# Schulze STV

Schulze STV is a proposed multi-winner ranked voting system designed to achieve proportional representation. It was invented by Markus Schulze, who developed the Schulze method for resolving ties using a Condorcet method. Schulze STV is similar to CPO-STV in that it compares possible winning candidate pairs and selects the Condorcet winner. It is named in analogy to the single transferable vote (STV), but only shares its aim of proportional representation, and is otherwise based on unrelated principles.
The system is based on Schulze's investigations into vote management and free riding. When a voter prefers a popular candidate, there is an advantage to first choosing a candidate who is unlikely to win ("Woodall free riding") or omitting his preferred candidate from his rankings ("Hylland free riding"). Schulze STV is designed to be as resistant to free riding as possible, without giving up the Droop proportionality criterion.

## Example

Each voter ranks candidates in their order of preference. In a hypothetical election, three candidates vie for two seats; Andrea and Carter represent the Yellow Party, and Brad represents the Purple Party. Andrea is a popular candidate, and has supporters who are not Yellow Party supporters. It is assumed that the Yellow Party can influence their own supporters, but not Andrea's.
There are 90 voters, and their preferences are
| Andrea's supporters | Andrea's supporters | Yellow Party supporters | Yellow Party supporters | Purple Party supporters |
| 12 | 26 | 12 | 13 | 27 |
| Andrea (Y) Brad (P) Carter (Y) | Andrea (Y) Carter (Y) Brad (P) | Andrea (Y) Carter (Y) Brad (P) | Carter (Y) Andrea (Y) Brad (P) | Brad (P) |
In the STV system, the initial tallies are:
* Andrea (Y): 50
* Carter (Y): 13
* Brad (P): 27
The quota is determined according to $({\rm {\mbox{valid votes cast}}})/({\rm {\mbox{seats to fill}}}+1)=90/(2+1)=30.$ Andrea is declared elected and her surplus, ${\rm {\mbox{Andrea's votes}}}-{\rm {\mbox{quota}}}=50-30=20$, is distributed with
${\mbox{Round}}\left({{\mbox{votes for next preference belonging to the original candidate}} \over {\mbox{total votes for the original candidate}}}\times {\mbox{surplus votes for original candidate}}\right).$
* Carter (Y): $13+{\mbox{Round}}\left({\frac {26+12}{50}}\times 20\right)=13+15=28$
* Brad (P): $27+{\mbox{Round}}\left({\frac {12}{50}}\times 20\right)=27+5=32$
Brad is also elected.
The Schulze STV system has three possible outcomes (sets of winners) in the election: Andrea and Carter, Andrea and Brad, and Carter and Brad. In this system, any candidate with more than the Droop quota of first choices will be elected. Andrea is certain to be elected, with two possible outcomes: Andrea and Carter, and Andrea and Brad.

### Resistance to vote management

In vote management, a party instructs its voters not to rank a popular party candidate first. If the Yellow Party's leaders instruct their supporters to choose Carter first (followed by Andrea), the balloting changes. Unlike STV, however, Schulze STV resists vote management.

## Potential for tactical voting

Proportional representation systems are much less susceptible to tactical voting than single-winner systems such as the first past the post system and instant-runoff voting (IRV), if the number of seats to be filled is sufficiently large. Schulze STV aims to have additional resistance to forms of tactical voting which are specific to single transferable voting methods, in particular a phenomenon that Schulze calls Hylland Free Riding. STV methods which make use of Meek's or Warren's method are resistant to what Schulze calls Woodall Free Riding, but are still vulnerable to Hylland Free Riding.
As Schulze STV reduces to the Schulze method in single winner elections, it fails the participation criterion, the later-no-harm criterion and the later-no-help criterion, whereas traditional forms of STV (that reduce to IRV in single winner elections) fulfill later-no-help and later-no-harm.

## Complexity

Schulze STV is no more complicated for the voter than other forms of STV; the ballot is the same, and candidates are ranked in order of preference. In calculating an election result, however, Schulze STV is significantly more complex than STV. In most applications, computer calculation would be required. The algorithm implementing Schulze STV requires exponentially many steps in the number of seats to be filled (roughly on the order of $m^{3k}$ steps when k out of m candidates are to be selected), making the computation difficult if this number is not very small. In particular, the rule does not have polynomial runtime.
Compared to CPO-STV, implementing Schulze STV might be somewhat faster, since it only compares outcomes differing by one candidate; CPO-STV compares all possible pairs.