[EM] it's pleocracy, not democracy

Sun Mar 4 12:29:44 PST 2007

On Mar 2, 2007, at 12:40 , Jobst Heitzig wrote:

> [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:

I'd maybe call this a specific kind of proportionality, serial  
proportionality or proportionality in time. Single winner at its  
purest is just electing one of a number of candidates, giving no  
consideration to if it was the same voters that last time got their  
way through. Basic single winner methods maybe have worse utility  
than ones that take distribute the power over a sequence of  
decisions, but I'd still allow use of word "democratic" to describe  
them. Sometimes it makes sense to just forget the history.

Maybe electing the best film of the year is one example. It doesn't  
matter much even if the same people liked the best film of this year  
and of last year. Another example could be starting a genocide. Even  
if few percent of the voters would like that, it maybe is not the  
best option to start it after a sufficient number of years/elections  
have passed.

Different elections may have different targets. Sometimes  
"consensus", "compromise", "least opposition", "acceptability" or  
"wide support" could be key criteria instead of "proportionality".

But this type of methods are good for places where proportionality is  
the way to go.

> 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
> ...

How would you count the actual and to-be-expected success rates of  
new voters that vote at round n but that have not voted before (if  
this method is expected to cover such cases)?

> 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.

Another strategy could be to vote one's worst competitor if one's own  
favourite candidate is already expected to win in any case with  
sufficient margin. This way one could try to collect more weight for  
the next election.

One approach is to have a formula that carries only positive weight  
to the next election.

And one more approach is to tie the carried weight to the parties/ 
candidates instead of the voters (if they and their support stay  
relatively stable from one election to the next).

These methods would work also in multi-winner elections (also and  
especially when the number of elected alternatives is small).

One more observation on possible alternative approaches. Divisor  
methods like e.g. d'Hondt provide and ordering of the candidates. And  
fractions quotas may be a good values to be carried between elections.

> 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.

Random ballot is a good method for some elections (but maybe not e.g.  
in the genocide example above).

> 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.

Here again the society needs to decide if they want proportionality  
(each voter group one day gets a winner that has similar opinions to  
those of this group of voters) or if they want to elect every time an  
alternative that represents as many of the voters as well as possible.

I'd be quite happy to accept also "semi-heuristic" methods where we  
somehow try to live between these targets. That could mean e.g.  
losing some of the "credits" in time (=old credits gradually lost).  
The major parties would get their candidates elected in an  
alternating pattern but extremists maybe would get only reasonable  
compromises (maybe a "more extreme" candidate of the closest party of  
reasonable size). The target utility functions could be quite  
different for different elections.

Juho

> 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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