UNIT III: COMPETITIVE STRATEGY

• Monopoly• Oligopoly• Strategic Behavior7/21

Strategic Behavior

• Nash Equilibrium (continued)• Mixed Strategies• Repeated Games• The Folk Theorem• Cartel Enforcement

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.

Is it always advisable to play a NE strategy?

What do we need to know about the other player?

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)

Mixed Strategies

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

=> q* = 5/9

and the expected payoff to Player 2:

EP2(L) = -2(p) + 2(1-p) = 2 – 4p EP2(R) = 4(p) – 1(1-p) = -1 + 5p

=> p* = 1/3

L R

L

R

(p)

(1-p)

(q) (1-q)

GAME 2.NE = {(1/3), (5/9)}

q

LEFT 1

5/9

RIGHT 0

0 1/3 1 p

q*(p)

p*(q)

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

Mixed Strategies

GAME 2.

2x2 Game

T1 T2

1. Prisoner’s Dilemma

2. Button – Button

3. Stag Hunt

4. Chicken

5. Battle of Sexes

x1,x2 w1, w2

z1,z2 y1, y2

S1

S2

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); ( , )}

(0,0)

(1,1)

Battle of the Sexes

(p, q); (p, q)

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

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

equity

efficiency

Bargaining power

Existence of Nash Equilibrium

Prisoner’s Dilemma Battle of the Sexes Button-Button

GAME 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).

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

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 others 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

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

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-)

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

Cartel Enforcement

Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by:

P = 7 – Q

Assume that the firms choose prices. If the two firms choose different prices, the one with the lower price gets all the customers; if they choose the same price, they split the market demand.

What is the Nash Equilibrium of this game?

Cartel Enforcement

Consider a market in which two identical firms can produce a good with a marginal cost of $1 per unit. The market demand function is given by:

P = 7 – Q

Now suppose that the firms compete repeatedly, and each firm attempts to maximize the discounted value of its profits ( < 1).

What if this pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)?

Cartel Enforcement

What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)?

P = 7 – Q; TCi = qi

Monopoly Bertrand Duopoly

= TR – TC Q = q1 + q2 = PQ – Q Pb = MC = 1; Qb = 6= (7-Q)Q - Q = 7Q - Q2 - Q

FOC: 7-2Q-1 = 0 => Qm = 3; Pm = 4

w/2 plants: q1 = q2 = 1.5 q1 = q2 = 31= 2 = 4.5 = 2 = 0

Cartel Enforcement

What if a pair of Bertrand duopolists try to behave as a monopolist (w/2 plants)?

Promise: I’ll charge Pm = 4, if you do.Threat: I’ll charge Pb = 1, forever, if you deviate.

4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 … 4.5 = 4.5/(1-)4.5 … 4.5 … 4.5 … 9 … 0 … 0 … 0

If is sufficiently high, the threat will be credible, and the pair of trigger strategies is a Nash equilibrium.

* = 0.5

Trigger Strategy

Current gain from deviation =

4.5

Future gain from cooperation =

(4.5)/(1-)

Next Time

UNIT IV: INFORMATION & WELFARE

7/26 Decision on Under Uncertainty

Pindyck & Rubenfeld, Ch. 5.

Besanko, Ch. 15