Delegating in eDemocracy, my Way!

The vote of a person who is present should weight more than the vote of a person absent, which is asking someone to vote for him. Also no one, independently to the number of votes, should have a vote higher than a certain value M. How much is M? I was suggesting that we should start with 1/630-ith of the number of participating people. So no one person should have more power than a parliamentary in the [Italian] House of Parliament today. Without explicitly listing them I was suggesting that the characteristics that we should use to decide the weight of a delegated vote were:

v(0)=1 (the weight of who does not receive any mandate should still be 1)

v(1)< 2 (the weight of who receive a single mandate should be lower than the weight of two people that vote directly).

$\lim_{x\rightarrow\infty}(v(x))=M$ (If the number of mandates that a person receives grow, the weight must tend to M)

$\dot{v}(x)\geq0$ per $x\geq0$ (as the number of mandates grows, so does the weight of the vote of the person receiving them)

$lim_{x\rightarrow\infty}(\dot{v}(x))=0$ (Each extra mandate grows the weight a bit less each time)

I even went as far as to say that there were an infinity of functions that satisfied those requirements. And in fact…

Few days later my friend Daniele Gewurz wrote me… Daniele is a great mathematician, very precise in his work; His blog is called L’Accademia dei Pignuoli (con la u!) [Hard to translate, maybe: Academy of Nitpickers?]. So I contacted Daniele, and after I wrote the blog post he came back to me with the answer.

[pause… silence… suspense… ]

The function, in fact the functions that we are looking for exist. In particular, each function of the form:

$v(x) = \frac{(M-1)x}{(x+(M-1)+k)}+1$ per $k\geq 0$

will satisfy those requirements.

As k grows the weight of the delegated person will go down. For K=0 the single mandate will give an extra 0.99, that is v(1)=1.99 (if I receive the mandate from one single person, my vote will then weight 1.99, we lose 1% mandating your vote to someone that does not receive other mandates). If we assume that no person should have a weight higher than 100, and we take the fastest growing function (K=0) we obtain $v(x) = \frac{99x}{(x+99)}+1$ . That you can see here on the side.

Notice how the function has an asymptote equal to a 100 (that is equal to M), and passes through the points (0,1) and (1,1.99). It is also monotonically growing (which means, it always grows), but, as it grows, it grows slower and slower. So it is better not to delegate to the same person that others ate also mandating, when possible. And instead distribute your mandate to other people. It is a function that fights against the creation of an elite!

In the second image you can see how the function effectively approaches 100 (actually M, with M=100, in the example) as x grows.

And finally in the last image you can see how for low values the function approximates very well the function y=x+1. Which is the function that is (implicitly) used when people delegate their vote. In other words who mandates someone which no one else delegated,, will see his vote transmitted nearly fully. While who mandates a very popular person will only change slightly the weight of their vote.

We still need to decide if we should let people delegate to someone else the mandate they received. The famous proxy voting. Also in this case the same reasoning holds: it would be better not, as each passage between the person mandating, and the person actually casting it increases the imprecision (the person voting, ending up voting something different from what he should have voted to correctly represent the person asking him to vote for them), but if we really cannot avoid it, we can use another function. At the end the principle is the same, so the simplest thing is to just use the same function v. So if x people delegate another person (who’s vote will then have a weight of v(x), and this person delegates another person, this second person vote will now weight v(v(x)). It should be noted that if a person sums the mandates from two people ( $x_1$ e $x_2$ ) the result will not be v( $x_1$ )+v( $x_2$ ), but v( $x_1$ + $x_2$ ).

Let’s see some examples:

5 people delegate Anthony. Anthony’s vote, at this point, will have a weight of v(5)=5.76. So about 5 and 3/4. Anthony votes with Carl, and together have a weight of v(5)+v(0)=6.76.

Later Anthony needs to leave, and mandates Carl. Carl’s vote will then have a weight of v(v(5))=6.44.

Notice that if all have delegated directly carl, his vote would have had a weight of v(6)=6.66.

At the end of the day, passing through Anthony costed to the group v(6)-v(v(5))=0.21 (Note, 6.66-6.44=0.22, but 0.21 is the result of better approximations). Not exactly a full vote, but a fifth of a vote. On the other side, society, all together, will pay the price of a lower precision on the result. Who knows if Anthony, delegating Carl, is really doing what those 5 people wants?

What would be a better strategy for this group? Obviously if they could all participate, they would have a combined weight of 7. But if, instead, they knew that only 2 people would participate, the best thing to do would have been to split the mandates between the participants, and obtain v(3)+v(2)=6.87; having lost only 0.13 points respect to the fact of having voted all together. (There is also the option to split your mandate among different people, but this we shall discuss in another article).

Considering all I think that permitting delegation using the function $v_{M,k}(x)$ expressed with an M well chosen, and maybe also with a K=0 would permit to everyone to participate, without going toward those excess that we have seen in the various liquid democracy systems, where few participants will obtain enough votes to dominate the decision. And not let anyone else decide. Note that M is defined at a 1/630-ith of the total number of participants, and if the number of participants, is equal to 630, M will be equal to 1. And the result will be

In other words, if M=1 it is not possible anymore to delegate anyone.

And if M<1. In this case the function decreases as the number of the people delegating a person. In other words, if you are part of such a small group of people, do not mandate your vote, but participate directly!