[EM] Monotony and consistency - 3 examples

[Joe Weinstein]

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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