UNIT III: MONOPOLY & OLIGOPOLY Monopoly Oligopoly Strategic Competition 7/28

Market Structure Perfect Comp Oligopoly Monopoly No. of Firms infinite (>)2 1 Output MR = MC = P ??? MR = MC < P ProfitNo ? Yes Efficiency Yes ? ???

Oligopoly We have no general theory of oligopoly. Rather, there are a variety of models, differing in assumptions about strategic behavior and information conditions. All the models feature a tension between: –Collusion: maximize joint profits –Competition: capture a larger share of the pie

Duopoly Models Cournot Duopoly Nash Equilibrium Leader/Follower Model Price Competition

Duopoly Models Cournot Duopoly Nash Equilibrium Stackelberg Duopoly Bertrand Duopoly

Monopoly Cyberstax is the only supplier of Vidiot, a hot new computer game. The market for Vidiot is characterized by the following demand and cost conditions: P = /6QTC = Q

Monopoly P = /6QTC = Q MR = /3QMC = 8 => Q* = 66 P* = 19  = TR – TC = PQ – (40 + 8Q) = (19)(66) – 40 -(8)(66)  = 686 $ 30 P* = 19 Q* = Q MC = 8 MR D

Duopoly Megacorp is thinking of moving into the Vidiot business with a clone which is indistinguishable from the original. It has access to the same production technology, reflected in the following total cost function: TC 2 = q 2 Will Megacorp enter the market? What is its profit maximizing level of output?

Duopoly If Megacorp (Firm 2) takes Cyberstax’s (Firm 1) output as given, its residual demand curve is P = /6Q Q = q 1 + q 2 ; q 1 = 66 P = /6(q 1 + q 2 ) P = /6q 2 $ q 2 = 0 q 1 = Q

Duopoly P = /6q 2 TC 2 = q 2 MR 2 = /3q 2 = MC 2 = 8 => q 2 * = 33 q 1 * = 66 P = 30 – 1/6(q 1 + q 2 ) P* = $13.50  2 = Before entry, P* = 19;  1 = 686 Now,  1 ’ = 323 ow,  C ‘ = 297 q 1 *+q 2 * = Q $ q 2 = 0 MC 2 = 8

Duopoly What will happen now that Cyberstax knows there is a competitor? Will it change its level of output? How will Megacorp respond? Where will this process end?

Cournot Duopoly Reaction curves (or best response curves) show each firm’s profit maximizing level of output as a function of the other firm’s output. q1qm q1qm q 2 q 2 R 1 : q 1 * = f(q 2 )

Cournot Duopoly To find R 1, set MR = MC. Now, MR is a (-) function not only of q 1 but also of q 2 : q q 2 P = /6(q 1 +q 2 ) TR 1 = Pq 1 = [30 - 1/6(q 1 +q 2 )]q 1 = 30q 1 - 1/6q /6q 2 q 1 MR 1 = 30 -1/3q 1 - 1/6q 2 = MC = 8 R 1 : q 1 * = 66 – 1/2q 2

Cournot Duopoly The outcome (q 1 *, q 2 *) is an equilibrium in the following sense: neither firm can increase its profits by changing its behavior unilaterally. q 1 q 1 * = 44 q 2 * = 44q 2 R 2 : q 2 * = /2q 1 R 1 : q 1 * = /2q 2 For the case of identical firms

Nash Equilibrium q 1 q 1 * = 44 q 2 * = 44q 2 R 2 : q 2 * = /2q 1 R 1 : q 1 * = /2q 2 For the case of identical firms A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *.

Nash Equilibrium q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 1 reduces its output while Firm 2 continues to produce q 2 *, the price rises and Firm 2’s profits increase. A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *.

Nash Equilibrium q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 2 reduces its output while Firm 1 continues to produce q 1 *, the price rises and Firm 1’s profits increase. A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *.

Nash Equilibrium q1q1* q1q1* q 2 * q 2 Is this the best they can do? If they can agree to restrict output, there are a range of outcomes to the SW that make both firms better off. A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *.

Stackelberg Duopoly Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower. q1 q1 q 2 R2R2 Firm 1 gets to search along Firm 2’s reaction curve to find the point that maximizes Firm 1’s profits.  1 = ?

Stackelberg Duopoly Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower. q 2 * = 33 q 2 R2R2 q 1 q 1 * = 66 MR 1 = MC 1 TR 1 = Pq 1 = [30-1/6(q 1 +q 2 *)]q 1 Find q 2 * from R 2 : q 2 * = /2q 1 = [30-1/6(q /2q 1 )]q 1 = 30q 1 -1/6q q 1 +1/12q 1 2 MR 1 =19 -1/6q 1 = MC 1 = 8 q 1 * = 66; q 2 * = 33

Stackelberg Duopoly Firm 1 is the dominant firm, or Leader, (e.g., GM) and moves first. Firm 2 is the subordinate firm, or Follower. q 2 * = 33 q 2 Firm 1 has a first mover advantage: by committing itself to produce q 1, it constraints Firm 2’s output decision. Firm 1 can employ excess capacity to deter entry by a potential rival. R2R2 q 1 q 1 * = 66

Bertrand Duopoly Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg). P q 1 If P 1 > P 2 => q 1 = 0 If P 1 = P 2 => q 1 = q 2 = ½ Q If P 1 q 2 = 0 Assumptions:

Bertrand Duopoly Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg). PP2 PP2 q 1 If P 1 > P 2 => q 1 = 0 If P 1 = P 2 => q 1 = q 2 = ½ Q If P 1 q 2 = 0 d1d1 Assumptions:

Bertrand Duopoly Under Bertrand duopoly, firms compete on the basis of price, not quantity (as in Cournot and Stackelberg). PP2 PP2 q 1 Eventually, price will be competed down to the perfect competition level. Not very interesting model (so far). d1d1

Duopoly Models If we compare these results, we see that qualitatively different outcomes arise out of the finer-grained assumptions of the models: P c Q Cournot Stackelberg Bertrand P = /6Q TC = q

Duopoly Models If we compare these results, we see that qualitatively different outcomes arise out of the finer-grained assumptions of the models: Cournot Stackelberg Bertrand Q c < Q s < Q b P c > P s > P b  1s >  1c >  1b  2c >  2s >  2b P P c P s P b =P pc c Q c Q s Q b = Q pc Q

Duopoly Models Summary Oligopolistic markets are underdetermined by theory. Outcomes depend upon specific assumptions about strategic behavior. Nash Equilibrium is strategically stable or self-enforcing, b/c no single firm can increase its profits by deviating. In general, we observe a tension between –Collusion: maximize joint profits –Competition: capture a larger share of the pie

Game Theory Game Trees and Matrices Games of Chance v. Strategy The Prisoner’s Dilemma Dominance Reasoning Best Response and Nash Equilibrium Mixed Strategies

Games of Chance Buy Don’t Buy (1000) (-1) (0) (0) Player 1 Chance You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1. What would you do?

Games of Chance Buy Don’t Buy (1000) (-1) (0) (0) Player 1 Chance You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1. The chance of your number being chosen is independent of your decision to buy the ticket.

Games of Strategy Buy Don’t Buy (1000,-1000) (-1,1) (0,0) (0,0) Player 1 Player 2 Player 2 chooses the winning number. What are Player 2’s payoffs?

Games of Strategy Advertise Don’t Advertise A D A D (10,5) (15,0) (6,8) (20,2) Firm 1 Firm 2 Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice. How should the game be played? Profits are in ( )

Games of Strategy Advertise Don’t Advertise A D A D (10,5) (15,0) (6,8) (20,2) Firm 1 Firm 2 Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice. How should the game be played? Backwards-induction

Games of Strategy Advertise Don’t Advertise A D A D (10,5) (15,0) (6,8) (20,2) Firm 1 Firm 2 Duopolists deciding to advertise. The 2 firms move simultaneously. (Firm 2 does not see Firm 1’s choice.) Imperfect Information. Information set

Matrix Games Advertise Don’t Advertise A D A D (10,5) (15,0) (6,8) (20,2) Firm 1 Firm 2 10, 5 15, 0 6, 8 20, 2 A D A D

Matrix Games Advertise Don’t Advertise A D A D (10,5) (15,0) (6,8) (20,2) Firm 1 Firm 2 10, 5 15, 0 6, 8 20, 2 A D A D

Games of Strategy Games of strategy require at least two players. Players choose strategies and get payoffs. Chance is not a player! In games of chance, uncertainty is probabilistic, random, subject to statistical regularities. In games of strategy, uncertainty is not random; rather it results from the choice of another strategic actor. Thus, game theory is to games of strategy as probability theory is to games of chance.

A Brief History of Game Theory Minimax Theorem 1928 Theory of Games & Economic Behavior 1944 Nash Equilibrium 1950 Prisoner’s Dilemma 1950 The Evolution of Cooperation 1984 Nobel Prize: Harsanyi, Selten & Nash 1994

The Prisoner’s Dilemma In years in jail Player 2 ConfessDon’t Confess Player 1 Don’t -10, -100, , 0 -1, -1 The pair of dominant strategies ( Confess, Confess ) is a Nash Eq. GAME 1.

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?

Dominance Definition Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T 1 T 2 T 3 0,2 4,3 3,3 4,0 5,4 5,6 3,5 3,5 2,3 0,2 4,3 3,3 4,0 5,4 5,3 3,5 3,5 2,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Sure Thing Principle: If you have a dominant strategy, use it!

Dominance Definition Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T 1 T 2 T 3 0,2 4,3 3,3 4,0 5,4 5,6 3,5 3,5 2,3 0,2 4,3 3,3 4,0 5,4 5,3 3,5 3,5 2,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Sure Thing Principle: If you have a dominant strategy, use it! (S 2,T 3 ) (S 2,T 2 )

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.

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) Button-Button 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) EP 1 (R) = 2(q) – 1(1-q) (Equalizers) q* = 5/9and for Player 2: p* = 1/3 EP 2 (L) = -2(p) + 2(1-p) EP 2 (R) = 4(p) – 1(1-p) L R LRLR (p) (1-p) (q) (1-q) NE = {(1/3), (5/9)} GAME 2.

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

2x2 Game T 1 T 2 1. Prisoner’s Dilemma 2. Button – Button 3. Stag Hunt 4. Chicken 5. Battle of Sexes S 1 S 2 x 1,x 2 w 1, w 2 z 1,z 2 y 1, y 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); (5/8, 3/8)} (0,0) (5/8,3/8) (1,1) Battle of the Sexes

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).

Next Time 8/4 Repeated Games Decision under Uncertainty Pindyck, Chs 5, 13. Besanko, Chs 14-16