[EM] Anyone got a good analysis on limitations of approval andrange voting?

[Jonathan Lundell]

On Nov 16, 2009, at 4:58 PM, Raph Frank wrote:

> The theorem states (from wiki) that there is no method which has the
> following properties:
> 
>    * If every voter prefers X over Y, then the group prefers X over Y.
>    * If every voter prefers X over Y, then adding Z to the slate
> won't change the group's preference of X over Y.
>    * There is no dictator.
> 
> All 3 of those conditions are met for range.  The only problem is that
> adding Z could cause renormalisation changes in how people vote.

The problem in applying the theorem to range isn't in the properties; it's in the formal definition of the method and the proof.

http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem#Formal_statement_of_the_theorem

The proof applies to strict social welfare functions, defined as functions mapping a set of linear orderings (by voters) to a social ordering. While a range ballot (without ties) can be interpreted as a linear ordering, the range counting rule is not a function of that ordering, but rather of the cardinal values on the ballot. Different range ballots could have the same ordering but lead to a different result.