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MBA 299 – Section Notes

4/25/03

Haas School of Business, UC Berkeley

Rawley

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AGENDA

Stackelberg duopoly

Repeated games

Problem set #3 and #4

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STACKELBERG DUOPOLOY (I)Game Set-up

Firm 1 moves first and chooses q1≥0

Firm 2 observes q1 and chooses q2≥0

Payoffs are (qi,qj)=qi[P(Q)-c]

P(Q) = a - Q

Q = q1 + q2

Solve by backwards induction

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STACKELBERG DUOPOLOY (II)Equilibrium Outcomes

Firm 2max 2(q1,q2) = max q2[a-q1-q2-c]q2≥0 q2≥0

q2=R2(q1) = [a - q1 - c]/2 (assuming q1 < a -c)

Firm 1anticipates Firm 2’s move

max 1(q1,R2(q1)) = max q1[a-q1-R2(q1)-c]q1≥0 =max q1[a-q1-c]/2

q1* = [a-c]/2 and R2(q1*) =q2* = [a-c]/4

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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

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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

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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?

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REPEATED GAMES: CREDIBLE THREATSAn 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)

1,1 5,0 0,0

0,5 4,4 0,0

0,0 0,0 3,3

U

M

D

L C R

The players anticipate that theround 2 outcome will be a NE . . .

But will it be U,L or D,R? Howwill the round 2 equilibrium outcome influence stage 1 strategies?

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. . . AN EXAMPLE OF A FINITELY REPEATED GAME

1,1 5,0 0,0

0,5 4,4 0,0

0,0 0,0 3,3

U

M

D

L C R

If each player believes the otherplayer will punish them with the lower NE in stage 2 if they fail toplay 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

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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

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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

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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 = xp = xt + xt+1 + xt+2 + . . . +

xS = xp =xt+1 + xt+2 + xt+3 + . . . +

S - xS = (1-x)S= xt

S = xt/(1-x)if x =

and p=0 then S = 1/(1- )and p=1 then S = /(1- )

p=t

infinity

infinity

p=t

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INFINITE PD WITH DISCOUNTING

3,3 6,2

2,6 4,4

D

C

D C

Proposed strategy for iplay C unless j plays Dthen 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)

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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

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TACIT COLLUSION

The repeated Bertrand game as the “oligopoly game”– Rather than sustaining p=MC why not sustain p=pM*?– strategy for i

• charge p=pM 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

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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

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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) + .92D/4(v-c) +

.93D/4(v-c) + .94 10D/4(v-c) . . . = 10D/4(v-c)[1/(1-.94)] + D/4(v-c)[.9/(1-.9)-.94/(1-.94)]=10D(v-c)*0.90426

(b) (collude at psummer) = 10D(ps-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

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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(ps – c) ≤ NPV colludeNPV collude = 10D/4(ps) + .9D/4(100) + .92D/4(100) +

.93D/4(100) + .94 10D/4(ps) . . . = 10D/4(ps)[1/(1-.94)] + D/4(100)[.9/(1-.9)-.94/(1-.94)]=10D(v-c)*0.90426

So 40ps ≤ 10ps x 2.908 + 100 x 7.092

=> ps = 64.95

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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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MBA 299 – Section Notes

4/18/03

Haas School of Business, UC Berkeley

Rawley

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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

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QUESTION 1: COURNOT EQUILIBRIUM

Q(p) = 2,000,000 - 50,000p

MC1 = MC2 = 10

a.) P(Q) = 40 - Q/50,000

=> q1 = q2 = (40-10)/3*50,000 = 500,000

b.) i = (p-c)*qi = (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

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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) + 0since 10/4 1 this is true, hence cooperation/collusion is sustainable