[EM] Pattanaik and Peleg's 'Regularity' is not better
Craig Carey
research at ijs.co.nz
Thu Dec 16 00:04:12 PST 1999
----------------
Errata in the (P1) defined message:
I wrote "No deletion of the A in the ABC papers was possible by B, so
all the 10 ABC papers must have been transformed into the 10 papers,
AB, ACB, AD."
It should read "so the 10 ABC papers changed into 10 of the 16 A...
papers. (ABC can change into A).
----------------
I suggest that this message will be uninteresting to most.
I comment on Mr Catchpole's "regularity" theory.
The definition of the theory, from an 8 December 1999 message, is quoted
below.
At 15:29 15.12.99 , David Catchpole wrote:
>On Tue, 14 Dec 1999, Craig Carey wrote:
...
>AV, FPTP, etc. as they are usually expressed are _deterministic_ election
>systems- they all share the same two-candidate system probability function
x A
(1-x) B
>
>f(x)= { 0 if x<1/2
> { 1/2 if x=1/2
> { 1 if x>1/2
>
>Maybe you're getting the drift now? I'm going to go through the slog of
...
The function f was not really explained. Perhaps it is this:
f(x) = probability that A wins and B loses.
So Mr Catchpole has garbed one ordinary preferential method (the 2
candidate 1 winner formula) in a probability form. Obviously that can be
done for all methods. Any disputes over tie probabilities can be excluded.
At 15:40 16.12.99 , David Catchpole wrote:
...
>> ---------------------------------------------------------------------
>> 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
>
>Abstract: This paper considers stochastic social choice rules which, for
>every feasible set of alternatives and every profile of individual
>orderings, specify social choice probabilities for the feasible
>alternatives. It is shown that if such a social choice rule satisfies:
>
> (i) a probabilistic counterpart of Arrow's independence of
> irrelevant alternatives;
> (ii) ex-post Pareto optimality; and
>> > (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);
>
>then the power structure under it is almost completely characterised by
>weighted random dictatorship.
>> ...
>> > ... 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 ...
>> ---------------------------------------------------------------------
"Assumption": They call regularity an assumption, which is something that
can be wrong without that being particularly significant.
The paper uses the word "regularity" without particularly connecting that
definition to the term "random dictatorship".
...
>> 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 ...
...
>> 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)
There is no derivation of those numbers.
>> 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. ... Again, we've demonstrated that a
>system exists which does not violate regularity! ...
The argument here would be that "regularity" holds since 4/11 <= 4/9.
-------------
Note that there is a hidden "all preferences after the first preference
are totally ignored" aspect to the calculation of probabalities.
How else could these probabilities have been derived?:
"A 4/11, B 2/11, C 5/11".
So a conclusion is that Mr Catchpole's regularity theory ignores the
methods it judges and it instead judges methods by "probability"
numbers that depend only on the data and not in any way at all, on
the method given to the "regularity" principle.
That is not what a good rule for judging methods would do.
Here is the original definition. Note that the formula used to calculate
the probabilities is undefined and unknown:
-------------------------------------------------------------------------
DEFINITION OF REGULARITY
At 15:39 08.12.99 , David Catchpole wrote:
>Can I call 'em the Metameucil systems, can I? Can I?
>
>Okay- "Regularity" is the name used earlier by Albert Langer (Craig might
>recognise the name ;) ) to describe the probabilistic analogue of IIA. It
>goes like this-
>
>The addition (removal) of a candidate does not, for any other candidate,
>increase (decrease) the probability of that other candidate winning.
>
-----------------
At 23:10 13.12.99 , David Catchpole wrote:
>On Mon, 13 Dec 1999, Craig Carey wrote:
...
>I don't approve of non-deterministic systems being used in real life
>(don't approve of Condorcet being used either, so you can't use that
>against me either), but that doesn't mean they should be dismissed as an
>analytical tool- hopefully, especially, as a vindication of
...
> ... The basic
>substance of the first message was this- in order for regularity to be
>satisfied, the probability of A winning versus B as a function of
>x=n(A>B)/( n(A>B)+n(B>A) ) has to obey the following restraints...
>
>p(1/2)=1/2
>
>p(2/3)=2/3
>
>p(3/4)=3/4
>
The definition of "x" seems clear but then the varible is not used.
Numbers are usually not probabilities unless they sum to 1. The number
(1/2 + 2/3 + 3/4) is (6/12+8/12+9/12), = 23/12. So those numbers there
can't be probabilities.
Anyway, the probability of 1/2 is not 1/2, any more than the probability
of any other real number is 1/2.
So I ask: is that all part of the defintion ("basic substance") of
regularity, and if so, is what is "p"?.
-------------------------------------------------------------------------
I ask Mr Catchpole: are there any other definitions of Catchpole
"regularity" or is the definition above, authoritative. This seems to
me to be the central issue, defintions are typically badly thought out.
Note: the definitions reproduced do not use an idea of "random
dictatorships".
Note that the definition contains the words "candidate winning", yet
the numerical probabilities Mr Catchpole gave in the last day included:
"A 4/11, B 2/11, C 5/11".
Mr Catchpole obtained those numbers without knowing or stating what the
method was.
How can the "regularity rule" apply to methods when it is able to find
out features of methods without knowing what the method is?. The first
"15:39 08.12.99" definition certainly seems to be worded as if it took
a method as a parameter (as input).
-----------------
>relevant. Witness the fact that regularity actually does have a quite
>simple, quite easily proved, consistent method in random
>dictatorship. Think! ...
What do the words "does have a ... method" mean. How many methods does
regularity have?.
If a person were to ask Catchpole to test their method against regularity,
then which is tested: the supplied method or one of the methods that
"regularity" has?.
I presume that the words "random dictatorship" is a reference to an
analogue of FPTP that supplies numbers that were "probabilities".
That raises a question: how does Mr Catchpole define ["random
dictatorship" type] analogues of any preferential voting method?.
That is a seemingly relevant matter to the discussion since it is a
line of consideration that permits the "regularity" consept to be found
to be an idea of some merit in preferential voting. [The published
paper appeared to not do that.]
Mr Catchpole informed me that something. about which nothing is known,
namely the existence of a reference (or a referring) to the field of
mathematics known as probability theory, was a defence against my
arguments made against the regularity definition. Is there any other
field of mathematics that needs to be taken down before regularity
theory can be unprotected against attempted, perhaps even faulty,
proofs of uselessness?
Craig Carey
Auckland
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