Arrow’s Impossibility Theorem

•Game Theory•Social Choice Theory

Jess Barak

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 offIndependence 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

Ice Cream Flavor Preference

Group Vanilla Chocolate Strawberry

X 1 2 3

Y 2 3 1

Z 3 1 2

Social preference for the three ice cream flavors, vanilla, chocolate and strawberry

• 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/wiki100k/docs/Arrow_s_impossibility_theorem.html

• http://www.econport.org/econport/request?page=man_gametheory_intro1

• http://www.academicroom.com/topics/social-choice-theory