[EM] recommendations

Jobst Heitzig heitzig-j at web.de
Tue Aug 3 02:04:30 PDT 2004

Currently, I prefer a simple method called "Random Order Acrobatic Chain
Climbing (ROACC)", posted by me on June 13. Before repeating its
definition, I will try to motivate it from first pinciples:

Every voter is to be treated equally and should be allowed to express
every binary preference s/he wants to express without having to express
any preference s/he doesn't want to express. In particular, when a voter
prefers some option to all other options, s/he must at least be allowed
to express that fact.

The decision method's goal is to find an option which is "optimal"
in some sense, given the sincere preferences of the voters. Therefore,
it must be able to use as much reliable information about the voters'
preferences as possible, but only such information whose interpretation
is unambigous. In particular, it must give voters as few incentives to
vote strategically as possible since insincere expressed preferences
give no reliable information about sincere preferences.

Whenever more than half of the voters unanimously express some
preference relation in which some option X is preferred to all other
options, X must be elected.

  (a) From (2) and (3) it follows that the method can only elect some
option X with certainty when no option Y is prefered to X by more than
half of the voters. Otherwise, those voters prefering Y to X would have
an incentive to elect Y by voting "Y > all others". In other words,
whenever each option is defeated by some other option with absolute
majority, the method cannot elect any option with certainty but must
choose randomly between at least two options.
  (b) More generally, (2) and (3) also imply the following: Assume that
the method elects X,Y,... with probabilities P(X),P(Y),... Let Y be any
option. Consider a voter V for which no option Z which V prefers to Y
has P(Z)>0, but V prefers Y to at least one option X with P(X)>0. Then,
for each Y, the number of those voters V must be less than half of all
voters since otherwise they would have an incentive to elect Y by voting
"Y > all others". In other words, the method must give enough options a
positive probability of election so that no option "dominates" all these
possible winners in more than half of all voters' preferences.

The elected option must be a sufficiently "stable" choice in some
sense. In particular, when X is elected and some group of voters prefers
Y to X, there must be some means to rebut this "binary argument". The
most natural way to rebut such binary arguments is to use similar
arguments to contradict it. A minimal requirement is that the elected
option should have a "beat path" to every other option, that is, belong
to the Smith set. However, these beat paths should also be as short or
as strong as possible. Some methods such as the beatpath method,
Tideman's method, and the river method optimise the *strength* of those
beat paths in the sense that every binary argument against the elected
option can be rebutted by a beat path at least that strong. However,
there are in general too few options with that "immunity" property too
simultaneously fulfill requirement (4) above. When we focus on *short*
rebutting beat paths instead of strong ones, we can easily find enough
possible winners: in general, every uncovered option (in particular,
every member of the Banks set and the Tournament Equilibrium Set) has a
beat path of length 1 or 2 to every other option.

The method must be based on a clear and plausible principle which can
easily be understood by all voters. It should be possible to demonstrate
the essential properties of the method such as Condorcet compliance and
monotonicity easily. Also, the result should only as much depend on
chance as is neccessary to fulfil (4). If possible, the method should
also be applicable easily so that it can easily be learned and controlled.

These essential considerations lead me to prefering the following method:

Def.: Random Order Acrobatic Chain Climbing (ROACC)
Starting with an empty "chain" of options, successively pick a new opion
uniformly at random from all options not yet picked, and add it to the
chain when it defeats all options already in the chain. When all options
have been picked, the last option added is elected.

This method is very easy to apply, monotone, and it fulfils (4b) and
(5). And it has a comparatively small set of possible winners, namely
the Banks set (in particular, it is a Condorcet method). Note that the
Banks set is also clone consistent.

I consider this the best method so far when strategical voting is to be
assumed. However, it still has some disadvantages: First of all, it does
not use the strengths of defeats. Second, it is only partly clone
consistent since the total probability of all clones can deviate from
the probability of the cloned option. This could perhaps be avoided by
picking the options in some other way than above. The size of the set of
possible winners could perhaps further be restricted to the Tournament
Equilibrium Set for which there is also a natural probability
distribution. The latter, however, is not as easily determined and it is
not yet known whether it is monotonous.



ad (1) and (2).  This implies that voters must not be allowed to express
"degrees of preference" as long as it is not clear what a "degree of
preference" could possibly mean. As for "approval cut-offs", I am also
quite doubtful whether I understand what "approval" is to be meaning.
Although I think it might be possible to give an operational definition
of "approval" I fear that definition will be too complex to be
understood by all voters. In contrast, binary preferences are one of the
most fundamental concepts of human thought and it is also very easy to
define them operationally.

ad (3).  Other than the Condorcet criterion, this axiom is probably the
weakest possible version of "majority rule".

ad (4).  Note that this does not imply the Condorcet criterion so far.
However, the easiest way to fulfil (4) without giving *every* option a
positive probability seems to be ROACC (see above).

ad (5).  Note that there does not always exist an option which has beat
paths which are *both* short and strong!

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