Expanding approvals rule

An expanding approvals rule (EAR) is a rule for multi-winner elections, which allows agents to express weak ordinal preferences (i.e., ranking with indifferences), and guarantees a form of proportional representation called proportionality for solid coalitions. The family of EAR was presented by Aziz and Lee.

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).