UNIT III: COMPETITIVE STRATEGY Monopoly Oligopoly Strategic Behavior 7/21

Strategic Behavior Nash Equilibrium (continued) Mixed Strategies Repeated Games The Folk Theorem Cartel Enforcement

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 ,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Nash Equilibrium: a set 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*. (S 3,T 3 )

Nash Equilibrium ,4 2,3 1,5 3,2 1,1 0,0 5,1 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Nash equilibrium need not be efficient.

Nash Equilibrium ,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Nash equilibrium need not be unique. A COORDINATION PROBLEM

Nash Equilibrium ,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Multiple and Inefficient Nash Equilibria. Is it always advisable to play a NE strategy? What do we need to know about the other player?

Nash Equilibrium ,1 0,0 0,-100 0,0 1,1 0,0 -100,0 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Multiple and Inefficient Nash Equilibria. Is it always advisable to play a NE strategy? What do we need to know about the other player?

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

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

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

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

Mixed Strategies -2, 2 4, -4 2, -2 -1, 1 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 LRLR GAME 2.

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

q LEFT 1 5/9 RIGHT p p*(q) Mixed Strategies Player 1’s best response function. GAME 2.

Mixed Strategies -2, 2 4, -4 2, -2 -1, 1 Then the expected payoff to Player 1: EP 1 (L) = -2(q) + 4(1-q) = 4 – 6q EP 1 (R) = 2(q) – 1(1-q) = q => q* = 5/9 and the expected payoff to Player 2: EP 2 (L) = -2(p) + 2(1-p) = 2 – 4p EP 2 (R) = 4(p) – 1(1-p) = p => p* = 1/3 L R LRLR (p) (1-p) (q) (1-q) GAME 2. NE = {(1/3), (5/9)}

q LEFT 1 5/9 RIGHT 0 01/3 1 p q*(p) p*(q) NE = {(1/3), (5/9)} Mixed Strategies GAME 2.

2x2 Game T 1 T 2 1. Prisoner’s Dilemma 2. Button – Button 3. Stag Hunt 4. Chicken 5. Battle of Sexes x 1,x 2 w 1, w 2 z 1,z 2 y 1, y 2 S 1 S 2

Stag Hunt T 1 T 2 S 1 S 2 5,5 0,3 3,0 1,1 also Assurance Game NE = {(S 1,T 1 ), (S 2,T 2 )} GAME 3.

Chicken T 1 T 2 S 1 S 2 3,3 1,5 5,1 0,0 also Hawk/Dove NE = {(S 1,T 2 ), (S 2,T 1 )} GAME 4.

Battle of the Sexes T 1 T 2 S 1 S 2 5,3 0,0 0,0 3,5 NE = {(S 1,T 1 ), (S 2,T 2 )} GAME 5.

P2530P P 1 GAME 5. NE = {(1, 1); (0, 0); (, )} (0,0) (1,1) Battle of the Sexes (p, q); (p, q)

P2530P P 1 GAME 5. NE = {(1, 1); (0, 0); (5/8, 3/8)} (0,0) (5/8,3/8) (1,1) Battle of the Sexes

P2530P P 1 GAME 5. NE = {(1, 1); (0, 0); (5/8, 3/8)} (0,0) (5/8,3/8) (1,1) Battle of the Sexes equity efficiency Bargaining power

Existence of Nash Equilibrium Prisoner’s DilemmaBattle of the SexesButton-Button GAME 1.GAME 5. (Also 3, 4)GAME p q10q10 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).

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

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

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

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)

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 others defects, then defect forever.

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) = = 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

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

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

Repeated Games Now draw the matrix form of this game: 1x T3,3 0,5 3,3 C 3,3 0,53,3 D 5,0 1,15,0 C D T

Repeated Games T 3,3 1-  1+  3,3 C 3,3 0,5 3,3 D 5,0 1,1 1+ ,1-  C D T If the game is repeated, ALWAYS DEFECT is no longer dominant. Time Average Payoffs

Repeated Games T 3,3 1-  1+  3,3 C 3,3 0,5 3,3 D 5,0 1,1 1+ ,1-  C D T … and TRIGGER achieves “a NE with itself.”

Repeated Games Time Average Payoffs T(emptation)> R(eward)> P(unishment)> S(ucker) T R,R P-  P +  R,R C R,R S,T R,R D T,S P,P P + , P -  C D T

Discounting The discount parameter, , is the weight of the next payoff relative to the current payoff. In a indefinitely repeated game,  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-  )). 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,  P for the second,  2 P for the third, and so on (Axelrod: 13-4): V (D/T) = T +  P +  2 P + … “The Shadow of the Future”

Discounting Writing this as V (D/T) = T +  P +   2 P +..., we have the following: V (D/D) = P +  P +  2 P + … = P/(1-  ) V (C/C) =R +  R +  2 R + … = R/(1-  ) V (T/T)=R +  R +  2 R + … = R/(1-  ) V (D/C)=T +  T +  2 T + … = T/(1-  ) V (D/T)=T +  P +  2 P + … = T+  P/(1-  ) V (C/D)=S +  S +  2 S + … = S/(1-  ) V (C/T)=R +  R +  2 R + … = R/(1-  ) V (T/D)=S +  P +  2 P + … = S+  P/(1-  ) V (T/C)=R +  R +  2 R + … = R/(1-  )

T C D Discounted Payoffs T > R > P > S 0 >  > 1 R /(1-  ) S /(1-  ) R /(1-  ) R /(1-  ) T /(1-  ) R /(1-  ) T /(1-  ) P /(1-  ) T +  P /(1-  ) S /(1-  ) P /(1-  ) S +  P /(1-  ) Discounting C D T R /(1-  ) S +  P /(1-  ) R /(1-  ) R /(1-  ) T +  P /(1-  ) R /(1-  )

T C D Discounted Payoffs T > R > P > S 0 >  > 1 T weakly dominates C R /(1-  ) S /(1-  ) R /(1-  ) R /(1-  ) T /(1-  ) R /(1-  ) T /(1-  ) P /(1-  ) T +  P /(1-  ) S /(1-  ) P /(1-  ) S +  P /(1-  ) Discounting C D T R /(1-  ) S +  P /(1-  ) R /(1-  ) R /(1-  ) T +  P /(1-  ) R /(1-  )

Discounting Now consider what happens to these values as  varies (from 0-1): V (D/D) = P +  P +  2 P + … = P/(1-  ) V (C/C) =R +  R +  2 R + … = R/(1-  ) V (T/T)=R +  R +  2 R + … = R/(1-  ) V (D/C)=T +  T +  2 T + … = T/(1-  ) V (D/T)=T +  P +  2 P + … = T+  P/(1-  ) V (C/D)=S +  S +  2 S + … = S/(1-  ) V (C/T)=R +  R +  2 R + … = R/(1-  ) V (T/D)=S +  P +  2 P + … = S+  P/(1-  ) V (T/C)=R +  R +  2 R + … = R/(1-  )

Discounting Now consider what happens to these values as  varies (from 0-1): V (D/D) = P +  P +  2 P + … = P+  P/(1-  ) V (C/C) =R +  R +  2 R + … = R/(1-  ) V (T/T)=R +  R +  2 R + … = R/(1-  ) V (D/C)=T +  T +  2 T + … = T/(1-  ) V (D/T)=T +  P +  2 P + … = T+  P/(1-  ) V (C/D)=S +  S +  2 S + … = S/(1-  ) V (C /T) = R +  R +  2 R + … = R/(1-  ) V (T/D)=S +  P +  2 P + … = S+  P/(1-  ) V (T/C)=R +  R +  2 R + … = R/(1-  ) V(D/D) > V(T/D) D is a best response to D

Discounting Now consider what happens to these values as  varies (from 0-1): V (D/D) = P +  P +  2 P + … = P+  P/(1-  ) V (C/C) =R +  R +  2 R + … = R/(1-  ) V (T/T)=R +  R +  2 R + … = R/(1-  ) V (D/C)=T +  T +  2 T + … = T/(1-  ) V (D/T)=T +  P +  2 P + … = T+  P/(1-  ) V (C/D)=S +  S +  2 S + … = S/(1-  ) V (C/T)=R +  R +  2 R + … = R/(1-  ) V (T/D)=S +  P +  2 P + … = S+  P/(1-  ) V (T/C)=R +  R +  2 R + … = R/(1-  ) ?

Discounting Now consider what happens to these values as  varies (from 0-1): For all values of  : V(D/T) > V(D/D) > V(T/D) V(T/T) > V(D/D) > V(T/D) Is there a value of  s.t., V(D/T) = V(T/T)? Call this  *. If  <  *, 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+  P/(1-  ) = R/(1-  ) T-  t+  P = R T-R =  (T-P)   * = (T-R)/(T-P) ?

Discounting Now consider what happens to these values as  varies (from 0-1): For all values of  : V(D/T) > V(D/D) > V(T/D) V(T/T) > V(D/D) > V(T/D) Is there a value of  s.t., V(D/T) = V(T/T)? Call this  *.  * = (T-R)/(T-P) If  >  *, the following ordering hold: V(T/T) > V(D/T) > V(D/D) > V(T/D) D is a best response to D; T is a best response to T; multiple NE.

Discounting V(T/T) = R/(1-  )  * 1 V TRV TR Graphically: The V(alue) to a player using ALWAYS DEFECT (D) against TRIGGER (T), and the V(T/T) as a function of the discount parameter (  ) V(D/T) = T +  P/(1-  )

The Folk Theorem (R,R) (T,S) (S,T) (P,P) The payoff set of the repeated PD is the convex closure of the points [( T,S ); ( R,R ); ( S,T ); ( P,P )].

The Folk Theorem (R,R) (T,S) (S,T) (P,P) The shaded area is the set of payoffs that Pareto-dominate the one-shot NE ( P,P ).

The Folk Theorem (R,R) (T,S) (S,T) (P,P) Theorem: Any payoff that pareto- dominates the one-shot NE can be supported in a SPNE of the repeated game, if the discount parameter is sufficiently high.

The Folk Theorem (R,R) (T,S) (S,T) (P,P) In other words, in the repeated game, if the future matters “enough” i.e., (  >  * ), there are zillions of equilibria!

The theorem tells us that in general, repeated games give rise to a very large set of Nash equilibria. In the repeated PD, these are pareto-rankable, i.e., some are efficient and some are not. In this context, evolution can be seen as a process that selects for repeated game strategies with efficient payoffs. “Survival of the Fittest” The Folk Theorem

Cartel Enforcement Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by: P = 7 – Q Assume that the firms choose prices. If the two firms choose different prices, the one with the lower price gets all the customers; if they choose the same price, they split the market demand. What is the Nash Equilibrium of this game?

Cartel Enforcement Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by: P = 7 – Q Now suppose that the firms compete repeatedly, and each firm attempts to maximize the discounted value of its profits (  < 1). What if this pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)?

Cartel Enforcement What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)? P = 7 – Q; TC i = q i Monopoly Bertrand Duopoly  = TR – TCQ = q 1 + q 2 = PQ – QP b = MC = 1; Q b = 6 = (7-Q)Q - Q = 7Q - Q 2 - Q FOC: 7-2Q-1 = 0 => Q m = 3; P m = 4 w/2 plants: q 1 = q 2 = 1.5q 1 = q 2 = 3  1 =  2 = 4.5    =  2 = 0

Cartel Enforcement What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)? Promise: I’ll charge P m = 4, if you do. Threat: I’ll charge P b = 1, forever, if you deviate. 4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 = 4.5/(1-  ) 4.5 … 4.5 … 4.5 … 9 … 0 … 0 … 0 If  is sufficiently high, the threat will be credible, and the pair of trigger strategies is a Nash equilibrium.  * = 0.5 Trigger Strategy Current gain from deviation = 4.5 Future gain from cooperation =  (4.5)/(1-  )

Next Time UNIT IV: INFORMATION & WELFARE 7/26Decision on Under Uncertainty Pindyck & Rubenfeld, Ch. 5. Besanko, Ch. 15