1. Arrow’s Impossibility Theorem
Jess Barak
Game Theory
Social Choice Theory

2. 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

3. 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

4. History
The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and popularized it in his book Social Choice and Individual Values

5. 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

6. Unrestricted Domain Non-dictatorship
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

7. Independence of Irrelevant Alternatives
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

8. 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

9. 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 Z

10. Implies that vanilla would be preferred to 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.

11. 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

12. References http://www.sjsu.edu/faculty/watkins/arrow.htm
_theorem
k/docs/Arrow_s_impossibility_theorem.html
man_gametheory_intro1
choice-theory