[EM] Pattanaik and Peleg's 'Regularity' is not better

[David Catchpole]

The definition given by Pattanaik and Peleg (and by myself, now in
the distant past in EM time) relates to election methods where it is taken
as given that only one option (candidate) will be selected. However,
regularity does persist for probabilistic multi-winner election systems,
and my intuition and my work with regularity seems to indicate, if
anything, a tendency towards proportionality in multiple-winner
systems.(more further down)

On Mon, 15 Feb 1999, Craig Carey wrote:

> 
> This message has little of interest to most.
> This regularity definition is taking a few more arguments than I had
>  expected. It is opposed to proportionality and instead favours unfair
>  outcomes.
> 
> 
> At 15:29 15.12.99 , David Catchpole wrote:
> >On Tue, 14 Dec 1999, Craig Carey wrote:
> ...
> >>  a connection and none exists. It is a weak form of rebuttal: to connect
> >>  one failing idea to another when they are in truth separate.
> >
> >You're using that awful circular logic again. Please stop. ...
> 
> There is no circular reasoning. To say there is when there isn't is to
>  make a misleading statement. I had written that 'regularity' prohibited
>  the closing of a B-wins region in a simple example and that that 
>  problem rejected the method instead. How could that be circular? (it is
>  not).

That was in a deterministic formalism, in which case a deterministic
paradox no longer exists. We're talking here about a _probabilistic_
formalism, in which the simplification principles you applied in your
quasi-disproof (also now in the dim distant past) are no longer
relevant. Witness the fact that regularity actually does have a quite
simple, quite easily proved, consistent method in random
dictatorship. Think!(more further down)

> 
> 
> >Maybe you're getting the drift now? I'm going to go through the slog of
> >sending the introduction of "Distribution of Power Under Stochastic 
> >Social Choice Rules" by Pattanaik and Peleg, Econometrica Vol. 54 No. 4
> > to you.
> >Hopefully then you'll understand-
> 
> That paper by Pattanaik and Peleg describes a dud idea named 'regularity'.
> 
> It is very similar to Mr Catchpole's regularity and it fails under the
>  same considerations, as indicated below

It _is_ Catchpole's regularity (please, somebody... Markus? Blake? Tell
him it's so!).(more further down)

> 
> ...
> 
> ---------------------------------------------------------------------
> At 16:49 15.12.99 , David Catchpole wrote:
> >Distribution of power under stochastic social choice rules
> >Prasanta K. Pattanaik and Bezalel Peleg
> >Econometrica Vol 54 No 4
> ...
> >  (iii) "regularity" (a "rationality" property postulating that given
> >      the individual preference orderings, if the feasible set of
> >      alternatives is expanded, then the social choice probability for 
> >      an initially feasible alternative cannot increase);
> ...
> >      ...                             Regularity implies that given the
> >profile of individual preferences, if one enlarges the feasible set of
> >alternatives by adding more alternatives, then the probability of the
> >society's choosing any of the alternatives figuring in the original
> >feasible set cannot increase after the feasible set is enlarged. This
> >seems to be the natural probabilistic counterpart of Sen's ...
> 
> > ... We show that these three assumptions, together with the
> >assumption that the universal set of alternatives has at least four
> >elements and that individual preference orderings are strict, imply that
> >for every proper subset of the universal set of alternatives, the
> >probabilistic social decision procedure must take the form of random ...
> ---------------------------------------------------------------------
> 
> Consider election examples [it seems that Mr Catchpole did read my
>  message that wrote off his regularity idea, so a bit of repetition
>  is perhaps appropriate]
> 
> What follows is a repetition by me...
> 
> --------- (1): ---------
> 
> Note that in the 1st of the 2 following examples, A could have won but
>  C won instead. Candidate B was then added and then candidate A won.
>  A was a 'feasible alternative' presumably, in the smaller election.
> 
> At 20:48 09.12.99 , Craig Carey wrote: ...
> 
> >>:   A. 2
> >>:   C. 3
> >--- ------ A loses after B is removed

Under random dictatorship, the probabilities of each
candidate winning read: A: 2/5, C: 3/5 (more further down)

> 
> >>>> A. 2
> >>>> BC 2
> >>>> CA 1
> >--- ------ A wins ... (The IFPP quota = 1.666, so STV=IFPP here)

Under random dictatorship, the probabilities of each candidate winning
read: A: 2/5, B: 2/5, C: 1/5.

Note how the system definitely fails to contradict regularity?

I believe I'm starting to understand the hackney logic you're using now-
IFPP fails regularity, therefore regularity=bad.(more further down)

> 
> 
> --------- (2): ---------
> 
> Note that if B is added to the first example of the next two, then the
>  feasible alternative A, becomes the winner.
> 
> 4 A
> 5 C        Now candidate A loses.   The IFPP/STV winner = C

Under random dictatorship, the probabilities of each candidate winning
are: A 4/9, C 5/9 (more further down)

> 
> 4 A
> 2 BA
> 5 C        IFPP/STV winner = A      (the IFPP 1/3 quota = 3.666)

Under random dictatorship, the probabilities of each candidate winning
are: A 4/11, B 2/11, C 5/11. Whoah! Again, we've demonstrated that a
system exists which does not violate regularity!

Really, I'm getting sick of this, Craig. I really don't like people who
allow ignorance to fuel some kind of personal vendetta...

-------------------------------------------
Nothing is foolproof given a talented fool.