Imperfect Competition: A Game-Theoretic Approach

1. Chapter 13 Imperfect Competition:A Game-Theoretic Approach -A non-passive environment, unlike PC and Monopoly 13-1

2. Chapter Outline • An Introduction to the Theory of Games • Some Specific Oligopoly Models • Competition When There are Increasing Returns to Scale • Monopolistic Competition • A Spatial Interpretation of Monopolistic Competition • Historical Note: Hotelling’s Hot Dog Vendors • Consumer Preferences and Advertising 13-2

3. Prisoner's Dilemma--difficulty of collusion even with few producers • Two prisoners are held in separate cells for a serious crime that they did in fact commit. The prosecutor has only enough hard evidence to convict them of a minor offense, for which the penalty is a year in jail. • Each prisoner is told that if one confesses while the other remains silent, the confessor will go scot-free while the other spends 20 years in prison. • If both confess, they will get an intermediate sentence 5 years. 13-3

4. Prisoner’s Dilemma • Dominant strategy--- the strategy in a game that produces better results irrespective of the strategy chosen by one’s opponent. • Yet when each confesses, each does worse {5 years each}than if each had not confessed {1 year for each}. Profits to Cooperation and Defection in a Prisoner’s Dilemma • The dominant strategy is for each firm to defect, for doing so, it earns higher profit no matter which option its rival chooses. • Yet when both defect, each earns marginally less {$49.50 each} than when each cooperates {$50 each} • Dominant strategy- the strategy in a game that produces better • results irrespective of the strategy chosen by one’s opponent. • Nash equilibrium:the combination of strategies in a game such that neither player has any incentive to change strategies given the strategy of his opponent. • A Nash equilibrium does not require both players to have a dominant strategy

5. A Game in which Firm 2 has no Dominant Strategy – a Maximin Approach • Firm 1’s dominant strategy is to advertise regardless of what Firm 2 does. • Firm 2 has no dominant strategy. Thus, if Firm 1 advertises, Firm 2 does best by advertising as well {Π1=$300, Π2=$200}. • BUT if Firm I doesn’t advertise, Firm 2 does best by not advertising as well{Π1=$500 • Π1=$400}. • Since Firm 2 doesn’t have a dominant strategy, its response is determined by (a) likelihood it assigns to Firm 1’s choices and (b) how its own payoffs are affected by (a). • One approach is for Firm 2 to take the maximin approach – choose the option that maximizes its lowest possible value of its own payoff. • If Firm 2 doesn’t advertise, its lowest payoff is $100 if Firm 1 advertises. • But if it chooses to advertise, the lowest payoff is $0 if Firm 1 doesn’t advertise. • Thus, if it follows a maximin strategy, Firm 2 will choose not to advertise. • Maximin strategy--choosing the option that makes the lowest payoff one can receive as large as possible

6. Tit-for-Tat Tit-for-tat strategy- The first time you interact with someone, you cooperate. In each subsequent interaction you simply do what that person did in the previous interaction. • Thus, if your partner defected on your first interaction, you would then defect on your next interaction with her. • If she then cooperates, your move next time will be to cooperate as well. • Requirement: there not be a known, fixed number of future interactions. Sequential Games • Sequential game: one player moves first, and the other is then able to choose his strategy with full knowledge of the first player’s choice. • Example - United States and the former Soviet Union (USSR) during much of the Cold War. • Strategic entry deterrence – they change potential rivals’ expectations about how the firm will respond when its market position is threatened. 13-6

7. Figure 13.1: Nuclear Deterrenceas a Sequential Game Nash Equilibrium if the USSR does attack initially. 1st move Points B & C are US response that depend on Soviet initial action “Doomsday” devise eliminates the bottom part. Nash Equilibrium if the USSR does not attack initially. • If the USSR attacked, the best response of the US is notto retaliate {Point E} • If the USSR doesn’t attack, the best US response is not to attack {Point G} • Since the USSR gets a higher payoff from attacking {Point E} than not attacking {Point G}, the US assumed (like the USSR) that the USSR would attack – reason to the Cold War built-up. • However, if the US maximizes its payoff, its threat to retaliate {Point D} is not credible since -50 > -100. 13-7

8. Nash Equilibrium if X knows Sears payoff Figure 13.2: The Decision to Buildthe Tallest Building Figure 13.3: Strategic Entry Deterrence • Assume that at construction, Sears had the option to build a platform that allows to create it to build a higher building if it so chose later. Cost of this is 10 units but the presence of a platform reduces building higher floors by 20 units. • Given this provision, X (a rival to Sears) knows that Sears can add floors if X enters. If X enters and Sears builds, the outcome is D {Sears =40; X =-50}. But if X enters and Sears does not build, the outcome is E {Sears = 30; X =60}. Problem – X is not sure of outcome E. • The Nash Equilibrium is C {Sears=90; X =0}, i.e. the existence of a platform has acted a strategic deterrence to X’s entry! Note that X doesn’t enter: 90 at C {Fig 13.3}= 100 at C{Fig.13.2} +(-10) 13-8

9. Figure 13.4: The Profit-Maximizing Cournot Duopolist The portion to the right of the vertical at Q1 is the demand curve for Firm , i.e. Residual demand • The Cournot Model--oligopoly model in which each firm assumes that rivals will continue producing their current output levels (assumes a naïve rival – not a convincing assumption) • Main assumption - each duopolist treats the other’s quantity as a fixed number, one that will not respond to its own production decisions. • Reaction function- a curve that tells the profit-maximizing level of output for one oligopolist for each amount supplied by another. • Suppose Market demand is: P = a – b(Q1 + Q2) with MC = 0; • Firm 1’s demand: P1=(a – bQ2) – bQ1 implies TR1 = P1Q1 = Q1(a – bQ2) – bQ12 • MR1 =dTR1/dQ1 = a – bQ2 – 2bQ1 and set MR1 = MC and solve for Q1 • a – bQ2 – 2bQ1 = 0 or Q1 = (a - bQ2)/2b = RN1 function for Firm 1. Similarly, Q2 = (a – bQ1)/2b =RN2 since these are symmetric. 13-9

10. Figure 13.5: Reaction Functionsfor the Cournot Duopolists 13-10

11. Figure 13.6: Deriving the Reaction Functions for Specific Duopolists P=56 –2Q and MC =20 P1 = 56 – 2Q1- 2Q2; TR1 = 56Q1 – 2Q12 – 2Q1Q2; MR1 = 56 – 4Q1 – 2Q2 Set MR1 = MC and solve for Q1 = 9 – ½ Q2. Similarly for Q2= 9 – ½ Q1 Q1 = 9 – ½ Q2 = 9 -½(9 – ½ Q1)  3Q1 = 18 or Q1 = 6 = Q2. Residual Demand for Firm 1 The Bertrand Model Bertrand model - oligopoly model in which each firm assumes that rivals will continue charging their current prices (again – a naïve assumption about pricing behavior of a rival) Example: Duopolist demand function: P =56 -2Q, MC =20. Set P =MC but note that industry output and price: S =MC and D So 20 = 56 – 2Q so that Q = 18 and since they share the market equally, each firm produces 9 units. Naturally, P = 56 – 2*18 = 56 -36 = 20 which is the MC. 13-11

12. Stackelberg Model Figure 13.7: The Stackelberg Leader’s Demand and Marginal Revenue Curves Figure 13.8: The Stackelberg Equilibrium 13-12

13. Comparison Of Outcomes Figure 13.9: Comparing Equilibrium Price and Quantity 13-13

14. Competition When There Are Increasing Returns To Scale • In markets for privately sold goods, buyers are often too numerous to organize themselves to act collectively • Where it is impractical for buyers to organize direct collective action, it may nonetheless be possible for private agents to accomplish much the same objective on their behalf. Figure 13.10: Sharing a Market with Increasing Returns to Scale • With 2 firms in the market, costs are higher than with 1 firm (AC’ versus AC0). • Despite lower costs for the natural monopolist (AC0), it doesn’t follow that the incumbent will successfully prevent entry or drive-off potential entrants into the market. • Reason- problem of collective action – many consumers are too difficult to organize to boycott the natural monopolist that charges higher prices – Mancur Olson: The Logic of Collective Action (1965). 13-14