Largest remainders method in the context of "Single transferable vote"

The single transferable vote (STV) or proportional-ranked choice voting (P-RCV), also known as PR-STV and "proportional representation by means of the single transferable vote", is a multi-winner electoral system in which each voter casts a single vote in the form of a ranked ballot. Voters have the option to rank candidates, and their vote may be transferred according to alternative preferences if their preferred candidate is eliminated or elected with surplus votes, so that their vote is used to elect someone they prefer over others in the running. STV aims to approach proportional representation based on votes cast in the district where it is used, so that each vote is worth about the same as another.

STV is a family of multi-winner proportional representation electoral systems. The proportionality of its results and the proportion of votes actually used to elect someone are equivalent to those produced by proportional representation election systems based on lists. STV systems can be thought of as a variation on the largest remainders method that uses candidate-based solid coalitions, rather than party lists. Surplus votes belonging to winning candidates (those in excess of an electoral quota) may be thought of as remainder votes. Surplus votes may be transferred from a successful candidate to another candidate and then possibly used to elect that candidate.

Vote-ratio, weight-ratio, or population-ratio monotonicity is a property of some apportionment methods. It says that if the entitlement for ${\displaystyle A}$ grows at a faster rate than ${\displaystyle B}$ (i.e. ${\displaystyle A}$ grows proportionally more than ${\displaystyle B}$), ${\displaystyle A}$ should not lose a seat to ${\displaystyle B}$. More formally, if the ratio of votes or populations ${\displaystyle A/B}$ increases, then ${\displaystyle A}$ should not lose a seat while ${\displaystyle B}$ gains a seat. An apportionment method violating this rule may encounter population paradoxes.

A particularly severe variant, where voting for a party causes it to lose seats, is called a no-show paradox. The largest remainders method exhibits both population and no-show paradoxes.