[EM] General Purpose Universal Domain Tie Breaking Order

[Forest Simmons]

A good tie breaking order has to be a good, conclusive, stand alone social
order method, in other words a good agenda setting method.

Here's one that takes Implicit Approval to its logical conclusion:

First, we classify the positions on RCV ballots from the bottom up:

An alternative is at level zero on a ballot iff it is ranked above no other
alternative.

A candidate is at level one on a ballot iff it is ranked above only level
zero alternatives on that ballot, and is not itself at level zero.

In general, an alternative is at level k on ballot B iff it is only ranked
ahead of alternatives of level j<k, at least one of which is at level j=k-1.

Note that if some alternative X is not ranked on B, then it is classified
as at level zero for ballot B, i.e. bottom status below all of B's ranked
alternatives.

For each candidate X let S(X,epsilon) be the sum (over k from zero to N) of
...

epsilon^k times the number of ballots on which X is ranked above level k.

S(X,epsilon) is a polynomial in epsilon that gives our tie breaking order:

Alternative X is greater than Y in the tie breaking order iff

 S(X,epsilon)>S(Y,epsilon),

where epsilon is treated as a positive infinitesimal.

[Two polynomials in infinitesimal epsilon are compared by comparing the
coefficients of the lowest degree terms that are different for the two
polynomials.]

So the only way a tie between X and Y would not be resolved by this tie
breaker is when for all k, alternatives X and Y have the same precise
number of ballots for which they have level k status.

In that rare (statistically impossible) eventuality a coin toss or a
supreme court decree would be justified.

Note that this tie breaking order is efficiently precinct summable.

Use it to set the agenda for SPE (done right) and you have a super decisive
Universal Domain, Landau efficient method.

-fws
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20230429/61947cb9/attachment.htm>