A voting system satisfies the Relaxed Majority Criterion if a majority faction of voters can express a non-zero "maximum support - 1" to a second choice candidate, and still guarantee that the majority faction's "maximum supported" first choice wins.
Early work in the field of voting theory identified criteria a given voting method may or may not pass. Most criteria define a certain kind of undesirable outcome, and say that good voting methods should make such outcomes impossible.
For example, the Majority Criterion states that "if one candidate is preferred by a majority (more than 50%) of voters, then that candidate must win." This criterion is used to dismiss Score and Approval voting, as running contrary to our basic notions of democracy. Further, this Majority Criterion failure is hypothesized to encourage factional bullet voting, because a majority offering any support at all to another candidate can contribute to the loss of the majority's preferred choice.
Contemporary voting theory work goes a step further: instead of asking “can a certain kind of problem ever happen?”, modern analytical methods ask, “how rarely do problems of all kinds happen, and with what impact?” To this end, we suggest a relaxed version of the Majority Criterion that recognizes degree of impact on representative accuracy and strategic voting.
For example, although STAR Voting doesn't satisfy the Majority Criterion, it fails in a dramatically less severe way than Approval and Score: with STAR the majority has to offer support for two additional candidates in order for the majority's top choice to have a chance of losing. For this reason, we expect voting systems that pass either the Majority Criterion or the Relaxed Majority Criterion will allow voters to much more confidently express support for second candidates without fearing harm of their first choices.
IRV, STAR, Unified Primary and 3-2-1 voting pass the Relaxed Majority Criterion. Score and Approval Voting do not.