Expanding approvals rule

In general, the EAR algorithm works as follows. Let n denote the number of voters, and k the number of seats to be filled. Initially, each voter is given 1 unit of virtual money. Groups of voters can use their virtual money to "buy" candidates, where the "price" of each candidate is ${\displaystyle n/k}$ (though the divisor can be slightly different; see highest averages method). The EAR goes rank by rank, starting at rank 1 which corresponds to the top candidates of the voters, and increasing the rank in each iteration. (This is where the term "expanding approvals" comes from: as the rank increases, the number of approved candidates expands.) For each rank r:

1. EAR checks if there is a candidate who can be afforded by all voters who rank this candidate r-th or better. If there is such a candidate, EAR selects one such candidate c (there are different variants regarding how to select this candidate), and adds c to the committee.
2. The "price" of n/k is deducted from the balance of voters who rank c r-th or better (there are different variants regarding how exactly the price is split among them).

Properties

Aziz and Lee prove that 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 can be computed in polynomial time and satisfies several weak candidate monotonicity properties.