A democracy problem

This idea follows from my last post about dinosaurs funnily enough. It relates to a common problem in democracies called the tyranny of the majority. This may sound like a strange term because surely having a government that reflects the majority opinion in a country is a good thing. But in fact it is problematic.

Let's take an example where 60% of a country is Christian and 40% is Muslim. In one general election we would expect a party with Christian-aligned policies to gain power. This is a reasonable outcome for a single election.

But over the period of 100 general elections, there is a good chance that all of them will be won by a Christian-aligned party, since 60-40 is a very large majority in politics. This leads to frustration by the minority population, disillusion, and instability.

The problem hinges on the fact that:

mean({a,b,c,..}) ≠ {mean(a), mean(b), mean(c), ...}

Where mean() and a,b,c can refer to many more aspects of democracy. For instance, mean(x) could be 

• the winning party in electorate x. 
• the elected party for each election year x.
• majority vote for each bill x.

In each case the fallacy is that a set of "mean" opinions is sufficient to be a mean set of opinions. 

But these are not the same thing. A set of mean opinions lacks the diversity that should exist in a mean set of opinions. 

For example, if each consistuency has a range of views on retirement age from 55-75, with the mean at 65, then the MPs representing the mean view of the consituencys will *all* vote for 65 as the retirement age. It will appear as though the country is united. If you are a subculture that occupies 10% of the vote wanting a retirement age of 55, none of the 200 MPs will be representing your view.

Ideally, a mean set of retirement ages that reflect the constituencies should have a diversity of views from 55-75. And a correct mean set does indeed reflect this diversity. But you cannot calculate it just by taking the set of the individual means.

It is interesting that this problem has been acknowledged, and some countries, such as New Zealand, use proportional representation to alleviate this problem. In this case 10% of the MPs will reflect the 10% of the population supporting 55 year old retirement. 

When the proportion is the same across districts, this is exactly what the 'mean set' gives (see last post), for distinct classes like parties, where the mean naturally becomes a mode.

However, when the proportion varies across districts, we get something different to standard proportional representation. 

For example, what if the proportion of a and b are 20% and 80% in France and 60% and 40% in Spain, and your representative set is one from element from France and one from Spain?

In this case, if the order is France,Spain then (a,a) has chance 0.12, (a,b) has chance 0.08, (b,a) has chance 0.48 and (b,b) has chance 0.32. Then with order symmetry, the chance of {a,a} is 0.12, {a,b} is 0.56 and {b,b} is 0.32. So {a,b} is the mean set. This is different than if we just added up all the probabilities to give 40% for a and 60% for b, then you would get 0.48 for {a,b}.

This is a proportional representation of the views of each of the constituencies, rather than a proportional representation of overall votes. It includes the individual constituency view back into the result. 

There are a million different PR schemes, so it would be interesting to see if this is one of them, or how it compares.

As mentioned in the bullets earlier, it would also make sense to use mean sets over multiple general elections. So if one party always gets 10% of the vote then over 10 elections it will get in once.