Industrial Organization or Imperfect Competition Limit Pricing

1
Industrial Organization or Imperfect
Competition Limit Pricing

• Univ. Prof. dr. Maarten Janssen
• University of Vienna
• Summer semester 2008
• Week 10 (May 27)

2
Limit Pricing an asymmetric information story
about entry deterrence

• Incumbent has private information about cost
• Cost can be either high or low
• Potential entrant does not know cost structure
  incumbent, but has some beliefs about it
• If incumbents cost is really low, entrant cannot
  compete and make positive profits if cost is
  high, entrant does make profit by entering
• Can incumbent prevent entry in all or some cases?

3
Limit Pricing basic idea

• A high-cost incumbent has an incentive to pretend
  to be low-cost if by so doing it can deter entry
• The entrant recognizes this incentive to
  masquerade as a low cost firm
• What can the entrant infer from observing low
  price knowing this may be a deception?
• In turn this depends on the probability that
  observing a low-price means that the incumbent is
  a low-cost firm
• First, need to develop game theory with private
  information

4
TOY GAME

• Student considers applying for a PhD program
• Student knows his own qualities (good, bad)
• University has to decide whether or not to accept
  student if (s)he applies
• University does not know quality of student, but
  believes that probability is 50-50
• Pay-offs student, university depend on whether
  student is good or not in case student is
  accepted.

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Game Structure Toy Game I
2, 2
Accept
U
Apply
-1 , 0
Reject
S
Dont Apply
0,0
GOOD
½
-2, -3
Accept
N
U
½
Apply
-1 , 0
Reject
S
BAD
Dont Apply
0,0
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What is a strategy?

• Rule that tells a player what to choose given the
  information (s)he has
• Action to be taken conditional on information
• Students strategy
• Example Apply if, and only if, I am a good
  student
• Four possible strategies (always apply, never
  apply, apply iff good, apply iff bad)
• Universitys strategy
• Just two possible accept, reject

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Equilibrium definition

• General Nash one strategy for each player such
  that no player has incentive to deviate given
  strategies of other players
• Here, updating of information possible
• Players may learn more about information other
  players have based on actions they take
• University may learn information about quality
  student given whether student applies.
• Update information using Bayes rule whenever
  possible
• P(A/B) P(B/A)P(A) P(B)

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Equilibria in Toy Game I

• Suppose student chooses apply iff good, what is
  optimal reaction of university
• Bayesian updating P(good/apply)
  P(apply/good)P(good)/P(apply) 1.½ / ½ 1
• Thus, P(bad/apply) 0
• University will rationally accept
• Given university will accept, above proposed
  strategy student is optimal
• Separating (or revealing) equilibrium
• Different types of students choose different
  actions
• Is there another (pooling) equilibrium?
• Student never applies, university rejects

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Game Structure Toy Game II
U
10
Equilibria in Toy Game II

• Separating equilibrium unaffected
• Student chooses apply iff good
• University chooses accept
• Pooling equilibrium unaffected
• Student never applies, university rejects
• But, it is strange now
• University is always better off to accept
  students
• Reject is what seems to be an incredible threat
• How to get rid of this incredible threat? Subgame
  perfection?
• What is value of P(good/apply)?
• Out-of-equilibrium beliefs (when certain
  information sets are not on the equilibrium path,
  Bayes rule cannot be applied)
• Perfect Bayes-Nash equilibrium impose as an
  additional restriction that given certain
  arbitrary off-the-equilibrium beliefs, strategies
  should be optimal

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Perfect Bayes-Nash Equilibria in Toy Game II

• Separating equilibrium unaffected
• There are no out-of-equilibrium beliefs
• Pooling equilibrium affected
• P(good/apply) should be defined. Let us say it
  equals µ, with 0 lt µ lt 1.
• For any value of µ, accept is the optimal
  strategy
• Therefore, in any perfect Bayes-Nash equilibrium,
  where students do not apply, university should
  accept.
• Given university accepts, good student will
  apply.
• No Pooling equilibrium exists

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Game Structure Toy Game III
U
13
Equilibria in Toy Game III

• Separating equilibrium unaffected
• Is a perfect Bayes-Nash equilibrium
• Pooling equilibrium Is a perfect Bayes-Nash
  equilibrium
• Student never applies, university rejects
• for values of µ with 2µ 3(1-µ) 0,
  P(good/apply) 5/6
• But, it (again) is strange
• Bad student is always better off not applying
• Why should the university be afraid of bad
  students applying?
• Domination-based beliefs if one type of player
  never has incentives to apply, other type may
  have an incentive to deviate (for certain
  reactions of opponent), then out-of-equilibrium
  beliefs should be such that all probability mass
  is given to player who may have incentive to
  deviate.
• Domination-based beliefs restrict the set of
  reasonable out-of-equilibrium beliefs
• Here, domination-based beliefs requires that µ
  1.
• Given µ 1, university will accept
• Pooling equilibrium does not satisfy
  domination-based beliefs requirement