[EM] Favorite Betrayal and Condorcet

[Forest Simmons]

Kris,

Your comments remind me that (if I remember correctly) there is supposed to
always exists a Nash equilibrium approval ballot set which elects the
sincere CW candidate when one exists.

But a DSV method that finds such an equilibrium (along with its concomitant
candidate) would have to satisfy the FBC, since any one voter defecting
from that equilibrium to approve her favorite F would get away with it ...
if the winner changed at all it would have to change to F.

So all we need is a constructive proof of the alleged Nash Equilibrium
existence.

Can someone clear up this mystery?

El lun., 18 de abr. de 2022 2:40 a. m., Kristofer Munsterhjelm <
km_elmet at t-online.de> escribió:

> On 18.04.2022 02:21, Forest Simmons wrote:
> > It is well known that Range Voting, no matter its level of resolution,
> > is strategically equivalent to Approval. In particular, this means that
> > under perfect information conditions there always exists an optimal
> > strategy that makes no use of any intermediate ratings. [However, as in
> > Linear Programming, existence of an optimal "corner" solution in no way
> > denies the possible existence of other equally optimal non-corner
> > solutions.]
> >
> > Not so well known, but equally true, is that every Condorcet compliant,
> > Universal Domain (i.e. RCV) method reduces to  Approval when voters vote
> > only at the extremes.
> >
> > Question 1. Does every perfect information UD Condorcet election have an
> > optimal strategy that makes no use of the intermediate rankings? This
> > certainly seems to be the tacit assumption of many Designated Strategy
> > Voting methods.
>
> As I understand it, in proper Condorcet methods, it's sometimes useful
> to use intermediate rankings because you can both express a preference
> for A over B and one for B over C at the same time.
>
> There are modifications of Condorcet that pass the FBC, e.g. Kevin's ICA
> and Mike Ossipoff's ICT and Symmetrical ICT. These work by making
> Approval strategy equally-optimal so that if (for the purpose of
> contradiction) your favorite is F and compromise is C, then F=C>... can
> be no worse then C>... But in so doing, they lose Condorcet efficiency.[1]
>
> Kevin's simulations of
>
> http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-June/114476.html
> seem to indicate that Condorcet methods (at least "advanced" ones like
> Schulze) have a low rate of FBC failure. The "Improved Condorcet"
> methods would presumably be the flipside of this coin, passing FBC
> absolutely but having some (low?) rate of Condorcet failure.
>
> I also seem to recall that MMPO doesn't reduce to Approval despite
> passing FBC, but my memory is not the clearest, so I could be mistaken.
>

It doesn't, in fact Kevin's first EM post twenty years ago (plus or minus a
few months) was MMPO in the context of Approval ballots. His example showed
that MMPO does not reduce to approval and (unlike Approval) fails both
Plurality and the ballot Condorcet criterion.

My example DSV method M below is a conversion of any RCV style ballot set
beta into an approval ballot set M(beta) such that the approval winner of
M(beta) is the only possibility for a CW of beta.

Kevin gave a  example to show that method M fails the FBC, and I showed
that changing the phrase "is not defeated by" to " is not majority defeated
by " makes the method FBC compliant and even preserves the property of the
CW getting maximum DSV approval ... but is not decisive because (analogous
to Copeland) there is an appreciable chance that the argmaxapproval set
will not be a singleton. If you break this DSV approval tie with something
like Plurality or Implicit approval, it seems that the CW is not guaranteed
to win. So the situation is very similar to ICT and MDDA.

-Forest

>
> -km
>
> [1] A similar trick applied to Borda is Mike's Summed Ranks method.
>
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