A while back, Warren D. Smith made the interesting observation that score voting satisfies a natural generalization of the Condorcet criterion: any candidate who would pairwise-beat every other candidate in a one-on-one race, must win the election. This leads to the question, are there other natural generalizations of the majoritarian criteria (e.g. majority, mutual majority, or Smith)? Smith is easy: because we have a total order over the reals, there's always a cardinal-Condorcet winner, so it reduces to the cardinal-Condorcet criterion. Majority can be extended too, if we think of a majority winner as a "super-Condorcet" winner. Define a majority winner as someone who could defeat every other candidate *put together *in an unholy Voltron-style mashup. We define X to be a majority winner if they could defeat a new candidate, X', who equals the "sum" of all other candidates. With ordinal ballots, we take sums of candidates by taking the minimum rank of each candidate (as in tropical geometry). Score voting satisfies the same criterion, where the natural sense of adding two candidates is adding their scores: if one candidate has more points then every other candidate put together (i.e. a majority of all points assigned by voters), they're guaranteed to win. Any other natural redefinitions?

On 2024-02-10 22:33, Closed Limelike Curves wrote: > A while back, Warren D. Smith made the interesting observation that > score voting satisfies a natural generalization of the Condorcet > criterion: any candidate who would pairwise-beat every other candidate > in a one-on-one race, must win the election. > > This leads to the question, are there other natural generalizations of > the majoritarian criteria (e.g. majority, mutual majority, or Smith)? > > Smith is easy: because we have a total order over the reals, > there's always a cardinal-Condorcet winner, so it reduces to the > cardinal-Condorcet criterion. > > Majority can be extended too, if we think of a majority winner as a > "super-Condorcet" winner. Define a majority winner as someone who could > defeat every other candidate /put together /in an unholy Voltron-style > mashup. We define X to be a majority winner if they could defeat a new > candidate, X', who equals the "sum" of all other candidates. With > ordinal ballots, we take sums of candidates by taking the minimum rank > of each candidate (as in tropical geometry). > > Score voting satisfies the same criterion, where the natural sense of > adding two candidates is adding their scores: if one candidate has more > points then every other candidate put together (i.e. a majority of all > points assigned by voters), they're guaranteed to win. > > Any other natural redefinitions? Because Score/Range passes IIA (with some serious practical caveats), the ordering is always transitive and everything pretty much collapses. The majority winner (by your definition) is the Range winner. The CW is the Range winner. The mutual majority set is a single candidate consisting of the Range winner, etc. Other concepts of majority lead to other sets. For instance, you could say that a majority candidate is a candidate who a majority of the voters rates higher than anybody else. As an example, in a post a while ago, I considered a Condorcet analog that might allow for the creation of cloneproof generalizations of STAR, while still taking intensity of preference into account. Let a cardinal election restricted to {A, B, C} be constructed by eliminating everbody but A, B, and C, and then normalizing every voter's ballot to unit lp norm. Then you could do something like: say that A is a "cardinal Condorcet winner" if, for all X and Y, in the election restricted to {A, X, Y}, A is the winner. Or A is never a loser. And then construct Smith sets, etc. from there. (E.g. something like A > B if in every election {A, X, B}, A's score is higher than B's) Two-candidate contests also need to be considered to have a chance of clone independence; I've omitted that here. -km