## Strategy-Free Criterion (SFC)

#### Definitions

A sincere vote is one with no falsified preferences or preferences left unspecified when the election method allows them to be specified (in addition to the preferences already specified).

One candidate is preferred over another candidate if, in a one-on-one competition, more voters prefer the first candidate than prefer the other candidate.

If one candidate is preferred over each of the other candidates, that candidate is the Ideal Democratic Winner (IDW).

#### Statement of Criterion

If an Ideal Democratic Winner (IDW) exists, and if a majority prefers the IDW to another candidate, then the other candidate should not win if that majority votes sincerely and no other voter falsifies any preferences.

#### Complying Methods

The Condorcet method complies with the Strategy-Free Criterion, but none of the other methods in the compliance table above comply.

#### Commentary

The reader may be wondering how the IDW, if one exists, could possibly not be preferred by a majority of voters over any other candidate. The key is that some voters may have no preference between a given pair of candidates. Out of 100 voters, for example, 45 could prefer the IDW over another particular candidate, and 40 could prefer the opposite, with the other 15 having no preference between the two. In that case, it is not true that a majority of voters prefer the IDW over the other candidate, and SFC does not apply.

In order to understand SFC, one must also understand that there are two types of insincere votes: false preferences and truncated preferences. Voters truncate by terminating their rank list before their true preferences are fully specified (note that the last choice is always implied, so leaving it out is not considered truncation). Voters falsify their preferences, on the other hand, by reversing the order of their true preferences or by specifying a preference they don't really have. Suppose, for example, that a voter's true preferences are (A,B,C,D). The vote (A) or (A,B) would be a truncated vote, and the vote (B,A,C) or (A,C,B) would be a falsified vote.

SFC requires that the majority of voters who prefer the IDW to another particular candidate vote sincerely (neither falsify nor truncate their preferences), and it also requires that no other voter falsifies preferences. SFC therefore implies that the minority that does not prefer the IDW to the other candidate cannot cause the other candidate to win by truncating their preferences. (In theory, that minority could cause the other candidate to win by falsifying their preferences, but that would be a very risky offensive strategy that is more likely to backfire than to succeed.) The significance of the SFC guarantee is that the majority has no need for defensive strategy, hence the name Strategy-Free Criterion.

The Condorcet election method was shown to comply with both the Condorcet and Generalized Condorcet Criteria (CC and GCC) above. Although compliance with CC and GCC are important, those criteria apply only in the theoretically ideal case in which all votes are sincere. The Strategy-Free criterion goes further and shows that, under certain reasonable conditions, a majority of voters have no incentive to vote insincerely. The fact that the Condorcet also complies with SFC therefore enhances the significance of CC and GCC considerably.