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Game Theory and Strategic Thinking
Lecture 2: Simultaneous-Move Games
Jackie So
Monash University
1 August 2023
Jackie So
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Games with Simultaneous Moves
A game is said to have simultaneous moves if each player must
move without knowing what other players have chosen to do.
This is true if players choose their actions at exactly the same time.
But it is also true in a situation in which players make choices at
different times but do not have any information about others’
moves.
It is quite common in the business world where firms constantly
choose what price to charge, which technology to adopt, what
marketing strategies to implement, without knowing what each
other are doing.
Simultaneous-move games are also called static game (e.g.
Gibbons, 1992) or strategic game (e.g. Osborne, 2004).
Jackie So
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Games with Simultaneous Moves
Robinson Crusoe and Friday are considering whether to work
together to build a dam on their island.
If both parties contribute their efforts, a good dam can be built.
If a party contribute his effort while the other shirks, a dam will still
be constructed, but it will be of low quality.
If both shirk, then no dam will be constructed.
A good dam is worth $7 to each party, while a low quality dam is
worth $2.
The cost of effort is $6 for each party.
Our job is to study whether Robinson and Friday will contribute
their efforts to build a dam.
Jackie So
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Games with Simultaneous Moves
The three elements of the game are:
A set of players or agents (i.e. Robinson and Friday)
A set of strategies (i.e. Contribute and Shirk)
A set of payoff functions (i.e. Value of Dam minus Effort)
The key is that the payoff of a player is not only a function of the
player’s action but also a function of the actions of other players.
Here is the payoffs of our example in table form
Robinson
Contribute
Shirk
Jackie So
Friday
Contribute Shirk
1, 1
-4, 2
2, -4
0, 0
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Dominant Strategy
Robinson
Contribute
Shirk
Friday
Contribute Shirk
1, 1
-4, 2
2, -4
0, 0
A strategy is said to be a dominant strategy for a player if it earns
a higher payoff than any other strategy, no matter how that player’s
opponents may play.
Shirk is a dominant strategy for both Robinson and Friday.
For example, regardless what Friday plays (Contribute or Shirk), it is
better for Robinson to play Shirk. It is because 2 > 1 when Friday
plays Contribute and 0 > −4 when Friday plays Shirk.
The same happens to Friday.
Thus it is reasonable to expect that both Robinson and Friday will
shirk.
Jackie So
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Successive Elimination of Dominated Strategies
There is strong reason for players to play the dominant strategies.
However, not all games have dominant strategies. For example, if
the value of good quality dam is increased to $10, then we have
Robinson
Contribute
Shirk
Friday
Contribute Shirk
4, 4
-4, 2
2, -4
0, 0
Shirking is no longer beneficial when the other player contributes.
Robinson should choose to contribute if Friday contributes and to
choose shirk if Friday shirks.
Similarly, Friday should choose to contribute if Robinson contributes
and to choose shirk if Robinson shirks.
Jackie So
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Successive Elimination of Dominated Strategies
A strategy is a dominated strategy for a player if it earns a lower
payoff than some other strategies, no matter how that player’s
opponents may play.
Consider the following game without a dominant strategy:
McDonald
High
Medium
Low
High
60, 60
70, 36
35, 36
KFC
Medium
36, 70
50, 50
35, 30
Low
36, 35
30, 35
25, 25
Low price is a strictly dominated strategy for McDonald as well as
for KFC because it is dominated by both high and medium price.
For example, if KFC plays High, the payoff from Low (i.e. 35) is less
than the payoffs from High (i.e. 60) and from Medium (i.e. 70).
The same happens for Mcdonald.
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Successive Elimination of Dominated Strategies
If we eliminate Low from the game,
McDonald
High
Medium
High
60, 60
70, 36
KFC
Medium
36, 70
50, 50
Then High will be dominated by Medium for both players.
After eliminating High from the game, the only possible outcome
left will be for both players choosing a medium price –
(Medium, Medium).
If successive elimination of dominated strategies does yield a unique
equilibrium, the game is said to be dominance solvable.
Jackie So
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Successive Elimination of Dominated Strategies
Consider one more example:
Row
Row
U
D
U
M
D
L
4, 11
3, 4
3, 10
Column
C
R
3, 6 5, 12
2, 8 4, 6
4, 6 3, 8
Column
L
R
4, 11 5, 12
3, 10 3, 8
U
U
D
Column
L
R
4, 11 5, 12
Jackie So
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L
4, 11
3, 10
Column
C
R
3, 6 5, 12
4, 6 3, 8
U
Column
R
5, 12
The Maximin Method
Consider the following football game which is not dominance
solvable
Offence
Run
Short Pass
Medium Pass
Long Pass
Run
2
6
6
10
Defence
Pass
5
5.6
4.5
3
Blitz
13
10.5
1
-2
Offence wants the outcome to be a cell with as high a number (yard
advanced) as possible, and Defence wants the outcome to be a cell
with as low a number as possible.
We call it a zero-sum game because one’s gain is equivalent to
another’s loss (the payoffs of the two players in each cell add up to
zero).
Jackie So
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The Maximin Method
The minimax method of solution relies on the view that players in
such games are pessimistic about their chances of achieving good
outcomes.
You expect that your opponent is going to do the best she can. In a
zero-sum game, this implies that she will effectively pick a strategy
that makes you as bad off as she can make you.
Similarly, your opponent figures that you will pick a strategy from
among your choices that minimizes her payoff from whatever
strategy she chooses.
Maximin means maximizing the minimum gain.
It is called minimax when minimizing the maximum loss.
Jackie So
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The Maximin Method
Offence
Run
Short Pass
Medium Pass
Long Pass
Run
2
6
6
10
max = 10
Defence
Pass
5
5.6
4.5
3
max = 5.6
Blitz
13
10.5
1
-2
max = 13
min
min
min
min
=
=
=
=
Offence figures that for each of her rows, Defence will choose the
column with the lowest number in that row.
Therefore, Offence should choose the row that gives her the highest
among these lowest numbers, or the maximum among the minima –
the maximin for short.
Jackie So
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2
5.6
1
-2
The Maximin Method
Offence
Run
Short Pass
Medium Pass
Long Pass
Run
2
6
6
10
max = 10
Defence
Pass
5
5.6
4.5
3
max = 5.6
Blitz
13
10.5
1
-2
max = 13
min
min
min
min
=
=
=
=
Defence reckons that for each of her columns, Row will choose the
row with the largest numbers in that column.
Defence should choose the column with the smallest number among
these largest ones, or the minimum among the maxima – the
minimax.
Jackie So
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2
5.6
1
-2
The Maximin Method
In the football game, the minimax method yields the outcome
(Short Pass, Pass).
It is worthwhile to remind you again that the minimax method is
only applicable to zero-sum games.
But not all zero-sum games can be solved by the maximin method,
e.g. Paper-Scissors-Rock
Row Player
Paper
Scissors
Rock
Paper
0, 0
1, -1
-1, 1
min = -1
Jackie So
Column Player
Scissors
Rock
-1, 1
1, -1
0, 0
-1, 1
1, -1
0, 0
min = -1 min = -1
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min = -1
min = -1
min = -1
Nash Equilibrium
Both the successive elimination of dominated strategies method and
the maximin method are applicable to some games only.
Thus we need a concept that is applicable to a very wide range of
games – Nash equilibrium.
Nash equilibrium is a more general concept because solutions
derived from successive elimination of dominated strategies and
maximin method are also Nash equilibria.
We start with two widely studied non zero-sum games that are not
dominance solvable – the battle of sexes and the chicken game.
Jackie So
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The Battle of Sexes
A husband and wife are supposed to choose between going to a
boxing match and going to a ballet.
The husband prefers the boxing match and the wife prefers ballet.
So both players want to go out, but prefer different events. The
husband might get a payoff of 2 and the wife a payoff of 1 from
going to a boxing match, and the other way around from going to a
ballet.
Wife
Ballet
Boxing
Jackie So
Husband
Ballet Boxing
2, 1
0, 0
0, 0
1, 2
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The Chicken Game
James and Dean take their cars to opposite ends of a street and
start to drive toward each other. The one who swerves to prevent a
collision is the ”chicken”, and the one who keeps going straight is
the winner.
If both maintains a straight course, there is a collision in which both
cars are damaged and both players injured.
James
Swerve (Chicken)
Straight (Tough)
Dean
Swerve (Chicken) Straight (Tough)
0, 0
-1, 1
1, -1
-2, -2
Jackie So
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John F. Nash Jr.
Completed his PhD at the age of 22. His 28 pages thesis
”Non-Cooperative Games” introduces the notion of Nash
equilibrium which vastly expanded the scope of game theory.
Shared the 1994 Nobel Prize with John Harsanyi and Reinhard
Seltan.
His life became the basis for the film A Beautiful Mind, in which
Nash was portrayed by Russell Crowe.
Died in 2015 at the age of 86.
Jackie So
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A quick review of sets and functions
Definition
A set is a collection of irreducible objects, which we call elements.
E.g. {Up, Down} is the set of actions that a row player can choose.
Let SR = {Up, Down} be a set as defined above.
We say Up is in SR (or Up ∈ SR ) if and only if Up is an element of
SR
We say Left is not in SR or Left ∈
/ SR if Left is not an element of
SR .
The order of elements in a set does not matter, thus
SR = {Up, Down} = {Down, Up}
Jackie So
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A quick review of sets and functions
Definition
The Cartesian product of two nonempty sets X and Y , denoted as
X × Y , is the set of all ordered pairs (x, y ) where x comes from X and y
comes from Y .
An ordered pair is an ordered list (x, y ) consisting of two objects x
and y .
Ordered pairs are not sets but elements (i.e. (x, y ) 6= {x, y }).
Since the order matters, (x, y ) 6= (y , x) unless x = y .
Recall the set SR = {Up, Down}, let SC = {Left, Right},
SR × SC = {(Up, Left), (Up, Right), (Down, Left), (Down, Right)}
Jackie So
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A quick review of sets and functions
Definition
A subset R of X × Y is called a (binary) relation from X to Y . If (x, y )
is in R, then we think of R as associating the object x with y .
Definition
A function f that maps X onto Y , denoted as f : X → Y , is a relation
f ⊂ X × Y such that for every x ∈ X , there exists exactly one y ∈ Y
such that (x, y ) ∈ f or xfy .
The function UR : SR × SC → R maps the joint actions of the Row
and Column players into a real number. For example, we write
UR (Up, Left) = 1 or ((Up, Left), 1) ∈ UR .
The function BR : SC → SR maps the actions of the Column player
onto an action of the Row player. For example, we write
BR (Left) = Up or (Left, Up) ∈ BR .
Jackie So
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A quick review of sets and functions
Definition
A set Y is a subset of another set X if and only if for every element of
Y is also an element of X .
Functions can be considered as sets.
Let BR = {(Left, Up), (Right, Up)} be the function BR : SC → SR ,
then we have BR ⊆ SC × SR .
Note that (Left, Down) and (Right, Down) are not in BR since they
are not related.
Similarly, we can define BC = {(Up, Left), (Down, Left)} ⊆ SR × SC
be the function BR : SR → SC .
Rewrite BC to BC0 = {(Left, Up), (Left, Down)} ⊆ SC × SR
The union of sets BR and BC0 , denoted by BR ∪ BC , is
{(Left, Up), (Right, Up), (Left, Down)}.
The intersect of BR and BC0 , denoted by BR ∩ BC , is {(Left, Up)}.
Jackie So
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Nash Equilibrium and Best Response Functions
Recall the game between Robinson and Friday
Robinson
Contribute
Shirk
Friday
Contribute Shirk
1, 1
-4, 2
2, -4
0, 0
Let A = {Robinson, Friday } be the set of players.
In this game, the set of actions available to Robinson is
S1 = {Contribute, Shirk}. The set of actions available to Friday is
S2 = {Contribute, Shirk}.
s = (s1 , s2 ) is called a strategy profile. In this case, the set of all
strategy profiles is the Cartesian product of S1 and S2 :
S1 × S2 = {(Contribute, Contribute), (Contribute, Shirk),
(Shirk, Contribute), (Shirk, Shirk)}
Jackie So
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Nash Equilibrium and Best Response Functions
The player’s payoff is determined by his action and the actions
chosen by other players. That is, a player’s payoff is a function of
the strategy profile chosen by the players.
Let U1 and U2 be the payoff functions of Robinson and Friday, we
have
U1 : S1 × S2 → R
and
U2 : S1 × S2 → R
Each function is mapping a strategy profile onto the real line.
For example, U1 (Contribute, Shirk) = 2 − 6 = −4 and
U2 (Contribute, Shirk) = 2 − 0 = 2.
Jackie So
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Nash Equilibrium and Best Response Functions
For any s2 , s1∗ is called a best response to s2 if and only if
U1 (s1∗ , s2 ) ≥ U1 (s10 , s2 ) ∀s10 ∈ S1
Robinson
Contribute
Shirk
Friday
Contribute Shirk
1, 1
-4, 2
2, -4
0, 0
In the above example, Shirk is Robinson’s best response to
Contribute since
U1 (Shirk, Contribute) = 2 > U1 (Contribute, Contribute) = 1
Shirk is also Robinson’s best response to Shirk since
U1 (Shirk, Shirk) = 0 > U1 (Contribute, Shirk) = −4
Jackie So
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Nash Equilibrium and Best Response Functions
The best response function gives the best response for each possible
strategy of other players. Let B1 and B2 be the best response
functions for Robinson and Friday, then we have B1 : S2 → S1 and
B2 : S1 → S2 .
Writing B1 as a set, we have
B1 = {(Contribute, Shirk), (Shirk, Shirk)} ⊂ S2 × S1
A dominant strategy happens when the best response function is a
constant. Since
B1 (Contribute) = B1 (Shirk) = Shirk
Shirk is a dominant strategy for Robinson.
A dominated strategy is a strategy which is not in the image of the
best response function. That is a dominated strategy does not
appear in the set of best responses.
Jackie So
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Nash Equilibrium and Best Response Functions
In a two player game, a strategy profile s ∗ = (s1∗ , s2∗ ) is a pure
strategy Nash equilibrium if and only if
U1 (s1∗ , s2∗ ) ≥ U1 (s10 , s2∗ ) ∀s10 ∈ S1
and
U2 (s1∗ , s2∗ ) ≥ U2 (s1∗ , s20 ) ∀s20 ∈ S2
In general, in an n-player game, Nash equilibrium is a strategy
profile such that each player’s strategy is a best response to the
strategies of all other players. That is the intersection of all players’
best response functions.
A Nash equilibrium is a strategy selection such that no player can
gain by deviating, given the strategy of his opponents.
When both Robinson and Friday are playing their dominant strategy
(i.e. Shirk), the game is at its Nash equilbiurm.
Jackie So
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The Chicken Game Revisited
James
Swerve (Chicken)
Straight (Tough)
Dean
Swerve (Chicken) Straight (Tough)
0, 0
-1, 1
1, -1
-2, -2
Let A = {J, D} be the set of agents.
Let S = {C , T } be the strategies of the two agents where C refers
to Swerve and T refers to Straight.
The strategy space is defined as S = S 2 which is the two-fold
Cartesian product of S (i.e. S = {(C , C ), (C , T ), (T , C ), (T , T )}
where the first entry of an element refers to the strategy plays by
James and the second entry refers to the strategy plays by Dean).
Jackie So
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The Chicken Game Revisited
James
Swerve (Chicken)
Straight (Tough)
Dean
Swerve (Chicken) Straight (Tough)
0, 0
-1, 1
1, -1
-2, -2
The payoff for each agent is a function Ui : S → R where
i ∈ A = {J, D}.
Observe that UJ (C , T ) ≥ UJ (T , T ) and UJ (T , C ) ≥ UJ (C , C ), and
UD (T , C ) ≥ UJ (T , T ) and UJ (C , T ) ≥ UJ (C , C )
A best response function Bi for each agent is then a subset of S
(i.e. BJ = {(C , T ), (T , C )} and BD = {(C , T ), (T , C )}).
The Nash equilibrium is the intersect of all agents’ best response
functions (i.e. BJ ∩ BD = {(C , T ), (T , C )}).
Jackie So
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Prisoner’s Dilemma
Consider the following game
Prisoner 1
Confess
Deny
Prisoner 2
Confess
Deny
-10, -10 -1, -25
-25, -1
-3, -3
The best response functions for them are both
{(Confess, Confess), (Confess, Deny )}.
The Nash equilibrium (which is also a dominant strategy for both
players) involves both of the players to choose Confess and each of
them will end up in prison for 10 years which yields a utility of -10.
However, if both of the players choose to Deny, then both of them
receive a higher utility (i.e. -3).
(Deny , Deny ) cannot be sustainable because the prisoner can cheat
and earn a higher playoff if the opponent plays Deny.
Jackie So
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Prisoner’s Dilemma
In public goods games, each N players can invest resources ci from
their endowment ei in a public good that is shared by everyone
P and
has a total per-unit value of m. Player i earns ei − ci + m( k ck ).
Assuming 1 > m > 1/N, the payoff-maximizing outcome is to
contribute nothing (ci = 0).
If everyone contributed, however, the players would collectively earn
the most.
Suppose there are 2 players with ei = 20 and m = 0.6, then we can
easily see that the public goods game is just a prisoner’s dilemma:
Player 1
c1 = 20
c2 = 0
Jackie So
Player 2
c1 = 20 c2 = 0
24, 24
12, 32
32, 12
20, 20
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Prisoner’s Dilemma
Prisoner’s dilemma game is non-cooperative – players make their
decisions and implement their choices individually.
If the two players could discuss, choose, and play their strategies
jointly – there would be no difficulty about their achieving the
outcome that both would prefer.
The essence of the question of whether, when, and how a prisoners’
dilemma can be resolved is the difficulty of achieving a cooperative
outcome through non-cooperative actions.
In experiments, players cooperate in prisoners’ dilemma games
about half the time and and contribute about half their endowments
in public good game (Behavioral Game Theory, Camerer (2003)).
Jackie So
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A quick review of differentiation and optimization
Rules of Differentiation
If f (x) = c, then f 0 (x) = 0.
If f (x) = cx, then f 0 (x) = c.
If f (x) = x n , then f 0 (x) = nx n−1 .
If f (x) = g (x) + h(x) then f 0 (x) = g 0 (x) + h0 (x).
If f (x) = g (x) − h(x) then f 0 (x) = g 0 (x) − h0 (x).
To solve the problem
max f (x)
x
we use the first order condition
f 0 (x) = 0
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Cournot Model
Two firms produce the same good. Suppose both of them
simultaneously choose the outputs to maximize their profits.
Consider the following demand and cost functions:
P = 40 − Q = 40 − (q1 + q2 )
c1 = 4q1
and c2 = 4q2
The strategy set for both players is R+ with payoff functions equal
to their profit functions,
π1 (q1 , q2 ) = (40 − q1 − q2 )q1 − 4q1
π2 (q2 , q1 ) = (40 − q1 − q2 )q2 − 4q2
Jackie So
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Cournot Model
Both firms choose their outputs to maximize profit given another
firm’s output,
Firm 1’s problem: max(40q1 − q12 − q2 q1 − 4q1 )
q1
Firm 2’s problem max(40q2 − q22 − q1 q2 − 4q2 )
q2
Thus the best response functions are simply the first order
conditions for the above two maximization problems:
∂π1
1
= 40 − 2q1 − q2 − 4 = 0 ⇒ q1∗ = B1 (q2 ) = 18 − q2
∂q1
2
∂π2
1
= 40 − q1 − 2q2 − 4 = 0 ⇒ q2∗ = B2 (q1 ) = 18 − q1
∂q2
2
Jackie So
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Cournot Model
Recall that Nash equilibrium is a strategy profile such that each
player’s strategy is a best response to the strategies of another
player.
Let (q1∗ , q2∗ ) be such strategy profile, then it must satisfy both
B1 (q2 ) and B2 (q1 ).
That can be found by solving the intersection of the two best
response functions,
1
1
q1 = 18 − q2 18 − q1
⇒ q1∗ = 12
2
2
and q2∗ = 12.
The profits for the two firms are
40(12) − 122 − (12)(12) − 4(12) = 144
Jackie So
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Cournot Model
q2
36
A q = B (q )
1 2
A 1
A
A
A
A
18 H
Nash Equilibrium
HHA
A
H
12
H
A H
A HH
HH q = B (q )
A
2 1
HH2
A
A
H
q1
12
18
Jackie So
36
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Cournot Model
Joint profit maximization
max Π = (40 − Q)Q − 4Q
Q
First-order condition:
∂π
= 0 ⇒ 40 − 2Q = 4 ⇒ Q = 18
∂Q
Consider the collusive agreement q1 = q2 = 9
If q2 = 9, firm 2’s best response will be
1
q2 = 18 − (9) = 13.5 > 9
2
The collusive agreement is not self-enforcing. Both firms will have
an incentive to deviate.
Jackie So
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Cournot Model
Social optimal occurs at the output level where P = MC
40 − Q = 4 ⇒ Q = 36, P = 4
To summarise
Joint Profit
Maximization
Cournot
Equilibrium
Social Optimum
Output
(9,9)
Price
22
Profits
(162,162)
Consumer Surplus
162
(12,12)
16
(144,144)
288
(18,18)
4
(0,0)
648
CS = (40 − P) ∗ Q/2
Jackie So
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Cournot Model
q2
36
q1 = B1 (q2 )
A
So
A
A
A
al
im
pt
lO
cia
A
A
18 H
HHA
HA
H
A H
A HH
q2 = B2 (q1 )
HH
A
HH
A
A
H
q1
y
ol
op
on
M
18
Jackie So
36
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Cournot Model
The Nash equilibrium of the Cournot game (sometimes referred to
as the Cournot Nash equilibrium) is worse than the collusive
outcome from the perspective of the firms.
It resembles a prisoner’s dilemma from the players’ perspective if the
game allows only for the collusive and the Cournot output levels.
Firm 1
Q=9
Q = 12
Firm 2
Q=9
Q = 12
162, 162 135, 180
180, 135 144, 144
However, the equilibrium is preferred to the collusive outcome from
the perspectives of the consumers.
Jackie So
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References
Dixit, Chapter 4 - 5
Jackie So
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