UNIT III: MONOPOLY & OLIGOPOLY

• Monopoly• Oligopoly• Strategic Competition7/20

Market Structure

Perfect Comp Oligopoly Monopoly

No. of Firms infinite (>)2 1

Output MR = MC = P ??? MR = MC < P

Profit No ? Yes

Efficiency Yes ? ???

Oligopoly

We have no general theory of oligopoly. Rather, there are a variety of models, differing in assumptions about strategic behavior and information conditions.

All the models feature a tension between:

– Collusion: maximize joint profits– Competition: capture a larger share of the pie

Game Theory

• Game Trees and Matrices• Games of Chance v. Strategy• The Prisoner’s Dilemma• Dominance Reasoning• Best Response and Nash

Equilibrium• Mixed Strategies

Games of Chance

Buy Don’t Buy

(1000) (-1) (0) (0)

Player 1

Chance

You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1.

What would you do?

Games of Chance

Buy Don’t Buy

(1000) (-1) (0) (0)

Player 1

Chance

You are offered a fair gamble to purchase a lottery ticket that pays $1000, if your number is drawn. The ticket costs $1.

The chance of your number being chosen is independent of your decision to buy the ticket.

Games of Strategy

Buy Don’t Buy

(1000,-1000) (-1,1) (0,0) (0,0)

Player 1

Player 2

Player 2 chooses the winning number.

What are Player 2’s payoffs?

Games of Strategy

Advertise Don’t

Advertise

A D A D

(10,5) (15,0) (6,8) (20,2)

Firm 1

Firm 2

Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice.

How should the game be played?

Profits are in ( )

Games of Strategy

Advertise Don’t

Advertise

A D A D

(10,5) (15,0) (6,8) (20,2)

Firm 1

Firm 2

Duopolists deciding to advertise. Firm 1 moves first. Firm 2 observes Firm 1’s choice and then makes its own choice.

How should the game be played?

Backwards-induction

Games of Strategy

Advertise Don’t

Advertise

A D A D

(10,5) (15,0) (6,8) (20,2)

Firm 1

Firm 2

Duopolists deciding to advertise. The 2 firms move simultaneously. (Firm 2 does not see Firm 1’s choice.)

Imperfect Information.

Information set

Matrix Games

Advertise Don’t

Advertise

A D A D

(10,5) (15,0) (6,8) (20,2)

Firm 1

Firm 2

10, 5 15, 0

6, 8 20, 2

A D

A

D

Matrix Games

Advertise Don’t

Advertise

A D A D

(10,5) (15,0) (6,8) (20,2)

Firm 1

Firm 2

10, 5 15, 0

6, 8 20, 2

A D

A

D

Games of Strategy

• Games of strategy require at least two players.

• Players choose strategies and get payoffs. Chance is not a player!

• In games of chance, uncertainty is probabilistic, random, subject to statistical regularities.

• In games of strategy, uncertainty is not random; rather it results from the choice of another strategic actor.

• Thus, game theory is to games of strategy as probability theory is to games of chance.

A Brief History of Game Theory

Minimax Theorem 1928

Theory of Games & Economic Behavior 1944

Nash Equilibrium 1950

Prisoner’s Dilemma 1950

The Evolution of Cooperation 1984

Nobel Prize: Harsanyi, Selten & Nash 1994

The Prisoner’s Dilemma

In years in jail Player 2

Confess Don’t

Confess

Player 1

Don’t

-10, -10 0, -20

-20, 0 -1, -1

The pair of dominant strategies (Confess, Confess)is a Nash Eq.

GAME 1.

The Prisoner’s Dilemma

Each player has a dominant strategy. Yet the outcome (-10, -10) is pareto inefficient.

Is this a result of imperfect information? What would happen if the players could communicate?

What would happen if the game were repeated? A finite number of times? An infinite or unknown number of times?

What would happen if rather than 2, there were many players?

DominanceDefinition

Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).

T1 T2 T3 T1 T2 T3

0,2 4,3 3,3

4,0 5,4 5,6 3,5 3,5 2,3

0,2 4,3 3,3

4,0 5,4 5,3 3,5 3,5 2,3

S1

S2

S3

S1

S2

S3

Sure Thing Principle: If you have a dominant strategy, use it!

DominanceDefinition

Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s).

T1 T2 T3 T1 T2 T3

0,2 4,3 3,3

4,0 5,4 5,6 3,5 3,5 2,3

0,2 4,3 3,3

4,0 5,4 5,3 3,5 3,5 2,3

S1

S2

S3

S1

S2

S3

Sure Thing Principle: If you have a dominant strategy, use it!

(S2,T3)(S2,T2)

Nash EquilibriumDefinitions

Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s.

-3 0 -10

-1 5 2

-2 -4 0

0,4 4,0 5,3

4,0 0,4 5,3 3,5 3,5 6,6

S1

S2

S3

S1

S2

S3

T1 T2 T3

Nash Equilibrium: a set of best response strategies (one for

each player), (s*, t*) such that s* is a best

response to t* and t* is a b.r. to s*.

(S3,T3)

Nash Equilibrium

-3 0 -10

-1 5 2

-2 -4 0

4,4 2,3 1,5

3,2 1,1 0,0 5,1 0,0 3,3

S1

S2

S3

S1

S2

S3

T1 T2 T3Nash equilibrium need not be Efficient.

Nash Equilibrium

-3 0 -10

-1 5 2

-2 -4 0

1,1 0,0 0,0

0,0 1,1 0,0 0,0 0,0 1,1

S1

S2

S3

S1

S2

S3

T1 T2 T3Nash equilibrium need not be unique.

A COORDINATION PROBLEM

Nash Equilibrium

-3 0 -10

-1 5 2

-2 -4 0

1,1 0,0 0,0

0,0 1,1 0,0 0,0 0,0 3,3

S1

S2

S3

S1

S2

S3

T1 T2 T3Multiple and Inefficient Nash Equilibria.

Nash Equilibrium

-3 0 -10

-1 5 2

-2 -4 0

1,1 0,0 0,-100

0,0 1,1 0,0 -100,0 0,0 3,3

S1

S2

S3

S1

S2

S3

T1 T2 T3Multiple and Inefficient Nash Equilibria.

Is it always advisable to play a NE strategy?

What do we need to know about the other player?

Button-Button

Left Right

L R L R

(-2,2) (4,-4) (2,-2) (-1,1)

Player 1

Player 2

Player 1 hides a button in his Left or Right hand. Player 2 observes Player 1’s choice and then picks either Left or Right.

How should the game be played?

GAME 2.

Button-Button

Left Right

L R L R

(-2,2) (4,-4) (2,-2) (-1,1)

Player 1

Player 2

Player 1 should hide the button in his Right hand. Player 2 should picks Right.

GAME 2.

Button-Button

Left Right

L R L R

(-2,2) (4,-4) (2,-2) (-1,1)

Player 1

Player 2

What happens if Player 2 cannot observe Player 1’s choice?

GAME 2.

Button-Button

Left Right

L R L R

(-2,2) (4,-4) (2,-2) (-1,1)

Player 1

Player 2

-2, 2 4, -4

2, -2 -1, 1

L R

L

R

GAME 2.

Mixed Strategies

-2, 2 4, -4

2, -2 -1, 1

Definition

Mixed Strategy: A mixed strategy is a probability distribution over all strategies available to a player.

Let (p, 1-p) = prob. Player 1 chooses L, R.(q, 1-q) = prob. Player 2 chooses L, R.

L R

L

R

GAME 2.

Mixed Strategies

-2, 2 4, -4

2, -2 -1, 1

Then the expected payoff to Player 1:

EP1(L) = -2(q) + 4(1-q) = 4 – 6qEP1(R) = 2(q) – 1(1-q) = -1 + 3q

Then if q < 5/9, Player 1’s best response is to always play L (p = 1)

L R

L

R

(p)

(1-p)

(q) (1-q)

GAME 2.

q

LEFT 1

5/9

RIGHT 0

0 1 p

p*(q)

Button-Button

Player 1’s best response function.

GAME 2.

Mixed Strategies

-2, 2 4, -4

2, -2 -1, 1

Then the expected payoff to Player 1:

EP1(L) = -2(q) + 4(1-q) EP1(R) = 2(q) – 1(1-q)

(Equalizers) q* = 5/9 and for Player 2:

p* = 1/3 EP2(L) = -2(p) + 2(1-p) EP2(R) = 4(p) – 1(1-p)

L R

L

R

(p)

(1-p)

(q) (1-q)

NE = {(1/3), (5/9)}

GAME 2.

q

LEFT 1

5/9

RIGHT 0

0 1/3 1 p

q*(p)

p*(q)

NE = {(1/3), (5/9)}

Button-Button

GAME 2.

2x2 Game

T1 T2

1. Prisoner’s Dilemma2. Button – Button3. Stag Hunt4. Chicken5. Battle of Sexes

S1

S2

x1,x2 w1, w2

z1,z2 y1, y2

Stag Hunt

T1 T2

S1

S2

5,5 0,3

3,0 1,1

also Assurance Game

NE = {(S1,T1), (S2,T2)}

GAME 3.

Chicken

T1 T2

S1

S2

3,3 1,5

5,1 0,0

also Hawk/Dove

NE = {(S1,T2), (S2,T1)}

GAME 4.

Battle of the Sexes

T1 T2

S1

S2

5,3 0,0

0,0 3,5

NE = {(S1,T1), (S2,T2)}

GAME 5.

P2

5

3

0

0 3 5 P1

GAME 5.NE = {(1, 1); (0, 0); (5/8, 3/8)}

(0,0)

(5/8,3/8)

(1,1)

Battle of the Sexes

Existence of Nash Equilibrium

Prisoner’s Dilemma Battle of the Sexes Button-ButtonGAME 1. GAME 5. (Also 3, 4) GAME 2.

0 1 0 1 0 1 p

q

1

0

There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w).

Strategic Competition

• Prisoner’s Dilemma• Repeated Games• Discounting• The Folk Theorem• Cartel Enforcement

Repeated Games

Some Questions:

• What happens when a game is repeated? • Can threats and promises about the future

influence behavior in the present?• Cheap talk• Finitely repeated games: Backward induction• Indefinitely repeated games: Trigger strategies

Repeated Games

Examples of Repeated Prisoner’s Dilemma

• Cartel enforcement• Transboundary pollution• Common property resources• Arms races

The Tragedy of the Commons

Free-rider Problems

Can threats and promises about future actions influence behavior in the present?

Consider the following game, played 2X:

C 3,3 0,5

D 5,0 1,1

Repeated Games

C D

See Gibbons: 82-104.

Repeated Games

Draw the extensive form game:

(3,3) (0,5) (5,0) (1,1)

(6,6) (3,8) (8,3) (4,4) (3,8)(0,10)(5,5)(1,6)(8,3) (5,5)(10,0) (6,1) (4,4) (1,6) (6,1) (2,2)

Repeated Games

Now, consider three repeated game strategies:

D (ALWAYS DEFECT): Defect on every move.

C (ALWAYS COOPERATE): Cooperate on every move.

T (TRIGGER): Cooperate on the first move, then

cooperate after the other cooperates. If the other defects, then defect forever.

Repeated Games

If the game is played twice, the V(alue) to a player using ALWAYS DEFECT (D) against an opponent using ALWAYS DEFECT(D) is:

V (D/D) = 1 + 1 = 2, and so on. . . V (C/C) = 3 + 3 = 6V (T/T) = 3 + 3 = 6V (D/C) = 5 + 5 = 10V (D/T) = 5 + 1 = 6V (C/D) = 0 + 0 = 0V (C/T) = 3 + 3 = 6

V (T/D) = 0 + 1 = 1V (T/C) = 3 + 3 = 6

Repeated Games

And 3x:

V (D/D) = 1 + 1 + 1 = 3 V (C/C) = 3 + 3 + 3 = 9V (T/T) = 3 + 3 + 3 = 9V (D/C) = 5 + 5 + 5 = 15V (D/T) = 5 + 1 + 1 = 7V (C/D) = 0 + 0 + 0 = 0V (C/T) = 3 + 3 + 3 = 9

V (T/D) = 0 + 1 + 1 = 2V (T/C) = 3 + 3 + 3 = 9

Repeated Games

Time average payoffs: n=3

V (D/D) = 1 + 1 + 1 = 3 /3 = 1V (C/C) = 3 + 3 + 3 = 9 /3 = 3V (T/T) = 3 + 3 + 3 = 9 /3 = 3V (D/C) = 5 + 5 + 5 = 15 /3 = 5V (D/T) = 5 + 1 + 1 = 7 /3 = 7/3V (C/D) = 0 + 0 + 0 = 0 /3 = 0V (C/T) = 3 + 3 + 3 = 9 /3 = 3

V (T/D) = 0 + 1 + 1 = 2 /3 = 2/3

V (T/C) = 3 + 3 + 3 = 9 /3 = 3

Repeated Games

Time average payoffs: n

V (D/D) = 1 + 1 + 1 + ... /n = 1V (C/C) = 3 + 3 + 3 + ... /n = 3V (T/T) = 3 + 3 + 3 + ... /n = 3V (D/C) = 5 + 5 + 5 + ... /n = 5V (D/T) = 5 + 1 + 1 + ... /n = 1 + V (C/D) = 0 + 0 + 0 + ... /n = 0V (C/T) = 3 + 3 + 3 + … /n = 3

V (T/D) = 0 + 1 + 1 + ... /n = 1 -

V (T/C) = 3 + 3 + 3 + ... /n = 3

Repeated Games Now draw the matrix form of this game:

1x

T 3,3 0,5 3,3

C 3,3 0,5 3,3

D 5,0 1,1 5,0

C D T

Repeated Games

T 3,3 1-1+ 3,3

C 3,3 0,5 3,3

D 5,0 1,1 1+,1-

C D T

If the game is repeated, ALWAYS DEFECTis no longer dominant.

Time Average

Payoffs

Repeated Games

T 3,3 1-1+ 3,3

C 3,3 0,5 3,3

D 5,0 1,1 1+,1-

C D T

… and TRIGGERachieves “a NE with itself.”

Repeated Games

Time Average

Payoffs

T(emptation) >R(eward)>P(unishment)>S(ucker)

T R,R P-P+ R,R

C R,R S,T R,R

D T,S P,P P+,P-

C D T

Discounting

The discount parameter, , is the weight of the next payoff relative to the current payoff.

In a indefinitely repeated game, can also be interpreted as the likelihood of the game continuing for another round (so that the expected number of moves per game is 1/(1-)).  

The V(alue) to someone using ALWAYS DEFECT (D) when playing with someone using TRIGGER (T) is the sum of T for the first move, P for the second, 2P for the third, and so on (Axelrod: 13-4): 

V (D/T) = T + P + 2P + …

“The Shadow of the Future”

Discounting

Writing this as V (D/T) = T + P + 2P +..., we have the following:

V (D/D) = P + P + 2P + … = P/(1-)

V (C/C) = R + R + 2R + … = R/(1-)

V (T/T) = R + R + 2R + … = R/(1-)

V (D/C) = T + T + 2T + … = T/(1-)

V (D/T) = T + P + 2P + … = T+ P/(1-)

V (C/D) = S + S + 2S + … = S/(1-)

V (C/T) = R + R + 2R + … = R/(1- )

V (T/D) = S + P + 2P + … = S+ P/(1-)

V (T/C) = R + R + 2R + … = R/(1- )

T

C

D

DiscountedPayoffs

T > R > P > S 0 > > 1

T weakly dominates C

R/(1-) S/(1-) R/(1-)

R/(1-) T/(1-) R/(1-)T/(1-) P/(1-) T + P/(1-)

S/(1-) P/(1-) S + P/(1-)

Discounting

C D T

R/(1-) S + P/(1-) R/(1- )

R/(1-) T + P/(1-) R/(1-)

Discounting

Now consider what happens to these values as varies (from 0-1):

V (D/D) = P + P + 2P + … = P/(1-)

V (C/C) = R + R + 2R + … = R/(1-)

V (T/T) = R + R + 2R + … = R/(1-)

V (D/C) = T + T + 2T + … = T/(1-)

V (D/T) = T + P + 2P + … = T+ P/(1-)

V (C/D) = S + S + 2S + … = S/(1-)

V (C/T) = R + R + 2R + … = R/(1- )

V (T/D) = S + P + 2P + … = S+ P/(1-)

V (T/C) = R + R + 2R + … = R/(1- )

Discounting

Now consider what happens to these values as varies (from 0-1):

V (D/D) = P + P + 2P + … = P+ P/(1-) V (C/C) = R + R + 2R + … = R/(1-)

V (T/T) = R + R + 2R + … = R/(1-)

V (D/C) = T + T + 2T + … = T/(1-)

V (D/T) = T + P + 2P + … = T+ P/(1-)

V (C/D) = S + S + 2S + … = S/(1-)

V (C/T) = R + R + 2R + … = R/(1- )

V (T/D) = S + P + 2P + … = S+ P/(1-) V (T/C) = R + R + 2R + … = R/(1- )

V(D/D) > V(T/D) D is a best response to D

Discounting

Now consider what happens to these values as varies (from 0-1):

V (D/D) = P + P + 2P + … = P+ P/(1-)

V (C/C) = R + R + 2R + … = R/(1-)

V (T/T) = R + R + 2R + … = R/(1-)

V (D/C) = T + T + 2T + … = T/(1-)

V (D/T) = T + P + 2P + … = T+ P/(1-)

V (C/D) = S + S + 2S + … = S/(1-)

V (C/T) = R + R + 2R + … = R/(1- )

V (T/D) = S + P + 2P + … = S+ P/(1-)

V (T/C) = R + R + 2R + … = R/(1- )

2

1

3

?

Discounting

Now consider what happens to these values as varies (from 0-1): 

For all values of : V(D/T) > V(D/D) > V(T/D) V(T/T) > V(D/D) > V(T/D)

 

Is there a value of s.t., V(D/T) = V(T/T)? Call this *.

If < *, the following ordering hold: 

V(D/T) > V(T/T) > V(D/D) > V(T/D)  

D is dominant: GAME SOLVED

V(D/T) = V(T/T)T+P(1-) = R/(1-) T-t+P = R T-R = (T-P)

* = (T-R)/(T-P)

?

Discounting

Now consider what happens to these values as varies (from 0-1): 

For all values of : V(D/T) > V(D/D) > V(T/D) V(T/T) > V(D/D) > V(T/D)

 

Is there a value of s.t., V(D/T) = V(T/T)? Call this *.

* = (T-R)/(T-P)

If > *, the following ordering hold: 

V(T/T) > V(D/T) > V(D/D) > V(T/D)  

D is a best response to D; T is a best response to T; multiple NE.

Discounting

V(T/T) = R/(1-)

* 1

V

TR

Graphically:

The V(alue) to a player using ALWAYSDEFECT (D) against TRIGGER (T), and the V(T/T) as a functionof the discount

parameter ()

V(D/T) = T + P/(1-)

The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

The payoff set of the repeated PD is the convex closure of the points [(T,S); (R,R); (S,T); (P,P)].

The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

The shaded area is the set of payoffs that Pareto-dominate the one-shot NE (P,P).

The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

Theorem: Any payoff that pareto-dominates the one-shot NE can be supported in a SPNE of the repeated game, if the discount parameter is sufficiently high.

The Folk Theorem

(R,R)

(T,S)

(S,T)

(P,P)

In other words, in the repeatedgame, if the future matters “enough”i.e., ( > *),there are zillions of equilibria!

• The theorem tells us that in general, repeated games give rise to a very large set of Nash equilibria. In the repeated PD, these are pareto-rankable, i.e., some are efficient and some are not.

• In this context, evolution can be seen as a process that selects for repeated game strategies with efficient payoffs.

“Survival of the Fittest”

The Folk Theorem

Next Time

7/22 Decision under Uncertainty

Pindyck, Chs 5, 13.

Besanko, Chs 14-16