# Talk:Plurality criterion

Why can't we just say "B should not be elected", rather than "A's probability of election must be no less than B's"? James Green-Armytage 15:56, 9 Jun 2005 (PDT)

We could, but this wouldn't behave exactly the same when the result isn't decisive. Kevin Venzke 20:41, 11 Jun 2005 (PDT)

## How does the Plurality set (and a subset of that) relate to anti-order reversal criteria?

The plurality criterion says that if an option X has less preferences of any kind (meaning other than last place) than option Y has first place preferences X shouldn't be more likely to win. I guess the following complies with that criterion:

Look at rankings (equal rankings allowed) and for each option simulate that it got the luckiest Approval voting treatment aside from order reversal. That means: For those ballots that give option X any other ranking than last place move the approval cutoff just behind X. For those ballots that have X in last place let approval be as stingy as possible so only approve their first place preferences.

Those options that would win under these conditions are the Plurality set (or better call it the set of possible Approval Winners). Are there methods that satisfy plurality but not independence of options outside this set?

• Update: The set of Possible Approval Winners is more discriminating than and therefore not equal to the Plurality set. See:Electorama posting by Chris Benham quoting an example by Kevin Venzke.

More discriminating: For those ballots that give option X any other ranking than last place move the approval cutoff just behind X. For those ballots that have X in last place let approval be as generous as possible so only disapprove their last place preference.

Is there a name for that more discriminating subset of possible Approval Winners yet? Which methods select from that set and which of these are independent of options outside this set? Is this somewhat related to Ossipoff's criteria about unneccessary order reversal if you are in a majority (or look at: http://alumnus.caltech.edu/~seppley/ for minimal defense, non-drastic defense, truncation resistance)? I think no method can satisfy those tactical criteria without satisfying Plurality plus something else.

Is repeated search for the more discriminating subset and elimination of those outside a good idea?

The innermost set should always contain the Condorcet Winner because an option can't even get outside the set of possible Approval winners without being beaten pairwise by at least one option (and even that isn't enough to be sure to not win under Approval).

Would that have problems with Mono-Raise or Reversal Symmetry like so many other elimination methods and if it does, could applying the elimination process in the opposite direction as well fix it (at least the symmetry bit I guess)? --R.H. 13:34, 14 September 2006 (PDT)