Thoughts Guiding Systems Corp.,

Ottawa, Canada.

e-mail:karasik@sympatico.ca

The recent events in Turkey have shown that democracy may turn out to be a dictatorship for a minority. The democratically elected leader that has backing of a majority of the population started ignoring the will of the minority and imposing the will of the majority on it. This is, of course, what democracy is not for. In a real democracy the will of a minority is also respected.

There are many ways of devising a democratic system, which would guarantee that the will of a minority would not be trumped. The simplest one is to require consensus by law. Such a system was implemented, for instance, in Poland in the 17th and 18th century, where every Pan had a representation in the Sejm with the veto power [1].

But requiring consensus for adopting decisions could be too onerous. That is why a different scheme was adopted in the US. Its primary concern was preventing urban population (which was a majority) from dictating to rural population (which was a minority). That was achieved by not proportional allocation of electors.

I am going to build upon this approach and propose a mathematical algorithm for balancing interests of various groups of population. The algorithm is based on clustering population by interests.

Suppose that population can be clustered into 4 groups G1, G2, G3, and G4 by proximity of their interests. This means that interests of people in every group are closer to each other than to interests of people in other groups. Suppose that interests of people in groups G2 and G3 are closer to each other than to interests of people in groups G1 or G4. Then groups G2 and G3 can be also clustered into super-group G2-3:

Suppose that the electoral college consists of X electors. Then to be democratic groups G1, G2-3, and G4 have to have equal number of electors, i.e. X/3 in this case. Groups G2 and G3 also have to have equal number of electors, i.e. X/6 in this case. Thus the minimum number of electors in electoral college has to be 6 and G1 and G4 should elect 2 electors each, whereas G2 and G3 should elect 1 electors each. The number of electors could, of course, be proportionally increased.

Now consider the problem in general terms. Suppose that the entire population can be clustered into a tree of depth N. Let the lowest leafs of the tree be level N and the root be level 1. Let n_{ij} be the number of children of node number i on level j.

Then the number of electors in a leaf is X/(n_{11}×n_{i1j1}× ...×n_{ikjk}),
where {(1,1),(i_{1}, j_{1}), ...(i_{k},j_{k})} is the route from the root of the tree to this leaf.
Then the minimum number of electors in electoral college is the Least Common Multiple (LCM) of all such numbers for routes of length N.

To implement this truely democratic scheme population interests have to be properly polled and clustered before elections.

To make the scheme even more democratic, clustering has to be not limited to trees only. It may turn out that other types of graphs are more suitable for clustering. In this case the matematics of assigning electors to clusters might become quite complex. It is an interesting area of research that may result in a mathematical theory of maximazing democracy.