Economcs 21 Bentz X02 Topic4

8/4/2019 Economcs 21 Bentz X02 Topic4

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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS ECONOMICS 21

Andreas Bentz page 1

Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202

Topic 4: Fun and GamesTopic 4: Fun and Games

Economics 21, Summer 2002

Andreas Bentz

Based Primarily on Shy Chapter 2

and Varian Chapter 27, 28

2

Review: Choices and OutcomesReview: Choices and Outcomes

Consumer theory:

From a given choice set (e.g. budget set), choose

the option (e.g. bundle of goods) that you most

prefer.

Under certainty, the outcome of choice is certain.

Choose the option that has an outcome that maximizesutility.

Under uncertainty, the probability distribution over

possible outcomes is known.

Choose the action (associated with a number of

outcomes, where each occurs with given probability) that

maximizes expected utility.


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3

Choices and Outcomes, contdChoices and Outcomes, contd

Producer theory - two extreme cases: Perfect competition:

From a range of possible prices, choose the price

that maximizes profit.

Under certainty, the outcome of choice is certain:

p > MC: zero demand,

p < MC: negative profit,

p = MC: zero profit.

Under uncertainty, the probability distribution over

possible outcomes is known.

Maximize expected profit (not covered).

4

Choices and Outcomes, contdChoices and Outcomes, contd

Monopolist:

From the price-quantity pairs given by the demand

curve, choose the one that maximizes profit.

Under certainty, the outcome of choice is certain:

= p x q(p) - c(q(p))

Under uncertainty, the probability distribution overpossible outcomes is known.

Maximize expected profit (not covered).


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5

Choices against NatureChoices against Nature

In these cases, choice is the choice of one agent, froma given set of alternatives that give certain (expected)

utility (or profit).

The agents choice is a game against nature:

The agent chooses an action (associated with a number of

outcomes). Then nature chooses the outcome that actually

occurs:

Under certainty, nature chooses the single outcome for sure.

Under uncertainty, nature chooses one of the possible

outcomes (with the probability of that outcome).

The agent cannot influence natures move in thisgame.

6

Modeling InteractionModeling Interaction

In general, in all social interaction, my choice

of action influences your choice (because my

action influences your payoff [utility, profit],

and your action influences mine).

Example (duopoly): How I choose my price

depends on how I expect you to choose your price,which depends on how you expect me to choose

my price, which depends on how I expect you to

choose because our profits depend on how we

both choose prices.

We call these social encounters games.


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9

A Classification of GamesA Classification of Games

Simultaneous move games (static games): All players make their choices at the same time.

Method of analysis: (usually) games in normal

(or, strategic) form.

Sequential move games (dynamic games):

Some players make their choices first, then other

players observe these choices and then make

theirs, etc.

Method of analysis: games in extensive form.


Normal Form GamesNormal Form Games

Simultaneous Move Games

in Normal (Strategic) Form


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11

Normal Form GamesNormal Form Games

Definition: A normal form game (or, strategicform game) is defined by:

the set ofplayers in the game;

the strategies (actions) that are available to each

player;

each player chooses one of her available strategies; a

strategy profile is a list of the strategies chosen by each

player;

thepayoffs for each player, depending on the

choice of action of every player; i.e. each players payoff depends on the strategy profile.

Analogy with parlor games: e.g. Pong.

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Example: The Price War GameExample: The Price War Game

Duopolists: player 1 (row), player 2 (column)

Player 2:

cut price dont cut

cut price (1, 1) (3, 0)Player 1:

dont cut (0, 3) (2, 2)

This normal form (or strategic form) of the

game captures all the information needed.


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13

The Price War Game, contdThe Price War Game, contd

The normal form captures all the informationthe definition requires:

Players: {1, 2}

Available strategies:

player 1: {cut price, dont cut}

player 2: {cut price, dont cut}

Strategy profiles: Payoffs: player 1 - player 2

(cut price, cut price) 1 1

(cut price, dont cut) 3 0

(dont cut, cut price) 0 3

(dont cut, dont cut) 2 2

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Equilibrium in GamesEquilibrium in Games

What is our prediction for the play of a game?

Which strategies will agents choose?

What is an appropriate definition of equilibrium in

games?

What do we want from an equilibrium

concept?

Existence: The equilibrium concept should yield a

prediction for all games.

Uniqueness: The equilibrium concept should yield a

unique prediction of equilibrium play in all games.


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15

Dominant StrategiesDominant Strategies

Suggestion 1:If a player has some strategy that gives her a

higher payoff than any other strategy she

could choose, regardless of what the other

players in the game do, she will choose that

strategy.

Such a strategy is called a dominant strategy.

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Dominant Strategies, contdDominant Strategies, contd

Equilibrium prediction: If every player has a

dominant strategy, every player will choose

that dominant strategy.

Definition: An equilibrium in dominant

strategies (ordominant strategy equilibrium) is

a strategy profile in which every player

chooses her dominant strategy.

This is an intuitively appealing and robust

equilibrium concept.


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What is the dominant strategy equilibrium?

Player 2:

cut price dont cut

cut price (1, 1) (3, 0)

Player 1:dont cut (0, 3) (2, 2)

The equilibrium strategy profile in dominant strategies

is (cut price, cut price).

In this equilibrium the payoffs are: (1, 1).

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Fun: Prisoners Dilemma GameFun: Prisoners Dilemma Game

Relabelling players and strategies in the price

war game, we get the prisoners dilemma

game (political philosophy, politics):

Prisoner 2:

confess lie

confess (1, 1) (3, 0)

Prisoner 1:lie (0, 3) (2, 2)


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19

The Advertising GameThe Advertising Game

Duopolists 1 and 2 decide on advertisingexpenditure.

2:

low med. high

low (1, 1) (0, 3) (0, 2)

1: med. (3, 0) (1, 1) (0, 3)

high (2, 0) (3, 0) (1, 1)

What is the dominant strategy equilibrium in

this game?

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The Advertising Game, contdThe Advertising Game, contd

In this game, no player has a dominant

strategy.

There is no dominant strategy equilibrium.

What should our equilibrium prediction be?

(Most games do not have a dominant strategyequilibrium.)


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21

Nash EquilibriumNash Equilibrium

Suggestion 2:If there is a (potential equilibrium) strategy

profile in which no player wishes to deviate

unilaterally (i.e. choose a different strategy

while all other players continue playing their

(potential equilibrium) strategies), this will be

the equilibrium of the game.

Definition: An equilibrium in which no player

wishes to deviate unilaterally is called a Nashequilibrium (John Nash, 1951).

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What is the Nash equilibrium in the price war

game?

Player 2:

cut price dont cut

cut price (1, 1) (3, 0)Player 1:

dont cut (0, 3) (2, 2)

Check each potential equilibrium strategy

profile.


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23

Nash E. and Dominant StrategiesNash E. and Dominant Strategies

Proposition: Every dominant strategyequilibrium is also a Nash equilibrium.

Proof: In a dominant strategy equilibrium, each

player is playing the strategy that gives them

the highest payoff regardless of what the other

players do. Therefore, no player would want to

deviate: all other strategies open to the players

are worse.

But: not every Nash equilibrium is a dominantstrategy equilibrium.

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The Advertising Game, contdThe Advertising Game, contd

What is the Nash equilibrium in the

advertising game?

2:

low med. high

low (1, 1) (0, 3) (0, 2)1: med. (3, 0) (1, 1) (0, 3)

high (2, 0) (3, 0) (1, 1)

The unique Nash equilibrium strategy profile is

(high, high).


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25

Existence of Nash EquilibriumExistence of Nash Equilibrium

Proposition (Nash): A Nash equilibrium(possibly in mixed strategies) exists in every

game.

Mixed strategies are strategies where players

randomize over strategies.

Example (mixed strategy): My advertising

expenditure is: low with probability 0.3, medium

with prob. 0.2, high with probability 0.5.

This course does not cover mixed strategies. Is the Nash equilibrium prediction unique?

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The Standards GameThe Standards Game

Duopolists decide simultaneously on the

standard for VCRs.

Sony (2):

VHS Beta

VHS (2, 1) (0, 0)JVC (1):

Beta (0, 0) (1, 2)

What is the Nash equilibrium in this game?

There are two Nash equilibria (in pure strat.).


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27

Fun: Battle of the Sexes GameFun: Battle of the Sexes Game

Lovers decide where to go on a Friday night:Her:

boxing ballet

boxing (2, 1) (0, 0)

Him:ballet (0, 0) (1, 2)

Although he prefers boxing, and she prefers

ballet, each would rather be with the otherthan on their own.

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Nash Equilibrium and UniquenessNash Equilibrium and Uniqueness

Nash equilibria are not unique.

Can we somehow trim down the number of

Nash equilibria?

Thomas Schelling The Strategy of Conflict:

Some equilibria in co-ordination games such as the battle

of the sexes game are salient. For instance, going

whereverhe prefers has been salient (is no longer?).

The overall conclusion is negative: there is no

uncontested way of paring down the number of

Nash equilibria.

Are multiple equilibria a feature of the world?


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Best ResponsesBest Responses

Another way of thinking about Nash equilibriais in terms of best responses:

Definition: A players best response to the

strategies played by the other players, is the

strategy that gives her the highest payoff,

given the strategies played by the other

players.

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Best Responses, contdBest Responses, contd

Example: the price war game:

Player 2:

cut price dont cut

cut price (1, 1) (3, 0)

Player 1:

dont cut (0, 3) (2, 2)

1s best response to 2 playing (cut price) is: (cut price).

1s best response to 2 playing (dont cut) is: (cut price).




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31

Best Responses, contdBest Responses, contd

Example: the standards game:Sony (2):

VHS Beta

VHS (2, 1) (0, 0)

JVC (1):Beta (0, 0) (1, 2)

1s best response to 2 playing (VHS) is: (VHS).

1s best response to 2 playing (Beta) is: (Beta).



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Nash and Best ResponsesNash and Best Responses

Proposition: In a Nash equilibrium, every

players equilibrium strategy is her best

response to the other players equilibrium

strategy.

Proof: In a Nash equilibrium, no player wishes

to deviate, given the other players continue to

play their Nash equilibrium strategies.

Therefore, her strategy must be the best

response to the other players equilibrium

strategies.


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Nash and Best Responses, contdNash and Best Responses, contd

Example: the price war game: 1s best response to 2 playing (cut price) is: (cut price).




Player 2:

cut price dont cut

cut price (1, 1) (3, 0)

Player 1: dont cut (0, 3) (2, 2)

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Nash and Best Responses, contdNash and Best Responses, contd

Example: the standards game:





Sony (2):

VHS Beta

VHS (2, 1) (0, 0)

JVC (1):Beta (0, 0) (1, 2)


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ApplicationsApplications

between monopoly and perfectcompetition ...


Market Structure III:Market Structure III:

An ApplicationAn Application

Simultaneous Price Setting:

The Bertrand Game (1883)

(Shy pp. 107-110)


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37

The Bertrand GameThe Bertrand Game

The game: Players:

two firms (duopolists), 1 and 2

Strategies:

players 1 and 2 set prices p1, p2 simultaneously

Payoffs:

players 1, 2 produce quantities y1, y2 of the same

homogeneous product, each at constant marginal cost c

inverse demand: p = a - bY, where Y = y1 + y2

assumption: if p1 < p2, then y1 = Y, y2 = 0 and vice versa assumption: if p1 = p2, then y1 = y2 = 1/2 Y

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The Bertrand Game, contdThe Bertrand Game, contd

Payoffs, contd:

player is profit: i = piyi - cyi, ori = (pi - c)yi, where i = 1, 2

for player 1:

player 1s profit when p1 < p2:

1 = (p1 - c) Y,

i.e. 1 = (p1 - c) (a - p1)/b

player 1s profit when p1 > p2:

1 = 0

player 1s profit when p1 = p2:

1 = (p1 - c) 1/2 Y,

i.e. 1 = (p1 - c) 1/2 (a - p1)/b

similarly for player 2.


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Solution: When prices can be chosen continuously,there is a simple and intuitive solution to the Bertrand

game:

Can a price less than marginal cost be optimal?

No: profits are negative.

Can a price greater than marginal cost be optimal?

Suppose player 1 were to charge a price above marginal cost.

Then player 2 could just undercut player 1s price and take the

entire market. Similarly for player 2.

The only price at which one player does not have to anticipate

being undercut by the other player is price = marginal cost.

The Nash equilibrium strategy profile in the Bertrand

game is for both players (i = 1, 2) to set pi = c.

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If oligopolists compete in prices (Bertrand

competition), the outcome will be efficient:

price = marginal cost.


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Market Structure IV:Market Structure IV:


Simultaneous Quantity Setting:

The Cournot Game (1838)

(Shy pp. 98-101; Varian Ch 27)

42

The Cournot GameThe Cournot Game

The game:

Players:


Strategies:

players 1 and 2 set quantities y1, y2 simultaneously

Payoffs: players 1, 2 produce quantities y1, y2 of the same

homogeneous product, each at constant marginal cost c

inverse demand: p = a - bY, where Y = y1 + y2


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43

The Cournot Game, contdThe Cournot Game, contd

Payoffs, contd: firm 1s profit when it sets quantity y1 and firm 2 sets

quantity y2:

1 (y1, y2) = p y1 - c y1, or:

1 (y1, y2) = (a - b(y1 + y2)) y1 - c y1, or:

1 (y1, y2) = ay1 - by12 - by2y1 - c y1, or:

1 (y1, y2) = - by12 + (a - by2 - c)y1.

Similarly for firm 2:

2 (y1, y2) = - by22 + (a - by1 - c)y2.

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So:

Firm 1 profit: 1(y1, y2) = - by12 + (a - by2 - c)y1.

Firm 2 profit: 2(y1, y2) = - by22 + (a - by1 - c)y2.

What is firm 1s best response (reaction) when firm 2

chooses y2?

Choose y1 to max

1 (y1, y2): 1(y1, y2) / y1 = - 2by1 + a - by2 - c = 0

that is: y1 = (a - by2 - c)/2b

This is firm 1s best response (or reaction) function.

What is firm 2s best response function?

y2 = (a - by1 - c)/2b


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45


Recall: In a Nash equilibrium, every playersequilibrium strategy is her best response to the other

players equilibrium strategy.

So we know that

y1 = (a - by2 - c)/2b and

y2 = (a - by1 - c)/2b

are both true.

Solve for y1:

y1 = (a - c)/3b

y2 = (a - c)/3b

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f1(y2) - firm 1s best response

(or, reaction) function

f2(y1) - firm 2s

reaction

function


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49

The Cournot Game: ComparisonThe Cournot Game: Comparison



f2(y1) - firm 2s

reaction

function

Nash equilibrium in

the Cournot game

Monopoly solution(firm 1 is monopolist)

Perfect Competition (assuming

linear demand and symmetry)

50


If oligopolists compete in quantities (Cournot

competition), the joint quantity is:

greater than the quantity in a monopoly,

but less than the quantity under perfect competition

(or under Bertrand competition).

Cournot Bertrand

quantity

Monopoly P. C.


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Extensive Form GamesExtensive Form Games

(Mostly) Sequential Move Games

in Extensive Form

52

Example: The Entry GameExample: The Entry Game

potential

entrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

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


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53

Extensive Form GamesExtensive Form Games

Definition: An extensive form game is: a game tree (one starting node, other decision

nodes, terminal nodes, and branches linking each

decision node to successor nodes);

the set ofplayers in the game;

at each decision node, the name of the player

making a decision at that node;

the actions available to players at each node;

a players strategyis a list of actions of that player at each

decision node where that player can take an action;

thepayoffs for each player at each terminal node.

54

Extensive Form Games, contdExtensive Form Games, contd

Note:

We now need to be careful about the distinction:

action - strategy:

An action at some decision node is a players decision of

what to do when that node is reached.

A strategy is a complete list of actions that a player plansto take at each decision node, whether or not that node is

actually reached.

Example (the entry game): if player 1 chooses to stay

out, player 2s decision node is not reached.


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55

The Entry Game and Nash Eq.The Entry Game and Nash Eq.

What is the Nash equilibrium in the entrygame?

Recall: In a Nash equilibrium, no player wishes

to deviate unilaterally.

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The Entry Game, contdThe Entry Game, contd

potential

entrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

(2, 2)

possible Nash

equilibria:

(enter, fight)

(enter, share)

(stay out, fight)

(stay out, share)

This game has two Nash

equilibria (in pure strategies).(-1, -1)


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We can convert this extensive form game into anormal (strategic) form game:

normal (strategic) form:potential

entrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

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

enter

stay out

sharefight

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

(0, 8) (0, 8)

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One of the two Nash equilibria in the entry game is

unreasonable: (stay out, fight)

The potential entrant only stays out because, if she were to

enter, the incumbent threatens to fight.

But consider what would happen if the entrant did enter: once

she has entered (i.e. once we are at player 2s decision node),

the incumbent would want to share the market (i.e. not fight). This Nash equilibrium is based on a non-credible threat.

This (overall) equilibrium is unreasonable because, once play

of the game has reached player 2s decision node,

subsequent play (i.e. play in the subgame that starts at player

2s decision node) is not an equilibrium.


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Multiple Nash EquilibriaMultiple Nash Equilibria

In extensive form games we can sometimeseliminate unreasonable Nash equilibria.

Remember: we want a unique prediction for the

play of the game.

We only admit reasonable Nash equilibria:

We want equilibrium play in a game to be such that

each players strategies are an equilibrium not only

in the overall game, but also at every decision

node, for the subsequent game (the subgame

starting at that decision node).

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Subgame Perfect EquilibriumSubgame Perfect Equilibrium

Definition: A subgame is the game that starts

at one of the decision nodes of the original

game; i.e. it is a decision node from the

original game along with the decision nodes

and terminal nodes directly following this node.

Definition: A Nash equilibrium with the

property that it induces equilibrium play at

every subgame is called a subgame perfect

equilibrium.


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potentialentrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

(2, 2)

1. What is theequilibrium in the

subgame starting

at player 2s

decision node?

2. Once we know

this, what is the

equilibrium in the

subgame starting

at the startingnode?

(-1, -1)

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There is a unique subgame perfect equilibrium

in the entry game.

Subgame perfection may help us trim down

the number of Nash equilibria in sequential-

move games in extensive form.

Subgame perfection is the solution concept we

will use for sequential-move games in

extensive form.


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63

Backward InductionBackward Induction

A method for finding subgame perfectequilibria is backward induction.

A subgame perfect equilibrium is a specification of

all players strategies such that play in every

subgame is a (Nash) equilibrium for that subgame.

In particular, this is true for the final subgame(s).

So we know what happens in the final subgame:

we can replace that subgame by the payoff that will

be reached in that subgame.

Then proceed similarly in this new reduced game,

until there is only one subgame left.

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Backward Induction, contdBackward Induction, contd

potential

entrant (1)

enter stay out

(0, 8)incumbent (2)

fight share

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


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Market Structure V:Market Structure V:


Entry Deterrence

Dixit (1982)AER

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Entry DeterrenceEntry Deterrence

Entry deterrence: the incumbent takes an action that

influences payoffs such that she can commit to the

threat of fighting a new entrant.

Remember: in the entry game, the threat to fight was non-

credible, and was therefore eliminated by subgame perfection.

Suppose before playing the entry game, the

incumbent can choose to incur a cost in readiness to

fight a price war.

Suppose this cost does not reduce payoffs if there is a price

war, but does reduce costs if there is no price war.

(In our example, this cost is 4.)

What is the subgame perfect equilibrium?

Solve by backward induction.


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67

The Entry Deterrence GameThe Entry Deterrence Game

potential

entrant (1)

incumbent (2)

enter stay out

(0, 4)

fight share

(2, -2)

incumbent (2)committed passive

(-1, -1)

potential

entrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

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

68

Entry Deterrence Game, contdEntry Deterrence Game, contd

The entry deterrence game in our example

has a unique subgame perfect equilibrium:

(stay out [at B], enter [at C]; committed [at A],

fight [at D], share [at E]).

(Remember: a players strategy lists an action for

each of that players decision nodes.)


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69

Entry Deterrence Game, contdEntry Deterrence Game, contd

We can convert this game into a normal (strategic)form game:

potential

entrant (1)

incumbent (2)

enter stay out

(0, 4)

fight share

(2, -2)

incumbent (2)

committed passive

(-1, -1)

potential

entrant (1)

incumbent (2)

enter stay out

(0, 8)

fight share

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

A

B C

D E

enter (B), enter (C)

enter (B), stay out (C)

stay out (B), enter (C)

stay out (B), stay out (C)

player 1: etc.


Market Structure VI:Market Structure VI:


Sequential Quantity Setting:

The Stackelberg Game


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The Stackelberg GameThe Stackelberg Game

The game: Players:


Strategies:

players 1 and 2 set quantities y1, y2

player 1 moves first (she is the Stackelberg leader)

player 2 observes 1s choice of y1, and then sets y2.

Payoffs:

players 1, 2 produce quantities y1, y2 of the same

homogeneous product, each at constant marginal cost c inverse demand: p = a - bY, where Y = y1 + y2

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The Stackelberg Game, contdThe Stackelberg Game, contd

Payoffs, contd:

firm 1s profit when it sets quantity y1 and firm 2 sets

quantity y2:

1 (y1, y2) = p y1 - c y1, or:

1 (y1, y2) = (a - b(y1 + y2)) y1 - c y1, or:

1 (y1, y2) = ay1 - by12 - by2y1 - c y1, or:

1 (y1, y2) = - by12 + (a - by2 - c)y1.

The combinations of y1 and y2 for which profit is constant

are firm 1s isoprofit curves. (Topic 4)

Similarly for firm 2:

2 (y1, y2) = - by22 + (a - by1 - c)y2.


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Solution: by backward induction: Player 2 chooses the quantity that is best for her,

after observing what player 1 has chosen,

i.e. player 2 plays her best response to player 1s

choice: player 2 chooses a point on her best

response function.

Knowing this, player 1 chooses the quantity that is

best for her, given that (after she has chosen),

player 2 will choose a point on her best response

function, i.e. player 1 chooses the point on player 2s best

response function that is best for her.

74


Firm 2 chooses the quantity that is best, after having

observed firm 1s choice of quantity y1.

Firm 2 chooses y2 to:

max 2 (y1, y2) = - by22 + (a - by1 - c)y2.

- 2by2 + a - by1 - c = 0, or

y2 = (a - by1 - c)/2b.

Knowing this, firm 1 chooses the quantity that is best.

Firm 1 chooses y1 to:

max 1 (y1, (a - by1 - c)/2b) =

= - by12 + (a - b((a - by1 - c)/2b) - c)y1.

- 2by1 + a - c - 0.5a + 0.5c + by1 = 0, or

y1 = (a - c) / 2b


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So firm 1 chooses y1 = (a - c) / 2b. Therefore firm 2 chooses y2 = (a - by1 - c)/2b,

or y2 = (a - b((a - c) / 2b) - c)/2b, or:

y2 = (a - c) / 4b


f2(y1) - firm 2s

reaction

function



Nash equilibrium in

the Cournot game

Subgame perfect equilibrium

in the Stackelberg game

Documents

Economcs 21 Bentz X02 Topic4