[EM] serious strategy problem in Condorcet but not in IRV?
Jobst Heitzig
heitzig-j at web.de
Mon Apr 19 00:16:02 PDT 2004
Another idea to solve the problem:
Def. COARSE<whatever>//APPROVAL:
1. Coarsen the information about majorities by rounding each majority's
magnitude s to the smallest element of {1/2, 2/3, 3/4, 4/5, ..., 1}
which is greater or equal to s.
2. To these "coarsened" magnitudes, apply some magnitudes-based method
without its tiebreaker, for example beatpath, Tideman or the river
method. This reduces the options to a set of "acceptable" options.
3. Pick the most approved acceptable option.
The rationale behind this is to consider majorities only to decide which
options are acceptable, without overinterpreting magnitudes.
This solves James' example at least when the original winner has more
than 1/3 of the voters as direct support.
The set {1/2, 2/3, ...} is of course chosen quite arbitrarily - the main
feature is the gap between the first and second values. Depending on how
much importance you want to give "Majorities" or "Approval", you could
also use {1/2,2/3,1}, or {1/2} united with the interval [2/3,1], and so on.
Steps one and two of "COARSE BEATPATH//APPROVAL" is one of the methods I
studied in "Social Choice Under Incomplete, Cyclic Preferences"
(http://www.mathpreprints.com/math/Preprint/heitzig/20020123/1). Also
full beatpath is one of those methods, of course.
Steve Eppley's "Immunity from Majority Complaints" criterion is
essentially what I had called "Immunity from Majority Arguments" or
simply "Immunity" in that paper.
Although it may fail full component consistency, I tend to prefer SD for
finding the acceptable alternatives in step 2, more precisely:
simultaneously remove ALL defeats which are of minimal magnitude in some
cycle. In other words: the acceptable options are exactly the "immune"
ones (but with coarsened magnitudes). That is also one of the methods
discussed in the paper. Call it "COARSE IMMUNITY//APPROVAL".
My feeling is that coarsening magnitudes should preserve most
monotonicity and other properties and that using approval to find the
optimal acceptable option could give a wonderful compromise between
supporters of Condorcet and approval...
Jobst
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