0 MBA 299 – Section Notes 4/25/03 Haas School of Business, UC Berkeley Rawley

1 AGENDA Stackelberg duopoly Repeated games Problem set #3 and #4

2 STACKELBERG DUOPOLOY (I) Game Set-up Firm 1 moves first and chooses q 1 ≥0 Firm 2 observes q 1 and chooses q 2 ≥0 Payoffs are  (q i,q j )=q i [P(Q)-c] P(Q) = a - Q Q = q 1 + q 2 Solve by backwards induction

3 STACKELBERG DUOPOLOY (II) Equilibrium Outcomes Firm 2 max  2 (q 1,q 2 ) = max q 2 [a-q 1 -q 2 -c] q2≥0 q 2 =R 2 (q 1 ) = [a - q 1 - c]/2 (assuming q 1 < a -c) Firm 1 anticipates Firm 2’s move max  1 (q 1,R 2 (q 1 )) = max q 1 [a-q 1 -R 2 (q 1 )-c] q1≥0 =max q 1 [a-q 1 -c]/2 q 1 * = [a-c]/2 andR 2 (q 1 *) =q 2 * = [a-c]/4

4 STACKELBERG DUOPOLOY (III) Analysis (I) Recall the Cornout duopoly outcome was [a-c]/3 for both firms so total output is higher in the Stackelberg game... and therefore prices are lower –Cornout total output = 2[a-c]/3 –Stackelberg total output = 3[a-c]/4 Firm 1 is better off in the Stackelberg game but Firm 2 is worse off –Firm 1 could have chosen the Cornout output level but did not therefore Firm 1 must be better off –Firm 2 produces less at lower prices so must be worse off

5 STACKELBERG DUOPOLOY (IV) Analysis (II) Observe the role of information –Firm 1 knows that Firm 2 will optimize it’s output based on what Firm 1 does –Firm 2 knows that Firm 1 knows that Firm 2 will optimize it’s output based on what Firm 1 does Order matters too –If Firm 1 went after Firm 2 {and Firm 2 chose [a- c]/4}, Firm 1’s optimal output would be 3[a-c]/8... so Firm 2’s choice would not be an equilibrium outcome... and we would end up back at the Cournot output level

6 STACKELBERG DUOPOLOY (V) Intuition Firm 1 is the dominant firm, in fact it acts like a monopolist, while Firm 2 takes the scraps off the table (acting as a monopolist in the “remaining” market) Firm 1 is the first mover in a market while Firm 2 is the late-comer Note the difference in equilibrium in a sequential move game versus a simultaneous move game Do you observe this effect in the CSG game? If you are entering a market already “held” by an incumbent do you assume they will re-capture their prior period sales?

7 REPEATED GAMES: CREDIBLE THREATS An Example Credible threats about the future can influence current behavior Observe the SPNE outcome of a the game below played twice without discounting (we will cover discounting in infinite games) 1,11,15,00,0 0,54,40,0 3,33,3 U M D LCR The players anticipate that the round 2 outcome will be a NE... But will it be U,L or D,R? How will the round 2 equilibrium outcome influence stage 1 strategies?

8... AN EXAMPLE OF A FINITELY REPEATED GAME 1,11,15,00,0 0,54,40,0 3,33,3 U M D LCR If each player believes the other player will punish them with the lower NE in stage 2 if they fail to play C or M in stage 1 (respectively) then (M,C), (D,R) is the SPNE to this 2-stage game Ex. what will happen if player 2 cheats and plays L in response to 1’s M in stage 1? Player 1 will play U in stage 2. Payoffs: 2 gets 5+1 = 6 vs. 4+3 = 7 in the SPNE outcome... so 2 has an incentive not to cheat in stage 1

9 COMMENTS ON REPEATED GAMES (I) Repetition of a NE of the stage game of a repeated game is always a SPNE of the repeated game A credible threat specifies that of a set of Nash Equilibria play the one such that the other player(s) is (are) worse off than in their preferred outcome (see previous ex.) If there are credible threats there are SPNE that differ from stage game NE outcomes in some rounds... but if there is only one NE in the stage game and the end of the game is known with certainty the unraveling results holds where the only SPNE is repeated play of the stage game NE Remember, only SPNE strategies are credible... think about this... recall Odysseus vs. the Sirens

10 COMMENTS ON REPEATED GAMES (II) Does the last period have to be a NE? Yes Cooperation can be achieved in all but the last round of a repeated game... this means the players may not be playing a stage game NE in every period except the last... still at every subgame (the current stage game plus all the future stage games) will be in NE and therefore the whole game is in SPNE Think about stage games, subgames, NE, SPNE and credible threats

11 DISCOUNTING IN INFINITELY REPEATED GAMES context: infinite “doom” trigger strategies - cooperate unless the other player defects, then defect every round forever discount factor is  = 1/[1+r] < 1 Discounting in infinitely repeated games S =  x p = x t + x t+1 + x t xS =  x p =x t+1 + x t+2 + x t S - xS = (1-x)S= x t S = x t /(1-x) if x =  and p=0 then S = 1/(1-  ) and p=1 then S =  /(1-  ) p=t infinity p=t

12 INFINITE PD WITH DISCOUNTING 3,33,36,2 2,64,44,4 D C DC Proposed strategy for i play C unless j plays D then play D forever When is this strategy a SPNE? 4[1/(1-  )] > 6 + 3[  /(1-  )] if  ≥2/3 Therefore, cooperating in every subgame is SPNE if  is large enough (players don’t discount too much)

13 THE FOLK THEOREM: MULTIPLE SPNE As long as gamma is big enough (players are sufficiently patient) any stage game outcome with payoffs in excess of the stage game Nash Equilibrium (singular) outcome can be supported as a cooperative SPNE Contrast this with the unraveling result This is a nice result because it tells us we can support cooperation with trigger strategies in long-period repeated games where players are patient as long as neither player knows when the game will end Problem: An embarrassment of riches

14 TACIT COLLUSION The repeated Bertrand game as the “oligopoly game” –Rather than sustaining p=MC why not sustain p=p M *? –strategy for i charge p=p M unless j charges a lower price, then charge p=c forever the punishment is a stage game NE so it is SP (see Odysseus) check the math (p.18 of Professor Hermalin’s notes), if  ≥ 1/2 the strategy is SPNE –Note that the minimum  increases as the number of players increase Tacit collusion = the oligopoly game = f(N,  ) –Impatient players do not make good tacit co-conspirators

15 PRICE WARS Punishment strategies do not have to be doom trigger strategies –Punishments don’t have to last forever to be effective... however, they do need to be deep to be effective if they are short Price wars are short, deep punishments (note: discounting still plays a role) When only imperfect monitoring of rivals exists trigger strategies are the only credible threat rivals will respect... they need not be doom triggers though

16 PROBLEM SET ON COURNOT, TACIT COLLUSION AND ENTRY DETERRENCE #3 (I) (a)Compare the payoff to undercutting in the summer (once only) to colluding forever NPV undercut = 10D(v-c) NPV collude = 10D/4(v-c) +.9D/4(v-c) D/4(v-c) D/4(v-c) D/4(v-c)... = 10D/4(v-c)[1/( )] + D/4(v-c)[.9/(1-.9)-.9 4 /( )] =10D(v-c)* (b)  (collude at p summer ) = 10D(p s -c)/4 = 10D[(v+9c)/10-c]/4 =D(v-c)/4 See problem #2 for how tacit collusion can be obtained at this level

17 PROBLEM SET ON COURNOT, TACIT COLLUSION AND ENTRY DETERRENCE #3 (II) (c) This explains seasonal sales as a mechanism for sustaining tacit collusion when demand is high... These are essentially ways to release tension in the tacit colluding process (d) 10D(p s – c) ≤ NPV collude NPV collude = 10D/4(p s ) +.9D/4(100) D/4(100) D/4(100) D/4(p s )... = 10D/4(p s )[1/( )] + D/4(100)[.9/(1-.9)-.9 4 /( )] =10D(v-c)* So 40p s ≤ 10p s x x => p s = 64.95

18 PROBLEM SET ON COURNOT, TACIT COLLUSION AND ENTRY DETERRENCE #4 (a)  1 = (v-c)*D (b)  2 = cD-K => if c>K/D player 2 will enter (c)E[c]*D – K, where E[c] = 3v/4*1/2 = 3v/8 since K<3v/8 player 2 should enter (d) If player 1 plays K/D in the first period in anticipation of keeping player 2 out she will earn (K/D –c)D + (v-c)D If she does not want to keep player 2 out she will price at v and earn (v-c)D Therefore, entry deterrence only makes sense if K/D>c (e) Limit pricing can work but only if costs are low enough

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20 MBA 299 – Section Notes 4/18/03 Haas School of Business, UC Berkeley Rawley

21 AGENDA Administrative CSG concepts –Discussion of demand estimation –Cournot equilibrium Multiple players Different costs –Firm-specific demand for differentiated products and how that can look very different than the demand faced by a monopolist Problem set on Cournot, Tacit Collusion and Entry Deterrence

22 QUESTION 1: COURNOT EQUILIBRIUM Q(p) = 2,000, ,000p MC 1 = MC 2 = 10 a.) P(Q) = 40 - Q/50,000 => q1 = q2 = (40-10)/3*50,000 = 500,000 b.)  i = (p-c)*q i = (40-1,000,000/50,000-10)*500,000 = $5M c.) Setting MR = MC => a – 2bQ = c Q* = (a-c)/2b =>  m = {a – b[(a-c)/2b]}*(a-c)/2b = (a-c) 2 /4b =>  m = (40-10)/4*50,000 =$11.25M

23 QUESTION 2: REPEATED GAMES AND TACIT COLLUSION Bertrand model set-up with four firms and  = 0.9  (cooperate) = D*(v – c)/4  (defect) = D*(v – c)  (punishment) = 0 Colluding is superior iff  {D*(v-c)/4*  t } = D*(v-c)/4*[1/(1-.9)]  D*(v-c) + 0 since 10/4  1 this is true, hence cooperation/collusion is sustainable