Cooperative games
Week 10
Ch 8
De nitions
Although agents can coordinate, they are still rational so maximise their own payo
Players N t nl
Core
an 75astiesasian
Or, an action of a grand coalition y is in the core if there exists no coalition S and action x that this
coalition can take such that
ie. it is Pareto e cient
Games in characteristic form
HSEN Us isthevalueofgroups
t aken
Action
bysix I pearls soti
I
ieanaction
coalition
cantareisadivision
ofthevalue
Core HEIR neVIN
ASEN v15
so
Two players
N1,2 v110.5 v120.6 v41.23 1
V11Ov120,411,2311
Core x Y sax1
x villa's
serial0.6
core
Mt x vill0
L exist
doesn't
Three players
11 12,3 vlil1tieN.ir13 4 v12 5 v1233 VIN110
t one sansa
I a sexy
amino axes t
x xe5
x x 4
ng a É
Three player with symmetric payo s
lilb vlijka.IN10
I
the'iastelementifenecoretobe canbeusedtotes ifcoreisempty
invalidated.seyfgYYy
, Allocation of objects
1141 n M1 m e strictpreferencesof ioverM
trade
coalition whenasubset players o bjects
is of
amatching MN Moo ieagent givena
either orn ot
house
tinsuchthat MilMl
Serial dictatorship algorithm
• Rank the agents
• The rst agent picks their preferred object,
• Remove their choice and the next agent chooses
Proof by contradiction that this is a core:
• Suppose agent 1 prefers object 1, which was unavailable to them, to the object 2 matched to
them
• The person who chose object 1
◦ Object 1 was the best object in their view, so giving them object 2 would decrease their
welfare
◦ Preferred another object that was unavailable to them
‣ This would require giving them the object unavailable to them
‣ The chain of improvements would continue until at least one person, the person ranked
rst, received their best choice
For any core allocation there is at least one person who is allocated their rst best choice, proof by
contradiction:
• Imagine a core allocation in which no agents have their favourite object, and they all indicate
whose object they’d prefer
• There is a trading cycle which would be a Pareto improvement
• This is a contradiction, as the initial allocation cannot be in the core if there can be a Pareto
improvement
Any Pareto e cient outcome can be generated by a serial dictatorship mechanism, proof:
• From a Pareto e cient allocation, select all agents whose receive their rst best choice (these
exist due to proof above) and assign them top rank
• Remove them and their objects, and the allocation is also Pareto e cient
• Repeat this with the remaining agents having their best of the objects remaining, until there are no
agents left
• Applying serial dictatorship allocation with the resulting ranking will obtain the allocation in
question
Allocation of objects with endowments- top trading cycles
N t n M 1 n InitialmatchingMo
Core
Misinthecoreif ISenFmforSMos tiesMitMit e.g
I
i
Top trading cycles: 4 5
s 7 Rounds
• Agents indicate their favourite objects
• Find all cycles and trade (cycles always exist due to graph theory)
3 6 rounds
• Remove everyone who traded 6 1
Week 10
Ch 8
De nitions
Although agents can coordinate, they are still rational so maximise their own payo
Players N t nl
Core
an 75astiesasian
Or, an action of a grand coalition y is in the core if there exists no coalition S and action x that this
coalition can take such that
ie. it is Pareto e cient
Games in characteristic form
HSEN Us isthevalueofgroups
t aken
Action
bysix I pearls soti
I
ieanaction
coalition
cantareisadivision
ofthevalue
Core HEIR neVIN
ASEN v15
so
Two players
N1,2 v110.5 v120.6 v41.23 1
V11Ov120,411,2311
Core x Y sax1
x villa's
serial0.6
core
Mt x vill0
L exist
doesn't
Three players
11 12,3 vlil1tieN.ir13 4 v12 5 v1233 VIN110
t one sansa
I a sexy
amino axes t
x xe5
x x 4
ng a É
Three player with symmetric payo s
lilb vlijka.IN10
I
the'iastelementifenecoretobe canbeusedtotes ifcoreisempty
invalidated.seyfgYYy
, Allocation of objects
1141 n M1 m e strictpreferencesof ioverM
trade
coalition whenasubset players o bjects
is of
amatching MN Moo ieagent givena
either orn ot
house
tinsuchthat MilMl
Serial dictatorship algorithm
• Rank the agents
• The rst agent picks their preferred object,
• Remove their choice and the next agent chooses
Proof by contradiction that this is a core:
• Suppose agent 1 prefers object 1, which was unavailable to them, to the object 2 matched to
them
• The person who chose object 1
◦ Object 1 was the best object in their view, so giving them object 2 would decrease their
welfare
◦ Preferred another object that was unavailable to them
‣ This would require giving them the object unavailable to them
‣ The chain of improvements would continue until at least one person, the person ranked
rst, received their best choice
For any core allocation there is at least one person who is allocated their rst best choice, proof by
contradiction:
• Imagine a core allocation in which no agents have their favourite object, and they all indicate
whose object they’d prefer
• There is a trading cycle which would be a Pareto improvement
• This is a contradiction, as the initial allocation cannot be in the core if there can be a Pareto
improvement
Any Pareto e cient outcome can be generated by a serial dictatorship mechanism, proof:
• From a Pareto e cient allocation, select all agents whose receive their rst best choice (these
exist due to proof above) and assign them top rank
• Remove them and their objects, and the allocation is also Pareto e cient
• Repeat this with the remaining agents having their best of the objects remaining, until there are no
agents left
• Applying serial dictatorship allocation with the resulting ranking will obtain the allocation in
question
Allocation of objects with endowments- top trading cycles
N t n M 1 n InitialmatchingMo
Core
Misinthecoreif ISenFmforSMos tiesMitMit e.g
I
i
Top trading cycles: 4 5
s 7 Rounds
• Agents indicate their favourite objects
• Find all cycles and trade (cycles always exist due to graph theory)
3 6 rounds
• Remove everyone who traded 6 1
