Transcript Arrow’s Impossibility Theorem

Arrow’s Impossibility Theorem
Jess Barak
•Game Theory
•Social Choice Theory
Game Theory
• Three basic elements of any game:
▫ Set of players or participants
▫ Moves or actions each player makes
▫ Scores or payoffs that each player earns at the end
Social Choice Theory
• The theory of analyzing a decision between a
collection of alternatives made by a collection
of n voters with separate opinions. Any choice
for the entire group should reflect the desires of
the individual voters to the extent possible.
• Kenneth Arrow's Social Choice and Individual
Values and Arrow's impossibility theorem are
acknowledged as the basis of the modern social
choice theory
History
• The theorem is named after economist Kenneth
Arrow, who demonstrated the theorem in his
Ph.D. thesis and popularized it in his 1951
book Social Choice and Individual Values
Arrow’s Impossibility Theorem
• When voters have three or more discrete
alternatives (options), no voting system can
convert the ranked preferences of individuals
into a community-wide ranking while also
meeting a certain set of criteria:
▫
▫
▫
▫
Unrestricted domain
Non-dictatorship
Pareto efficiency
Independence of irrelevant alternatives
Unrestricted Domain
• All preferences of all voters (but no other
considerations) are allowed
Non-dictatorship
• Results can’t mirror that of any single person's
preferences without consideration of the other
voters
Pareto Efficiency
• State of allocation of resources in which it is
impossible to make any one individual better off
without making at least one individual worse off
Independence of Irrelevant
Alternatives
• Social preferences between multiple options
depend only on the individual preferences
between those options
• The theorem proves that no voting system can be
designed that satisfies these three "fairness"
criteria:
▫ If every voter prefers alternative X over alternative
Y, then the group prefers X over Y
▫ If every voter's preference between X and Y
remains unchanged, then the group's preference
between X and Y will also remain unchanged
▫ There is no "dictator": no single voter possesses
the power to always determine the group's
preference
Social preference for the three ice cream flavors,
vanilla, chocolate and strawberry
Ice Cream Flavor Preference
Group
Vanilla
Chocolate
Strawberry
X
1
2
3
Y
2
3
1
Z
3
1
2
• In a choice between vanilla and chocolate, X votes
for vanilla, Y votes for vanilla and Z votes for
chocolate. Vanilla is socially preferred to chocolate.
• In a choice between chocolate and strawberry X
votes for chocolate, Y votes for strawberry and Z
votes for chocolate. Chocolate is preferred to
strawberry.
▫ Implies that vanilla would be preferred to
strawberry.
 Choice between vanilla and strawberry, X votes for
vanilla, Y votes for strawberry and Z votes for
strawberry. So strawberry is socially preferred to
vanilla.
• Thus we have the irrational result that socially
vanilla is preferred to chocolate and chocolate is
preferred to strawberry but strawberry is
preferred to vanilla
▫ Transitivity does not work
References
• http://www.sjsu.edu/faculty/watkins/arrow.htm
• http://en.wikipedia.org/wiki/Arrow's_impossibility
_theorem
• http://www.princeton.edu/~achaney/tmve/wiki100
k/docs/Arrow_s_impossibility_theorem.html
• http://www.econport.org/econport/request?page=
man_gametheory_intro1
• http://www.academicroom.com/topics/socialchoice-theory