DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Topic 4: Fun and Games
Economics 21, Summer 2002
Andreas Bentz
Based Primarily on Shy Chapter 2
and Varian Chapter 27, 28
Review: 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 maximizes
utility.
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.
2
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Choices and Outcomes, cont’d
„
„
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).
3
Choices and Outcomes, cont’d
„
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:
‹
»
‹
π = p x q(p) - c(q(p))
Under uncertainty, the probability distribution over
possible outcomes is known.
»
Maximize expected profit (not covered).
4
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Choices against Nature
„
„
In these cases, choice is the choice of one agent, from
a given set of alternatives that give certain (expected)
utility (or profit).
The agent’s 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 nature’s move in this
“game”.
5
Modeling 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”.
6
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Game Theory
„
„
Game theory is the study of such “games”: social
interactions between rational agents.
“All of economics is a branch of game theory”
— (Robert Aumann)
„
We have already analyzed some “games”:
‹
In monopoly, there is no (real) interaction:
‹
In perfect competition, there is no need to model interaction:
»
»
‹
There is only one agent (and nature).
The number of agents is so large that the action of one agent
has no effect on the other agents’ payoffs.
Principal-Agent analyses:
»
A (non-trivial) example of game-theoretic analysis.
7
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Game Theory
John von Neumann and Oskar
Morgenstern (1944) Theory of Games
and Economic Behavior
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
A 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.
‹
9
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Normal Form Games
Simultaneous Move Games
in Normal (Strategic) Form
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Normal Form Games
„
Definition: A normal form game (or, strategic
form game) is defined by:
the set of players in the game;
‹ the strategies (actions) that are available to each
player;
‹
»
‹
the payoffs for each player, depending on the
choice of action of every player;
»
„
each player chooses one of her available strategies; a
strategy profile is a list of the strategies chosen by each
player;
i.e. each player’s payoff depends on the strategy profile.
Analogy with “parlor” games: e.g. Pong.
11
Example: The “Price War” Game
„
Duopolists: player 1 (row), player 2 (column)
Player 2:
Player 1:
„
cut price
don’t cut
cut price
(1, 1)
(3, 0)
don’t cut
(0, 3)
(2, 2)
This “normal form” (or “strategic form”) of the
game captures all the information needed.
12
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Price War” Game, cont’d
„
The normal form captures all the information
the definition requires:
Players: {1, 2}
‹ Available strategies:
‹
»
»
‹
player 1: {cut price, don’t cut}
player 2: {cut price, don’t cut}
Strategy profiles:
»
»
»
»
Payoffs: player 1 - player 2
(cut price, cut price)
(cut price, don’t cut)
(don’t cut, cut price)
(don’t cut, don’t cut)
1
3
0
2
1
0
3
2
13
Equilibrium 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.
‹
14
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dominant 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.
15
Dominant Strategies, cont’d
„
„
„
Equilibrium prediction: If every player has a
dominant strategy, every player will choose
that dominant strategy.
Definition: An equilibrium in dominant
strategies (or dominant strategy equilibrium) is
a strategy profile in which every player
chooses her dominant strategy.
This is an intuitively appealing and robust
equilibrium concept.
16
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Price War” Game, cont’d
„
What is the dominant strategy equilibrium?
Player 2:
Player 1:
„
„
cut price
don’t cut
cut price
(1, 1)
(3, 0)
don’t 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).
17
Fun: “Prisoners’ Dilemma” Game
„
Relabelling players and strategies in the “price
war” game, we get the “prisoners’ dilemma”
game (political philosophy, politics):
Prisoner 2:
Prisoner 1:
confess
lie
confess
(1, 1)
(3, 0)
lie
(0, 3)
(2, 2)
18
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Advertising” Game
„
Duopolists 1 and 2 decide on advertising
expenditure.
1:
„
2:
low med.
low (1, 1) (0, 3)
med. (3, 0) (1, 1)
high (2, 0) (3, 0)
high
(0, 2)
(0, 3)
(1, 1)
What is the dominant strategy equilibrium in
this game?
19
The “Advertising” Game, cont’d
„
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 strategy
equilibrium.)
20
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Nash 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 Nash
equilibrium (John Nash, 1951).
21
The “Price War” Game, cont’d
„
What is the Nash equilibrium in the “price war”
game?
Player 2:
cut price
don’t cut
Player 1:
„
cut price
(1, 1)
(3, 0)
don’t cut
(0, 3)
(2, 2)
Check each potential equilibrium strategy
profile.
22
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Nash E. and Dominant Strategies
„
„
„
Proposition: Every dominant strategy
equilibrium 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 dominant
strategy equilibrium.
23
The “Advertising” Game, cont’d
„
What is the Nash equilibrium in the
“advertising” game?
1:
„
2:
low med.
low (1, 1) (0, 3)
med. (3, 0) (1, 1)
high (2, 0) (3, 0)
high
(0, 2)
(0, 3)
(1, 1)
The unique Nash equilibrium strategy profile is
(high, high).
24
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Existence 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?
25
The “Standards” Game
„
Duopolists decide simultaneously on the
standard for VCRs.
Sony (2):
VHS
Beta
JVC (1):
„
„
VHS
(2, 1)
(0, 0)
Beta
(0, 0)
(1, 2)
What is the Nash equilibrium in this game?
There are two Nash equilibria (in pure strat.).
26
© Andreas Bentz
page 13
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Fun: “Battle of the Sexes” Game
„
Lovers decide where to go on a Friday night:
Her:
boxing
ballet
Him:
„
boxing
(2, 1)
(0, 0)
ballet
(0, 0)
(1, 2)
Although he prefers boxing, and she prefers
ballet, each would rather be with the other
than on their own.
27
Nash 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
wherever he 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?
28
© Andreas Bentz
page 14
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Best Responses
„
„
Another way of thinking about Nash equilibria
is in terms of “best responses”:
Definition: A player’s 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.
29
Best Responses, cont’d
„
Example: the “price war” game:
Player 2:
cut price
don’t cut
Player 1:
cut price
(1, 1)
(3, 0)
don’t cut
(0, 3)
(2, 2)
„
1’s best response to 2 playing (cut price) is: (cut price).
„
1’s best response to 2 playing (don’t cut) is: (cut price).
2’s best response to 1 playing (cut price) is: (cut price).
2’s best response to 1 playing (don’t cut) is: (cut price).
„
„
30
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Best Responses, cont’d
„
Example: the “standards” game:
Sony (2):
VHS
Beta
JVC (1):
VHS
(2, 1)
(0, 0)
Beta
(0, 0)
(1, 2)
„
1’s best response to 2 playing (VHS) is: (VHS).
„
1’s best response to 2 playing (Beta) is: (Beta).
2’s best response to 1 playing (VHS) is: (VHS).
2’s best response to 1 playing (Beta) is: (Beta).
„
„
31
Nash and Best Responses
„
„
Proposition: In a Nash equilibrium, every
player’s equilibrium strategy is her best
response to the other player’s 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.
32
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Nash and Best Responses, cont’d
„
Example: the “price war” game:
‹
‹
‹
‹
1’s best response to 2 playing (cut price) is: (cut price).
1’s best response to 2 playing (don’t cut) is: (cut price).
2’s best response to 1 playing (cut price) is: (cut price).
2’s best response to 1 playing (don’t cut) is: (cut price).
Player 2:
Player 1:
cut price
don’t cut
cut price
(1, 1)
(3, 0)
don’t cut
(0, 3)
(2, 2)
33
Nash and Best Responses, cont’d
„
Example: the “standards” game:
‹
‹
‹
‹
1’s best response to 2 playing (VHS) is: (VHS).
1’s best response to 2 playing (Beta) is: (Beta).
2’s best response to 1 playing (VHS) is: (VHS).
2’s best response to 1 playing (Beta) is: (Beta).
Sony (2):
JVC (1):
VHS
Beta
VHS
(2, 1)
(0, 0)
Beta
(0, 0)
(1, 2)
34
© Andreas Bentz
page 17
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Applications
… between monopoly and perfect
competition ...
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Market Structure III:
An Application
Simultaneous Price Setting:
The Bertrand Game (1883)
(Shy pp. 107-110)
© Andreas Bentz
page 18
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Bertrand Game
„
The game:
‹
Players:
»
‹
Strategies:
»
‹
two firms (duopolists), 1 and 2
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
37
The Bertrand Game, cont’d
‹
Payoffs, cont’d:
»
»
»
»
»
»
player i’s profit: πi = piyi - cyi, or πi = (pi - c)yi, where i = 1, 2
for player 1:
player 1’s profit when p1 < p2:
• π1 = (p1 - c) Y,
• i.e. π1 = (p1 - c) (a - p1)/b
player 1’s profit when p1 > p2:
• π1 = 0
player 1’s profit when p1 = p2:
• π1 = (p1 - c) 1/2 Y,
• i.e. π1 = (p1 - c) 1/2 (a - p1)/b
similarly for player 2.
38
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Bertrand Game, cont’d
„
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?
»
‹
Can a price greater than marginal cost be optimal?
»
‹
„
No: profits are negative.
Suppose player 1 were to charge a price above marginal cost.
Then player 2 could just undercut player 1’s 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.
39
The Bertrand Game, cont’d
„
If oligopolists compete in prices (“Bertrand
competition”), the outcome will be efficient:
price = marginal cost.
40
© Andreas Bentz
page 20
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Market Structure IV:
An Application
Simultaneous Quantity Setting:
The Cournot Game (1838)
(Shy pp. 98-101; Varian Ch 27)
The Cournot Game
„
The game:
‹
Players:
»
‹
Strategies:
»
‹
two firms (duopolists), 1 and 2
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
42
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Cournot Game, cont’d
‹
Payoffs, cont’d:
»
»
firm 1’s profit when it sets quantity y1 and firm 2 sets
quantity y2:
• π1 (y1, y2) = p y1 - c y1, or:
• π1 (y1, y2) = (a - b(y1 + y2)) y1 - c y1, or:
• π1 (y1, y2) = ay1 - by12 - by2y1 - c y1, or:
• π1 (y1, y2) = - by12 + (a - by2 - c)y1.
Similarly for firm 2:
• π2 (y1, y2) = - by22 + (a - by1 - c)y2.
43
The Cournot Game, cont’d
„
So:
‹
‹
„
What is firm 1’s best response (“reaction”) when firm 2
chooses y2?
‹
‹
‹
‹
„
Firm 1 profit: π1(y1, y2) = - by12 + (a - by2 - c)y1.
Firm 2 profit: π2(y1, y2) = - by22 + (a - by1 - c)y2.
Choose y1 to max π1 (y1, y2):
∂π1(y1, y2) / ∂y1 = - 2by1 + a - by2 - c = 0
that is: y1 = (a - by2 - c)/2b
This is firm 1’s best response (or “reaction”) function.
What is firm 2’s best response function?
‹
y2 = (a - by1 - c)/2b
44
© Andreas Bentz
page 22
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Cournot Game, cont’d
„
„
Recall: In a Nash equilibrium, every player’s
equilibrium strategy is her best response to the other
player’s 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
45
The Cournot Game, cont’d
f1(y2) - firm 1’s best response
(or, “reaction”) function
f2(y1) - firm 2’s
reaction
function
46
© Andreas Bentz
page 23
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Cournot Game: Equilibrium?
f1(y2) - firm 1’s best response
(or, “reaction”) function
f2(y1) - firm 2’s
reaction
function
47
The Cournot Game: Equilibrium!
f1(y2) - firm 1’s best response
(or, “reaction”) function
Nash equilibrium
f2(y1) - firm 2’s
reaction
function
48
© Andreas Bentz
page 24
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Cournot Game: Comparison
f1(y2) - firm 1’s best response
(or, “reaction”) function
Perfect Competition (assuming
linear demand and symmetry)
Nash equilibrium in
the Cournot game
f2(y1) - firm 2’s
reaction
function
Monopoly solution
(firm 1 is monopolist)
49
The Cournot Game, cont’d
„
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
Monopoly
Bertrand
quantity
P. C.
50
© Andreas Bentz
page 25
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Extensive Form Games
(Mostly) Sequential Move Games
in Extensive Form
Example: The “Entry” Game
potential
entrant (1)
enter
stay out
incumbent (2)
(0, 8)
fight
(-1, -1)
share
(2, 2)
52
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Extensive 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 of players 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 player’s strategy is a list of actions of that player at each
decision node where that player can take an action;
the payoffs for each player at each terminal node.
53
Extensive Form Games, cont’d
„
Note:
‹
We now need to be careful about the distinction:
action - strategy:
»
»
»
An action at some decision node is a player’s decision of
what to do when that node is reached.
A strategy is a complete list of actions that a player plans
to 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 2’s decision node is not reached.
54
© Andreas Bentz
page 27
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Entry” Game and Nash Eq.
„
„
What is the Nash equilibrium in the “entry”
game?
Recall: In a Nash equilibrium, no player wishes
to deviate unilaterally.
55
The “Entry” Game, cont’d
potential
entrant (1)
possible Nash
equilibria:
enter
stay out
incumbent (2)
(0, 8)
(enter, fight)
(enter, share)
(stay out, fight)
fight
(-1, -1)
share
(2, 2)
(stay out, share)
This game has two Nash
equilibria (in pure strategies).
56
© Andreas Bentz
page 28
DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Entry” Game, cont’d
„
We can convert this extensive form game into a
normal (strategic) form game:
potential
entrant (1)
normal (strategic) form:
enter
stay out
incumbent (2)
(0, 8)
fight
share
(2, 2)
(-1, -1)
enter
stay out
fight
share
(-1, -1)
(2, 2)
(0, 8)
(0, 8)
57
The “Entry” Game, cont’d
„
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 2’s 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 2’s decision node,
subsequent play (i.e. play in the subgame that starts at player
2’s decision node) is not an equilibrium.
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Multiple Nash Equilibria
„
In extensive form games we can sometimes
eliminate “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 player’s 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).
59
Subgame 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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© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Entry” Game, cont’d
potential
entrant (1)
enter
stay out
incumbent (2)
(0, 8)
fight
(-1, -1)
share
(2, 2)
1. What is the
equilibrium in the
subgame starting
at player 2’s
decision node?
2. Once we know
this, what is the
equilibrium in the
subgame starting
at the starting
node?
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The “Entry” Game, cont’d
„
„
„
There is a unique subgame perfect equilibrium
in the “entry” game.
Subgame perfection may help us trim down
the number of Nash equilibria in sequentialmove games in extensive form.
Subgame perfection is the solution concept we
will use for sequential-move games in
extensive form.
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Backward Induction
„
A method for finding subgame perfect
equilibria 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.
‹
63
Backward Induction, cont’d
potential
entrant (1)
enter
stay out
incumbent (2)
(0, 8)
fight
(-1, -1)
share
(2, 2)
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Market Structure V:
An Application
Entry Deterrence
Dixit (1982) AER
Entry Deterrence
„
Entry deterrence: the incumbent takes an action that
influences payoffs such that she can “commit” to the
threat of fighting a new entrant.
‹
„
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?
‹
© Andreas Bentz
Remember: in the “entry” game, the threat to fight was noncredible, and was therefore eliminated by subgame perfection.
Solve by backward induction.
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The “Entry Deterrence” Game
incumbent (2)
committed
passive
potential
entrant (1)
enter
stay out
incumbent (2)
fight
share
(-1, -1)
(2, -2)
(0, 4)
potential
entrant (1)
enter
stay out
incumbent (2)
fight
(-1, -1)
(0, 8)
share
(2, 2)
67
“Entry Deterrence” Game, cont’d
„
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 player’s strategy lists an action for
each of that player’s decision nodes.)
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
“Entry Deterrence” Game, cont’d
We can convert this game into a normal (strategic)
form game:
„
A
incumbent (2)
committed
B
share
(-1, -1)
(2, -2)
(0, 4)
E
stay out
incumbent (2)
fight
(-1, -1)
player 1:
potential
entrant (1)
enter
stay out
incumbent (2)
fight
C
potential
entrant (1)
enter
D
passive
share
(2, 2)
(0, 8)
etc.
enter (B), enter (C)
enter (B), stay out (C)
stay out (B), enter (C)
stay out (B), stay out (C)
69
Dartmouth College, Department of Economics: Economics 21, Summer
‘02
Summer‘02
Market Structure VI:
An Application
Sequential Quantity Setting:
The Stackelberg Game
© Andreas Bentz
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Stackelberg Game
„
The game:
‹
Players:
»
‹
Strategies:
»
»
»
‹
two firms (duopolists), 1 and 2
players 1 and 2 set quantities y1, y2
player 1 moves first (she is the Stackelberg leader)
player 2 observes 1’s 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
71
The Stackelberg Game, cont’d
‹
Payoffs, cont’d:
»
»
»
firm 1’s profit when it sets quantity y1 and firm 2 sets
quantity y2:
• π1 (y1, y2) = p y1 - c y1, or:
• π1 (y1, y2) = (a - b(y1 + y2)) y1 - c y1, or:
• π1 (y1, y2) = ay1 - by12 - by2y1 - c y1, or:
• π1 (y1, y2) = - by12 + (a - by2 - c)y1.
The combinations of y1 and y2 for which profit is constant
are firm 1’s isoprofit curves. (Topic 4)
Similarly for firm 2:
• π2 (y1, y2) = - by22 + (a - by1 - c)y2.
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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Stackelberg Game, cont’d
„
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 1’s
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 2’s best
response function that is best for her.
‹
73
The Stackelberg Game, cont’d
„
Firm 2 chooses the quantity that is best, after having
observed firm 1’s choice of quantity y1.
‹
‹
‹
‹
„
Firm 2 chooses y2 to:
max π2 (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 π1 (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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DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS
ECONOMICS 21
The Stackelberg Game, cont’d
„
„
„
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
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The Stackelberg Game, cont’d
f1(y2) - firm 1’s best response
(or, “reaction”) function
Nash equilibrium in
the Cournot game
Subgame perfect equilibrium
in the Stackelberg game
f2(y1) - firm 2’s
reaction
function
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