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S Ananthanarayanan: Elections and the higher math

An individual's influence does matter, even in larger voting groups

S Ananthanarayanan  |  New Delhi 

The principle of "one person-one vote" is unquestionably fair and just. But in the groups that usually participate in voting, in national, international or even large company elections, this principle alone need not guarantee equality of "influence over decisions".
Using mathematics to analyse electoral systems leads us to some surprising conclusions.
Power of a voter
In an election where a decision is supported by 90 per cent of the voters, clearly, individual voters are not particularly important. But if the votes were equally split, then the individual vote becomes pivotal. Mathematicians have studied how a voter's influence changes as the number of voters increases.
On the face of it, it would seem that a voter becomes less important at the same rate as the number of people in the group increases. This seems reasonable "" an individual has only half the influence over a group of 100 than over a group of 50.
Thus, if people in a group are electing a representative to a national Assembly, then the representatives should have power proportionate to the strength of the groups they represent.
In this way, the less influence that an individual has in a larger group would be compensated by the greater weight of the representative of the larger groups.
But it is not this way
This thinking assumes that a person in a group of 100 has half the influence of a person in a group of 50. Or in a group of a million, over a group of half a million.
Mathematicians find that this is not quite true. A person's influence in a group is really related to the chances that the vote in that group would be evenly split.
Until this happens, the person is irrelevant. Now, the chance of an even split, as the group gets larger, it turns out, does not reduce in step with the size of the group, but a little slower.
The reason is that if voting behaviour is taken as being random, then, 50-50 division of votes is generally more likely than marked polarisation. This is like the temperature of the air in a room is usually the same everywhere, rather than warm in one part and chill in another. As we go to larger and larger numbers, this greater probability of even distribution gets overpowering.
The implication on elections is that when a group is larger, the individual's influence is not proportionately less than what it would have been in a smaller group. Lionel Penrose, father of Roger Penrose, the physicist, was a geneticist and mathematician. He worked out that the individual voter's influence reduces in a large group of say "N" persons, not according to "N" but only the square root of "N".
This is to say that if there were a group of 1,000 and a group of 1,00,000, which is 100 times larger, then the voter's influence in the larger group is not 100 times less but only 10 times less (approximately).
The European Union
This means the representatives elected by larger groups should not have weightage according to the population of their group, but only according to the square root of the population. When the rules of were framed, something like this was built in, based purely on "impressions" of what would be fair. As more countries join in, changes to give representatives "proportionate" powers have been proposed.
Polish mathematicians Karol Zyczkowski and Slomczynski point out that this would give the German, French, English and Italian citizen more influence in the Union than people from smaller countries.
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