Pareto Optimality
An outcome is said to be Pareto optimal (or Pareto efficient) if there is no other outcome that makes one agent better off without making another agent worse off.
If an outcome is Pareto optimal, then at least one agent will be reluctant to move away from it (because this agent will be worse off).
If an outcome ! is not Pareto optimal, then there is another outcome !1 that makes everyone as happy, if not happier, than !.
“Reasonable” agents would agree to move to !1 from ! if ! is not Pareto optimal and !1 is.
c -Trenn, King’s College London 2
Look
q:
a
X3
X
,
× .
her
> chop
Pareto Optimality
This game has one Pareto efficient outcome, pD, Dq. DC
¥j i3
1 2
0 1
0
There is no solution in which either agent does better.
D5
C
2
c -Trenn, King’s College London 3
Pareto Optimality
This next game has two Pareto effic8ient outcomes, pC, Dq and pD, Cq. j
DC D14
i31 C11
41
Note that Pareto efficiency doesn’t necessarily mean fair.
Just that you can’t move away and make one agent better off without making the other worse off.
Different way of thinking about this: Ignore players and imagine you are buying socks online. All options cost the same
i is the number of stars (rating) and j is the number of pairs you get.
Trade-off: all reasonable options are Pareto efficient
c -Trenn, King’s College London 4
Pareto optimal?
Consider this scenario (again):
j
CD A
1 2
4 3
2 3
3 2
i
B Are there any Pareto optimal outcomes?
c -Trenn, King’s College London
5
Pareto optimal?
Consider this scenario (again):
j
CD A
1 2
4 3
2 3
3 2
i
B
Are there any Pareto optimal outcomes? (A,D)
c -Trenn, King’s College London
6
Pareto Optimality
Pareto optimality is a rather weak concept.
100 0 99
I 2
(coolfunpedia.blogspot.co.uk)
What is the Pareto optimal way to divide a pile of money between player i and player j?
98 973
÷ 4020
c -Trenn, King’s College London
7
Social Welfare
The social welfare of an outcome ! is the sum of the utilities that each agent gets from !:
ÿ uip!q iPAg
Think of it as the “total amount of money in the system”.
As a solution concept, may be appropriate when the whole system (all agents) has a single owner (then overall benefit of the system is important, not individuals).
c -Trenn, King’s College London 8
Social Welfare
As a solution concept it doesn’t consider the benefits to individuals.
(telegraph.co.uk)
A very skewed outcome can maximise social welfare.
c -Trenn, King’s College London 9
Social Welfare
In both these games, pC, Cq maximises social welfare. j
DC D
i
C
2 2
1 1
3 3
4 4
DC D
i
C
j
2 2
1 1
3 3
9 0
c -Trenn, King’s College London
10