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Schulze method

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Single-winner electoral system

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TheSchulze method (/ˈʃʊltsə/), also known as thebeatpath method, is asingle winner ranked-choice voting rule developed by Markus Schulze. The Schulze method is aCondorcet completion method, which means it will elect amajority-preferred candidate if one exists. In other words, if most people rankA aboveB,A will defeatB (whenever this is possible). Schulze's method breakscyclic ties by using indirect victories. The idea is that ifAlice beats Bob, and Bob beats Charlie, then Alice (indirectly) beats Charlie; this kind of indirect win is called a "beatpath".

The Schulze method is used by several organizations includingDebian,Ubuntu,Gentoo,Pirate Party political parties andmany others. It was also used byWikimedia prior to their adoption ofscore voting.

	$d[*,A]$	$d[*,B]$	$d[*,C]$	$d[*,D]$	$d[*,E]$
$d[A,*]$		20	26	30	22
$d[B,*]$	25		16	33	18
$d[C,*]$	19	29		17	24
$d[D,*]$	15	12	28		14
$d[E,*]$	23	27	21	31

To From	A	B	C	D	E
A	—	A-(30)-D-(28)-C-(29)-B	A-(30)-D-(28)-C	A-(30)-D	A-(30)-D-(28)-C-(24)-E	A
B	B-(25)-A	—	B-(33)-D-(28)-C	B-(33)-D	B-(33)-D-(28)-C-(24)-E	B
C	C-(29)-B-(25)-A	C-(29)-B	—	C-(29)-B-(33)-D	C-(24)-E	C
D	D-(28)-C-(29)-B-(25)-A	D-(28)-C-(29)-B	D-(28)-C	—	D-(28)-C-(24)-E	D
E	E-(31)-D-(28)-C-(29)-B-(25)-A	E-(31)-D-(28)-C-(29)-B	E-(31)-D-(28)-C	E-(31)-D	—	E
	A	B	C	D	E	From To

	$p[*,A]$	$p[*,B]$	$p[*,C]$	$p[*,D]$	$p[*,E]$
$p[A,*]$		28	28	30	24
$p[B,*]$	25		28	33	24
$p[C,*]$	25	29		29	24
$p[D,*]$	25	28	28		24
$p[E,*]$	25	28	28	31

	E	A	C	B	D
E		1	−3	9	17
A	−1		7	−5	15
C	3	−7		13	−11
B	−9	5	−13		21
D	−17	−15	11	−21

Criterion Method	Majority winner	Majority loser	Mutual majority	Condorcet winner^{[Tn 1]}	Condorcet loser	Smith^{[Tn 1]}	Smith-IIA^{[Tn 1]}	IIA/LIIA^{[Tn 1]}	Cloneproof	Monotone	Consistency	Participation	Reversal symmetry	Homogeneity	Later-no-harm^{[Tn 1]}	Later-no-help^{[Tn 1]}	No favorite betrayal^{[Tn 1]}	Ballot type
First-past-the-post	Yes	No	No	No	No	No	No	No	No	Yes	Yes	Yes	No	Yes	Yes	Yes	No	Single mark
Anti-plurality	No	Yes	No	No	No	No	No	No	No	Yes	Yes	Yes	No	Yes	No	No	Yes	Single mark
Two round system	Yes	Yes	No	No	Yes	No	No	No	No	No	No	No		Yes	Yes	Yes	No	Single mark
Instant-runoff	Yes	Yes	Yes	No	Yes	No	No	No	Yes	No	No	No	No	Yes	Yes	Yes	No	Ranking
Coombs	Yes	Yes	Yes	No	Yes	No	No	No	No	No	No	No		Yes	No	No	Yes	Ranking
Nanson	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	Yes		No	No	No	Ranking
Baldwin	Yes	Yes	Yes	Yes	Yes	Yes	No	No	No	No	No	No	No		No	No	No	Ranking
Tideman alternative	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	No	No	No	No		No	No	No	Ranking
Minimax	Yes	No	No	Yes^{[Tn 2]}	No	No	No	No	No	Yes	No	No	No		No^{[Tn 2]}	No	No	Ranking
Copeland	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	No	Yes	No	No			No	No	No	Ranking
Black	Yes	Yes	No	Yes	Yes	No	No	No	No	Yes	No	No	Yes	Yes	No	No	No	Ranking
Kemeny	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	No	Yes	No	No	Yes	Yes	No	No	No	Ranking
Ranked pairs	Yes	Yes	Yes	Yes	Yes	Yes	Yes	LIIA Only	Yes	Yes	No	No^{[Tn 3]}	Yes	Yes	No	No	No	Ranking
Schulze	Yes	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	No	No^{[Tn 3]}	Yes	Yes	No	No	No	Ranking
Borda	No	Yes	No	No	Yes	No	No	No	No	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Ranking
Bucklin	Yes	Yes	Yes	No	No	No	No	No	No	Yes	No	No	No	Yes	No	Yes	No	Ranking
Approval	Yes	No	No	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	Approvals
Majority Judgement	No	No^{[Tn 5]}	No^{[Tn 6]}	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	No	No^{[Tn 3]}		Yes	No	Yes	Yes	Scores
Score	No	No	No	No	No	No	No	Yes^{[Tn 4]}	Yes	Yes	Yes	Yes	Yes	Yes	No	Yes	Yes	Scores
STAR	No	Yes	No	No	Yes	No	No	No	No	Yes	No	No	No		No	No	No	Scores
Quadratic	No	No	No	No	No	No	No	No	No	Yes		Yes			N/A	N/A	No	Credits
Random ballot^{[Tn 7]}	No	No	No	No	No	No	No	Yes	Yes	Yes		Yes		Yes	Yes	Yes	Yes	Single mark
Sortition^{[Tn 8]}	No	No	No	No	No	No	No	Yes	No	Yes		Yes		N/A	Yes	Yes	Yes	None
Table Notes	^^a ^b ^c ^d ^e ^f ^gCondorcet's criterion is incompatible with theconsistency,independence of irrelevant alternatives,participation,later-no-harm,later-no-help, andsincere favorite criteria. ^^a ^bA variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm. ^^a ^b ^cIn Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate without grading any less-preferred candidate above any more-preferred one. ^^a ^b ^cApproval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their ownabsolute scale. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates. ^Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters. ^Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade. ^A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria. ^Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.

Number of voters	Order of preference
5	ACBED
5	ADECB
8	BEDAC
3	CABED
7	CAEBD
2	CBADE
7	DCEBA
8	EBADC

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Description of the method

Beatpath explanation

Iterative description

Example

Implementation

Ties and alternative implementations

Satisfied and failed criteria

Satisfied criteria

Failed criteria

Comparison table

Difference from ranked pairs

History

Usage

Government

Political parties

Student government and associations

Organizations

Generalizations

Notes

External links