# Homogeneity criterion

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Homogeneity is a common property for [voting systems](/source/Voting_system). The property is satisfied if, in any election, the result depends only on the proportion of ballots of each possible type. That is, if every ballot is replicated the same number of times, then the result should not change.[1][2][3]

## Complying methods

Any voting method that counts voter preferences proportionally satisfies homogeneity, including voting methods such as [Plurality voting](/source/Plurality_voting), [Two-round system](/source/Two-round_system), [Single transferable vote](/source/Single_transferable_vote), [Instant Runoff Voting](/source/Instant_Runoff_Voting), [Contingent vote](/source/Contingent_vote), [Coombs' method](/source/Coombs'_method), [Approval voting](/source/Approval_voting), [Anti-plurality voting](/source/Anti-plurality_voting), [Borda count](/source/Borda_count), [Range voting](/source/Range_voting), [Bucklin voting](/source/Bucklin_voting), [Majority Judgment](/source/Majority_Judgment), [Condorcet methods](/source/Condorcet_method) and others.

## Noncomplying methods

A voting method that determines a winner by eliminating candidates not having a *fixed* number of votes, rather than a *proportional* or a *percentage* of votes, may not satisfy the homogeneity criterion.

[Dodgson's method](/source/Dodgson's_method) does not satisfy homogeneity.[4][5]

## Example of Proportional Preference Profiles

The following four voter preference profiles show rankings of candidates by voters that are proportional.

**Profile 1**

# of voters Preferences 6 A > B > C 3 B > A > C 3 C > B > A

**Profile 2**

Ratio of voters Preferences .5 A > B > C .25 B > A > C .25 C > B > A

**Profile 3**

Percent of voters Preferences 50% A > B > C 25% B > A > C 25% C > B > A

**Profile 4**

Fraction of voters Preferences 1 2 {\displaystyle {\tfrac {1}{2}}} A > B > C 1 4 {\displaystyle {\tfrac {1}{4}}} B > A > C 1 4 {\displaystyle {\tfrac {1}{4}}} C > B > A

A voting method satisfying homogeneity will return the same election results for each of the four preference profiles.

## References

1. **[^](#cite_ref-1)** Smith, John H. (November 1973). "Aggregation of Preferences with Variable Electorate". *Econometrica*. **41** (6): 1027–1041. [doi](/source/Doi_(identifier)):[10.2307/1914033](https://doi.org/10.2307%2F1914033). [JSTOR](/source/JSTOR_(identifier)) [1914033](https://www.jstor.org/stable/1914033).

1. **[^](#cite_ref-2)** Woodall, Douglas R. (1996). ["Monotonicity and single-seat election rules"](http://www.votingmatters.org.uk/ISSUE6/P4.HTM). *Voting matters*. **6**: 9–14.

1. **[^](#cite_ref-3)** [*Homogeneity and monotonicity of distance-rationalizable voting rules*](https://dl.acm.org/doi/abs/10.5555/2031678.2031735). 2 May 2011. pp. 821–828. [ISBN](/source/ISBN_(identifier)) [978-0-9826571-6-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-9826571-6-4).

1. **[^](#cite_ref-4)** Fishburn, Peter C. (November 1977). "Condorcet Social Choice Functions". *SIAM Journal on Applied Mathematics*. **33** (3): 469–489. [doi](/source/Doi_(identifier)):[10.1137/0133030](https://doi.org/10.1137%2F0133030).

1. **[^](#cite_ref-5)** Brandt, Felix (August 2009). "Some Remarks on Dodgson's Voting Rule". *Mathematical Logic Quarterly*. **55** (4): 460–463. [doi](/source/Doi_(identifier)):[10.1002/malq.200810017](https://doi.org/10.1002%2Fmalq.200810017). [S2CID](/source/S2CID_(identifier)) [2208925](https://api.semanticscholar.org/CorpusID:2208925).

v t e Electoral systems Part of the politics and Economics series Single-winner Approval voting Combined approval voting Unified primary Borda count Bucklin voting Condorcet methods Copeland's method Dodgson's method Kemeny method Minimax Condorcet method Nanson's method Ranked pairs Schulze method Exhaustive ballot First-past-the-post voting Instant-runoff voting Coombs' method Contingent vote Supplementary vote Simple majoritarianism Plurality Positional voting system Score voting STAR voting Two-round system Graduated majority judgment Proportional Systems Mixed-member Mixed single vote Party-list Proportional approval voting Rural–urban Sequential proportional approval voting Single transferable vote CPO-STV Hare–Clark Schulze STV Spare vote Indirect single transferable voting Allocation Highest averages method Webster/Sainte-Laguë D'Hondt Largest remainders method Quotas Droop quota/Hagenbach-Bischoff quota Hare quota Imperiali quota Mixed Parallel voting MMP Additional member system Alternative vote plus Mixed single vote Mixed ballot transferable vote Scorporo Vote linkage mixed system Semi-proportional Single non-transferable vote Limited voting Cumulative voting Satisfaction approval voting Criteria Condorcet winner criterion Condorcet loser criterion Consistency criterion Independence of clones Independence of irrelevant alternatives Independence of Smith-dominated alternatives Later-no-harm criterion Majority criterion Majority loser criterion Monotonicity criterion Mutual majority criterion Participation criterion Plurality criterion Resolvability criterion Reversal symmetry Smith criterion Seats-to-votes ratio Other Ballot Center squeeze Election threshold First-preference votes Liquid democracy Nomination rules Sham election Spoilt vote Sortition Unseating Wasted vote Comparison Comparison of electoral systems Electoral systems by country Portal — Project

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