Nash equilibrium

Defn: A strategy profile * is a Nash equilibrium if for every player

*, * , * i i i i i i i i

s i

U s s U s s s S

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

Consider conjectures Two players; each player makes some conjecture about what the other will do and optimizes

2

2 1 1 1 2

2 2 1 1

is what player (1) thinks (2) is doing

given , (1) chooses optimally; is a BR to

In equilibrium, we require that beliefs be correct:

A strategy profile

c

c c

c c

s

s s s s

s s s s

s

is a Nash Equilibrium if

, , i i i i i i i i

* i N

U s* s * U s s * s S

In equilibrium - every player is best responding to the other - conjectures about other player’s moves are correct

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

BetaVHS

adopt VHS

stick with Beta

stick with VHS

100, 50 30, 20

adopt Beta 0, 0 60, 90

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

1 2: VHS, VHS : Beta, Beta NE NE

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

doesn’t

21

turn don’t turn

turn 0, 0 0, 4

don’t turn 4, 0 -4, -4We 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

1 2: not, turn ; : turn, not NE NE

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 P>MC- if firms compete by choosing price then P=MC

The Cournot Game

1 2

Two firms: strategies: 0 (assume price equates supply and demand)

assume constant

firm makes a conjecture about and maxes its profits

Max

i

j

q

P a Q Q q q MC AC c

i q

,

: 2 0 by symmetry 2 2

BR functions given : . Any 2

i

ci i j i i i j i i

q

ci j i

c cjc i

i j i j

cjc

j i

q q Pq cq a q q q cq

a q q c q

a c q a c qFOC a q q c q q

a c qq q other choice of would result in lower profits. iq

Nash Equilibrium

2 2 1 1

2 11 2

1 2

1

, and each player best responds to the other

; 2 2

Two linear equations, two unknowns, sub in and solve *3

* *

c cq q q q

a c q a c qq q

a cq q *

Q q

2

2 1 1

2 2 ; ;

3 3 9i

a c a c a cq * P* a π* P* q * cq*

The fixed point of the BR functions

1 2 *3

a cq q *

This is the point where the BR functions have a fixed point.

1q

2q

a c

a c

2a c

2a c

1b

2b

Cournot best response functions have other player’s quantities in equation

In equilibrium, Strategies are Functions of Parameters

Once we get strategies as functions of parameters then we can do comparative statics

Review Cournot solution

: ,3 3

a c a cNE

2 11 2;

2 2a c q a c q

q q

22 * ;

3 9

if , then and or if , then and

i

a c a cQ π*

a Q c Q

Cournot model for N firms

1

1 1

1 2

Consider n firms where ...

... ...

: All firms are identical: ...2

Substitute in and solve:

i

n

n n i iq

jj i

i n

P a q q

Max q q a q q q cq

a c q

FOC q q q q q

a cq*

2 2

2 2

; ; *

1 1 1

* * ; Industry profits

1 1

Comparative statics: as then and

i i

i i

n a c n a cP* a Q* nq

n n n

a c n a cπ* P c q nπ*

n n

n π* nπ*

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

, ,

, if then (1) gets entire mkt demand

= if then (1) gets half the mkt demand2

= 0

i i j i i i j

i i j i i j

ii j

Max p p p c D p p

D p p D p p p

D pp p

if then (1) gets zero mkt demand

In equilibrium, each correctly conjectures other's price - each firm in equilbrium sets .

i j

i

p p

p c

Proof

1 2 1 2

1

1 2

11

Suppose . Then firm 1 has 0 demand and 0 . If firm 1 charges

then it can get the entire market. Therefore * is not a BR.

Suppose

Firm (1)'s profit is 2

Agai

p* p* c p p *

p

p* p* c

D p *p * c

1 1

11 1 1

n if (1) charges , its profit is

, which for small enough , is greater than 2

p p *

D p *D p * p * c p * c

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.

Proof

1 2

1

1 1

Suppose . Then both firms earn 0

Is this an equilbrium? No because (2) could raise its to

Suppose . Then the firm gets a negative profit; it can do better with

We hav

p* p* c

p p *

p c p c

e ruled out everything except both set price equal to marginal cost. We still have to show that

both pricing at marginal cost is a NE. This is easy though; a firm cannot profit by deviating from

as

p c

long as its competitor sets

.

This is the same structure we had with Cournot:

2 firms, constant average costs , linear inverse demand

We have different conduct that results in different lev

p c

QED

c P a Q

els of profits

0 0 i iP a Q Q a c n

Weakly dominated strategies

Defn: A strategy is weakly dominated if there exists another strategy

such that for all

, ,

with the inequality being strict for at

i i

i i i i

i i i i i i

s S

s S s S

U s s U s s

1

least one : weakly dominates

In the Bertrand game if we eliminate weakly dominated strategies there is no NE.

-players want to undercut each other but can't set

-they can

i i is s s

p c

1

just get arbitrarily close.

With more players:

the equilibrium requires that at least two set , while the others don't have to.

Sometime eliminating weakly dominated strategies helps prediction.

p c

Chairman’s ParadoxConsider a voting game – Outcomes {a, b, c} Players {1, 2, 3}

The player’s 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 player’s 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 doesn’t matter what the remaining player does -so voting for a is a BR

One way to proceed is to eliminate weakly dominated strategies

Elimination of strategies

Suppose preferences are as follows: (1) a>b>c (2) b>a>c (3) c>b>a

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 didn’t 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 winThe prediction is (a, b, b).

The Chairman’s Paradox is that even with more power, his preferred outcome does not win.

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= Px+Py

Suppose the demand for systems is Q= – Ps= – (Px+Py)

Denote the number of computer and monitors sold by X and Y respectively

Assume the two goods are always sold as a system: Q=X=Y

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

x

2

x

The computer firm solves

: 2 0 Similarly, 2 2

In equilibrium, 3 3 9

2Price of a system is

3

xx x x y

p

y xx y x y x y

x y y

s x y

Max p x p p p

p pFOC p p BR p p p

p p Q x y

p p p

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

2

m

Now suppose that the two firms merge

: 2 2 4s

m m m m ms s s

pMax p p p Q X Y

Mixed Strategies

Defn: Given player ' finite pure strategy set , a mixed strategy for player

: 0,1 assigns to each pure strategy

a probability 0 that it will be played, where 1

Pla

i i

i

i i i i

i i i is S

i s S i

S s S

s s

11

yer ' set of possible mixed strategies is denoted by Δ

Δ { ,..., : 0 for all 1,..., and 1

Note that pure strategies (the ' ) are just degenerate mixed strategies,

where al

i

MM

i M m mm

i

i s S

S σ σ σ m M

s s

l the probability is assigned to one pure strategy

- this is a generalization of a pure strategy

Randomization

When players randomize, the outcome is random, and so are the payoffs.

We assume each player ' normal form payoff

( ) is of the Von Neumann-Morgenstern (VNM) type

Abusing the (.) notation s

i

i s

U s

U

1 1 2 2

lightly, player ' VNM utility

from the mixed strategy profile is denoted by

... ( )i n n is S

i s

U s s s U s

Generalized definition

All of the definitions – strictly dominant, strictly dominated, weakly dominant, and weakly dominated can be generalized in a straightforward way.

iA strategy is strictly dominated for player

if there exists another strategy s.t. for all ,

, , ( strictly dominates )

i

i i -i jj i

i i -i i i -i i i

S i

σ S σ S

U σ σ U σ σ σ σ

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 possiblewe just need to check against all possible profiles of opponents’ pure strategies-not their mixed ones.

Prop: Player ' pure strategy is strictly dominated

iff there exists another strategy s.t.

for all

i i

i i

i i i i i i i i

i s s S

σ S

U σ ,s U s ,s s S