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Known Mathematical Political Results

There are many thousands of Political Scientist with PhDs employed all over the world. But, when you study the work they do, you find most all their work is associated with how our present systems run. They don’t study new different kinds of systems. There is a branch of Political Theory, which you would imagine would consider new different systems. But in fact this subject is associated with the ancient Greek philosophers. So very few people study new different problems or systems. Thus there are very few results.
            The only 4 clear-cut results, I know of, are the following.



1)         The division of the cake between N people


This might appear to be a trivial problem to study. But, if you consider that the cake could be the lands of the world, and we are asking how these lands should be fairly divided between all the peoples of the world, then clearly the problem is no longer trivial. (But most people would consider that this particular problem to be extremely esoteric.)
            The accepted solution to this problem is a generalisation of the two-person solution, where one person cuts and the other person chooses.
            In the N person case, one person starts the process and he cuts himself a slice. But, before he has he can have his slice, all the other N-1 people can successively choose his slice, but they must reduce the slice by a small amount (and return it to the cake). The last person to accept the slice - receives the slice. So, if no person other person accepted the slice, then the original person would still have the slice.
            This process is then repeated with the remaining cake (together with all the little cut off pieces) between the remaining N-1 people.
            This process is repeated between N-2 people, and then this is continued until there are only two people left (where one person cuts and the other chooses).

            Most people’s reaction to this accepted solution is that – surely there must a simpler and easier method than this.
            What this problem teaches us is that this problem is extremely hard, because some people can collude together against other people. And all the simpler methods cannot overcome the many wicked colluding possibilities. So a colluding majority of people could diddle a minority of people out of their fair share.
            And this sort of problem applies to all political problems. Devising any political system, which is fair to all people, is simply very damn hard. It can be done - but it isn’t easy. This is a very important lesson for us all to learn.
            In any political system, you must not expect too much.



2)         Normal Voting Systems cannot be completely rational


A mathematician, called Arrow, has proved that no normal ordinal voting system can obey 5 sensible conditions that you would expect of such a system. (An ordinal voting system is a system where all voters order all their candidates in their preference.)
            I won’t go through the details because it is too complex. The standard case, which we all are aware of, occurs when two people stand with very similar aims in an election. In most ordinal systems these two candidates will be disadvantaged because these candidates would share the voting between them. So the view-point they represent is less likely to be represented. This is unfair.
            Certain ordinal voting systems can overcome this problem. But then one of the other sensible conditions will be broken.
            This point again illustrates that we cannot expect too much of a voting system. There are fundamental problems in all political systems. (In my Society of Choice book I overcome this problem by not using an ordinal voting system. But my system is rather complex.)



3)         The Lack of Incentive to Vote Well


This is by far the worst problem of all. In the literature on this subject, this effect is sometimes referred to as “ The Apathy Problem”. I don’t like this term because it implies that it is the voters who are at fault. It is just a fundamental problem, which we all ought to know and recognise. But all people simply hate to think about the problem and it is usually just ignored.
            The effect can be estimated in terms of:
                             The value (i.e. importance) of the decision to a voter
                                                               divided by
                          the number of people voting on the subject.
This little calculation gives an estimate of the value, which a voter will derive from voting in the correct manner. The important item to note is that, when the number of people voting is large, then this value will be negligible. So then a voter will have no personal financial incentive to vote well. So - why should a voter bother? – particularly when it is really hard to know all the basic facts concerning a particular voting decision.


This diagram should give you some idea

how I overcome the problem in my


This problem becomes much worse as soon a political entity becomes very large. So this condition tells us that we should only construct large political entities when they are essential. And this is what I do in all the political entities I devise (besides various other carefully thought-out procedures).
            The other aspect, which this condition tells us, is that the concept of “secret ballot” is a mixed blessing. If all voting can be seen and recorded then all voters will know that any really bad decisions, they might make, will be seen by future generations. So voters then have an incentive to vote well because they could then lose the respect of their children and grand-children. This circumstance becomes very important when people are voting on something like Climate Change.









4)         The Individual Ownership of Land


The Ownership of Land is the hardest problem of all. But there is one very sensible suggestion on the subject, which all people should be aware of.
            Henry George, in his book “Progress and Poverty” suggested that there should be no private ownership of land. Then the renting of all the nation’s land (in the form of competitive leases) should be the one and only form of tax in the nation. This is a sensible system, which supports private enterprise, but yet it is completely fair to all people.
            The private ownership of land is full of problems. The world’s resources, in the form of land, are essential to all life on Earth. So to either sell or give this land permanently to any one person means that a private person can expect to hold this land forever. Then, if the rich people buy up most of the land (which can happen very easy), then the poorer people of the world and all wildlife are completely dependent on these rich people for their very existence. This is an intolerable situation.
            So Henry George’s simple solution to the problem should always be remembered.


I think that it is essential that all sensible people remember these four well-known facts.



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