Arrows Impossibility Theorm

This used to be at "http://arp.anu.edu.au/arp/guy/appendixarrowsimpossib_431.html".
I would be overjoyed if the original author could contact me!

Created 6/12/95 by g.howe@student.anu.edu.au

Re-HTMLified and re-worked by me.

Arrow's Impossibility Theorem was published in an essay called A Difficulty in The Concept of Social Welfare. It demonstrates a profound and a priori lack of reliability of joint decision systems and a lack of coherence of any notion such as the will of the people.

Let Prefersi(a,b) mean that person i prefers a to b. Let Prefers be some joint decision procedure that, thus, generates either Prefers(a,b) or Prefers(b,a) for any a, b in some decision set, Set.

Then Arrow's impossibility Theorem says that the following 5 reasonable conditions on the joint preference relation Prefers cannot all be met by any single decision process:

  1. Prefers is independent of irrelevant alternatives
    that is to say, the ordering of any 2 items in Prefers is a function only of their ordering with respect to each other within each of the Prefersi.
  2. Prefers is non-dictatorial
    that is to say, Prefers is not necessarily identical to Prefersi for some i.
  3. Prefers is pareto-inclusive
    that is to say, Prefers will rank 2 elements of Set in a particular order if all Prefersi do.
  4. Prefers is transitive.
  5. Prefers is a complete ordering on Set.
A Brief Note on Consequences

Note that the first 3 conditions are different from the last 2. The first 3 are what might be called the morality conditions. They say that a joint decision system should respect the individual wills of those elements of which it is composed. The last 2 conditions are what might be called rationality conditions. They say that a joint decision process should display consistent behaviour (which is really what rational means.).

So what Arrow's Impossibility Theorem says is that any joint decision process which is in a reasonable sense democratic and respecting of individuality is also irrational or if you prefer a less loaded term, unreliable. It is likely to display behaviour where its decisions can be controlled by control of its order of making of parts of decisions; or where its behaviour does not respect the independence (in ethics and metaphysics, it's called the freedom) of its elements; or where it is capable of ignoring the unanimous will of its elements.

In other words, you can't trust bureaucracy. This, I think, if there is one, is the reason why traditional state-run-enterprise socialism fails. There are also a great many other interesting consequences of Arrow's Impossibility Theorem, many of them obtained when we abandon our instinctive prejudices regarding what a decision system is, and consequently what Arrow's Impossibility Theorem is about.

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