[EM] Monotony and consistency - 3 examples
Joe Weinstein
jweins123 at hotmail.com
Wed Jan 30 01:32:01 PST 2002
Here are three examples. In all cases, ‘monotone’ may be interpreted
precisely as ‘basic monotone’, defined in my recent posting.
IRV: NEITHER MONOTONE NOR CONSISTENT?
Yes! Here’s a simple example that displays BOTH failures: an election among
three candidates X,Y and Z, run in two precincts P1 and P2, with 8 and 9
voters. There are six factions (i.e., kinds of fully marked ballot)
possible: for simplicity, notation omits the last-place candidate. Thus, XZ
denotes the faction which votes X first, Z second (and then necessarily Y
last).
For each of the two precincts and each faction, the table below gives the
number of votes (in that precinct, from that faction).
Faction: XY XZ YX YZ ZX ZY
-------------------------------------------------------------------
Precinct:
P1 1 2 0 3 2 0
P2 1 4 0 1 3 0
Overall 2 6 0 4 5 0
In each precinct, the IRV winner is X. However, the overall IRV winner is
Z, contra consistency.
Moreover, suppose both XY voters change their ballots to vote YX. Overall,
the result is to make X the overall winner, despite X having been disfavored
in the ballot swap, contra monotony.
MONOTONY WITHOUT CONSISTENCY?
Yes! Here’s a simple example: an election between two candidates X and Y,
run in two precincts P1 and P2, each with three voters, using a CR-type
ballot with MAXGRADE=3. Each candidate’s score is his median grade, and
highest median wins (allegedly, ‘Bucklin’s Method’).
For each of the four precinct-candidate combinations and each possible
grade, the table below gives the number of votes (in that precinct which
award the candidate the given grade):
Grade: 0 1 2 3
----------------------------------------------
Precinct & candidate:
P1 - X 1 2
P1 - Y 2 1
P2 - X 1 2
P2 - Y 2 1
X wins each of the precincts: for X and Y respective medians are 3 and 2 in
P1 and are 1 and 0 in P2. However, overall, Y wins: overall X- and
Y-medians are 1 and 2.
CONSISTENCY WITHOUT MONOTONY?
YES! (?) Richard Moore et al have asked whether we can find a non-monotone
consistent method. An example of his, and other examples below, show that
the answer is YES - IF, for the scoring of the marked ballots, we allow a
method which overtly subverts voters’ intuitive ‘monotone’ expectations of
what they should be able to achieve by marking candidate X ‘higher’.
Consider any election method M for one or several winners which satisfies
two conditions. First, M uses a CR-type ballot: every candidate is rated
one of a fixed ordered set of (at least two) grades (‘ratings’). Second, M
uses some instance of a generalized usual CR approach to scoring. Namely,
each candidate’s overall score is got by summing (or, if you prefer,
averaging) over the individual ballots a quantity which for each ballot is
based just on the ballot’s grade for the candidate and possibly also on the
overall frequency distribution of grades on that ballot. Candidates with
the highest score are the winners.
Because scores are got by summing over the ballots a quantity which for each
ballot depends just on what’s on that ballot, the method M must be
CONSISTENT.
One such method M1 uses the Five-Slot ballot: MAXGRADE=4, and numerical
grades may be coded by A=4, B=3, C=2, D=1, F=0. M1 awards a score to each
candidate as follows: 3 for each A, 4 for each B, 2 for each C, 1 for each
D, 0 for each F. This method is (formally, anyhow) NOT monotone, because an
initial winner X can be made a loser by being voted nominally higher on one
or more ballots - namely by grade switch from B to A.
However, it’s also clear that the scoring method grossly and obviously
violates voter expectations of what voting ‘higher’ is supposed to mean
operationally. At the very least, voters expect that if they all
unanimously award grade A (numerically 4) to candidate X and grade B
(numerically 3) to candidate Y, then X will be a winner - and moreover, in
a 1-winner contest, Y will not even tie for winner. More generally, in the
case of unanimity, for each candidate a higher common grade should imply a
higher score.
Another example method M2 may seem less outrageous. M2 softens M1's obvious
violation of voter’s expectations about scoring, but at the cost of using
randomization which voters might find irksomely gratuitous, because it is
not needed or used to break ties. Namely, M2 uses the usual CR Five-Slot
scoring in nine elections out of ten (A=4, B=3, etc.) but uses the above M1
scoring in one election out of ten. In order to deflect voter wrath from
obvious use of M1, for each election an automated randomizing device - whose
outcome is made known only after the voting - is used to decide whether
scoring is by M1 or M2.
The real issue becomes just which sorts of these anti-monotony devices we
rule out as being too grossly (or irksomely) subversive of reasonable voter
expectation that a higher grade on the given voter’s ballot will likely help
rather than hurt a candidate. Conceivably, we may be able to prove that
monotony will follow from consistency plus sufficiently strong but clearly
reasonable restrictions against these devices. Or, conceivably we may be
able to agree that M2 is a counterexample which already satisfies enough
such restrictions. Or, though we might disagree on M2, we might still
agree that some other counterexample satisfies enough restrictions.
Joe Weinstein
Long Beach CA USA
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