lecture 12: game theory // nash equilibriummauricio-romero.com/pdfs/ecoiv/20201/lecture12.pdflecture...
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![Page 1: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/1.jpg)
Lecture 12: Game Theory // Nash equilibrium
Mauricio Romero
![Page 2: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/2.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 3: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/3.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 4: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/4.jpg)
Beauty contest
I Consider the following game among 100 people. Each individual selects a number,si , between 20 and 60.
I Let a−i be the average of the number selected by the other 99 people. i.e.a−i =
∑j 6=i
sj99 .
I The utility function of the individual i is ui (si , s−i ) = 100− (si − 32a−i )
2
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Beauty contest
I Each individual maximizes his utility, FOC:
−2(si −3
2a−i ) = 0
I Individuals would prefer to select a number that is exactly equal to 1.5 times theaverage of the others
I That is they would like to choose si = 32a−i
I but a−i ∈ [20, 60]
I Therefore si = 20 is dominated by si = 30
![Page 6: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/6.jpg)
Beauty contest
I Each individual maximizes his utility, FOC:
−2(si −3
2a−i ) = 0
I Individuals would prefer to select a number that is exactly equal to 1.5 times theaverage of the others
I That is they would like to choose si = 32a−i
I but a−i ∈ [20, 60]
I Therefore si = 20 is dominated by si = 30
![Page 7: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/7.jpg)
Beauty contest
I Each individual maximizes his utility, FOC:
−2(si −3
2a−i ) = 0
I Individuals would prefer to select a number that is exactly equal to 1.5 times theaverage of the others
I That is they would like to choose si = 32a−i
I but a−i ∈ [20, 60]
I Therefore si = 20 is dominated by si = 30
![Page 8: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/8.jpg)
Beauty contest
I Each individual maximizes his utility, FOC:
−2(si −3
2a−i ) = 0
I Individuals would prefer to select a number that is exactly equal to 1.5 times theaverage of the others
I That is they would like to choose si = 32a−i
I but a−i ∈ [20, 60]
I Therefore si = 20 is dominated by si = 30
![Page 9: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/9.jpg)
Beauty contest
I Each individual maximizes his utility, FOC:
−2(si −3
2a−i ) = 0
I Individuals would prefer to select a number that is exactly equal to 1.5 times theaverage of the others
I That is they would like to choose si = 32a−i
I but a−i ∈ [20, 60]
I Therefore si = 20 is dominated by si = 30
![Page 10: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/10.jpg)
Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
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Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
![Page 12: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/12.jpg)
Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
![Page 13: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/13.jpg)
Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
![Page 14: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/14.jpg)
Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
![Page 15: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/15.jpg)
Beauty contest
I The same goes for any number between 20 (inclusive) and 30 (not included)
I Knowing this, all individuals believe that everyone else will select a numberbetween 30 and 60 (i.e., a−i ∈ [30, 60])
I Playing a number between 30 and 45 (not including) would be strictly dominatedby playing 45
I Knowing this, all individuals believe that everyone else will select a numberbetween 45 and 60 (i.e., a−i ∈ [45, 60])
I 60 would dominate any other selection and therefore all the players select 60.
I The solution by means of iterated elimination of dominated strategies is(60, 60, ..., 60)︸ ︷︷ ︸
100 times
![Page 16: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/16.jpg)
Lecture 12: Game Theory // Nash equilibrium
DominanceWeakly dominated strategies
Nash equilibrium
Some examples
Relationship to dominance
ExamplesCournot CompetitionCartels
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a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I There is no strictly dominated strategy
I However, C always gives at least the same utility to player 1 as B
I It’s tempting to think player 1 would never play C
I However, if player 1 is sure that player two is going to play a he would becompletely indifferent between playing B or C
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a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I There is no strictly dominated strategy
I However, C always gives at least the same utility to player 1 as B
I It’s tempting to think player 1 would never play C
I However, if player 1 is sure that player two is going to play a he would becompletely indifferent between playing B or C
![Page 19: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/19.jpg)
a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I There is no strictly dominated strategy
I However, C always gives at least the same utility to player 1 as B
I It’s tempting to think player 1 would never play C
I However, if player 1 is sure that player two is going to play a he would becompletely indifferent between playing B or C
![Page 20: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/20.jpg)
a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I There is no strictly dominated strategy
I However, C always gives at least the same utility to player 1 as B
I It’s tempting to think player 1 would never play C
I However, if player 1 is sure that player two is going to play a he would becompletely indifferent between playing B or C
![Page 21: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/21.jpg)
Definitionsi weakly dominates s ′i if for all opponent pure strategy profiles, s−i ∈ S−i ,
ui (si , s−i ) ≥ ui (s′i , s−i )
and there is at least one opponent strategy profile s ′′−i ∈ S−i for which
ui (si , s′′−i ) > ui (s
′i , s′′−i ).
![Page 22: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/22.jpg)
I Given the assumptions we have, we can not eliminate a weakly dominated strategy
I Rationality is not enough
I Even so, it sounds “logical” to do so and has the potential to greatly simplify agame
I There is a problem, and that is that the order in which we eliminate the strategiesmatters
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I Given the assumptions we have, we can not eliminate a weakly dominated strategy
I Rationality is not enough
I Even so, it sounds “logical” to do so and has the potential to greatly simplify agame
I There is a problem, and that is that the order in which we eliminate the strategiesmatters
![Page 24: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/24.jpg)
I Given the assumptions we have, we can not eliminate a weakly dominated strategy
I Rationality is not enough
I Even so, it sounds “logical” to do so and has the potential to greatly simplify agame
I There is a problem, and that is that the order in which we eliminate the strategiesmatters
![Page 25: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/25.jpg)
I Given the assumptions we have, we can not eliminate a weakly dominated strategy
I Rationality is not enough
I Even so, it sounds “logical” to do so and has the potential to greatly simplify agame
I There is a problem, and that is that the order in which we eliminate the strategiesmatters
![Page 26: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/26.jpg)
a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I If we eliminate B (C dominates weakly), then a weakly dominates b and we caneliminate b and therefore player 1 would never play A. This leads to the result(C , a).
I If on the other hand, we notice that A is also weakly dominated by C then we caneliminate it in the first round, and this would eliminate a in the second round andtherefore B would be eliminated. This would result in (C , b).
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a b
A 3, 4 4, 3
B 5, 3 3, 5
C 5, 3 4, 3
I If we eliminate B (C dominates weakly), then a weakly dominates b and we caneliminate b and therefore player 1 would never play A. This leads to the result(C , a).
I If on the other hand, we notice that A is also weakly dominated by C then we caneliminate it in the first round, and this would eliminate a in the second round andtherefore B would be eliminated. This would result in (C , b).
![Page 28: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/28.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 29: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/29.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 30: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/30.jpg)
Remember the definition of competitive equilibrium in a market economy.
DefinitionA competitive equilibrium in a market economy is a vector of prices and baskets xisuch that: 1) xi maximizes the utility of each individual given the price vector i.e.
xi = arg maxp cdotxi≤p·wi
u(xi )
2) the markets empty. ∑i
xi =∑i
wi
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I 1) means that given the prices, individuals have no incentive to demand adifferent amount
I The idea is to extend this concept to strategic situations
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I 1) means that given the prices, individuals have no incentive to demand adifferent amount
I The idea is to extend this concept to strategic situations
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Best response
We denote BRi (s−i ) (best response) as the set of strategies of individual i thatmaximize her utility given that other individuals follow the strategy profile s−i .Formally,
DefinitionGiven a strategy profile of opponents s−i , we can define the best response of player i :
BRi (s−i ) = arg maxs′i ∈Si
ui (s′i , s−i ).
I si ∈ BRi (s−i ) if and only if ui (si , s−i ) ≥ ui (s′i , s−i ) for all s ′i ∈ Si
I There could be multiple strategies in BRi (s−i ) but all such strategies give thesame utility to player i if the opponents are indeed playing according to s−i
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Best response
We denote BRi (s−i ) (best response) as the set of strategies of individual i thatmaximize her utility given that other individuals follow the strategy profile s−i .Formally,
DefinitionGiven a strategy profile of opponents s−i , we can define the best response of player i :
BRi (s−i ) = arg maxs′i ∈Si
ui (s′i , s−i ).
I si ∈ BRi (s−i ) if and only if ui (si , s−i ) ≥ ui (s′i , s−i ) for all s ′i ∈ Si
I There could be multiple strategies in BRi (s−i ) but all such strategies give thesame utility to player i if the opponents are indeed playing according to s−i
![Page 35: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/35.jpg)
Best response
We denote BRi (s−i ) (best response) as the set of strategies of individual i thatmaximize her utility given that other individuals follow the strategy profile s−i .Formally,
DefinitionGiven a strategy profile of opponents s−i , we can define the best response of player i :
BRi (s−i ) = arg maxs′i ∈Si
ui (s′i , s−i ).
I si ∈ BRi (s−i ) if and only if ui (si , s−i ) ≥ ui (s′i , s−i ) for all s ′i ∈ Si
I There could be multiple strategies in BRi (s−i ) but all such strategies give thesame utility to player i if the opponents are indeed playing according to s−i
![Page 36: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/36.jpg)
Nash equilibrium
DefinitionSuppose that we have a game (I = {1, 2, . . . n},S1, . . . ,Sn, u1, . . . , un). Then astrategy profile s∗ = (s∗1 , . . . , s
∗n) is a pure strategy Nash equilibrium if for every i and
for every si ∈ Si ,ui (s
∗i , s∗−i ) ≥ ui (si , s
∗−i ).
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Nash equilibrium
DefinitionSuppose that we have a game (I = {1, 2, . . . , n}, S1, . . . ,Sn, u1, . . . , un). Then astrategy profile s∗ = (s∗1 , . . . , s
∗n) is a pure strategy Nash equilibrium if for every i ,
s∗i ∈ BRi (s∗−i ).
I Analogous to that of a competitive equilibrium in the sense that nobody hasunilateral incentives to deviate
I once this equilibrium is reached, nobody has incentives to move from there
I This is a concept of stability, but there is no way to ensure, or predict, that thegame will reach this equilibrium
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Nash equilibrium
DefinitionSuppose that we have a game (I = {1, 2, . . . , n}, S1, . . . ,Sn, u1, . . . , un). Then astrategy profile s∗ = (s∗1 , . . . , s
∗n) is a pure strategy Nash equilibrium if for every i ,
s∗i ∈ BRi (s∗−i ).
I Analogous to that of a competitive equilibrium in the sense that nobody hasunilateral incentives to deviate
I once this equilibrium is reached, nobody has incentives to move from there
I This is a concept of stability, but there is no way to ensure, or predict, that thegame will reach this equilibrium
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Nash equilibrium
DefinitionSuppose that we have a game (I = {1, 2, . . . , n}, S1, . . . ,Sn, u1, . . . , un). Then astrategy profile s∗ = (s∗1 , . . . , s
∗n) is a pure strategy Nash equilibrium if for every i ,
s∗i ∈ BRi (s∗−i ).
I Analogous to that of a competitive equilibrium in the sense that nobody hasunilateral incentives to deviate
I once this equilibrium is reached, nobody has incentives to move from there
I This is a concept of stability, but there is no way to ensure, or predict, that thegame will reach this equilibrium
![Page 40: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/40.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 41: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/41.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 42: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/42.jpg)
Beauty contest
I Consider the following game among 2 people. Each individual selects a number,si , between 20 and 60.
I Let s−i be the number selected by the other individual.
I The utility function of the individual i is ui (si , s−i ) = 100− (si − 32s−i )
2
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Beauty contest
I Consider the following game among 2 people. Each individual selects a number,si , between 20 and 60.
I Let s−i be the number selected by the other individual.
I The utility function of the individual i is ui (si , s−i ) = 100− (si − 32s−i )
2
![Page 44: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/44.jpg)
Beauty contest
I Consider the following game among 2 people. Each individual selects a number,si , between 20 and 60.
I Let s−i be the number selected by the other individual.
I The utility function of the individual i is ui (si , s−i ) = 100− (si − 32s−i )
2
![Page 45: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/45.jpg)
Beauty contest
The best response of an individual is given by
si (s−i )∗ =
{32s−i if s−i ≤ 40
60 if s−i > 40
The Nash equilibrium is where both BR functions intersect (i.e., when both play 60)
![Page 46: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/46.jpg)
Prisoner’s dilemma
C NC
C 5,5 0,10
NC 10,0 2,2
The best response functions are:
BRi (s−i ) =
{NC if s−i = C
NC if s−i = NC
The Nash equilibrium is where both BR functions intersect (i.e., when both play NC,i.e., (NC ,NC ))
![Page 47: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/47.jpg)
Prisoner’s dilemma
C NC
C 5,5 0,10
NC 10,0 2,2
The best response functions are:
BRi (s−i ) =
{NC if s−i = C
NC if s−i = NC
The Nash equilibrium is where both BR functions intersect (i.e., when both play NC,i.e., (NC ,NC ))
![Page 48: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/48.jpg)
Prisoner’s dilemma – A trick
Best response of 1 to 2 playing C
C NC
C 5,5 0,10
NC 10,0 2,2
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Prisoner’s dilemma – A trick
Best response of 1 to 2 playing NC
C NC
C 5,5 0,10
NC 10,0 2,2
![Page 50: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/50.jpg)
Prisoner’s dilemma – A trick
Best response of 2 to 1 playing C
C NC
C 5,5 0,10
NC 10,0 2,2
![Page 51: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/51.jpg)
Prisoner’s dilemma – A trick
Best response of 2 to 1 playing NC
C NC
C 5,5 0,10
NC 10,0 2,2
When underlined for both players, it is a Nash equilibrium (both are doing their BR)
![Page 52: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/52.jpg)
Battle of the sexes
G P
G 2,1 0,0
P 0,0 1,2
![Page 53: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/53.jpg)
Battle of the sexes
G P
G 2,1 0,0
P 0,0 1,2
BRi (s−i ) =
{G if s−i = G
P if s−i = P
Thus, (G ,G ) y (P,P) are both Nash equilibrium
![Page 54: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/54.jpg)
Battle of the sexes
G P
G 2,1 0,0
P 0,0 1,2
BRi (s−i ) =
{G if s−i = G
P if s−i = P
Thus, (G ,G ) y (P,P) are both Nash equilibrium
![Page 55: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/55.jpg)
Matching pennies (Pares o Nones) – Simultaneous
1 2
1 (1000,-1000) (-1000,1000)2 (-1000,1000) (1000,-1000)
![Page 56: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/56.jpg)
Matching pennies (Pares o Nones) – Simultaneous
1 2
1 (1000,-1000) (-1000,1000)2 (-1000,1000) (1000,-1000)
BR1(s2) =
{1 if s2 = 1
2 if s2 = 2
BR2(s1) =
{2 if s1 = 1
1 if s2 = 2
There is no Nash equilibrium in pure strategies
![Page 57: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/57.jpg)
Matching pennies (Pares o Nones) – Simultaneous
1 2
1 (1000,-1000) (-1000,1000)2 (-1000,1000) (1000,-1000)
BR1(s2) =
{1 if s2 = 1
2 if s2 = 2
BR2(s1) =
{2 if s1 = 1
1 if s2 = 2
There is no Nash equilibrium in pure strategies
![Page 58: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/58.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 59: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/59.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 60: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/60.jpg)
Nash equilibrium survive IDSDS
TheoremEvery Nash equilibrium survives the iterative elimination of strictly dominated strategies
![Page 61: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/61.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 62: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/62.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 63: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/63.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 64: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/64.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 65: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/65.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 66: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/66.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 67: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/67.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−i
I And this is a contradiction!
![Page 68: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/68.jpg)
Proof
By contradiction:
I Suppose it is not true
I Then we must have eliminated some strategy in the Nash equilibrium s∗
I Lets zoom in in the round where we first eliminate a strategy that is part of s∗
I Without loss of generality say we eliminated the strategy s∗i of individual i
I It must have been that
ui (s∗i , s−i ) < ui (si , s−i )∀s−i ∈ S−i
.
I In particularui (s
∗i , s−i∗) < ui (si , s
∗−i )
I But this means s∗i is not the best response of individual i to s∗−iI And this is a contradiction!
![Page 69: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/69.jpg)
Nash equilibrium survive IDSDS
TheoremIf the process of IDSDS comes to a single solution, that solution is a Nash Equilibriumand is unique.
![Page 70: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/70.jpg)
Proof
First let’s proof its a Nash Equilibrium. The fact that is unique is trivial by theprevious theorem.
Proof.By contradiction:
I Suppose that the results from IDSDS (s∗) is not a Nash Equilibrium
I For some individual i there exits si such that
ui (si , s∗−i ) > ui (s
∗i , s∗−i )
I But then si could not have been eliminated
I And this is a contradiction!
![Page 71: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/71.jpg)
Proof
First let’s proof its a Nash Equilibrium. The fact that is unique is trivial by theprevious theorem.
Proof.By contradiction:
I Suppose that the results from IDSDS (s∗) is not a Nash Equilibrium
I For some individual i there exits si such that
ui (si , s∗−i ) > ui (s
∗i , s∗−i )
I But then si could not have been eliminated
I And this is a contradiction!
![Page 72: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/72.jpg)
Proof
First let’s proof its a Nash Equilibrium. The fact that is unique is trivial by theprevious theorem.
Proof.By contradiction:
I Suppose that the results from IDSDS (s∗) is not a Nash Equilibrium
I For some individual i there exits si such that
ui (si , s∗−i ) > ui (s
∗i , s∗−i )
I But then si could not have been eliminated
I And this is a contradiction!
![Page 73: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/73.jpg)
Proof
First let’s proof its a Nash Equilibrium. The fact that is unique is trivial by theprevious theorem.
Proof.By contradiction:
I Suppose that the results from IDSDS (s∗) is not a Nash Equilibrium
I For some individual i there exits si such that
ui (si , s∗−i ) > ui (s
∗i , s∗−i )
I But then si could not have been eliminated
I And this is a contradiction!
![Page 74: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/74.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 75: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/75.jpg)
Lecture 12: Game Theory // Nash equilibrium
Dominance
Nash equilibrium
Some examples
Relationship to dominance
Examples
![Page 76: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/76.jpg)
Lecture 12: Game Theory // Nash equilibrium
DominanceWeakly dominated strategies
Nash equilibrium
Some examples
Relationship to dominance
ExamplesCournot CompetitionCartels
![Page 77: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/77.jpg)
Cournot Competition
I We will apply the concept of pure Nash equilibrium to analyze oligopoly markets
I Suppose that there are two firms that produce the same product have zeromarginal cost of production.
I If firm 1 and 2 produce q1 and q2 units of the commodity respectively, the inversedemand function is given by:
P(Q) = 120− Q,Q = q1 + q2.
I Strategy space is Si = [0,+∞)
I The utility function of player i is given by:
π1(q1, q2) = (120− (q1 + q2))q1,
π2(q1, q2) = (120− (q1 + q2))q2.
![Page 78: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/78.jpg)
Cournot Competition
I We will apply the concept of pure Nash equilibrium to analyze oligopoly markets
I Suppose that there are two firms that produce the same product have zeromarginal cost of production.
I If firm 1 and 2 produce q1 and q2 units of the commodity respectively, the inversedemand function is given by:
P(Q) = 120− Q,Q = q1 + q2.
I Strategy space is Si = [0,+∞)
I The utility function of player i is given by:
π1(q1, q2) = (120− (q1 + q2))q1,
π2(q1, q2) = (120− (q1 + q2))q2.
![Page 79: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/79.jpg)
Cournot Competition
I We will apply the concept of pure Nash equilibrium to analyze oligopoly markets
I Suppose that there are two firms that produce the same product have zeromarginal cost of production.
I If firm 1 and 2 produce q1 and q2 units of the commodity respectively, the inversedemand function is given by:
P(Q) = 120− Q,Q = q1 + q2.
I Strategy space is Si = [0,+∞)
I The utility function of player i is given by:
π1(q1, q2) = (120− (q1 + q2))q1,
π2(q1, q2) = (120− (q1 + q2))q2.
![Page 80: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/80.jpg)
Cournot Competition
I We will apply the concept of pure Nash equilibrium to analyze oligopoly markets
I Suppose that there are two firms that produce the same product have zeromarginal cost of production.
I If firm 1 and 2 produce q1 and q2 units of the commodity respectively, the inversedemand function is given by:
P(Q) = 120− Q,Q = q1 + q2.
I Strategy space is Si = [0,+∞)
I The utility function of player i is given by:
π1(q1, q2) = (120− (q1 + q2))q1,
π2(q1, q2) = (120− (q1 + q2))q2.
![Page 81: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/81.jpg)
Cournot Competition
I We will apply the concept of pure Nash equilibrium to analyze oligopoly markets
I Suppose that there are two firms that produce the same product have zeromarginal cost of production.
I If firm 1 and 2 produce q1 and q2 units of the commodity respectively, the inversedemand function is given by:
P(Q) = 120− Q,Q = q1 + q2.
I Strategy space is Si = [0,+∞)
I The utility function of player i is given by:
π1(q1, q2) = (120− (q1 + q2))q1,
π2(q1, q2) = (120− (q1 + q2))q2.
![Page 82: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/82.jpg)
Cournot Competition
I Are there any strictly dominant strategies?
The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 83: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/83.jpg)
Cournot Competition
I Are there any strictly dominant strategies?
The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 84: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/84.jpg)
Cournot Competition
I Are there any strictly dominant strategies? The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 85: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/85.jpg)
Cournot Competition
I Are there any strictly dominant strategies? The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 86: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/86.jpg)
Cournot Competition
I Are there any strictly dominant strategies? The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 87: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/87.jpg)
Cournot Competition
I Are there any strictly dominant strategies? The answer is no, why?
I Are there any strictly dominated strategies?
I The strategies qi ∈ (120,+∞) are strictly dominated by the strategy 0
I Are there any others? given q−i ,
dπidqi
(120− qi − q−i )qi = 120− 2qi − q−i
I Therefore 60 strictly dominates any qi ∈ (60, 120]
![Page 88: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/88.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I for any qi ∈ [0, 60], there exists some q−i ∈ [0,+∞) such that BRi (q−i ) = qi
I Such a qi can never be strictly dominated
I After one round of deletion of strictly dominated strategies, we are left with:Si = [0, 60]
![Page 89: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/89.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I for any qi ∈ [0, 60], there exists some q−i ∈ [0,+∞) such that BRi (q−i ) = qi
I Such a qi can never be strictly dominated
I After one round of deletion of strictly dominated strategies, we are left with:Si = [0, 60]
![Page 90: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/90.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I for any qi ∈ [0, 60], there exists some q−i ∈ [0,+∞) such that BRi (q−i ) = qi
I Such a qi can never be strictly dominated
I After one round of deletion of strictly dominated strategies, we are left with:Si = [0, 60]
![Page 91: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/91.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I for any qi ∈ [0, 60], there exists some q−i ∈ [0,+∞) such that BRi (q−i ) = qi
I Such a qi can never be strictly dominated
I After one round of deletion of strictly dominated strategies, we are left with:Si = [0, 60]
![Page 92: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/92.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [0, 60]
I Therefore qi ∈ [0, 30) are strictly dominated by qi = 30
I After two rounds of deletion of strictly dominated strategies, we are left with:Si = [30, 60]
![Page 93: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/93.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [0, 60]
I Therefore qi ∈ [0, 30) are strictly dominated by qi = 30
I After two rounds of deletion of strictly dominated strategies, we are left with:Si = [30, 60]
![Page 94: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/94.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [0, 60]
I Therefore qi ∈ [0, 30) are strictly dominated by qi = 30
I After two rounds of deletion of strictly dominated strategies, we are left with:Si = [30, 60]
![Page 95: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/95.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [0, 60]
I Therefore qi ∈ [0, 30) are strictly dominated by qi = 30
I After two rounds of deletion of strictly dominated strategies, we are left with:Si = [30, 60]
![Page 96: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/96.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [30, 60]
I 45 strictly dominates all strategies qi ∈ (45, 60]
I After three rounds of deletion of strictly dominated strategies, we are left with:Si = [30, 45]
![Page 97: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/97.jpg)
Cournot Competition
I
BRi (q−i ) =120− q−i
2.
I q−i = [30, 45]
I 37.5 strictly dominates all strategies qi ∈ [30, 37.5]
I After four rounds of deletion of strictly dominated strategies, we are left with:Si = [37.5, 45]
![Page 98: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/98.jpg)
Cournot Competition
I After (infinitely) many iterations, the only remaining strategies are Si = 40
I The unique solution by IDSDS is q∗1 = q∗2 = 40.
![Page 99: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/99.jpg)
Cournot Competition
I There will also be a unique Nash equilibrium
I
BRi (q−i ) =120− q−i
2.
I At any Nash equilibrium, we must have: q∗1 ∈ BR1(q∗2) and q∗2 ∈ BR2(q∗1).
I
q∗1 =120− q∗2
2, q∗2 =
120− q∗12
.
I We can solve for q∗1 and q∗2 to obtain:
q∗1 = 40, q∗2 = 40,Q∗ = 80,Π∗1 = Π∗2 = 1600.
![Page 100: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/100.jpg)
Cournot Competition
I There will also be a unique Nash equilibrium
I
BRi (q−i ) =120− q−i
2.
I At any Nash equilibrium, we must have: q∗1 ∈ BR1(q∗2) and q∗2 ∈ BR2(q∗1).
I
q∗1 =120− q∗2
2, q∗2 =
120− q∗12
.
I We can solve for q∗1 and q∗2 to obtain:
q∗1 = 40, q∗2 = 40,Q∗ = 80,Π∗1 = Π∗2 = 1600.
![Page 101: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/101.jpg)
Cournot Competition
I There will also be a unique Nash equilibrium
I
BRi (q−i ) =120− q−i
2.
I At any Nash equilibrium, we must have: q∗1 ∈ BR1(q∗2) and q∗2 ∈ BR2(q∗1).
I
q∗1 =120− q∗2
2, q∗2 =
120− q∗12
.
I We can solve for q∗1 and q∗2 to obtain:
q∗1 = 40, q∗2 = 40,Q∗ = 80,Π∗1 = Π∗2 = 1600.
![Page 102: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/102.jpg)
Cournot Competition
I There will also be a unique Nash equilibrium
I
BRi (q−i ) =120− q−i
2.
I At any Nash equilibrium, we must have: q∗1 ∈ BR1(q∗2) and q∗2 ∈ BR2(q∗1).
I
q∗1 =120− q∗2
2, q∗2 =
120− q∗12
.
I We can solve for q∗1 and q∗2 to obtain:
q∗1 = 40, q∗2 = 40,Q∗ = 80,Π∗1 = Π∗2 = 1600.
![Page 103: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/103.jpg)
Cournot Competition
I There will also be a unique Nash equilibrium
I
BRi (q−i ) =120− q−i
2.
I At any Nash equilibrium, we must have: q∗1 ∈ BR1(q∗2) and q∗2 ∈ BR2(q∗1).
I
q∗1 =120− q∗2
2, q∗2 =
120− q∗12
.
I We can solve for q∗1 and q∗2 to obtain:
q∗1 = 40, q∗2 = 40,Q∗ = 80,Π∗1 = Π∗2 = 1600.
![Page 104: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/104.jpg)
Cournot Competition vs Monopoly (cartel)
I In a perfectly competitive market, price equals marginal cost and the totalquantity produced will be Q = 120.
I A monopolist would solve the following maximization problem:
maxQ
(120− Q)Q ⇒ Q∗ = 60,P∗ = 60,Πm = 3600.
I The profits to each firm in the Cournot Competition is less than half of themonopoly profits
I In a duopoly, externalities are imposed on the other firm
![Page 105: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/105.jpg)
Cournot Competition vs Monopoly (cartel)
I In a perfectly competitive market, price equals marginal cost and the totalquantity produced will be Q = 120.
I A monopolist would solve the following maximization problem:
maxQ
(120− Q)Q ⇒ Q∗ = 60,P∗ = 60,Πm = 3600.
I The profits to each firm in the Cournot Competition is less than half of themonopoly profits
I In a duopoly, externalities are imposed on the other firm
![Page 106: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/106.jpg)
Cournot Competition vs Monopoly (cartel)
I In a perfectly competitive market, price equals marginal cost and the totalquantity produced will be Q = 120.
I A monopolist would solve the following maximization problem:
maxQ
(120− Q)Q ⇒ Q∗ = 60,P∗ = 60,Πm = 3600.
I The profits to each firm in the Cournot Competition is less than half of themonopoly profits
I In a duopoly, externalities are imposed on the other firm
![Page 107: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/107.jpg)
Cournot Competition vs Monopoly (cartel)
I In a perfectly competitive market, price equals marginal cost and the totalquantity produced will be Q = 120.
I A monopolist would solve the following maximization problem:
maxQ
(120− Q)Q ⇒ Q∗ = 60,P∗ = 60,Πm = 3600.
I The profits to each firm in the Cournot Competition is less than half of themonopoly profits
I In a duopoly, externalities are imposed on the other firm
![Page 108: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/108.jpg)
Cournot Competition - General case
I n firms are competing a la Cournot
I The inverse demand function is given by:
P(q1 + q2 + · · · qn).
I Suppose that the cost function is ci (qi ) for firm i
I To simplify notation, let Q−i =∑
j 6=i qj
Imaxqi
p(qi + Q−i )qi − ci (qi )
![Page 109: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/109.jpg)
Cournot Competition - General case
I n firms are competing a la Cournot
I The inverse demand function is given by:
P(q1 + q2 + · · · qn).
I Suppose that the cost function is ci (qi ) for firm i
I To simplify notation, let Q−i =∑
j 6=i qj
Imaxqi
p(qi + Q−i )qi − ci (qi )
![Page 110: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/110.jpg)
Cournot Competition - General case
I n firms are competing a la Cournot
I The inverse demand function is given by:
P(q1 + q2 + · · · qn).
I Suppose that the cost function is ci (qi ) for firm i
I To simplify notation, let Q−i =∑
j 6=i qj
Imaxqi
p(qi + Q−i )qi − ci (qi )
![Page 111: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/111.jpg)
Cournot Competition - General case
I n firms are competing a la Cournot
I The inverse demand function is given by:
P(q1 + q2 + · · · qn).
I Suppose that the cost function is ci (qi ) for firm i
I To simplify notation, let Q−i =∑
j 6=i qj
Imaxqi
p(qi + Q−i )qi − ci (qi )
![Page 112: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/112.jpg)
Cournot Competition - General case
I n firms are competing a la Cournot
I The inverse demand function is given by:
P(q1 + q2 + · · · qn).
I Suppose that the cost function is ci (qi ) for firm i
I To simplify notation, let Q−i =∑
j 6=i qj
Imaxqi
p(qi + Q−i )qi − ci (qi )
![Page 113: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/113.jpg)
Cournot Competition - General case
maxqi
p(qi + Q−i )qi − ci (qi )
I First order condition implies:
qidP
dQ(qi + Q−i ) + P(qi + Q−i ) =
dcidqi
(qi )
qidP
dQ(Q) + P(Q) =
dcidqi
(qi )
P(Q)− dcidqi
(qi ) = −qidP
dQ(Q)
P(Q)− dcidqi
(qi )
P(Q)= −qi
Q
Q
P(Q)
dP
dQ(Q)
P(Q)− dcidqi
(qi )
P(Q)= −qi
Q
1
εQ,P(Q)
![Page 114: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/114.jpg)
Cournot Competition - General case
P(Q)− dcidqi
(qi )
P(Q)= −qi
Q
1
εQ,P(Q)
I Therefore in a pure strategy Nash equilibrium (q∗1 , q∗2 , . . . , q
∗n) with
Q∗ = q∗1 + q∗2 + · · · q∗n, we must have:
P(Q∗)− dc1dq1
(q∗1)
P(Q∗)= − q∗1
Q∗1
εQ,P(Q∗),
P(Q∗)− dc2dq2
(q∗2)
P(Q∗)= − q∗2
Q∗1
εQ,P(Q∗),
...
P(Q∗)− dcndqn
(q∗n)
P(Q∗)= − q∗n
Q∗1
εQ,P(Q∗).
![Page 115: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/115.jpg)
Cournot Competition - General case
I Suppose that all firms have exactly the same cost function c
P(Q∗)− dcdq1
(q∗1)
P(Q∗)= − q∗1
Q∗1
εQ,P(Q∗),
P(Q∗)− dcdq2
(q∗2)
P(Q∗)= − q∗2
Q∗1
εQ,P(Q∗),
...
P(Q∗)− dcdqn
(q∗n)
P(Q∗)= − q∗n
Q∗1
εQ,P(Q∗).
![Page 116: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/116.jpg)
Cournot Competition - General case
I Let us conjecture that there exists a pure strategy Nash equilibrium that issymmetric, in which q∗1 = q∗2 = · · · q∗n = q∗
I In this case Q∗ = nq∗
P(nq∗)− dcdq1
(q∗)
P(nq∗)= −1
n
1
εQ,P(nq∗)
I Rewriting
P(Q∗) =1
1 + 1n
1εQ,P(Q∗)
∂c
dq
(Q∗
n
).
![Page 117: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/117.jpg)
Cournot Competition - General case
I Let us conjecture that there exists a pure strategy Nash equilibrium that issymmetric, in which q∗1 = q∗2 = · · · q∗n = q∗
I In this case Q∗ = nq∗
P(nq∗)− dcdq1
(q∗)
P(nq∗)= −1
n
1
εQ,P(nq∗)
I Rewriting
P(Q∗) =1
1 + 1n
1εQ,P(Q∗)
∂c
dq
(Q∗
n
).
![Page 118: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/118.jpg)
Cournot Competition - General case
I Let us conjecture that there exists a pure strategy Nash equilibrium that issymmetric, in which q∗1 = q∗2 = · · · q∗n = q∗
I In this case Q∗ = nq∗
P(nq∗)− dcdq1
(q∗)
P(nq∗)= −1
n
1
εQ,P(nq∗)
I Rewriting
P(Q∗) =1
1 + 1n
1εQ,P(Q∗)
∂c
dq
(Q∗
n
).
![Page 119: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/119.jpg)
Lecture 12: Game Theory // Nash equilibrium
DominanceWeakly dominated strategies
Nash equilibrium
Some examples
Relationship to dominance
ExamplesCournot CompetitionCartels
![Page 120: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/120.jpg)
CartelsI Suppose there are three firms who face zero marginal costI The inverse demand function is given by:
p(q1 + q2 + q3) = 1− q1 − q2 − q3 = 1− Q
I The first order condition gives
1− 2qi − Q−i = 0 =⇒ qi =1− Q−i
2=⇒ BRi (Q−i ) =
1− Q−i2
.
I In a Nash equilibrium we must have:
q∗1 =1− q∗2 − q∗3
2
q∗2 =1− q∗1 − q∗3
2
q∗3 =1− q∗1 − q∗2
2.
![Page 121: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/121.jpg)
CartelsI Suppose there are three firms who face zero marginal costI The inverse demand function is given by:
p(q1 + q2 + q3) = 1− q1 − q2 − q3 = 1− Q
I The first order condition gives
1− 2qi − Q−i = 0 =⇒ qi =1− Q−i
2=⇒ BRi (Q−i ) =
1− Q−i2
.
I In a Nash equilibrium we must have:
q∗1 =1− q∗2 − q∗3
2
q∗2 =1− q∗1 − q∗3
2
q∗3 =1− q∗1 − q∗2
2.
![Page 122: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/122.jpg)
CartelsI Suppose there are three firms who face zero marginal costI The inverse demand function is given by:
p(q1 + q2 + q3) = 1− q1 − q2 − q3 = 1− Q
I The first order condition gives
1− 2qi − Q−i = 0 =⇒ qi =1− Q−i
2=⇒ BRi (Q−i ) =
1− Q−i2
.
I In a Nash equilibrium we must have:
q∗1 =1− q∗2 − q∗3
2
q∗2 =1− q∗1 − q∗3
2
q∗3 =1− q∗1 − q∗2
2.
![Page 123: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/123.jpg)
Cartels
I The easiest way to solve this first, let us add the three equations to get:
Q∗ =3
2− Q∗ =⇒ Q∗ =
3
4.
I Note that
q∗1 =1
2− q∗2 − q∗3
2=⇒ q∗1
2=
1
2− Q∗
2=⇒ q∗1 =
1
4.
I q∗1 = q∗2 = q∗3 = 14
I Price is p∗ = 1/4 and all firms get the same profits of 1/16
![Page 124: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/124.jpg)
Cartels
I The easiest way to solve this first, let us add the three equations to get:
Q∗ =3
2− Q∗ =⇒ Q∗ =
3
4.
I Note that
q∗1 =1
2− q∗2 − q∗3
2=⇒ q∗1
2=
1
2− Q∗
2=⇒ q∗1 =
1
4.
I q∗1 = q∗2 = q∗3 = 14
I Price is p∗ = 1/4 and all firms get the same profits of 1/16
![Page 125: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/125.jpg)
Cartels
I The easiest way to solve this first, let us add the three equations to get:
Q∗ =3
2− Q∗ =⇒ Q∗ =
3
4.
I Note that
q∗1 =1
2− q∗2 − q∗3
2=⇒ q∗1
2=
1
2− Q∗
2=⇒ q∗1 =
1
4.
I q∗1 = q∗2 = q∗3 = 14
I Price is p∗ = 1/4 and all firms get the same profits of 1/16
![Page 126: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/126.jpg)
Cartels
I The easiest way to solve this first, let us add the three equations to get:
Q∗ =3
2− Q∗ =⇒ Q∗ =
3
4.
I Note that
q∗1 =1
2− q∗2 − q∗3
2=⇒ q∗1
2=
1
2− Q∗
2=⇒ q∗1 =
1
4.
I q∗1 = q∗2 = q∗3 = 14
I Price is p∗ = 1/4 and all firms get the same profits of 1/16
![Page 127: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/127.jpg)
Cartels
I Two of the firms merge into firm A, while one of the firms remains single, callthat firm B
I Each firm then again faces the profit maximization problem:
maxqi
(1− qi − q−i )qi =⇒ BRi (q−i ) =1− q−i
2.
I Therefore
q∗A =1− q∗B
2
q∗B =1− q∗A
2.
![Page 128: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/128.jpg)
Cartels
I Two of the firms merge into firm A, while one of the firms remains single, callthat firm B
I Each firm then again faces the profit maximization problem:
maxqi
(1− qi − q−i )qi =⇒ BRi (q−i ) =1− q−i
2.
I Therefore
q∗A =1− q∗B
2
q∗B =1− q∗A
2.
![Page 129: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/129.jpg)
Cartels
I Two of the firms merge into firm A, while one of the firms remains single, callthat firm B
I Each firm then again faces the profit maximization problem:
maxqi
(1− qi − q−i )qi =⇒ BRi (q−i ) =1− q−i
2.
I Therefore
q∗A =1− q∗B
2
q∗B =1− q∗A
2.
![Page 130: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/130.jpg)
Cartels
I Solving this:
q∗A = q∗B =1
3.
I The price is then p∗ = 1/3
I If the profits are shared equally among firms 1 and 2 who have merged, thenprofits of firms 1 and 2 are 1/18 whereas firm 3 obtains a profit of 1/9
I Firms 1 and 2 suffered, while firm 3 is better off!
I Firm 3 is obtaining a disproportionate share of the joint profits (more than 1/3)
![Page 131: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/131.jpg)
Cartels
I Solving this:
q∗A = q∗B =1
3.
I The price is then p∗ = 1/3
I If the profits are shared equally among firms 1 and 2 who have merged, thenprofits of firms 1 and 2 are 1/18 whereas firm 3 obtains a profit of 1/9
I Firms 1 and 2 suffered, while firm 3 is better off!
I Firm 3 is obtaining a disproportionate share of the joint profits (more than 1/3)
![Page 132: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/132.jpg)
Cartels
I Solving this:
q∗A = q∗B =1
3.
I The price is then p∗ = 1/3
I If the profits are shared equally among firms 1 and 2 who have merged, thenprofits of firms 1 and 2 are 1/18 whereas firm 3 obtains a profit of 1/9
I Firms 1 and 2 suffered, while firm 3 is better off!
I Firm 3 is obtaining a disproportionate share of the joint profits (more than 1/3)
![Page 133: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/133.jpg)
Cartels
I Solving this:
q∗A = q∗B =1
3.
I The price is then p∗ = 1/3
I If the profits are shared equally among firms 1 and 2 who have merged, thenprofits of firms 1 and 2 are 1/18 whereas firm 3 obtains a profit of 1/9
I Firms 1 and 2 suffered, while firm 3 is better off!
I Firm 3 is obtaining a disproportionate share of the joint profits (more than 1/3)
![Page 134: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/134.jpg)
Cartels
I Solving this:
q∗A = q∗B =1
3.
I The price is then p∗ = 1/3
I If the profits are shared equally among firms 1 and 2 who have merged, thenprofits of firms 1 and 2 are 1/18 whereas firm 3 obtains a profit of 1/9
I Firms 1 and 2 suffered, while firm 3 is better off!
I Firm 3 is obtaining a disproportionate share of the joint profits (more than 1/3)
![Page 135: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/135.jpg)
Cartels
I You might expect that 3 may want to join the cartel as well...
I In the monopolist problem, we solve:
maxQ
(1− Q)Q =⇒ Q∗ =1
2.
I Total profits then are given by 14 which means that each firm obtains a profit of
112 <
19
I Firm 3 clearly wants to stay out
![Page 136: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/136.jpg)
Cartels
I You might expect that 3 may want to join the cartel as well...
I In the monopolist problem, we solve:
maxQ
(1− Q)Q =⇒ Q∗ =1
2.
I Total profits then are given by 14 which means that each firm obtains a profit of
112 <
19
I Firm 3 clearly wants to stay out
![Page 137: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/137.jpg)
Cartels
I You might expect that 3 may want to join the cartel as well...
I In the monopolist problem, we solve:
maxQ
(1− Q)Q =⇒ Q∗ =1
2.
I Total profits then are given by 14 which means that each firm obtains a profit of
112 <
19
I Firm 3 clearly wants to stay out
![Page 138: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/138.jpg)
Cartels
I You might expect that 3 may want to join the cartel as well...
I In the monopolist problem, we solve:
maxQ
(1− Q)Q =⇒ Q∗ =1
2.
I Total profits then are given by 14 which means that each firm obtains a profit of
112 <
19
I Firm 3 clearly wants to stay out
![Page 139: Lecture 12: Game Theory // Nash equilibriummauricio-romero.com/pdfs/EcoIV/20201/Lecture12.pdfLecture 12: Game Theory // Nash equilibrium Dominance Nash equilibrium Some examples Relationship](https://reader030.vdocuments.us/reader030/viewer/2022041014/5ec4b08bf9fcea01e44db2d6/html5/thumbnails/139.jpg)
Cartels
There are many ifficulties associated with sustaining collusive agreements (e.g., theOPEC cartel)