# Voting criteria

There are a number of different criteria which can be used for voting systems in an election, including the following

## Condorcet criterion and similar criteria
### Condorcet criterion

A Condorcet (French:, English: /kɒndɔːrˈseɪ/) winner is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condorcet winner criterion. The Condorcet winner criterion extends the principle of majority rule to elections with multiple candidates.
The Condorcet winner is also called a majority winner, a majority-preferred candidate, a beats-all winner, or tournament winner (by analogy with round-robin tournaments). A Condorcet winner may not necessarily always exist in a given electorate: it is possible to have a rock, paper, scissors-style cycle, when multiple candidates defeat each other (Rock < Paper < Scissors < Rock). This is called Condorcet's voting paradox, and is analogous to the counterintuitive intransitive dice phenomenon known in probability. However, the Smith set, a generalization of the Condorcet criteria that is the smallest set of candidates that are pairwise unbeaten by every candidate outside of it, will always exist.
If voters are arranged on a sole 1-dimensional axis, such as the left-right political spectrum for a common example, and always prefer candidates who are more similar to themselves, a majority-rule winner always exists and is the candidate whose ideology is most representative of the electorate, a result known as the median voter theorem. However, in real-life political electorates are inherently multidimensional, and the use of a one- or even two-dimensional model of such electorates would be inaccurate. Previous research has found cycles to be somewhat rare in real elections, with estimates of their prevalence ranging from 1-10% of races.

### Condorcet loser criterion

In single-winner voting system theory, the Condorcet loser criterion (CLC) is a measure for differentiating voting systems. It implies the majority loser criterion but does not imply the Condorcet winner criterion.

### Smith criterion

The Smith set, sometimes called the top-cycle or Condorcet winning set, generalizes the idea of a Condorcet winner to cases where no such winner exists. It does so by allowing cycles of candidates to be treated jointly, as if they were a single Condorcet winner. Voting systems that always elect a candidate from the Smith set pass the Smith criterion. The Smith set and Smith criterion are both named for mathematician John H. Smith.

## Consistency criterion

A voting system satisfies join-consistency (also called the reinforcement criterion) if combining two sets of votes, both electing A over B, always results in a combined electorate that ranks A over B. It is a stronger form of the participation criterion. Systems that fail the consistency criterion (such as Instant-runoff voting or Condorcet methods) are susceptible to the multiple-district paradox, a pathological behavior where a candidate can win an election without carrying even a single precinct. Conversely, it can be seen as allowing for a particularly egregious kind of gerrymander: it is possible to draw boundaries in such a way that a candidate who wins the overall election fails to carry even a single electoral district.

## Independence criteria
### Independence of irrelevant alternatives

Independence of irrelevant alternatives (IIA) is an axiom of decision theory which codifies the intuition that a choice between $A$ and $B$ should not depend on the quality of a third, unrelated outcome C{\textstyle C}. There are several different variations of this axiom, which are generally equivalent under mild conditions. As a result of its importance, the axiom has been independently rediscovered in various forms across a wide variety of fields, including economics, cognitive science, social choice, fair division, rational choice, artificial intelligence, probability, and game theory. It is closely tied to many of the most important theorems in these fields, including Arrow's impossibility theorem, the Balinski–Young theorem, and the money pump arguments.

### Independence of clones criterion

In social choice theory, the independence of (irrelevant) clones criterion says that adding a clone, i.e. a new candidate very similar to an already-existing candidate, should not spoil the results. It can be considered a weak form of the independence of irrelevant alternatives (IIA) criterion that nevertheless is failed by a number of voting rules. A method that passes the criterion is said to be clone independent.
A group of candidates are called clones if they are always ranked together, placed side-by-side, by every voter; no voter ranks any of the non-clone candidates between or equal to the clones. In other words, the process of cloning a candidate involves taking an existing candidate C, then replacing them with several candidates C1, C2... who are slotted into the original ballots in the spot where C previously was, with the clones being arranged in any order. If a set of clones contains at least two candidates, the criterion requires that deleting one of the clones must not increase or decrease the winning chance of any candidate not in the set of clones.
Ranked pairs, the Schulze method, and systems that unconditionally satisfy independence of irrelevant alternatives are clone independent. Instant-runoff voting passes as long as tied ranks are disallowed. If they are allowed, its clone independence depends on specific details of how the criterion is defined and how tied ranks are handled.
Rated methods like range voting or majority judgment that are spoilerproof under certain conditions are also clone independent under those conditions.
The Borda count, minimax, Kemeny–Young, Copeland's method, plurality, and the two-round system all fail the independence of clones criterion. Voting methods that limit the number of allowed ranks also fail the criterion, because the addition of clones can leave voters with insufficient space to express their preferences about other candidates. For similar reasons, ballot formats that impose such a limit may cause an otherwise clone-independent method to fail.

### Independence of Smith-dominated alternatives

Independence of Smith-dominated alternatives (ISDA, also known as Smith-IIA) is a voting system criterion which says that the winner of an election should not be affected by candidates who are not in the Smith set.

## Later-no criteria
### Later-no-harm criterion

Voting system :
| Name | Comply? |
| Plurality | Yes |
| Two-round system | Yes |
| Partisan primary | Yes |
| Instant-runoff voting | Yes |
| Minimax Opposition | Yes |
| DSC | Yes |
| Anti-plurality | No |
| Approval | N/A |
| Borda | No |
| Dodgson | No |
| Copeland | No |
| Kemeny–Young | No |
| Ranked Pairs | No |
| Schulze | No |
| Score | No |
| Majority judgment | No |
Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose. It is a common property in the plurality-rule family of voting systems.
For example, say a group of voters ranks Alice 2nd and Bob 6th, and Alice wins the election. In the next election, Bob focuses on expanding his appeal with this group of voters, but does not manage to defeat Alice—Bob's rating increases from 6th-place to 3rd. Later-no-harm says that this increased support from Alice's voters should not allow Bob to win.
Later-no-harm may be confused as implying center squeeze, since later-no-harm is a defining characteristic of first-preference plurality (FPP) and instant-runoff voting (IRV), and descending solid coalitions (DSC), systems that have similar mechanics that are based on first preference counting. These systems pass later-no-harm compliance by making sure the results either do not depend on lower preferences at all (plurality) or only depend on them if all higher preferences have been eliminated (IRV and DSC), and thus exhibit a center squeeze effect. However, this does not mean that methods that pass later-no-harm must be vulnerable to center squeezes. The properties are distinct, as Minimax opposition also passes later-no-harm.

### Later-no-help criterion
## Majority winner and loser
### Majority winner criterion

The majority criterion is a voting system criterion applicable to voting rules over ordinal preferences required that if only one candidate is ranked first by over 50% of voters, that candidate must win.
Some methods that comply with this criterion include any Condorcet method, instant-runoff voting, Bucklin voting, plurality voting, and approval voting.

### Majority loser criterion

The majority loser criterion is a criterion to evaluate single-winner voting systems. The criterion states that if a majority of voters give a candidate no support, i.e. do not list that candidate on their ballot, that candidate must lose (unless no candidate is accepted by a majority of voters).

## Monotonicity criterion

In social choice, the negative response, perversity, or additional support paradox is a pathological behavior of some voting rules where a candidate loses as a result of having too much support (or wins because of increased opposition). In other words, increasing (decreasing) a candidate's ranking or rating causes that candidate to lose (win), respectively. Electoral systems that do not exhibit perversity are sometimes said to satisfy the monotonicity criterion.
Perversity is often described by social choice theorists as an exceptionally severe kind of electoral pathology, as such rules can have "backwards" responses to voters' opinions, where popularity causes defeat while unpopularity leads to a win. Similar rules treat the well-being of some voters as "less than worthless". These issues have led to constitutional prohibitions on such systems as violating the right to equal and direct suffrage. Negative response is often cited as an example of a perverse incentive, as rules with perverse response can incentivize politicians to take extreme or unpopular positions in an attempt to shed excess votes.
Most ranked methods (including Borda and all common round-robin rules) satisfy positive response, as do all common rated voting methods (including approval, highest medians, and score).
Negative responsiveness occurs in instant-runoff voting (IRV), the single transferable vote, and the two-round system. Some quota-based apportionment methods also violate the rule, as can the randomized Condorcet method in cases of cyclic ties.

## Proportionality for solid coalitions

Proportionality for solid coalitions (PSC) is a criterion of proportionality for ranked voting systems. It is an adaptation of the quota rule to voting systems in which there are no official party lists, and voters can directly support candidates. The criterion was first proposed by the British philosopher and logician Michael Dummett.

## Participation criterion

In social choice, a no-show paradox is a surprising behavior in some voting rules, where a candidate loses an election as a result of having too many supporters. More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob. Voting systems without the no-show paradox are said to satisfy the participation criterion.
In systems that fail the participation criterion, a voter turning out to vote could make the result worse for them; such voters are sometimes referred to as having negative vote weights, particularly in the context of German constitutional law, where courts have ruled such a possibility violates the principle of one man, one vote.
Positional methods and score voting satisfy the participation criterion. All deterministic voting rules that satisfy pairwise majority-rule can fail in situations involving four-way cyclic ties, though such scenarios are empirically rare, and the randomized Condorcet rule is not affected by the pathology. The majority judgment rule fails as well. Ranked-choice voting (RCV) and the two-round system both fail the participation criterion with high frequency in competitive elections, typically as a result of a center squeeze.

## Plurality criterion

Woodall's plurality criterion is a voting criterion for ranked voting. It is stated as follows:
    If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference [other than last], then A's probability of winning must be no less than B's.
Woodall has called the plurality criterion "a rather weak property that surely must hold in any real election" opining that "every reasonable electoral system seems to satisfy it."
Among Condorcet methods which permit truncation, whether the plurality criterion is satisfied depends often on the measure of defeat strength. When winning votes is used as the measure of defeat strength, plurality is satisfied. Plurality is failed when margins is used. Minimax using pairwise opposition also fails plurality.
When truncation is permitted under Borda count, the plurality criterion is satisfied when no points are scored to truncated candidates, and ranked candidates receive no fewer votes than if the truncated candidates had been ranked. If truncated candidates are instead scored the average number of points that would have been awarded to those candidates had they been strictly ranked, or if Nauru's modified Borda count is used, the plurality criterion is failed.

## Resolvability criteria

A voting system is called decisive, resolvable, or resolute if it ensures a low probability of tied elections. There are two different criterion that formalize this.
* In Nicolaus Tideman's version of the criterion, adding one extra vote (with no tied ranks) should make the winner unique.
* Douglas R. Woodall's version requires that the probability of a tied vote under an impartial culture model gives a tie approaches zero as the number of voters increases toward infinity.

## Reversal symmetry

In social choice theory, the best-is-worst paradox occurs when a voting rule declares the same candidate to be both the best and worst possible winner. The worst candidate can be identified by reversing each voter's ballot (to rank candidates from worst-to-best), then applying the voting rule to the reversed ballots find a new "anti-winner".
Rules that never exhibit a best-is-worst paradox are said to satisfy the reversal criterion, which states that if every voter's opinions on each candidate are perfectly reversed (i.e. they rank candidates from worst to best), the outcome of the election should be reversed as well, meaning the first- and last- place finishers should switch places. In other words, the results of the election should not depend arbitrarily on whether voters rank candidates from best to worst (and then select the best candidate), or whether we ask them to rank candidates from worst to best (and then select the least-bad candidate).

## Unrestricted domain

In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem.