[EM] it's pleocracy, not democracy

[Jobst Heitzig]

[sorry if this comes twice, but it didn't seem to get thru the first time]

Dear folks,

some clarification because in recent posts democracy and majority rule 
were confused quite often...

In a dictatorial system, almost all people have no power.
In a majoritarian system, up to half of the people have no power.
In a democratic system, ALL people HAVE some power, that is, "the people rule".

Hence, majoritarian systems in which a majority of 50% + 1 voter can
make all decisions are NOT democratic. The greeks called them "pleocratic".

Can a system be democratic?
Can it even be democratic without using significant randomization?

If we are faced with a whole sequence of decisions instead of only one,
we could distribute the power over all decisions in the sequence:

Naive solution: assign each decision to a (different) single voter so
that each voter decides something in turn and hence all people have some
power. Obviously, there are many deep problems with this.

More sophisticated solution:

Remember for each voter in what fraction of the decisions so far the
voter's then-favourite option has been elected; call this that voter's
"actual success rate".

Also remember for each voter the average (over all decisions so far)
fraction of voters that had the same then-favourite as the voter at
hand; call this that voter's "to-be-expected success rate".

Now, in each decision, elect that option which minimizes the sum of
squared errors between the voters' current to-be-expected success rate
(including the current decision) and the voters' resulting actual
success rate if that option were elected. In the long run, this sum of
squared errors should converge to zero (remains to be proven), so this
method can be called "asymptotically" democratic.

For example: Assume a sequence of A/B-decisions, voter 1 votes always A
and voters 2-4 vote always B. Then the following would happen:

        to-be-expected             actual success       sum of
round   success rates     winner   rates afterwards     squared errors
1       .25 .75 .75 .75   B        0   1   1   1        1/4
2       .25 .75 .75 .75   A        .5  .5  .5  .5       1/4
3       .25 .75 .75 .75   B        .33 .67 .67 .67      1/36
4       .25 .75 .75 .75   B        .25 .75 .75 .75      0
5       .25 .75 .75 .75   B        .2  .8  .8  .8       1/100
6       .25 .75 .75 .75   A        .33 .67 .67 .67      1/36
7       .25 .75 .75 .75   B        .29 .71 .71 .71      1/196
8       .25 .75 .75 .75   B        .25 .75 .75 .75      0
...

A little mathematics shows that this method is equivalent to a kind of
"weighted plurality" in which each voters vote is weighted with the
following (not necessarily positive) history-dependent weight:
(current to-be-expected successes) - (earlier actual successes) - 1/2

The latter indicates a potential problem: Knowing my success rates so
far, I may deduce that my vote in the current decision is actually
negative, in which case I may have incentive to vote for the strongest
competitor of my favourite instead of for my favourite.

So far, we see that an asymptotically democratic method without
randomization is possible when there is a whole sequence of decisions,
but this method suffers from strong incentives for strategic voting.

Of course, WITH randomization allowed, there is a perfectly democratic
and absolutely strategy-proof method: random ballot.

However, both methods have another problem: They do not easily support
cooperation between voters since it is either optimal to vote for the
favourite or for the strongest competitor, while there is no incentive
to vote for compromise options. Therefore, the results are "just" but
not particularly "efficient" with respect to utility.

The method D2MAC aims to improve upon this. It is: Draw two ballots at 
random; the winner is the most approved option of those approved on 
both ballots, if such an option exists, or else the top option on the 
first ballot.

Yours, Jobst

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