Production | Games 1
Monopolistic Competition
Monopolistic competition arises when there are a large number of price-setting flrms in an industry with free entry.
Suppose there are n flrms,In the short run,each flrm faces an nth of the market demand curve.
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0
p
y
ACMC
p?
y?
pn(y)
MRn
…
Short run analysis is not very difierent to that of the monopolist,The flrm charges p? for an output of y? and
makes … proflt,The only difierence is that demand is given by pn(y),an nth of total demand.
However,in the long run other flrms are free to enter if incumbents are making a proflt or leave if they make a loss.
Production | Games 2
The Long Run
Firms will continue to enter until there is no incentive for further entry,That is,when proflts are driven to zero.
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0
p
y
ACMC
p?
y?
pm(y)
MRm
As more flrms enter demand is reduced to each of the incumbent flrms,Suppose once m flrms are in the industry
proflts for each flrm are zero,They will produce y? each at a price of p? in the long run.
Each flrm is proflt maximising,but receiving zero proflt,They are average cost pricing,This is still ine–cient,A
perfectly competitive flrm would produce at p = MC,which is a lower price and higher output.
Production | Games 3
Strategic Behaviour
The problem with this analysis is that flrms ignore the behaviour of their competitors.
Consider the short run and suppose there were just two flrms,Suppose D(p)=1? p and MC =0 for simplicity.
If flrm A operates in monopolistic competition,they think they face half the demand,DA(p)=(1? p)=2,They
produce at MR = MC,The inverse demand curve is p =1?2yA,So MR =1?4yA,They produce at 1?4y?A =0.
This gives,y?A =1=4,Suppose flrm B takes this into consideration,Then the actual inverse demand B faces is
1? y =1? yA? yB =1?1=4? yB and not 1?2yB,Proflt maximisation will yield a difierent quantity.
In fact,flrm B would set MR = MC which yields,3=4?2yB =0 giving y?B =3=8.
Notice if flrm B took monopolistic competition proflts they would get …B =1=8.
With strategic behaviour however,flrm B gets …B =(1?1=4?3=8)3=8=9=64,This is better.
Consider the long run,Is there a way to prevent other flrms from entering and hence maintaining positive proflts?
Production | Games 4
Games
Game theory is the study of strategic behaviour,A game is three things:
1,Players,The individuals (consumers or flrms etc.) involved.
2,Strategies,Each player’s choice set | what they can do.
3,Payofis,The utilities or proflts which depend on the strategies all of the players take.
Anything which has these three features can be described as a game.
Game theory is a useful language for describing all kinds of situations | but that is not all.
Production | Games 5
Some Examples
One of the most famous games is the prisoners’ dilemma,This has two players with two strategies each and four
associated payofis for each player,It can be represented by the two-by-two matrix below.
C D
C 3
3
5
0
D 0
5
1
1
This is called the normal form or strategic form of the game.
All discrete action space games can be represented in such matrices,Action spaces need not be discrete | for
example,flrms set quantities and consumers choose bundles of goods.
Most of the games presented will be simultaneous move games,All players make their decision without observing
the strategies played by other agents.
In many interesting games players do observe some of the actions of others,Firms may make decisions sequentially.
Such dynamic games will be considered very brie y.
Production | Games 6
Best Responses
Suppose a player’s opponent in a game plays a particular strategy,What would the player do? The strategy that
maximises the player’s payofi given their opponent’s play is a best response.
The collection of best responses for each possible action of their opponent is called a best response function | or a
reaction function (or even a best reply function),Plotting this gives the reaction curve.
What are the best responses in the prisoners’ dilemma?
C D
C 3
3
5
0
D 0
5
1
1
Common practice is to underline the payofis corresponding to best responses in such a normal form game.
Production | Games 7
Nash Equilibrium
A best response to a best response to a best response to a,.,is called a Nash equilibrium.
If all the payofis in a particular cell are underlined then the strategies that correspond to that cell are a Nash
equilibrium,Notice equilibrium is a collection of strategies,Sometimes this can be very complicated,Chess?
In the prisoners’ dilemma the Nash equilibrium is fD;Dg.
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Player 2
Player 1
BR2
BR1
C
D
C D
In the above graph the best responses are plotted for each player,These are reaction curves,The Nash equilibrium
is where the two reaction curves intersect,This is a useful method for flnding equilibria.
Along the axis are the possible actions taken by players 1 and 2,BR1 is the best reply function of player 1 and
BR2 is player 2s,They intersect at fD;Dg,the equilibrium,What are the dotted lines for?
Production | Games 8
Mixed Strategies
Suppose a player could \toss a coin" to decide which action to take,What would their opponent do?
Such an action is called a mixed strategy,For example,in the prisoners’ dilemma,with a fair coin,the opponent
would expect to see C half the time and D half the time,They calculate a best response.
If they choose to play C,half the time they would get 3,and half the time they would get 0,If they choose to play
D half the time they would get 5,and half the time they would get 1,Their expected payofis are:
312 +012 = 32 < 3=512 +112
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.,,,,,,,,,,,,,,,,,,2 mixes
half-half
Player 2
Player 1
BR2
BR1
C
D
C D
Therefore they would play D in response to such a mixture,In fact,they would play D in response to any mixture.
Hence the dotted lines can be fllled in as above,Mixed strategies are beyond the level required here.
Production | Games 9
The Stag Hunt
Best responses in the stag hunt are again underlined,Below it is a best response diagram.
S R
S 5
5
4
0
R 0
4
3
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Player 2
Player 1
BR2
BR1
S
R
S R
The dashed lines indicate BR2,Notice there are three Nash equilibria | two pure strategy Nash equilibria and one
\hidden" mixed strategy equilibrium,In any discrete action space game there is at least one Nash equilibrium.
Production | Games 10
Road Congestion
How can game theory be applied to economic problems? Consider the following road congestion game,n students
go to lectures,Should they drive (D) or take the bus (B),given they want to get there as fast as possible?
If they all drive,tra–c congestion implies it takes 45 minutes to get to the lecture.
If they all take the bus,the road is clear and it only takes 30 minutes.
If everyone else takes the bus and a single student takes the car,the one in the car ies to work (no other cars on
the road) in 15 minutes,If everyone else takes the car and a single student takes the bus,the one on the bus takes
60 minutes | all those cars and all those bus-stops.
Everyone Else
Me
B D
B 30 min 60 min
D 15 min 45 min
This is a generalisation of the prisoners’ dilemma,The student will drive,In fact it is dominant to do so,All the
other students will arrive at the same decision,Everyone takes the car.
Wouldn’t it be nice if everyone took the bus?
Production | Games 11
Dynamic Games
Suppose an incumbent monopolist is facing a possible entrant to a market,Such an action would reduce the
monopolist’s proflt,This is an example of a dynamic game,The entrant makes a decision whether to enter before
the monopolist can react,A simple game to model this situation is the entry deterrence game below.
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E
M
Fight
Don’t Fight
Enter
Don’t Enter (0;5)
(1;2)
(?1;0)
The monopolist,M,makes a \threat",If the new flrm,E enters,M threatens to \flght",(Perhaps ood the market
with cheap goods),The entrant will get no proflt and will lose the cost of starting up (1).
If M \doesn’t flght",they split the market equally,E still loses a start up cost of 1.
If E stays out of the market,they make zero proflt,M would make monopoly proflts of 5.
But this is not a credible threat,Why not?
Production | Games 12
Oligopoly
An industry with a few price-setting flrms is an oligopoly,If there are just two flrms,it is a duopoly.
Four difierent examples will be considered (with two flrms for simplicity):
1,Cournot,Firms simultaneously choose quantities to maximise proflts.
2,Bertrand,Firms simultaneously choose prices to maximise proflts.
3,Stackelberg,Firms sequentially choose quantities to maximise proflts.
4,Price Leadership,Firms sequentially choose prices to maximise proflts.
Unlike in the non-strategic models so far,the variable that is chosen can have a striking efiect upon the outcome.
Each can be represented as a game,The flrst two are simultaneous move games,the second two dynamic games.
Monopolistic Competition
Monopolistic competition arises when there are a large number of price-setting flrms in an industry with free entry.
Suppose there are n flrms,In the short run,each flrm faces an nth of the market demand curve.
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0
p
y
ACMC
p?
y?
pn(y)
MRn
…
Short run analysis is not very difierent to that of the monopolist,The flrm charges p? for an output of y? and
makes … proflt,The only difierence is that demand is given by pn(y),an nth of total demand.
However,in the long run other flrms are free to enter if incumbents are making a proflt or leave if they make a loss.
Production | Games 2
The Long Run
Firms will continue to enter until there is no incentive for further entry,That is,when proflts are driven to zero.
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.
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.............................
0
p
y
ACMC
p?
y?
pm(y)
MRm
As more flrms enter demand is reduced to each of the incumbent flrms,Suppose once m flrms are in the industry
proflts for each flrm are zero,They will produce y? each at a price of p? in the long run.
Each flrm is proflt maximising,but receiving zero proflt,They are average cost pricing,This is still ine–cient,A
perfectly competitive flrm would produce at p = MC,which is a lower price and higher output.
Production | Games 3
Strategic Behaviour
The problem with this analysis is that flrms ignore the behaviour of their competitors.
Consider the short run and suppose there were just two flrms,Suppose D(p)=1? p and MC =0 for simplicity.
If flrm A operates in monopolistic competition,they think they face half the demand,DA(p)=(1? p)=2,They
produce at MR = MC,The inverse demand curve is p =1?2yA,So MR =1?4yA,They produce at 1?4y?A =0.
This gives,y?A =1=4,Suppose flrm B takes this into consideration,Then the actual inverse demand B faces is
1? y =1? yA? yB =1?1=4? yB and not 1?2yB,Proflt maximisation will yield a difierent quantity.
In fact,flrm B would set MR = MC which yields,3=4?2yB =0 giving y?B =3=8.
Notice if flrm B took monopolistic competition proflts they would get …B =1=8.
With strategic behaviour however,flrm B gets …B =(1?1=4?3=8)3=8=9=64,This is better.
Consider the long run,Is there a way to prevent other flrms from entering and hence maintaining positive proflts?
Production | Games 4
Games
Game theory is the study of strategic behaviour,A game is three things:
1,Players,The individuals (consumers or flrms etc.) involved.
2,Strategies,Each player’s choice set | what they can do.
3,Payofis,The utilities or proflts which depend on the strategies all of the players take.
Anything which has these three features can be described as a game.
Game theory is a useful language for describing all kinds of situations | but that is not all.
Production | Games 5
Some Examples
One of the most famous games is the prisoners’ dilemma,This has two players with two strategies each and four
associated payofis for each player,It can be represented by the two-by-two matrix below.
C D
C 3
3
5
0
D 0
5
1
1
This is called the normal form or strategic form of the game.
All discrete action space games can be represented in such matrices,Action spaces need not be discrete | for
example,flrms set quantities and consumers choose bundles of goods.
Most of the games presented will be simultaneous move games,All players make their decision without observing
the strategies played by other agents.
In many interesting games players do observe some of the actions of others,Firms may make decisions sequentially.
Such dynamic games will be considered very brie y.
Production | Games 6
Best Responses
Suppose a player’s opponent in a game plays a particular strategy,What would the player do? The strategy that
maximises the player’s payofi given their opponent’s play is a best response.
The collection of best responses for each possible action of their opponent is called a best response function | or a
reaction function (or even a best reply function),Plotting this gives the reaction curve.
What are the best responses in the prisoners’ dilemma?
C D
C 3
3
5
0
D 0
5
1
1
Common practice is to underline the payofis corresponding to best responses in such a normal form game.
Production | Games 7
Nash Equilibrium
A best response to a best response to a best response to a,.,is called a Nash equilibrium.
If all the payofis in a particular cell are underlined then the strategies that correspond to that cell are a Nash
equilibrium,Notice equilibrium is a collection of strategies,Sometimes this can be very complicated,Chess?
In the prisoners’ dilemma the Nash equilibrium is fD;Dg.
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
....
.......................................................................................................................................
.......,......,......,......,......,......,......,......,......
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
......
.
Player 2
Player 1
BR2
BR1
C
D
C D
In the above graph the best responses are plotted for each player,These are reaction curves,The Nash equilibrium
is where the two reaction curves intersect,This is a useful method for flnding equilibria.
Along the axis are the possible actions taken by players 1 and 2,BR1 is the best reply function of player 1 and
BR2 is player 2s,They intersect at fD;Dg,the equilibrium,What are the dotted lines for?
Production | Games 8
Mixed Strategies
Suppose a player could \toss a coin" to decide which action to take,What would their opponent do?
Such an action is called a mixed strategy,For example,in the prisoners’ dilemma,with a fair coin,the opponent
would expect to see C half the time and D half the time,They calculate a best response.
If they choose to play C,half the time they would get 3,and half the time they would get 0,If they choose to play
D half the time they would get 5,and half the time they would get 1,Their expected payofis are:
312 +012 = 32 < 3=512 +112
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.,,,,,,,,,,,,,,,,,,2 mixes
half-half
Player 2
Player 1
BR2
BR1
C
D
C D
Therefore they would play D in response to such a mixture,In fact,they would play D in response to any mixture.
Hence the dotted lines can be fllled in as above,Mixed strategies are beyond the level required here.
Production | Games 9
The Stag Hunt
Best responses in the stag hunt are again underlined,Below it is a best response diagram.
S R
S 5
5
4
0
R 0
4
3
3
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...........................................................................................................
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......
.
......
.
......
........,......,......,......,......,......,.....
......,......,.
Player 2
Player 1
BR2
BR1
S
R
S R
The dashed lines indicate BR2,Notice there are three Nash equilibria | two pure strategy Nash equilibria and one
\hidden" mixed strategy equilibrium,In any discrete action space game there is at least one Nash equilibrium.
Production | Games 10
Road Congestion
How can game theory be applied to economic problems? Consider the following road congestion game,n students
go to lectures,Should they drive (D) or take the bus (B),given they want to get there as fast as possible?
If they all drive,tra–c congestion implies it takes 45 minutes to get to the lecture.
If they all take the bus,the road is clear and it only takes 30 minutes.
If everyone else takes the bus and a single student takes the car,the one in the car ies to work (no other cars on
the road) in 15 minutes,If everyone else takes the car and a single student takes the bus,the one on the bus takes
60 minutes | all those cars and all those bus-stops.
Everyone Else
Me
B D
B 30 min 60 min
D 15 min 45 min
This is a generalisation of the prisoners’ dilemma,The student will drive,In fact it is dominant to do so,All the
other students will arrive at the same decision,Everyone takes the car.
Wouldn’t it be nice if everyone took the bus?
Production | Games 11
Dynamic Games
Suppose an incumbent monopolist is facing a possible entrant to a market,Such an action would reduce the
monopolist’s proflt,This is an example of a dynamic game,The entrant makes a decision whether to enter before
the monopolist can react,A simple game to model this situation is the entry deterrence game below.
..................
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..................
..................
..................
..................
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................?
E
M
Fight
Don’t Fight
Enter
Don’t Enter (0;5)
(1;2)
(?1;0)
The monopolist,M,makes a \threat",If the new flrm,E enters,M threatens to \flght",(Perhaps ood the market
with cheap goods),The entrant will get no proflt and will lose the cost of starting up (1).
If M \doesn’t flght",they split the market equally,E still loses a start up cost of 1.
If E stays out of the market,they make zero proflt,M would make monopoly proflts of 5.
But this is not a credible threat,Why not?
Production | Games 12
Oligopoly
An industry with a few price-setting flrms is an oligopoly,If there are just two flrms,it is a duopoly.
Four difierent examples will be considered (with two flrms for simplicity):
1,Cournot,Firms simultaneously choose quantities to maximise proflts.
2,Bertrand,Firms simultaneously choose prices to maximise proflts.
3,Stackelberg,Firms sequentially choose quantities to maximise proflts.
4,Price Leadership,Firms sequentially choose prices to maximise proflts.
Unlike in the non-strategic models so far,the variable that is chosen can have a striking efiect upon the outcome.
Each can be represented as a game,The flrst two are simultaneous move games,the second two dynamic games.
