Nash equilibrium

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Nash equilibrium

Nash equilibrium is defined in terms of
strategies, not payoffs Every player is best
responding simultaneously (everyone optimizes)
This is a natural generalization of
single-person optimization solutions we keep the
idea that everyone is optimizing, but allow for
strategic interdependence
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Consider conjectures
Two players each player makes some conjecture
about what the other will do and optimizes
In equilibrium - every player is best
responding to the other - conjectures
about other players moves are correct
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Competitive Strategy
Consider the problem of standardization -when
new products are introduced, different firms
products might be incompatible VCRs Beta, VHS,
take different cassettes Suppose both types of
firms are in the market

• 2 equilibria both adopt VHS
• both adopt Beta
• VHS firms prefer the equilibrium where VHS is
  adopted
• Beta firms prefer the equilibrium where Beta is
  adopted
• In general, there is not a unique Nash
  equilibrium
• This is both a strength and a weakness of
  equilibrium analysis
• a standard will emerge, but we do not know which
  one
• which outcome occurs depends on strategies
  outside this game

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Chicken

• Two drivers drive toward each other
• - if no one turns they crash and
  burn
• - if one turns then the
  other wins
• - if both turn then no
  one wins
• Again there are two
  equilibria
• - one turns and the
  other doesnt

We can characterize the equilibrium even when it
is not unique We can say one player will turn,
but we do not know which player Which gets
played might depend on pre-play signaling,
communication, reputation, commitment devices In
order to do comparative statics it is nice to
have a unique equilibrium
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Cournot Competition
Two firms, identical products, each firm decides
how much to produce Comparing the Cournot and
Bertrand games shows that behavior matters in
industries In otherwise identical markets -
if firms compete by choosing quantities then
PgtMC - if firms compete by choosing price then
PMC
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The Cournot Game
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Nash Equilibrium
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The fixed point of the BR functions
This is the point where the BR functions have a
fixed point.
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Review Cournot solution
Cournot best response functions have other
players quantities in equation In
equilibrium, Strategies are Functions of
Parameters Once we get strategies as functions
of parameters then we can do comparative
statics
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Cournot model for N firms
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Bertrand Competition
Seemingly innocent changes in assumptions about
strategy sets can change predictions Consider
the exact same case as above, but assume firms
have different choice variables 2 firms,
identical products, constant mc, but firms
simultaneously choose prices not quantities
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Proof
Claim the unique Nash Equilibrium is each firm
sets price equal to marginal costs We cannot
use calculus to prove this because the demand
function is discontinuous We use a suppose not
argument to show that something is not a NE we
need to show that at least one player is not best
responding.
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Proof
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Weakly dominated strategies
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Chairmans Paradox
Consider a voting game Outcomes a, b,
c Players 1, 2, 3 The players vote, and the
outcome is chosen by majority rule. -if there is
a majority ex (a, a, b) then a wins -the Chair
(player 1) is allowed to break ties ex (a, b, c)
then a wins Note no matter what players
preferences are, we get the following NE (there
are others). (a, a, a) (b, b, b) (c, c, c)
a wins b wins c wins There is no clear
prediction with the best response logic -if
the other 2 players vote for a, then it doesnt
matter what the remaining player does -so
voting for a is a BR One way to proceed is to
eliminate weakly dominated strategies
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Elimination of strategies
Suppose preferences are as follows (1)
agtbgtc (2) bgtagtc (3) cgtbgta For (2) c is
weakly dominated by b -if (1) and (3) choose
the same thing, (2) is indifferent between voting
for a, b, or c -if (1) and (3) choose different
things then (2) might be the deciding vote -
but he would choose b over c in order to ensure
that c didnt win Similarly, for (3) a is a
weakly dominate by c For (1) both b and c are
weakly dominated by a -if (2) and (3) choose
the same thing, (1) is indifferent between voting
for a, b, c -if (2) and (3) choose different
things, then since (1) breaks ties, (1) will vote
for a The remaining NE (1) must play a. We
get (a, a, c ) a wins or (a, b, b) b wins
After another round of eliminating weakly
dominated strategies, since c can never win The
prediction is (a, b, b). The Chairmans Paradox
is that even with more power, his preferred
outcome does not win.
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Horizontal Merger
Among Firms Producing Complementary
Products Demand for systems -consider a market
for computer systems. -a computer system is
defined as a combination of two complementary
products - computers and monitors Px is the
price of a computer Py is the price of a
monitor The price of the system is Ps
PxPy Suppose the demand for systems is Q?
Ps? (PxPy) Denote the number of computer and
monitors sold by X and Y respectively Assume
the two goods are always sold as a system QXY
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Complementary Products
Consider 2 separate firms one makes computers
the other makes monitors Suppose firms choose
prices and for simplicity assume that costs are 0
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Systems Monopolist
Note the price of a system is cheaper, more
systems are sold, and firm profits are higher -
social welfare unambiguously improves -
integration removes the externality each firm
imposes on the other - a rise in the price of
one component lowers the demand for both Think
in terms of externalities With two firms if
firm (2) raises its price, then firm (1)s demand
falls -firm (2) does not consider how
increasing Py affects firm (1)s profits. The
monopolist takes into account both markets and
internalizes the externality
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Mixed Strategies
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Randomization
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Generalized definition
All of the definitions strictly dominant,
strictly dominated, weakly dominant, and weakly
dominated can be generalized in a
straightforward way.
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Pure Strategy
A pure strategy might be dominated by a mixed
strategy Playing a mixed strategy is like
spinning a roulette wheel or flipping a coin to
determine which pure strategy to play
This proposition says -to test whether a pure
strategy si is dominated when randomized play is
possible we just need to check against all
possible profiles of opponents pure
strategies -not their mixed ones.