4820–9
Information
Geir B.
Asheim
Information and strategic behavior
Introduction
Static
competition
4820–9
Dynamic
competition
Geir B. Asheim
Department of Economics, University of Oslo
ECON4820
Spring 2010
Last modified: 2010.03.16
Asymmetric information . . .
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
. . . gives opportunity for signaling
and forming a reputation
Such firm behavior is strategic
if it changes the competitors’ beliefs
about the private information of the firm.
Outline
(1) Static competition under asymmetric information
(a) Bayesian equilibrium
(b) A simple model of price competition
(2) Dynamic competition under asymmetric information
(a) Perfect Bayesian equilibrium
(b) The Milgrom-Roberts (Ecma 1982) model of limit pricing
(c) Multi-market reputation
Equilibrium concept for
static games of asymmetric information
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Bayesian
equilibrium
Price
competition
Dynamic
competition
Definition (Bayesian equilibrium)
A Bayesian equilibrium is a profile of type-contingent strategies
{ai∗ (ti )}ni=1 such that each player maximizes his expected utility
contigent on his type and taking the other players’
type-contingent strategies as given: ai = ai∗ (ti ) maximizes
pi (t−i |ti )Πi (a1∗ (t1 ), . . . , ai , . . . , an∗ (tn ), t1 , . . . , ti , . . . , tn )
t−i
Example: Firm 2 does not know the cost of firm 1. But firm 2
has a probability distribution over what costs firm 1 may have.
And firm 1 knows what probability distribution firm 2 has.
How are these interactive beliefs formed?
A simple model of price competition (1)
4820–9
Information
Geir B.
Asheim
Structure: (1) Prices set (2) Demand determines quantities
Firms 1 and 2 have constant unit cost c1 and c2 and set prices
p1 and p2 simultaneously. Sales are given by
Introduction
Static
competition
Bayesian
equilibrium
Price
competition
Dynamic
competition
Di (pi , pj ) = a − bpi + dpj
which is the demand function for firm i.
max(pi − ci )Di (pi , pj )
pi
Firm 1’s unit cost c1 is c L with probability x and c H with
probability 1 − x, where c L < c H . Write c e ≡ xc L + (1 − x)c H .
Firm 2’s unit cost c2 is c.
Firm 1 knows its own cost; firm 2 believes that firm 1’s cost is
c L with probability x and c H with probability 1 − x.
A simple model of price competition (2)
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Bayesian
equilibrium
Price
competition
Dynamic
competition
Results:
The low cost type of firm 1 charges a higher price than it
would have done if information were symmetric. Why?
The high cost type of firm 1 charges a lower price than it
would have done if information were symmetric. Why?
Suppose firm 1 could report verifiable information
about c1 costlessly before the firms compete.
Would such information be spread
if firm 1 wants to accommodate entry?
Would such information be spread
if firm 1 wants to deter entry?
Equilibrium concept for
dynamic games of asymmetric information
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
Definition (Perfect Bayesian equilibrium)
A Perfect Bayesian equilibrium is defined by the following two
properties:
(1) Given beliefs and the strategies of the other players,
each player’s strategy must satisfy sequential rationality.
(2) Beliefs are obtained from strategies and observed actions
using Bayes’ rule whenever possible.
Problems:
(i) Many equilibria (How to coordinate? How to predict?)
(ii) How are interactive belief about player types formed?
Limit pricing
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
Can an incumbent firm deter entry
by setting a low pre-entry price?
(Limit pricing, Bain, 1949)
Only if a low price before entry signals a low price after entry.
Before: A large K is an aggressive commitment which has
the unintentional affect of leading to a low price.
Now: A low p signals a harsh post-entry environment.
Milgrom-Roberts (Emca 1982) model of limit pricing
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
Model:
Two periods
Two firms
Structure:
t =1
Firm 1 is the incumbent,
has private information about own cost,
and sets price (p1 ) in period 1.
t =2
Firm 2 observes p1 and makes an entry decision.
After entry, firm 2 observes firm 1’s true cost
before the firms compete in period 2.
Assumptions
4820–9
Information
Geir B.
Asheim
x: Probability that firm 1’s cost = L
1 − x: Probability that firm 1’s cost = H
Introduction
Static
competition
M1t (p1 ): Monopoly profit for firm 1, given p1 and cost = t
Dynamic
competition
M1t = max M1t (p1 )
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
t = arg max M t (p )
pm
1 1
D1t : Duopoly profit for firm 1, given cost = t
D2t : Duopoly profit for firm 2, given cost = t
D2H > 0 > D2L
M1t > D1t , t = L, H
Separating equilibrium
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
Result
The incumbent manipulates his price, but the entrant is not
fooled. He learns the incumbent’s cost perfectly. Entry occurs
exactly when it would have occurred under symmetric info.
Result
Even though the incumbent doesn’t fool the entrant, he engages
in limit pricing: the low-cost type would be mistaken for the
high-cost type if it did not sacrifice short-run profits to signal.
Result
Social welfare is higher than under symmetric information. 2nd
period welfare is not affected as entry is not affected. 1st period
welfare is increased because the low-cost type reduces its price.
Pooling equilibrium
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
No pooling equilibrium if xD2L + (1 − x)D2H > 0.
Assume xD2L + (1 − x)D2H < 0.
Result
The incumbent manipulates the price without revealing
information. The entrant does not enter when appropriate.
Result
The low-cost type charges its monopoly price. The high-cost
engages in limit pricing to deter entry.
Result
Ambiguous welfare consequences. Higher in 1st period as type H
sets lower price. Lower in 2nd period due to less entry.
Multi-market reputation
Resolving the “chain store paradox”
4820–9
Information
Geir B.
Asheim
Introduction
Static
competition
A firm that is established in many markets will wish to fight
entrants aggressively (even at a loss in each market) to
improve its reputation as a tough competitor in other
markets. This will deter further entry.
Dynamic
competition
Perfect
Bayesian
equilibrium
Limit pricing
Separating
equilibrium
Pooling
equilibrium
Multi-market
reputation
Not a subgame-perfect equilibrium, since the incumbent will
not fight in the ultimate market, and hence, not in the
penultimate market etc. (Chain store paradox)
The chain store paradox is resolved by introducing
asymmetric information (Kreps & Wilson, Milgrom &
Roberts, JET 1982)