6230 swann stackelberg 2010 (l5)

8/6/2019 6230 Swann stackelberg 2010 (L5)

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Recap:

Last class (January 21, 2010) Examples of games with continuous action sets

Tragedy of the commons

Duopoly models: Cournot and Bertrand

Today (January 26, 2010) Duopoly models

Comparison of duopoly models with Monopoly

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& comparison with Cournot, Bertrand, and

Monopoly

Multistage games with observed actions

Stackelberg Model

Two competing firms, selling a homogeneous good

The mar inal cost of roducin each unit of the ood:c1 and c2

Firm 1 moves first and decides on the quantity to sell:q1

Firm 2 moves next and after seeing q1, decides on thequantity to sell: q2

Q= q1+q2 total market demand

The market rice P is determined b inverse market

2

demand: P=a-bQ if a>bQ, P=0 otherwise.

Both firms seek to maximize profits


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Stackelberg Model

Qj: the space of feasible qjs, j=1,2

trateg es o rm :

s2: Q1 Q2(That is, the strategy of firm 2 can depend on firm 1s decision)

Strategies of firm 1: q1Q1

3

(q1

, q2

) = (q1

, s2(q1

))

j(q1, q2) = [a-b(q1+ q2)- cj] qj

Stackelberg Model: Strategy of Firm 2

Suppose firm 1 produces q1 , 22 = (P-c)q2 = [a-b(q1+ q2)]q2 c2q2

= (Residual) revenue Cost

First order conditions:

d 2/dq2= a - 2bq2 bq1 c2 =

4

q2=(a-c2)/2b q1/2= R

2(q1)

s2 = R2(q1) Strategy of firm 2


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Stackelberg Model: Firm 1s decision

Firm 1s profits, if it produces q1 are:

= P-c = a-b + c

(Now, what is different from the Cournot game?)

We know that from the best response of Firm 2:q2=(a-c2)/2b q1/2

Substitute q into :

5

1 = [a-b(q1+ (a-c2)/2b- q1/2)]q1 c1q1

= [(a+ c2)/2-(b/2) q1-c1]q1 From FOC:

d1/dq1= (a+ c2)/2-b q1- c1 = 0

q1= (a-2c1+c2)/2b

Stackelberg Equilibrium

We have Firm 1s profits, if it produces q1:

1= - 1 2

And firm 2s best response

q2=(a-c2)/2b q1/2

Therefore:

q2=(a+2c1-3c2)/4b

6

1 2 q1= (a-c)/2b

q2= (a-c)/4b

Q = 3(a-c)/4b

Recall: qci = (a-c)/3b


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Cournot vs. Stackelberg vs. Bertrand

Bertrand Stackelberg Cournot Monopoly

Price c (a+3c)/4 (a+2c)/3 (a+c)/2

Quantity(a-c)/b

3(a-c)/4b((a-c)/2b+(a-c)/4b)

2(a-c)/3b (a-c)/2b

Total Firm- 2 - 2 - 2

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Profits

Example: Stackelberg Competition

P = 130-(q1+q2), so a=130, b=1

1 = 2 = =

Firm 2: q2=(a-c2)/2b q1/2 = 60 - q1/2

Firm 1:

1 = [a-b(q1+q2)]q1 c1q1 1 = [(a+c2)/2-(b/2)q1]q1c1q1 1 = [70-q1/2]q1 c1q1

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q_1 = 60

Market price and demand

Q=90 P=40


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Monopoly vs. Cournot vs. Bertrand vs. Stackelberg

Bertrand Stackelber Cournot Mono ol

Price10 40 50 70

Quantity120

90(60+30)

80 60

Total FirmProfits 0

27001800+900

3200 3600

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Firm profits and prices:


Stackelberg competitionP

130

P=130-Q

Consumer surplus=4050

Firm profits=2700

10

q

MC=10

130

40

90

Deadweight loss=450


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Monopoly vs. Cournot vs. Bertrand vs. Stackelberg


Consumersurplus

7200 4050 3200 1800

Deadweightloss

0 450 800 1800

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Total FirmProfits 0

2700(1800+900)

3200 3600

Stackelberg and Information

Does player 2 do better or worse in thistac e erg game compare to t e ournot

game?

Does player 2 have more or less information inthe Stackelberg game compared to the Cournotgame?

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Multi-Stage Games with Observed Actions

These games have stages such that

tage + s p aye sequent a y a ter stage

In each stage k, every player knows all theactions (including those by Nature) that weretaken at any previous stage Players can move simultaneously in each stage k

Some players may be limited to action set donothing in some stages

Each la er moves at most once within a iven

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stage

Players payoffs are common knowledge

Stackelberg game

Stage 1

Firm 1 chooses its quantity q1; Firm 2 doesnothing

Stage 2

Firm 2, knowing q1, chooses its own quantityq2; Firm 1 does nothing

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Multi-Stage Games with ObservedActions

k kh stageofstartat theHistory:

k

ki

is

hk

ihA

aaa

i specifiesthatplayerforstrategyPure:

historygivenstagein

playertoavailableactionsofSet:)(

,...,,,...,, ==

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k

ki

h

khAa

historyeach

andeachfor)(actionan

Q: why would the strategy need to specific an action foreach stage and history?

Finite games of perfect information

A multistage game has perfect information if each la er knows all revious moves when makin

a decision

for every stage k and history hk, exactly one playerhas a nontrivial action set, and all other playershave one-element action set do nothing

In a finite game of perfect information, thenumber of stages and the number of actionsat an sta e are finite.

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Theorem (Zermelo 1913; Kuhn 1953): Afinite game of perfect information has a pure-strategy Nash equilibrium (Note, this is a little stronger than the Nash result!)


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Backward induction

Determine the optimal action(s) in the finalsta e K for each histor hK

For each stage j=K-1,,1

Determine the optimal action(s) in stage j foreach possible hj given the optimal actionsdetermined for stages j+1,,K.

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The strategy profile constructed by backward

induction is a Nash Equilibrium.Each players actions are optimal at every

possible history.

Example: Stackelberg competition

P = 130-(q1+q2), c1 = c2 = c = 10

Backward induction Firm 2 strategy: s2(q1)= q2= 60 - q1/2

Firm 1 strategy: q1= 60

The outcome (60,30) is a Nash equilibrium(Stackelberg outcome)

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Is (60,30) the unique equilibrium in this game?

Lets consider the Cournot equilibrium (40,40) forthe Stackelberg game s2(q1)= 40 q1= 40


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Subgame-perfect equilibrium

Denote:

G(hk): game from stage k on with history hk

For each playerj, sj|hk is the restriction ofstrategies sj to the histories consistent with hk

A strategy profile s of a multistage game withobserved actions is a subgame-perfect equilibriumif, for every hk, the restriction s|hk is a Nash

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equilibrium of subgame G(hk).

Classroom exercise: Strategic investment

Firm 1 does not invest

= - = -

q2=(a-c2)/2b q1/2 = 6 - q1/2 (q1, q2)=(4,4)

Payoffs: (16,16)

Firm 1 does invest q1=(a-c1)/2b q2/2 = 7 - q2/2

q2=(a-c2)/2b q1/2 = 6 - q1/2 =

22

, ,

Payoffs: (256/9-f,100/9)

Firm 1 choice:

Invest if 256/9-f>16, i.e., if f


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Example: Stackelberg competition

P = 130-(q1+q2), c1 = c2 = c = 10

By backward induction the outcome (60,30)is a subgame-perfect equilibrium

The outcome (40,40) is NOT subgame

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per ec , ecause e s ra egy s q1 =does not induce a Nash equilibrium in

stage 2 for player 2, for histories otherthan q1= 40

Extensive form of a game

The set of players

The order of moves

The players payoffs as a function of themoves that were made

The set of actions available to the playerswhen they move

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ac p ayer s n orma on w en e ma eshis move

The probability distributions over anyexogenous events (Nature)


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Example 11

U D

L R L R

2 2

2 1 0 0 - 3 2

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, , , ,

Player 1 moves first. After observing player 1s action, player 2 moves

Player 1 action set: {U,D} Player 2 action set:{L,R}Player 1 strategies: {U,D}

Player 2 strategies: {(L,L), (L,R), (R,L), (R,R)}

(Strategies specify a complete plan of action for all contingencies)

Normal form representation of

extensive-form games

Player 2

(L,L) (L,R) (R,L) (R,R)

U 2,1 2,1 0,0 0,0

D -1,1 3,2 -1,1 3,2Player1

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contingent plan made in advance

We usually do not use normal-formrepresentation of an extensive-form game


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Example 2

1 2

U D

L R L R

2 2

2 1 (0,0) -1 1 3 2

L R

U D U D

1 1

2 1 -1.1 0 0 3 2

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,

Player 1 moves first, player 2

moves next. Player 2 does not

know player 1s action when

he chooses his action

,

Player 2 moves first, player 1

moves next. Player 1 does not

know player 2s action when

he chooses his action

Classroom exercise

1

U D

(2,0)2

L R

(1,1)1

28

X Y

(3,0) (0,2)


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Example to Draw

Pla er 1 chooses an action from thefeasible set {L,R}

Player 2 observes player 1s actionand then chooses an action fromthe feasible set {L,R}

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the history of actions is (R,R) and

then chooses an action from thefeasible set {L,R}

Example (cont.)

1

L R L R

2 2

3 3

L R

30

L R L R L R L R

6230 swann stackelberg 2010 (l5)

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