We try to give a short analysis of the relations between the representations we described. We define what it means for one representation to be finer than another.
Definition 4.2.1.We say that a representationSisfinerthan a representation T if there are functions fnS,T : Sn → Tn satisfying fnS,T(fnS(P)) = fnT(P) for
anyn∈Nandnvoter profileP.
Notice that the definition implies thatfS,T
n is onto. It also follows from the
definition that, naturally, the set of profiles P is the finest representation. If also follows that ifA is finer than B and B is finer thanC, we get that A is finer thanC, since we can definefnA,C =fnB,C◦fnA,B, and we get that for any
nandP:
fnA,C(fnA(P)) =fnB,C(fnA,B(fnA(P))) =fnB,C(fnB(P)) =fnC(P)
From this, together with some relations we will soon prove, we will get Diagram 4.1, where an arrow from S to T means that S is finer than T (we present only arrows between adjacent representations, and not arrows we get by composition).
We prove the relations in the diagram hold:
Proposition 4.2.2. We have the following relations between the representation introduced so far:
1. P is finer than AP 2. AP is finer than ABOVE.
P
AP ABOVE
WMG WSM
PSR
Figure 4.1: Hierarchy of representations for common voting rules. An arrow fromAtoB means that Ais finer thanB.
3. ABOVE is finer than WMG. 4. ABOVE is finer than WSM. 5. WSM is finer than PSR.
Proof. As we said, P is always the finest representation, so Claim 1 is trivial. We prove the other cases. In all cases, our function fA,B
m will simply be fnB◦
gA
n, and so we have to show that for any profile P, fnB(gAn(fnA(P))) = fnB(P).
Alternatively, we can show that whenever two profiles P, P0 satisfy fA n(P) =
fA
n(P0), we havefnB(P) =fnB(P0). To see that this is equivalent, denoteP0 =
gA
n(fnA(P)). From the definition of representation it follows that fnA(P0) =
fA
n(P), and from this follows the equivalence.
2. It’s simple to see that given a profileP, the order of the ballots has no affect onabove(P, Z, x), and so fnABOV E(gnAP(fnAP(P))) =fnABOV E(P). 3. We need to show that forP, P0 such thatfnABOV E(P) =fnABOV E(P0) we
haveN(x, y, P) =N(x, y, P0). This is the case, because
N(x, y, P) = X Z⊂X, x∈Z above(P, Z, y) = X Z⊂X, x∈Z above(P0, Z, y) =N(x, y, P0)
4. It’s enough to show that for any two profilesP, P0such thatabove(P, Z, xi) =
above(P0, Z, xi) for all xi, Z, we haveN(xi, j, P) =N(xi, j, P0). But
N(xi, j, P) = X Z⊂X, |Z|=j−1 above(P, Z, xi) = X Z⊂X, |Z|=j−1 above(P0, Z, xi) =N(xi, j, P0)
5. Take a scoring vectorhv1, . . . , vmi. We have to show that for anyP, P0such
that fnW SM(P) =fnW SM(P0) we get the same score for every candidate. For a candidatexi, the score ofxi is:
X 1≤j≤m N(xi, j, P)vj= X 1≤j≤m N(xi, j, P0)vj
so we have the claim.
We further mention that we cannot add additional arrows to Figure 4.1 (beyond arrows which we get using composition). For example, WSM is not finer than WMG. Consider the profiles
P = ( x1> x2> x3> x4 x3> x4> x1> x2 P 0 = ( x3> x2> x1> x4 x1> x4> x3> x2
The profiles satisfyfW SM
2 (P) =f2W SM(P0) butf2W M G(P)6=f2W M G(P0). Sim- ilarly, take: P = ( x1> x2> x3> x4 x4> x3> x2> x1 P 0 = ( x2> x1> x4> x3 x3> x4> x1> x2
These profiles satisfyf2W M G(P) =f2W M G(P0) but f2W SM(P)6=f2W SM(P0), so
WMG is not finer than WSM.
We still need to say something about how sufficient and necessary these representations are, and with respect to which voting rules. Connecting this to the hierarchy we displayed, we want to find for any ruleF the representation which is sufficient to computeF and it is the least fine representation that does that (that would be the closest to a necessary representation).
We know that ABOVE is sufficient and necessary for STV, and some new results by Chevaleyre et al. point out that it might also be sufficient for other rules [12]. WMG is sufficient for many Condorcet consistent rules, as we saw in the case of compilation complexity, though it is not necessary for all.
WSM is sufficient for Bucklin, though it is not necessary (a Bucklin winner has to be found within the firstbm
2c+ 1 columns, and so any two squares that
identify on those are equivalent for Bucklin). However, PSR representations are not fine enough for Bucklin, and we can also think of Bucklin variations for which that representation is necessary (one where the Bucklin score is computed not according to where the candidate gets more than half the votes, but rather when they get all the votes, for example).
PSR are sufficient and necessary for all positional scoring rules.
To conclude this section, we have shown how the information utilized by voting rules can be put into different structures, and those structure form a hierarchy of a sort. The study of the representations and the relations between them can assist in estimating measures such as compilation complexity, which we can try to get either from counting the elements of the representation (as we do for ABOVE in Appendix C, to get an upper bound for the compilation complexity of STV), or from looking at another representation which is close to it in the hierarchy whose size is known, and using the mapping between the two representations to estimate the size of the unknown representation (we elaborate
on this in the final discussion). Another advantage is that it gives a more formal way of looking at voting rules, thus seeing similarities and differences between different rules.
In the next section we present another formalism for voting rules, which has elements related to both compilation complexity and communication complex- ity.