Steph wrote:
While I think this is a valid definition, I don't think it represents EITHER the "spirit" of one person, one vote, OR the "equal voter power" idea that you are striving for.Suppose two sets, S1 the set of voters and S2 the set of candidates. Suppose an electoral method that produces scores for each candidate. If you can split S1 in |S2| subsets each of a cardinality equal to the score obtained by the corresponding candidate, you can link these two sets using a bijective mapping. Each voter contributes to one and only one candidate. If an electoral method produces scores that verify this property, it respects reciprocal fairness.
The most simple counter example of this I can think of would be a "fractional voting" system. By this, I mean an election method where each voter gets one vote, which they may split among as many candidates as they like. So, for example, I could give half my vote to one candidate, and a quarter of my vote to two other candidates.
Clearly, such a method would violate your "reciprocal fairness" criterion, and yet it seems obvious that each voter has the same total voting power - namely, one vote. So I don't think your reciprocal fairness criterion really represents the idea of one person, one vote in a meaningful way.
As I've said before, it's very hard to make a meaningful and applicable criterion from 1p1v, because you're really talking about the machinery of the method rather than the results. Your stab at a reciprocal fairness criterion sort of reveals this -- while it's clear that plurality passes and approval and Borda fail, it's unclear (to me, anyway) how you would apply this criteria to IRV or to Condorcet. (If IRV can be construed to pass, then so should approval - since it has been shown that approval can be implemented as a plurality runoff. Again, your criterion seems very dependent on the machinery of the method as oppose to the result.) And I've already shown that the criterion gives unintuitive results for other methods.
Alex already noted that the original idea behind 1p1v is captured by Markus's anonymity criterion, which is passed by any non-probabilistic method. Just to re-state that criterion, it's just that, "swapping voters should not change the result of the election." So every voter, upon stepping into the voting booth, has the same opportunity to make an impact on the election.
Now, I agree that 1p1v can be extended beyond this. But I can't formally state my impression of the "spirit of 1p1v" in a meaningful way. I'll take a stab at a heuristic definition. I stress that this is NOT a formal criterion and I would not expect to be able to consistently apply this to new methods. Anyway, it goes like this:
**There is a maximum impact (called "one vote") that a single voter can have on the final result of an election. If a voter knows the votes of every other voter, the voter can always vote in such a way as to have this maximum impact.**
Again, this is not a formal criterion in any sense. But this is what I think when I hear one person, one vote. So, let's look at some common election methods, and list the "maximum impact" strategy:
plurality: vote for either of the two front-runners.
approval: vote for either, but not both, of the two front-runners.
cardinal rankings: give one front-runner a maximum rating, and one a minimum rating.
IRV: include at least one of the two front-runners in your rankings.
Borda: Fully rank the candidates. Rank one of the two front-runners in last place on your rankings, and one in first place. (Borda violates 1p1v if not all precincts have the same candidate lists, or if some precincts allow write-ins while others do not. Intuitively, I feel Borda DOES violate 1p1v, but formally I can't show it. Again, results versus machinery).
Condorcet: Fully rank the candidates. (I really am not sure how you can apply the concept of 1p1v to Condorcet, given the possibility of circular ties in the preferences. In the end, I think the only way to look at Condorcet in the context of 1p1v is to view each pairwise election individually. With this in mind, fully ranking allows the voter to cast a vote in every pairwise contest.)
So, by my reckoning, every commonly discussed single-winner election method passes 1p1v, although Borda sort of teeters on the edge, and Condorcet doesn't really fit rules of 1p1v at all. Well, that's the best I can do, and I don't think it's particularly meaningful or applicable. Can anyone do better?
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