Symmetry (social choice)

Requirement of a social choice rule

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A joint Politics and Economics series

Social choice and electoral systems

* Social choice
* Mechanism design
* Comparative politics
* Comparison
* List (By country)

Single-winner methods

Single vote – plurality methods

* First preference plurality (FPP)
* Two-round (US: Jungle primary)
* Partisan primary
* Instant-runoff
* UK: Alternative vote (AV)
* US: Ranked-choice (RCV)
* Party block voting
* Plurality block voting

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Condorcet methods

* Condorcet-IRV
* Round-robin voting
* Minimax
* Kemeny
* Schulze
* Ranked pairs
* Maximal lottery

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Positional voting

* Plurality (el. IRV)
* Borda count (el. Baldwin, Mdn. Bucklin)
* Antiplurality (el. Coombs)

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Cardinal voting

* Score voting
* Approval voting
* Majority judgment
* STAR voting

Proportional representation

Party-list

* Apportionment
* Highest averages
* Largest remainders
* National remnant
* Biproportional
* List type
* Closed list
* Open list
* Panachage
* List-free PR
* Localized list

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Quota-remainder methods

* Hare STV
* Schulze STV
* CPO-STV
* Quota Borda

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Approval-based committees

* Thiele's method
* Phragmen's method
* Expanding approvals rule
* Method of equal shares

* * *

Fractional social choice

* Direct representation
* Interactive representation
* Liquid democracy
* Fractional approval voting
* Maximal lottery
* Random ballot

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Semi-proportional representation

* Cumulative
* SNTV
* Limited voting

Mixed systems

By results of combination

* Mixed-member majoritarian
* Mixed-member proportional

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By mechanism of combination

* Non-compensatory
* Parallel (superposition)
* Coexistence
* Conditional
* Fusion (majority bonus)
* Compensatory
* Seat linkage system
* UK: 'AMS'
* NZ: 'MMP'
* Vote linkage system
* Negative vote transfer
* Mixed ballot
* Supermixed systems
* Dual-member proportional
* Rural–urban proportional
* Majority jackpot

* * *

By ballot type

* Single vote
* Double simultaneous vote
* Dual-vote

Paradoxes and pathologies

Spoiler effects

* Spoiler effect
* Cloning paradox
* Frustrated majorities paradox
* Center squeeze

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Pathological response

* Perverse response
* Best-is-worst paradox
* No-show paradox
* Multiple districts paradox

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Strategic voting

* Lesser evil voting
* Exaggeration
* Truncation
* Turkey-raising
* Wasted vote

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Paradoxes of majority rule

* Tyranny of the majority
* Discursive dilemma
* Conflicting majorities paradox

Social and collective choice

Impossibility theorems

* Arrow's theorem
* Majority impossibility
* Moulin's impossibility theorem
* McKelvey–Schofield chaos theorem
* Gibbard's theorem

* * *

Positive results

* Median voter theorem
* Condorcet's jury theorem
* May's theorem
* Condorcet dominance theorems
* Harsanyi's utilitarian theorem
* VCG mechanism
* Quadratic voting

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Mathematics portal

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In economics and social choice, a function satisfies anonymity, neutrality, or symmetry if the rule does not discriminate between different participants ahead of time. For example, in an election, a voter-anonymous function is one where it does not matter who casts which vote, i.e. all voters' ballots are equal ahead of time. Formally, this is defined by saying the rule returns the same outcome (whatever this may be) if the votes are "relabeled" arbitrarily, e.g. by swapping votes #1 and #2.[1][2] Similarly, outcome-neutrality says the rule does not discriminate between different outcomes (e.g. candidates) ahead of time. Formally, if the labels assigned to each outcome are permuted arbitrarily, the returned result is permuted in the same way.[1][2]

Some authors reserve the term anonymity for agent symmetry and neutrality for outcome-symmetry,[1][2] but this pattern is not perfectly consistent.[3]: 75

## Examples

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Most voting rules are anonymous and neutral by design. For example, plurality voting is anonymous and neutral, since only counts the number of first-preferences for each candidate, regardless of who cast these votes.

Any rule that uses a secret ballot must be voter-anonymous, since they do not know which voter cast which vote. However, the converse is not true (as in e.g. roll call votes).

### Non-examples

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An example of a non-neutral rule is a rule which says that, in case of a tie, the alternative X is selected. This is particularly prominent in cases where X is the status quo option: parliamentary procedures often specify that the status quo unless there is a strict majority against it. Other rules are non-anonymous in the case of a tied vote, e.g. when a chairman is allowed to break ties.

However, not all violations of anonymity and neutrality are due to tied votes. For example, many motions require a supermajority to pass, and other rules can give certain stakeholders a veto. The United States' electoral college is a well-known example of a non-anonymous voting rule, as the results of the election depend not just on the votes for each candidate, but also on their physical arrangement across space.

Weighted voting rules are non-anonymous, as they give some voters a higher weight than others, for example, due to their expertise or entitlement.

## See also

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* One man, one vote
* Fair election

## References

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1. ^ a b c Bogomolnaia, Anna; Moulin, Hervé; Stong, Richard (2005-06-01). "Collective choice under dichotomous preferences" (PDF). Journal of Economic Theory. 122 (2): 165–184\. doi:10.1016/j.jet.2004.05.005. ISSN 0022-0531.
2. ^ a b c Felix Brandt (2017-10-26). "Roling the Dice: Recent Results in Probabilistic Social Choice". In Endriss, Ulle (ed.). Trends in Computational Social Choice. Lulu.com. ISBN 978-1-326-91209-3.
3. ^ Pukelsheim, Friedrich (2017), Pukelsheim, Friedrich (ed.), "Divisor Methods of Apportionment: Divide and Round", Proportional Representation: Apportionment Methods and Their Applications, Cham: Springer International Publishing, pp. 71–93, doi:10.1007/978-3-319-64707-4_4, ISBN 978-3-319-64707-4, retrieved 2021-09-01

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