Expanding approvals rule

For the single-winner variant, see Highest median voting rules.

The expanding approvals rule (EAR) is a rule for multi-winner elections that guarantees a form of proportional representation called proportionality for solid coalitions. It is a generalization of the highest median rules to include multiwinner elections and participatory budgeting. When working with ranked ballots, it is sometimes called the Bucklin transferable vote. However, the rule can be more effectively implemented using rated ballots, which are easier to use and provide additional cardinal utility information that can be used for better decision-making.

Procedure

Say there are n voters and k seats to be filled. Each voter has one vote. Groups of voters can expend their vote to elect a candidate, where the cost to elect a candidate is given by an electoral quota, most often assumed to be the Hare quota of $n/k$.

EAR sets an approval threshold that advances grade by grade, starting at the highest grade and lowering the bar at each iteration. As this bar is lowered, the number of approved candidates expands. When advancing to a new rating of $r$:

1. EAR checks if there is a candidate who can be afforded by all voters who rate this candidate $r$-th or better. (If there are multiple such candidates, there exist different tiebreaking rules for selecting among them; see highest median voting rules.)
2. The cost of $n/k$ is deducted from the balance of voters who assign this a grade of $r$ or higher (there are different variants regarding how exactly the price is split among them).

Properties

EAR satisfies generalized proportionality for solid coalitions (GPSC): a property for ordinal weak preferences that generalizes both proportionality for solid coalitions (for strict preferences) and proportional justified representation (for dichotomous preferences). Further, EAR satisfies several weak candidate monotonicity properties.

Related rules

The method of equal shares (MES) can be seen as a special case of EAR, in which, in step 1, the elected candidate is a candidate that can be purchased in the smallest price (in general, it is the candidate supported by the largest number of voters with remaining funds), and in step 2, the price is deducted as equally as possible (those who have insufficient budget pay all their remaining budget, and the others pay equally).

Single transferable vote (STV) can also be seen as a variant of EAR, in which voters always approve only their top candidate (r=1); however, if no candidate can be "purchased" by voters ranking it first, the candidate whose supporters have the fewest leftover votes is removed, bringing a new candidate to the top position of these voters. Like EAR, STV satisfies proportionality for solid coalitions. However, EAR has better candidate monotonicity properties.