I'm not sure the above writeup gives the full story, so allow me to jump in.
As stated above, Arrow puts forth 4 burdens
on an ideal voting system for a democracy
- Universal Domain - The system should be able to contain any possible social choice
- Independence of Irrelevent Alternatives - An individual preference stated in binary terms (one over the other) shouldn't affect his/her or other's ordering of other preferences. Ex: Liberal voters in America considering the choice between Ralph Nader and Al Gore shouldn't have to worry about inadvertently electing George W. Bush with their vote for Nader.
- Pareto Optimality - If every individual votes the same way, the society is decisive.
- Nondictatorship - More than one person's vote should be used in determining social choice
The Arrow Theorem, in simple terms, says that no voting system can satisfy all 4 above criteria
. The only way voting systems work, then, is by relaxing one of the criteria (which in practice is usually universal
The proof of the theorem
is rather simple and elegant
. It makes use of two principles:
The problem? If each larger group is subdivided into smaller groups of identical preference
, the second principle says that group can be subdivided as well. Thus, when the process is over we arrive at the vote of one person
. This person is the dictator
of the system, thus fail
ing the fourth criterion.
Sen, Amartya. Rationality and Social Choice. American Economic Review, March 1995. Vol 85, No. 1