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

[David Catchpole]

On Thu, 16 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

Yes! (more further down)

> 
> >
> >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.

Yes! (more further down)

> 
> 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.

..Gah? "Assumption" simply means in this case a rule whose implications
they are investigating. The choice of the word "assumption" is in no way
loaded with any implicit values. It's just ordinary talk in mathematical
economics papers. (more further down)

> 
> The paper uses the word "regularity" without particularly connecting that
>  definition to the term "random dictatorship".

The paper claims to have discovered that regularity, when coupled with
pareto ex-part (which boils down to f(0)=0 and f(1)=1 in the two-candidate
case, and regularity does the rest), implies the system is random
dictatorial.(more further down)

> 
> ...
> >> 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.

Simple garden random dictatorship looks like this: f(x)=x (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. ... 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.

Yes. (more further down)

> 
> -------------
> 
> Note that there is a hidden "all preferences after the first preference
>  are totally ignored" aspect to the calculation of probabalities.

No, there isn't generally- but this is indeed the case for random
dictatorship. Whatr the examples above are supposed to demonstrate to you
is that there does exist a system which satisfies reglarity. (more further
down)

> 
> 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.

Uh... no.(more further down)

> 
> 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:

Of course it's arbitrary (undefined and unknown)- it's a rule on an
arbitrary group of systems! Grrr! (more further down)

> 
> -------------------------------------------------------------------------
>                        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.

We're talking here about the conditions on the function for the two
candidate case, in order for regularity to be satisfied for all
transitions between 1, 2, and 3-candidate cases.
They are indeed probabilities. p(1/2)+p(1/2)=1/2+1/2=1,
p(2/3)+p(1/3)=2/3+1/3=1, p(3/4)+p(1/4)=3/4+1/4=1.

Note the basic axiom for the two-candidate case: p(x)+p(1-x)=1 (PS "p" is
the same as "f" and "g" used at various times during this one-sided
contest ;) ) (more further down)


> 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.

You can have a look at Pattanaik and Peleg, which states precisely the
same but in a formal setting with use of K's, B's and C's.

> 
> Note: the definitions reproduced do not use an idea of "random
>  dictatorships".

And?(more further down)

> 
> 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.
>

I did indeed! I said, "assuming random dictatorship"- Grrr!(more further
down)
  
> 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).

What use is a rule if it cannot be used to establish some rough features
of the methods which satisfy it? Example- when I say that a function in
odd (f(x)=-f(-x)) and continuously differentiable, this implies directly
that f can be expressed as x multipled by a continuously differentiable 
function of x^2. I don't actually state what the function is- indeed-
there is an infinite number of functions which satisfy this rule!(more
further down)

> 
> -----------------
> 
> >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?.

think! According to my intuition, there's an infinite family of regular
systems, including those in which candidates are randomly drawn, random
dictatorship, an, if Peleg and Pattanaik are a little misled, a whole lot
of others. According to Peleg and Pattanaik, the family is smaller, and if
we assume all candidates and voters are equal and that the system is
Pareto ex-post, the family only has one member.(more further down)

> 
> I presume that the words "random dictatorship" is a reference to an
>  analogue of FPTP that supplies numbers that were "probabilities".

Witness P+P's introduction, where they refer to earlier work on aggregate
point-scoring systems. Probabilities could be proportional to points, so
one example would be an analogue of Borda, where the Borda score gave the
probabilities.(more further down)

> 
> 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?

You need to couch any disproof of regularity's usefulness in a
probabilistic formalism- that is, you have to actually consider
probabilistic, not deterministic functions. A good start would be to rock
on down to wherever and pick up books on-

- game theory and chance
- stochastic mathematics
- something on general theories of probability and statistics

> 
> 
> Craig Carey
> Auckland 
> 
> 

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Nothing is foolproof given a talented fool.