Repeated Games: The Prisoner's Dilemmachristosaioannou.com/Repeated Games The Prisoners Dilemma...In...
Transcript of Repeated Games: The Prisoner's Dilemmachristosaioannou.com/Repeated Games The Prisoners Dilemma...In...
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Repeated Games: ThePrisoner’s Dilemma
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Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
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Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
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Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
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Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
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Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
-
Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
-
Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
-
Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
-
Examples of Prisoner’s Dilemmas
Christos A. Ioannou2/20
• In the Prisoner’s Dilemma,• both players have an incentive to
cheat, and• everyone is better off if no one
cheats.
• Consider firms contemplating whetherto advertise,
• or individuals contemplating whether touse steroids,
• or firms contemplating whether topollute.
Can cooperation emerge without external enforcement?
Cam
el
Malboro
Don
’tA
dver
tise
Don’t Advertise
4,4 1,7-c
7-c,1 4-c,4-c
Sos
a
McGwire
Cle
anS
tero
ids
Clean Steroids
4,4 1,7-c
7-c,1 4-c,4-c
Cou
ntry
1
Country 2
Don
’tP
ollu
te
Don’t Pollute
4,4 1-c,7-c
7-c,1-c 4-2c,4-2c
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Repeated Game Example
Christos A. Ioannou3/20
• The unique Nash equilibrium is (D,D).
• Consider the strategy Grim-Trigger (GT) where,if the other player chooses D once, then, youplay D forever, otherwise you play C.
• What should player 2 do if player 1 plays GT?
C D
C
D
2 2 0 3
3 0 1 1
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Discounting
• Players discount their payoffs with discount factorδ ∈ (0, 1).
• This says, that I care more about my payoff in period 1than in period 1,000.
• Let(a1, a2, . . . , aT
)be the choices for T periods; the
player’s discounted payoff is then,
ui(a1)+δui
(a2)+δ2ui
(a3)+· · ·+δT−1ui
(aT)=
T∑t=1
δt−1ui(at).
Christos A. Ioannou4/20
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Discounting
• Players discount their payoffs with discount factorδ ∈ (0, 1).• This says, that I care more about my payoff in period 1
than in period 1,000.
• Let(a1, a2, . . . , aT
)be the choices for T periods; the
player’s discounted payoff is then,
ui(a1)+δui
(a2)+δ2ui
(a3)+· · ·+δT−1ui
(aT)=
T∑t=1
δt−1ui(at).
Christos A. Ioannou4/20
-
Discounting
• Players discount their payoffs with discount factorδ ∈ (0, 1).• This says, that I care more about my payoff in period 1
than in period 1,000.
• Let(a1, a2, . . . , aT
)be the choices for T periods; the
player’s discounted payoff is then,
ui(a1)+δui
(a2)+δ2ui
(a3)+· · ·+δT−1ui
(aT)=
T∑t=1
δt−1ui(at).
Christos A. Ioannou4/20
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Normalized Discounted Payoffs
• What is the sum of2 + 2δ + 2δ2 + 2δ3 + · · ·
• The normalized discounted payoffs for action sequence(a1, a2, . . . , aT
)is
Ui(a1, a2, . . . , aT
)= (1− δ)
T∑t=1
δt−1ui(at).
Christos A. Ioannou5/20
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Normalized Discounted Payoffs
• What is the sum of2 + 2δ + 2δ2 + 2δ3 + · · ·
• The normalized discounted payoffs for action sequence(a1, a2, . . . , aT
)is
Ui(a1, a2, . . . , aT
)= (1− δ)
T∑t=1
δt−1ui(at).
Christos A. Ioannou5/20
-
Normalized Discounted Payoffs
• What is the sum of2 + 2δ + 2δ2 + 2δ3 + · · ·
• The normalized discounted payoffs for action sequence(a1, a2, . . . , aT
)is
Ui(a1, a2, . . . , aT
)= (1− δ)
T∑t=1
δt−1ui(at).
Christos A. Ioannou5/20
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Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,• the set of terminal histories is the set of sequences(
a1, a2, . . . , aT)
of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
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Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,
• the set of terminal histories is the set of sequences(a1, a2, . . . , aT
)of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
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Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,• the set of terminal histories is the set of sequences(
a1, a2, . . . , aT)
of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
-
Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,• the set of terminal histories is the set of sequences(
a1, a2, . . . , aT)
of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
-
Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,• the set of terminal histories is the set of sequences(
a1, a2, . . . , aT)
of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
-
Repeated GameDefinitionLet G be a strategic game. Denote the set of players by Nand the set of actions and payoff function of each player i byAi and ui, respectively. The T -period repeated game of Gfor discount factor δ is the extensive game with perfectinformation and simultaneous moves in which
• the set of players is N ,• the set of terminal histories is the set of sequences(
a1, a2, . . . , aT)
of action profiles in G,
• the player function assigns all players to all histories,
• the set of actions for player i after any history is Ai, and
• each player i evaluates terminal history according tonormalized discounted payoff Ui
(a1, a2, . . . , aT
).
Christos A. Ioannou6/20
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Finitely Repeated Prisoner’s Dilemma
Christos A. Ioannou7/20
• Consider the Prisoner’s Dilemma gamefor T = 2.
• How many terminal histories arethere?
• How many non-terminal histories arethere?
• How many strategies are there?• What are the the Nash equilibria?• What are the subgame Perfect Nash
equilibria?
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Finitely Repeated Prisoner’s Dilemma
Christos A. Ioannou8/20
• Consider the Prisoner’s Dilemma gamefor T > 2.
• How many terminal histories arethere?
• How many non-terminal histories arethere?
• How many strategies are there?• What are the the Nash equilibria?• What are the subgame Perfect Nash
equilibria?
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Strategies in the Infinitely-Repeated
Games• We need to specify an action after every history.
• Recall the Grim-Trigger strategy; thus,
si(a1, . . . , at
)=
C h = ∅C if (a1j , . . . , a
tj) = (C, . . . , C)
D otherwise.
• It is often useful to represent strategies as a finiteautomaton; that is,
C DStart
C
D
Any
Christos A. Ioannou9/20
-
Strategies in the Infinitely-Repeated
Games• We need to specify an action after every history.• Recall the Grim-Trigger strategy; thus,
si(a1, . . . , at
)=
C h = ∅C if (a1j , . . . , a
tj) = (C, . . . , C)
D otherwise.
• It is often useful to represent strategies as a finiteautomaton; that is,
C DStart
C
D
Any
Christos A. Ioannou9/20
-
Strategies in the Infinitely-Repeated
Games• We need to specify an action after every history.• Recall the Grim-Trigger strategy; thus,
si(a1, . . . , at
)=
C h = ∅C if (a1j , . . . , a
tj) = (C, . . . , C)
D otherwise.
• It is often useful to represent strategies as a finiteautomaton; that is,
C DStart
C
D
Any
Christos A. Ioannou9/20
-
Strategies in the Infinitely-Repeated
Games• We need to specify an action after every history.• Recall the Grim-Trigger strategy; thus,
si(a1, . . . , at
)=
C h = ∅C if (a1j , . . . , a
tj) = (C, . . . , C)
D otherwise.
• It is often useful to represent strategies as a finiteautomaton; that is,
C DStart
C
D
Any
Christos A. Ioannou9/20
-
Strategies in the Infinitely-Repeated
Games (Cont.)• Next, we look at a limited punishment strategy.
C D D DStart
C
D Any Any
Any
• Now, consider the popular Tit-For-Tat.
C DStart
C
D
C
D
Christos A. Ioannou10/20
-
Strategies in the Infinitely-Repeated
Games (Cont.)• Next, we look at a limited punishment strategy.
C D D DStart
C
D Any Any
Any
• Now, consider the popular Tit-For-Tat.
C DStart
C
D
C
D
Christos A. Ioannou10/20
-
Strategies in the Infinitely-Repeated
Games (Cont.)• Next, we look at a limited punishment strategy.
C D D DStart
C
D Any Any
Any
• Now, consider the popular Tit-For-Tat.
C DStart
C
D
C
D
Christos A. Ioannou10/20
-
Strategies in the Infinitely-Repeated
Games (Cont.)• Next, we look at a limited punishment strategy.
C D D DStart
C
D Any Any
Any
• Now, consider the popular Tit-For-Tat.
C DStart
C
D
C
D
Christos A. Ioannou10/20
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Examples of Tit-for-Tat
Christos A. Ioannou11/20
High Low
High
Low
20 20 0 35
35 0 10 10
• The low price guarantee scheme works as follows.
1 I set a high price in the first period,
2 if you lower your price, then, I will lower my price too,
3 otherwise, if you keep your price high, I will keep my price high.
-
Feasible Payoffs
Christos A. Ioannou12/20
C D
C
D
2 2 0 3
3 0 1 1
What payoffs are possibleas a Nash equilibrium?
-
Feasible Payoffs
Christos A. Ioannou12/20
C D
C
D
2 2 0 3
3 0 1 1
What payoffs are possibleas a Nash equilibrium?
-
Minmax Payoff
DefinitionPlayer i’s minmax payoff in a strategic game is
ui = mina−i∈A−i
(maxai∈Ai
ui (ai, a−i)
).
• Alternatively, we have
ui = mina−i∈A−i
(BRi (a−i)) .
• If player 2 tries to punish player 1 forever, this is the bestplayer 1 can do.
Christos A. Ioannou13/20
-
Minmax Payoff
DefinitionPlayer i’s minmax payoff in a strategic game is
ui = mina−i∈A−i
(maxai∈Ai
ui (ai, a−i)
).
• Alternatively, we have
ui = mina−i∈A−i
(BRi (a−i)) .
• If player 2 tries to punish player 1 forever, this is the bestplayer 1 can do.
Christos A. Ioannou13/20
-
Minmax Payoff
DefinitionPlayer i’s minmax payoff in a strategic game is
ui = mina−i∈A−i
(maxai∈Ai
ui (ai, a−i)
).
• Alternatively, we have
ui = mina−i∈A−i
(BRi (a−i)) .
• If player 2 tries to punish player 1 forever, this is the bestplayer 1 can do.
Christos A. Ioannou13/20
-
Minmax Payoff
DefinitionPlayer i’s minmax payoff in a strategic game is
ui = mina−i∈A−i
(maxai∈Ai
ui (ai, a−i)
).
• Alternatively, we have
ui = mina−i∈A−i
(BRi (a−i)) .
• If player 2 tries to punish player 1 forever, this is the bestplayer 1 can do.
Christos A. Ioannou13/20
-
Example
Christos A. Ioannou14/20
• What would happen in this game?
• In this game:
• Action A is strictly dominant for Player 1.
• Action C is strictly dominant for Player 2.
• Players will repeatedly play (A,C).
C D
A
B
3,3
1,2
2,1
0,0
-
Example
Christos A. Ioannou14/20
• What would happen in this game?
• In this game:
• Action A is strictly dominant for Player 1.
• Action C is strictly dominant for Player 2.
• Players will repeatedly play (A,C).
C D
A
B
3,3
1,2
2,1
0,0
-
Example
Christos A. Ioannou14/20
• What would happen in this game?
• In this game:
• Action A is strictly dominant for Player 1.
• Action C is strictly dominant for Player 2.
• Players will repeatedly play (A,C).
C D
A
B
3,3
1,2
2,1
0,0
-
Example
Christos A. Ioannou14/20
• What would happen in this game?
• In this game:
• Action A is strictly dominant for Player 1.
• Action C is strictly dominant for Player 2.
• Players will repeatedly play (A,C).
C D
A
B
3,3
1,2
2,1
0,0
-
Example
Christos A. Ioannou14/20
• What would happen in this game?
• In this game:
• Action A is strictly dominant for Player 1.
• Action C is strictly dominant for Player 2.
• Players will repeatedly play (A,C).
C D
A
B
3,3
1,2
2,1
0,0
-
Example (Cont.)
Christos A. Ioannou15/20
1 2 3
1
2
3
Player 1 Payoff
Pla
yer
2P
ayoff
(3, 3)
(2, 1)
(1, 2)
(0, 0)
Feasible Payoffs
Individually Rational Payoffs
Folk Theorem Payoffs
Experimental Data
-
Example (Cont.)
Christos A. Ioannou15/20
1 2 3
1
2
3
Player 1 Payoff
Pla
yer
2P
ayoff
(3, 3)
(2, 1)
(1, 2)
(0, 0)
Feasible Payoffs
Individually Rational Payoffs
Folk Theorem Payoffs
Experimental Data
-
Example (Cont.)
Christos A. Ioannou15/20
1 2 3
1
2
3
Player 1 Payoff
Pla
yer
2P
ayoff
(3, 3)
(2, 1)
(1, 2)
(0, 0)
Feasible Payoffs
Individually Rational Payoffs
Folk Theorem Payoffs
Experimental Data
-
Example (Cont.)
Christos A. Ioannou15/20
1 2 3
1
2
3
Player 1 Payoff
Pla
yer
2P
ayoff
(3, 3)
(2, 1)
(1, 2)
(0, 0)
Feasible Payoffs
Individually Rational Payoffs
Folk Theorem Payoffs
Experimental Data
-
Example (Cont.)
Christos A. Ioannou15/20
1 2 3
1
2
3
Player 1 Payoff
Pla
yer
2P
ayoff
(3, 3)
(2, 1)
(1, 2)
(0, 0)
Feasible Payoffs
Individually Rational Payoffs
Folk Theorem Payoffs
Experimental Data
-
Minmax examples
Christos A. Ioannou16/20
C D
C
D
2 2 0 3
3 0 1 1
C D
C
D
3 3 1 4
4 1 0 0
C D
C
D
1 1 2 4
4 2 1 1
-
Folk Theorem
Christos A. Ioannou17/20
C D
C
D
2 2 0 3
3 0 1 1
Folk Theorem
Any payoff (v1, v2) where both v1 > u1 and v2 > u2,can be supported as a Nash equilibrium if players aresufficiently patient.
-
Folk Theorem
Christos A. Ioannou17/20
C D
C
D
2 2 0 3
3 0 1 1
Folk Theorem
Any payoff (v1, v2) where both v1 > u1 and v2 > u2,can be supported as a Nash equilibrium if players aresufficiently patient.
-
Folk Theorem
Christos A. Ioannou17/20
C D
C
D
2 2 0 3
3 0 1 1
Folk Theorem
Any payoff (v1, v2) where both v1 > u1 and v2 > u2,can be supported as a Nash equilibrium if players aresufficiently patient.
-
Folk Theorem (Cont.)
Christos A. Ioannou18/20
C D
C
D
3 3 1 4
4 1 0 0
-
Folk Theorem (Cont.)
Christos A. Ioannou19/20
C D
C
D
1 1 2 4
4 2 1 1
-
Folk Theorem (Example)
C C D
D
Start
DD
D
C C
Any
C
C C C
D
Start
DD
C
C C
Any
D
Christos A. Ioannou20/20
-
Folk Theorem (Example)
C C D
D
Start
DD
D
C C
Any
C
C C C
D
Start
DD
C
C C
Any
D
Christos A. Ioannou20/20