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Imperfect Competition: A Game-Theoretic Approach

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Chapter 13 Imperfect Competition: A Game-Theoretic Approach-A non-passive environment, unlike PC and Monopoly 13-* – PowerPoint PPT presentation

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Title: Imperfect Competition: A Game-Theoretic Approach


1
Chapter 13
  • Imperfect CompetitionA 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 Hotellings 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
Prisoners Dilemma
  • Dominant strategy--- the strategy in a game that
    produces better results irrespective of the
    strategy chosen by ones opponent.
  • Yet when each confesses, each does worse 5 years
    eachthan if each had not confessed 1 year for
    each.

Prisoner Y Prisoner Y
Strategy Confess Dont Confess
Prisoner X Confess 5 years for X 5 years for Y 0 for X 20 for Y
Prisoner X Dont Confess 20 for X 0 for Y 1 year for X 1 year for Y
Profits to Cooperation and Defection in a
Prisoners Dilemma
Firm 1 Firm 1
Strategy Cooperate (P10) Defect (P9)
Firm 2 Cooperate (P10) ?150 ?150 ?199 ?20
Firm 2 Defect (P9) ?10 ?299 ?149.50 ?249.50
  • 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
    ones 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 Firm 1
Strategy Dont Advertise Advertise
Firm 2 Dont Advertise ?1500 ?1400 ?1750 ?2100
Firm 2 Advertise ?1200 ?20 ?1300 ?2200
  • Firm 1s 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 ?1300, ?2200.
  • BUT if Firm I doesnt advertise, Firm 2 does best
    by not advertising as well?1500
  • ?1400.
  • Since Firm 2 doesnt have a dominant strategy,
    its response is determined by (a) likelihood it
    assigns to Firm 1s 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 doesnt advertise, its lowest payoff is
    100 if Firm 1 advertises.
  • But if it chooses to advertise, the lowest payoff
    is 0 if Firm 1 doesnt 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 players 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 not to retaliate Point E
  • If the USSR doesnt 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 gt -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 Sears90 X 0, i.e.
    the existence of a platform has acted a strategic
    deterrence to Xs entry! Note that X doesnt
    enter 90 at C Fig 13.3 100 at CFig.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
    others 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 1s 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
P56 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
56 218 56 -36 20 which is the MC.
13-11
12
Stackelberg Model
Figure 13.7 The Stackelberg Leaders 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 doesnt 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
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