Say a vector v in **R**^{n} is nontrivial if

- v
_{i}≠ v_{j}for all i ≠ j - v
_{i}- v_{j}≠ v_{I}- v_{J}for all i,j,I,J so long as i ≠ j, I ≠ J, (i,j) ≠ (I,J).

**R**

^{n}, say v ~ w if for all i,j,I,J,

v

_{i}- v

_{j}< v

_{I}- v

_{J}

if and only if

w

_{i}- w

_{j}< w

_{I}- w

_{J}

The intuition is that v is a scalar vote, supposedly indicating how much the voter likes each of the n candidates, but obviously vulnerable to strategic voting — if you really used **R**, it would be a game of name the biggest number. Range voting, the topic of an interesting (but mildly repetitive) talk I just saw, limits this to some interval. Nonetheless, it is vulnerable to strategic exageration, and reduces, with a completely strategic electorate, to approval voting. I got curious about the properties of systems that make dishonesty harder, but I haven't checked the literature, so I may be naively reinventing wheels here.

The idea of the equivalence relation ~ is that it throws away a lot of information, but not as much as ordinal voting as found in Condorcet and other systems. It retains the ordering of candidates, and also an ordering of the gaps between candidates. Some questions:

- [probably just some combinatorics] How many ~-equivalence classes are there in
**R**^{n}? Doing some hand calculations I count 1,1,2,12,240 for n=0..4, which sadly I didn't find in the encyclopedia of integer sequences. - What's a better way to define these classes intrinsically instead of by reference to vectors in
**R**? - Are there any voting systems that take in one such ~-equivalence class from each voter, and output a social choice of the same type, and satisfy suitable IIR, monotonicity, and no-dictator axioms?

ETA: Man, I wish I knew more combinatorics. this stuff by Brignall, Ruskuc, and Vatter looks really neat.