Imperfect Competition:

A Game-Theoretic Approach

-A non-passive environment, unlike PC and Monopoly

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Chapter 13

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

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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.

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Prisoner Y Strategy Confess Don’t

ConfessPrisoner X Confess 5 years for

X5 years for Y

0 for X20 for Y

Don’t Confess

20 for X0 for Y

1 year for X1 year for Y

Prisoner’s Dilemma

Firm 1Strategy Cooperat

e(P=$10)

Defect(P=$9)

Firm 2 Cooperate(P=$10)

Π1=$50Π1=$50

Π1=$99Π2=$0

Defect(P=$9)

Π1=$0Π2=$99

Π1=$49.50Π2=$49.50

Profits to Cooperation and Defection in a 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}.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

Firm 1Strategy Don’t

AdvertiseAdvertise

Firm 2 Don’t Advertise

Π1=$500Π1=$400

Π1=$750Π2=$100

Advertise Π1=$200Π2=$0

Π1=$300Π2=$200

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=$500Π1=$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.1.If Firm 2 doesn’t advertise, its lowest payoff is $100 if Firm 1 advertises.2.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

Tit-for-TatTit-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.

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Figure 13.1: Nuclear Deterrenceas a Sequential Game

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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.

Nash Equilibrium if the USSR does attack initially.

If the USSR attacked, the best response of the US is not to 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.

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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)

Figure 13.4: The Profit-Maximizing Cournot Duopolist

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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) – bQ1

2

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.

The portion to the right of the vertical at Q1 is the demand curve for Firm , i.e. Residual demand

Figure 13.5: Reaction Functionsfor the Cournot Duopolists

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Figure 13.6: Deriving the Reaction Functions for Specific Duopolists

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The Bertrand ModelBertrand 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 DSo 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.

P=56 –2Q and MC =20

Residual Demand for Firm 1

P1 = 56 – 2Q1- 2Q2; TR1 = 56Q1 – 2Q12 –

2Q1Q2; MR1 = 56 – 4Q1 – 2Q2Set 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.

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Figure 13.7: The Stackelberg Leader’s Demand and Marginal Revenue Curves

Figure 13.8: The Stackelberg Equilibrium

Stackelberg Model

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Figure 13.9: Comparing Equilibrium Price and Quantity

Comparison Of Outcomes

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.

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Figure 13.10: Sharing a Market with Increasing Returns to ScaleWith 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).