29. The Prisoner’s Dilemma The pair of dominant strategies (Confess, Confess) is a Nash Eq. In years in jail Player 2 Confess Don’t Confess Player 1 Don’t -10, -10 0, -20 -20, 0 -1, -1 GAME 1.

  30. The Prisoner’s Dilemma Each player has a dominant strategy. Yet the outcome (-10, -10) is pareto inefficient. Is this a result of imperfect information? What would happen if the players could communicate? What would happen if the game were repeated? A finite number of times? An infinite or unknown number of times? What would happen if rather than 2, there were many players?

  31. Repeated Games Some Questions: • What happens when a game is repeated? • Can threats and promises about the future influence behavior in the present? • Cheap talk • Finitely repeated games: Backward induction • Indefinitely repeated games: Trigger strategies

  32. Repeated Games Examples of Repeated Prisoner’s Dilemma • Cartel enforcement • Transboundary pollution • Common property resources • Arms races The Tragedy of the Commons Free-rider Problems

  33. Repeated Games Can threats and promises about future actions influence behavior in the present? Consider the following game, played 2X: C D C3,3 0,5 D5,0 1,1

  34. Repeated Games Draw the extensive form game: (3,3) (0,5) (5,0) (1,1) (6,6) (3,8) (8,3) (4,4) (3,8)(0,10)(5,5)(1,6)(8,3) (5,5)(10,0) (6,1) (4,4) (1,6) (6,1) (2,2)

  35. Repeated Games Now, consider three repeated game strategies: D (ALWAYS DEFECT): Defect on every move. C (ALWAYS COOPERATE): Cooperate on every move. T (TRIGGER): Cooperate on the first move, then cooperate after the other cooperates. If the other defects, then defect forever.

  36. Repeated Games If the game is played twice, the V(alue) to a player using ALWAYS DEFECT (D) against an opponent using ALWAYS DEFECT(D) is: V (D/D) = 1 + 1 = 2, and so on. . . V (C/C) = 3 + 3 = 6 V (T/T) = 3 + 3 = 6 V (D/C) = 5 + 5 = 10 V (D/T) = 5 + 1 = 6 V (C/D) = 0 + 0 = 0 V (C/T) = 3 + 3 = 6 V (T/D) = 0 + 1 = 1 V (T/C) = 3 + 3 = 6

  37. Repeated Games And 3x: V (D/D) = 1 + 1 + 1 = 3 V (C/C) = 3 + 3 + 3 = 9 V (T/T) = 3 + 3 + 3 = 9 V (D/C) = 5 + 5 + 5 = 15 V (D/T) = 5 + 1 + 1 = 7 V (C/D) = 0 + 0 + 0 = 0 V (C/T) = 3 + 3 + 3 = 9 V (T/D) = 0 + 1 + 1 = 2 V (T/C) = 3 + 3 + 3 = 9

  38. Repeated Games Time average payoffs: n=3 V (D/D) = 1 + 1 + 1 = 3 /3 = 1 V (C/C) = 3 + 3 + 3 = 9 /3 = 3 V (T/T) = 3 + 3 + 3 = 9 /3 = 3 V (D/C) = 5 + 5 + 5 = 15 /3 = 5 V (D/T) = 5 + 1 + 1 = 7 /3 = 7/3 V (C/D) = 0 + 0 + 0 = 0 /3 = 0 V (C/T) = 3 + 3 + 3 = 9 /3 = 3 V (T/D) = 0 + 1 + 1 = 2 /3 = 2/3 V (T/C) = 3 + 3 + 3 = 9 /3 = 3

  39. Repeated Games Time average payoffs: n V (D/D) = 1 + 1 + 1 + ... /n = 1 V (C/C) = 3 + 3 + 3 + ... /n = 3 V (T/T) = 3 + 3 + 3 + ... /n = 3 V (D/C) = 5 + 5 + 5 + ... /n = 5 V (D/T) = 5 + 1 + 1 + ... /n = 1 + e V (C/D) = 0 + 0 + 0 + ... /n = 0 V (C/T) = 3 + 3 + 3 + … /n = 3 V (T/D) = 0 + 1 + 1 + ... /n = 1 - e V (T/C) = 3 + 3 + 3 + ... /n = 3

  40. Repeated Games Now draw the matrix form of this game: 1x C D T C3,3 0,5 3,3 D5,0 1,15,0 T 3,3 0,5 3,3

  41. Repeated Games Time Average Payoffs C D T C3,3 0,53,3 If the game is repeated, ALWAYS DEFECT is no longer dominant. D5,0 1,11+e,1-e T 3,3 1-e,1+e3,3

  42. Repeated Games C D T C3,3 0,53,3 … and TRIGGER achieves “a NE with itself.” D5,0 1,11+e,1-e T 3,3 1-e,1+e3,3

  43. Repeated Games Time Average Payoffs T(emptation) > R(eward) > P(unishment)> S(ucker) C D T CR,R S,TR,R DT,S P,PP+e,P-e T R,R P-e,P+eR,R

  44. Discounting The discount parameter, d, is the weight of the next payoff relative to the current payoff. In a indefinitely repeated game, d can also be interpreted as the likelihood of the game continuing for another round (so that the expected number of moves per game is 1/(1-d)). The V(alue) to someone using ALWAYS DEFECT (D) when playing with someone using TRIGGER (T) is the sum of T for the first move, d P for the second, d2P for the third, and so on (Axelrod: 13-4): V (D/T) = T + dP + d2P + … “The Shadow of the Future”

  45. Discounting Writing this as V (D/T) = T + dP + d2P +..., we have the following: V (D/D) = P + dP + d2P + … = P/(1-d) V (C/C) = R + dR + d2R + … = R/(1-d) V (T/T) = R + dR + d2R + … = R/(1-d) V (D/C) = T + dT + d2T + … = T/(1-d) V (D/T) = T + dP + d2P + … = T+ dP/(1-d) V (C/D) = S + dS + d2S + … = S/(1-d) V (C/T) = R + dR + d2R + … = R/(1- d) V (T/D) = S + dP + d2P + … = S+ dP/(1-d) V (T/C) = R + dR + d2R + … = R/(1- d)

  46. Discounting C D T R/(1-d) S/(1-d) R/(1-d) R/(1-d) T/(1-d) R/(1-d) C Discounted Payoffs T > R > P > S 0 > d > 1 T weakly dominates C T/(1-d) P/(1-d) T + dP/(1-d) S/(1-d)P/(1-d) S + dP/(1-d) D R/(1-d) S + dP/(1-d) R/(1- d) R/(1-d)T + dP/(1-d) R/(1-d) T

  47. Discounting • Now consider what happens to these values as d varies (from 0-1): • V (D/D) = P + dP + d2P + … = P/(1-d) • V (C/C) = R + dR + d2R + … = R/(1-d) • V (T/T) = R + dR + d2R + … = R/(1-d) • V (D/C) = T + dT + d2T + … = T/(1-d) • V (D/T) = T + dP + d2P + … = T+ dP/(1-d) • V (C/D) = S + dS + d2S + … = S/(1-d) • V (C/T) = R + dR + d2R + … = R/(1- d) • V (T/D) = S + dP + d2P + … = S+ dP/(1-d) • V (T/C) = R + dR + d2R + … = R/(1- d)

  48. Discounting • Now consider what happens to these values as d varies (from 0-1): • V (D/D) = P + dP + d2P + … = P+ dP/(1-d)V (C/C) = R + dR + d2R + … = R/(1-d) • V (T/T) = R + dR + d2R + … = R/(1-d) • V (D/C) = T + dT + d2T + … = T/(1-d) • V (D/T) = T + dP + d2P + … = T+ dP/(1-d) • V (C/D) = S + dS + d2S + … = S/(1-d) • V (C/T) =R + dR + d2R + … = R/(1- d) • V (T/D) = S + dP + d2P + … = S+ dP/(1-d)V (T/C) = R + dR + d2R + … = R/(1- d) V(D/D) > V(T/D) D is a best response to D

  49. Discounting • Now consider what happens to these values as d varies (from 0-1): • V (D/D) = P + dP + d2P + … = P+ dP/(1-d) • V (C/C) = R + dR + d2R + … = R/(1-d) • V (T/T) = R + dR + d2R + … = R/(1-d) • V (D/C) = T + dT + d2T + … = T/(1-d) • V (D/T) = T + dP + d2P + … = T+ dP/(1-d) • V (C/D) = S + dS + d2S + … = S/(1-d) • V (C/T) = R + dR + d2R + … = R/(1- d) • V (T/D) = S + dP + d2P + … = S+ dP/(1-d) • V (T/C) = R + dR + d2R + … = R/(1- d) 2 1 3 ?

  50. Discounting Now consider what happens to these values as d varies (from 0-1): For all values of d: V(D/T) > V(D/D) > V(T/D) V(T/T) > V(D/D) > V(T/D) Is there a value of d s.t., V(D/T) = V(T/T)? Call this d*. If d < d*,the following ordering hold: V(D/T) > V(T/T) > V(D/D) > V(T/D) D is dominant: GAME SOLVED ? V(D/T) = V(T/T) T+dP/(1-d) = R/(1-d) T-dt+dP = R T-R = d(T-P) d* = (T-R)/(T-P)

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