 | This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(January 2024) (Learn how and when to remove this message) |
| A jointPolitics andEconomics series |
| Social choice andelectoral systems |
|---|
 |
By results of combination By mechanism of combination By ballot type |
 Mathematics portal |
Anexpanding approvals rule (EAR) is a rule formulti-winner elections, which allows agents to express weakordinal preferences (i.e., ranking with indifferences), and guarantees a form of proportional representation calledproportionality for solid coalitions. The family of EAR was presented by Aziz and Lee.[1][2]
In general, the EAR algorithm works as follows. Letn denote the number of voters, andk 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 (though the divisor can be slightly different; seehighest 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 rankr:
Aziz and Lee[1] prove that EAR satisfies generalized proportionality for solid coalitions (GPSC): a property for ordinal weak preferences that generalizes bothproportionality for solid coalitions (for strict preferences) andproportional justified representation (for dichotomous preferences). Further, EAR can be computed in polynomial time and satisfies several weak candidate monotonicity properties.
Aziz and Lee[2] extended EAR to the setting ofcombinatorial participatory budgeting.
Themethod 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).[3]
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 (this brings a new candidate to the top position of these voters). Like EAR, STV satisfiesproportionality for solid coalitions. However, EAR allows weak rankings, whereas STV works only with strict rankings. Moreover, EAR has better candidate monotonicity properties. This addressed an open question byWoodall,[4] who asked if there are rules with the same political properties as STV, which are more monotonic.
