Homogeneity criterion

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

Complying methods

Any voting method that counts voter preferences proportionally satisfies homogeneity, including voting methods such as Plurality voting, Two-round system, Single transferable vote, Instant Runoff Voting, Contingent vote, Coombs' method, Approval voting, Anti-plurality voting, Borda count, Range voting, Bucklin voting, Majority Judgment, Condorcet methods 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 does not satisfy homogeneity.[3][4]

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
${\tfrac {1}{2}}$	A > B > C
${\tfrac {1}{4}}$	B > A > C
${\tfrac {1}{4}}$	C > B > A

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

References

Smith, John H. (November 1973). "Aggregation of Preferences with Variable Electorate". Econometrica. 41 (6): 1027–1041. doi:10.2307/1914033. JSTOR 1914033.
Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994
Fishburn, Peter C. (November 1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030.
Brandt, Felix (August 2009). "Some Remarks on Dodgson's Voting Rule". Mathematical Logic Quarterly. 55 (4): 460–463. doi:10.1002/malq.200810017. S2CID 2208925.

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[1] Smith, John H. (November 1973). "Aggregation of Preferences with Variable Electorate". Econometrica. 41 (6): 1027–1041. doi:10.2307/1914033. JSTOR 1914033.

[2] Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994

[3] Fishburn, Peter C. (November 1977). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 469–489. doi:10.1137/0133030.

[4] Brandt, Felix (August 2009). "Some Remarks on Dodgson's Voting Rule". Mathematical Logic Quarterly. 55 (4): 460–463. doi:10.1002/malq.200810017. S2CID 2208925.