Preference Cycles and Voting Properties

The following theorems establish that failures of the Favorite Betrayal and Later-No-Harm properties in head-to-head (Condorcet) elections require the voters' original preferences, together with those of the remaining electorate, to produce a preference cycle. These proofs assume a standard Condorcet completion method in which an undefeated head-to-head candidate is always elected, and a cycle resolution procedure is invoked only when no such candidate exists.

Favorite Betrayal

Theorem

Changing ballots from A > B > C to B > A > C can only change the election winner from C if the voters' original preferences, together with those of the remaining electorate, produce a preference cycle.

Proof
  1. Changing a ballot from A > B > C to B > A > C changes only the pairwise contest between A and B.
  2. Therefore, every pairwise contest involving C is unchanged.

    A vs. C is unchanged.

    B vs. C is unchanged.

  3. Therefore, if C was the Condorcet winner before the change, C remains the Condorcet winner afterward.
  4. Therefore, if the election winner changes from C to another candidate, C could not have been the Condorcet winner before the change.
  5. A Condorcet method invokes a preference cycle resolution only when no Condorcet winner exists. Therefore, C must have been elected through a preference cycle resolution. Thus, the voters' original preferences, together with those of the remaining electorate, must have produced a preference cycle.

Later-No-Harm (Reordering Later Preferences)

Theorem

Changing ballots from A > B > C to A > C > B can only harm A if the voters' original preferences, together with those of the remaining electorate, produce a preference cycle.

Proof
  1. Changing a ballot from A > B > C to A > C > B changes only the pairwise contest between B and C.
  2. Therefore, every pairwise contest involving A is unchanged.

    A vs. B is unchanged.

    A vs. C is unchanged.

  3. Therefore, if A was the Condorcet winner before the change, A remains the Condorcet winner afterward.
  4. Therefore, if A is harmed by the change, A could not have been the Condorcet winner before the change.
  5. A Condorcet method invokes a preference cycle resolution only when no Condorcet winner exists. Therefore, A must have been elected through a preference cycle resolution. Thus, the voters' original preferences, together with those of the remaining electorate, must have produced a preference cycle.

Later-No-Harm (Adding a Later Preference)

Theorem

Changing ballots from A to A > B can only harm A if the voters' original preferences, together with those of the remaining electorate, produce a preference cycle.

Proof
  1. Changing a ballot from A to A > B changes only the pairwise contests involving B and candidates not ranked on the original ballot other than A.

    B vs. C

    B vs. D

    ...

    B vs. Z

  2. Therefore, every pairwise contest involving A is unchanged.

    A vs. B is unchanged.

    A vs. C is unchanged.

    ...

    A vs. Z is unchanged.

  3. Therefore, if A was the Condorcet winner before the change, A remains the Condorcet winner afterward.
  4. Therefore, if A is harmed by the change, A could not have been the Condorcet winner before the change.
  5. A Condorcet method invokes a preference cycle resolution only when no Condorcet winner exists. Therefore, A must have been elected through a preference cycle resolution. Thus, the voters' original preferences, together with those of the remaining electorate, must have produced a preference cycle.

Participation Paradox

Theorem

Adding ballots that rank A above every other candidate can only harm A if the original electorate produced a preference cycle.

Proof
  1. If A were the Condorcet winner before the additional ballots were added, A would win every pairwise contest:

    A defeats B

    A defeats C

    ...

    A defeats Z

  2. Adding ballots that rank A above every other candidate can only strengthen A in each pairwise contest. Therefore, all of these victories would remain unchanged.
  3. Therefore, if adding these ballots harms A, A could not have been the Condorcet winner before the change.
  4. A Condorcet method invokes a preference cycle resolution only when no Condorcet winner exists. Therefore, A must have been elected through a preference cycle resolution. Thus, the voters' original preferences must have produced a preference cycle.

Comments

This site does not manage comment accounts. Comment data is handled by a third-party discussion service.