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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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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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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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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.
Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
Normal Form GamesNormal Form Games
Simultaneous Move Games
in Normal (Strategic) Form
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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.
12
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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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
14
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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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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The Price War Game, contdThe Price War Game, contd
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).
18
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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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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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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The Price War Game, contdThe Price War Game, contd
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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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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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?
26
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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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.
28
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.
30
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).
2s best response to 1 playing (cut price) is: (cut price).
2s best response to 1 playing (dont cut) is: (cut price).
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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).
2s best response to 1 playing (VHS) is: (VHS).
2s best response to 1 playing (Beta) is: (Beta).
32
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).
1s best response to 2 playing (dont cut) is: (cut price).
2s best response to 1 playing (cut price) is: (cut price).
2s best response to 1 playing (dont cut) is: (cut price).
Player 2:
cut price dont cut
cut price (1, 1) (3, 0)
Player 1: dont cut (0, 3) (2, 2)
34
Nash and Best Responses, contdNash and Best Responses, contd
Example: the standards game:
1s best response to 2 playing (VHS) is: (VHS).
1s best response to 2 playing (Beta) is: (Beta).
2s best response to 1 playing (VHS) is: (VHS).
2s best response to 1 playing (Beta) is: (Beta).
Sony (2):
VHS Beta
VHS (2, 1) (0, 0)
JVC (1):Beta (0, 0) (1, 2)
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Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
ApplicationsApplications
between monopoly and perfectcompetition ...
Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
Market Structure III:Market Structure III:
An ApplicationAn Application
Simultaneous Price Setting:
The Bertrand Game (1883)
(Shy pp. 107-110)
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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
38
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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The Bertrand Game, contdThe Bertrand Game, contd
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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The Bertrand Game, contdThe Bertrand Game, contd
If oligopolists compete in prices (Bertrand
competition), the outcome will be efficient:
price = marginal cost.
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Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
Market Structure IV:Market Structure IV:
An ApplicationAn Application
Simultaneous Quantity Setting:
The Cournot Game (1838)
(Shy pp. 98-101; Varian Ch 27)
42
The Cournot GameThe Cournot Game
The game:
Players:
two firms (duopolists), 1 and 2
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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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.
44
The Cournot Game, contdThe Cournot Game, contd
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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The Cournot Game, contdThe Cournot Game, contd
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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The Cournot Game, contdThe Cournot Game, contd
f1(y2) - firm 1s best response
(or, reaction) function
f2(y1) - firm 2s
reaction
function
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The Cournot Game: ComparisonThe Cournot Game: Comparison
f1(y2) - firm 1s best response
(or, reaction) function
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)
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The Cournot Game, contdThe Cournot Game, contd
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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Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
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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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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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.
56
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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The Entry Game, contdThe Entry Game, contd
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)
58
The Entry Game, contdThe Entry Game, contd
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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The Entry Game, contdThe Entry Game, contd
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)
62
The Entry Game, contdThe Entry Game, contd
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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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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Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
Market Structure V:Market Structure V:
An ApplicationAn Application
Entry Deterrence
Dixit (1982)AER
66
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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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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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.
Dartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, SummerDartmouth College, Department of Economics: Economics 21, Summer020202
Market Structure VI:Market Structure VI:
An ApplicationAn Application
Sequential Quantity Setting:
The Stackelberg Game
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The Stackelberg GameThe Stackelberg Game
The game: Players:
two firms (duopolists), 1 and 2
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
72
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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The Stackelberg Game, contdThe Stackelberg Game, contd
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
The Stackelberg Game, contdThe Stackelberg Game, contd
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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The Stackelberg Game, contdThe Stackelberg Game, contd
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
The Stackelberg Game, contdThe Stackelberg Game, contd
f2(y1) - firm 2s
reaction
function
f1(y2) - firm 1s best response
(or, reaction) function
Nash equilibrium in
the Cournot game
Subgame perfect equilibrium
in the Stackelberg game