unit iii: competitive strategy monopoly oligopoly strategic behavior 7/21
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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.
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)}
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
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-)
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