[EM] Re: Some hard example for Approval Voting

[Araucaria Araucana]

On 22 Mar 2005 at 14:04 UTC-0800, Rob LeGrand wrote:
> Jobst wrote:
>> Unfortunately, I get the impression that in the following example
>> there is no such equilibrium:
>>
>> 3 D>C>A>B
>> 3 D>A>B>C
>> 5 A>B>C>D
>> 4 C>B>D>A
>>
>> So, can anybody forecast what will happen with these preferences
>> under Approval Voting??
>
> Interesting example.  Bucklin gives B, IRV gives D, Borda gives A
> and most methods popular here (beatpath, River, Ranked Pairs) give
> C.  There is no Condorcet winner, so there is no Approval
> equilibrium; any leader will be quickly toppled if everyone uses
> strategy A (which is always sincere in the sense you give above).
> Strategy A allows individual voters to move the current result in
> the most advantageous direction with no notion of being part of a
> new majority coalition; new coalitions emerge naturally from the
> smart strategic moves.  Declared Strategy Voting in ballot-by-
> ballot mode running for many rounds using Approval and strategy A
> elects them with approximate probabilities A 25.05%, B 12.99%, C
> 27.54% and D 34.42%.
>

It is indeed an interesting example.  Consider Definite Majority
Choice ("DMC", aka Ranked Approval Voting) as an alternative:

All Approval cutoffs at 1st place: Approval order D,A,C (B=0).  
==> D wins.

All Approval cutoffs at 2nd place: Approval order B,A,C,D.
==> A wins.

All Approval cutoff at 3rd place: Approval order C=B, A, D.  
==> A wins.

Rob's voting calculator page shows that it isn't just Borda that gives
an A win, it's Borda, Bucklin, Copeland, Nanson, and many others.  I
think this reflects the effect of the Approval (cumulative higher
ranking) bias in DMC.

Plurality and IRV (and wv RP/Beatpath/River) would have picked a
winner with less than 50% approval -- in fact the (sincerely)
least-approved of all candidates.

Under DMC, the only voting block that could win by bullet-approval
cutoff is the "3:D>A>B>C" group.  But if any other block uses a more
generous cutoff, A will win (or possibly C in one or two cases).  So
there is no clear advantage for D>A>B>C to bullet-approve.  Just the
opposite, in fact.

With sincere approval cutoff at 2nd place, the set P (candidates not
defeated by any higher-approved opponent) contains A and B.  A wins
with a solid 11>4 victory over B, but with weak approval -- barely
over 50%.  But overall, B loses quite respectably with higher
approval.  B's faction could win the next election by winning over 4
of the 11 A>B voters (26.7% of the electorate).  And in the meantime,
A will be working *very* hard to avoid that reversal.

A centrist winner who pays attention to issues of concern to many.
Isn't that the outcome we're striving for here?

Ted
-- 
araucaria dot araucana at gmail dot com