[EM] Use of Sincere Confirmation Ballots to Detect and Correct Burial of the Sincere CW.
Forest Simmons
forest.simmons21 at gmail.com
Wed Mar 16 19:11:23 PDT 2022
Consider the following ballot profile ...
40 ABC
35 BCA
25 CAB
Who would be elected under River, CSSD, Ranked Pairs, Borda, MinMax, or IRV?
Answer: Everyone of them would elect candidate A except Borda, which would
elect B.
Question: which candidate should be elected?
Answer ... that depends on the voters' sincere preferences. It is possible
that the ballot profile is completely sincere ... possible, but highly
unlikely .... and impossible under the assumption that the three factions'
preferences were determined by their geometric proximities to each other.
So let's consider in turn the three most likely scenarios for giving rise
to this ballot profile:
40 ABC [sincere ACB]
35 BCA
25 CAB
In this first scenario, C is the sincere CW, and given sincere ballots, C
would be elected by any of the above mentioned methods except IRV.
So IRV gives no incentive to bury C, but only because its center squeeze
already eliminates the CW.
In the next scenario, the B faction buries A the sincere CW ... to no
avail, since A still wins, except under Borda, which rewards the burial by
electing B.
40 ABC
35 BCA[sincere BAC]
25 CAB
In the final scenario, the C faction buries its second choice B in order to
get its last choice A, elected. So this burial will not happen with
rational voters.
40 ABC
35 BCA
25 CAB[sincere CBA]
So the first scenario is the most likely.
Is there a single procedure that would result in the election of the
sincere CW in all three scenarios under all of the basic methods mentioned?
Wouldn't such a procedure require the ballot counters to magically divine
the sincere preferences of the voters?
Not if the election officials make judicious use of a second set of ballots
for a sincerity check.
"Judicious" entails zero incentive for insincere voting on the second set
of ballots.
The procedure takes as input the finish order of the voting method as well
as the pairwise win/loss/tie matrices for both ballot sets ... the
strategic, as well as the pure.
First use the strategic matrix to apply the chain climbing algorithm to the
finish order. Let MC be the resulting maximal totally ordered chain.
So let MC be the chain X2, X1 where X1 is last in the finish order and X2
is the candidate that is not pairwise beaten by X1 according to the
strategic matrix.
Now we use the pure, sincere matrix to check this supposed fact. Does X2
sincerely defeat X1?
If so, the winner of the strategic finish order should be elected.
If not, then X1 is the sincere CW, and should be elected.
Let's check the validity of these two statements in detail in each of the
three scenarios.
All three scenarios are based on the same ballot profile for which Borda
gives BCA as the finish order, while the other methods agree on ABC.
For both both finish orders we get X1=C and X2=B.
In the first scenario the sincerity check (X2>X1?) fails because X1=C
sincerely defeats X2=B. So the sincere CW is elected. (Success so far!)
The same two finish orders come up in the second scenario, so again X1=C,
and X2=B. But this time the sincerity check confirms that X2 pairwise
defeats X1, since B defeats C in the sincere order. So the strategic finish
order is upheld ... ABC in all of the methods except Borda, and BAC in the
case of Borda.
So Borda is the only method to reward a faction with a better than sincere
result.
In the third scenario, the sincerity check again confirms B defeats C, but
this time Borda voters have no burial incentive because Borda elects B in
either case.
Beyond the three burial cases, whenever the strategic ballots are actually
sincere, X2 is both the sincere and ballot CW, so X2 is not beaten by X1,
so the method finish order will be respected ... i.e. the sincere CW in all
cases except for IRV in the first scenario which elects A instead of C.
In summary, we see that the confirmation procedure is 100 percent
effective for burial protection and restoration with standard Condorcet
methods in the given scenarios ... and as good or better than nothing in
the case of Borda and IRV.
Coming up ... a more elaborate example involving five candidates ...
-Forest
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