Reinforcing Voting Rule (COMSOC)

Definition

Reinforcing Voting Rule ( COMSOC)

A voting rule satisfies reinforcement if agreement between two separate electorates (groups of voters) is preserved when their ballots are combined. Let $R_1,R_2$ be profiles of disjoint voter sets $V_1,V_2$ over the same alternatives. The merged profile $R_1\uplus R_2$ contains all ballots from $V_1\cup V_2$, retaining identical ballots submitted by different voters. The rule must be defined across electorate sizes, and the condition must hold for every such pair of profiles.

Single-winner output

$$f(R_1)=f(R_2)=x⟹f(R_1\uplus R_2)=x.$$

If both electorates elect $x$, their combination must elect $x$ too.

Set-valued output

$$F(R_1)\cap F(R_2)\ne \varnothing ⟹F(R_1\uplus R_2)=F(R_1)\cap F(R_2).$$

When the electorates share winners, the combined winners are exactly their common winners. This applies to sets of alternatives or sets of collective rankings, depending on the rule’s output. It imposes no condition when the original winner sets are disjoint.

Kemeny Rankings

Let $F(R)$ be the set of Kemeny consensuses of $R$. For any proposed preference ranking $\succ$, its Kemeny score is additive across electorates:

$$\mathrm{KT}\text{-}\mathrm{dist}(\succ ,R_1\uplus R_2)=\mathrm{KT}\text{-}\mathrm{dist}(\succ ,R_1)+\mathrm{KT}\text{-}\mathrm{dist}(\succ ,R_2).$$

Suppose a ranking is optimal for both profiles. Writing $s_1,s_2$ for their respective minimum scores, it attains $s_1+s_2$ after merging. Every proposal has score at least $s_1$ on the first profile and at least $s_2$ on the second. It attains their sum exactly when it attains both minima. Thus the merged optimal rankings are exactly $F(R_1)\cap F(R_2)$.

In particular, if both profiles have the same unique consensus,

$$F(R_1)=F(R_2)=\{{\succ }^{\ast }\}⟹F(R_1\uplus R_2)=\{{\succ }^{\ast }\}.$$

This establishes reinforcement for whole consensus rankings. Agreement on a first-place alternative alone does not imply agreement on an optimal ranking, so the argument does not automatically establish reinforcement for the alternative-valued Kemeny rule.