ECO290E: Game Theory
Lecture 4
Applications in Industrial
Organization
Review of Lecture 2
• Outcomes of games, i.e., Nash
equilibria may not be Pareto efficient.
(e.g., Prisoners’ Dilemma)
• There can be multiple equilibria. (e.g.,
Battle of the sexes)
• One equilibrium can be less efficient
than (Pareto dominated by) the other
equilibrium. (e.g., Coordination game)
 Coordination failure
Review of Lecture 3
• When players are rational and share the
correct belief about the future play, NE
will emerge.
• In some cases, however, NE can be
reached only by rationality.
 Dominant strategy (e.g., PD)
 Focal Point (e.g., Class room
experiment)
 Iterated elimination of strictly
dominated strategies (e.g., Spatial
competition model)
Spatial Competition
Model
• Players: Two ice cream shops
• Strategies: Shop location along a beach (any
integer between 0 and 100)
• Payoffs: Profits=The number of customers
Assumptions:
• Customers are located uniformly on the
beach.
• Each customer goes to the nearest shop
(and buys exactly one ice dream).
• If both shops choose the same location, each
receives half of the customers.
Nash Equilibrium
• There is a unique NE in which both shops
open at the middle.
Why?
• Choosing separate locations never becomes
a NE.
• Choosing the same locations other than the
middle point also fails to be a NE.
• If both shops choose the middle, then no one
has an incentive to change the location.
Solution by Iterated
Elimination
• Step 1: A rational player never takes the
edges, since 0 (100) is strictly dominated by
1 (99).
• Step 2: 1 and 99 are never chosen if the
players know their rival is rational.
• Step 3: 2 and 98 are never chosen if the
players know that their rival knows that you
are rational.
• Step 50: 49 and 51 are never chosen if the
players know that their rival knows that …
 Both players choose 50 in the end!
Common Knowledge
• Each step requires a further assumption
about what the players know about each
other’s rationality.
• We need to assume not only that all the
players are rational, but also that all the
players know that all the players are rational,
and that all the players know that all the
players know that all the players are rational,
and so on.
• For an arbitrary number of steps, we need to
assume that it is common knowledge that the
players are rational.
Weak Predictive Power
• The process often produces a very imprecise
prediction about the play of the game.
• Nash equilibrium is a stronger solution
concept than iterated elimination of strictly
dominated strategies, in the sense that the
players’ strategies in a Nash equilibrium
always survive during the process, but the
converse is not true.
 If the elimination processes pick up a unique
strategy profile (e.g., Spatial competition
model), then that must be a NE.
Bertrand Model
• Players: Two firms
• Strategies: Prices they will charge
• Payoffs: Profits
Assumptions:
• A linear demand function: P=a-bQ
• Common marginal cost, c.
• The firm with lower price must serve the
entire market demand.
• If the firms choose the same price, then each
firm sells the half of the market demand.
Bertrand-Nash
Equilibrium
• There is a unique NE in which both firms
charge the price equal to their (common)
marginal cost.
Why?
• Choosing different prices never becomes a
NE.
• Choosing the same price other than the
marginal cost also fails to be a NE.
• If both firms choose p=c, then no firm has an
(strict) incentive to change the price.
Cournot Model
• Players: Two firms
• Strategies: Quantities they will charge
• Payoffs: Profits
Assumptions:
• A linear demand function: P=a-bQ
• Common marginal cost, c.
• Firms cannot decide their prices to charge,
but the unique market price is determined so
as to clear the market.
Important Remarks
• Bertrand and Cournot models are different
games, i.e., price competition vs. quality
competition.
• The unique equilibrium concept (=NE) can
explain different market outcomes depending
on the models.
• That is, we don’t need different assumptions
about firms’ behaviors.
 Once a model is specified, then Nash
equilibrium gives us the result of the game.