In which I try to explain how elections aren't fair...whatever system you use!

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Arrow’s Theorem seeks to explore – in the context of social welfare economics – the nature of voting systems. In essence the theorem and its proofs show that:

“…no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from.”

The criteria that Arrow said must be met for the voting to be ‘fair’ were:

Unrestricted domain: Everybody’s vote (and nothing else) counts
Non-dictatorship: no single individual has a preference that always prevails
Non-imposition: all the possible ranking options are available to the voter
Independence: adding further choices must not affect the preferences for existing choices
Monotonicity: no vote for a choice should make that choice less likely

So I guess you’re all asking what on earth this all means! It doesn’t mean that some voting systems aren’t “fairer” than others – although “fairness” isn’t very easy to either define or to measure. You can see that the different criteria are all relevant to our understanding of “fairness”:

1. We want our vote to be counted and to count
2. We don’t want the choice of one individual to always prevail as this is dictatorship not democracy
3. We want to be able to make a choice from the full list of options not just from a restricted selection of these choices
4. We would see it as being unfair for the addition of further choices to unequally benefit one existing option
5. We would object to a vote FOR something reducing the prospects of that something being the choice or the electorate

Under forced run-off systems like the alternative vote (AV), not only does not every vote count (or get counted) but there is the possibility of a vote for an option damaging that options chances of success. It is clear that the second choices of those opting for the leading candidate(s) are discounted – under AV voting Conservative first and UKIP second is likely to mean the second choice is not counted whereas those voting UKIP first and Conservative second will have all their votes counted.

Also – as is shown here – there are also risks of non-monotonic results under forced run-off systems. Shifts in votes from less popular options can affect vote distribution meaning that a candidate can increase their first preferences and then lose. The likelihood of this occurring has been set at 1/4000 – a sufficiently common frequency for us to see it as a concern.

Single transferable vote (STV) systems have fewer distortions – although, as with AV, not every vote is fully counted and there is a very small probability of non-monotonicity. Depending on the system used to set quota (the point at which a choice is selected) STV will give a closer outcome to the actual rankings chosen by voters. If STV is used to make a single choice it is no different in outcome from forced run-off systems.

Under Arrow’s Theorem first-past-the-post is not less fair than these alternative systems – there is a strong suggestion that not every vote counts (although a strict interpretation would deny this argument) and, more significantly, the system restricts the options available to the voter – you can only vote for one option, once. This problem applies to list systems since they constrain choice to a given list of parties or similar. And all these systems are affected by the number of places to be filled – fewer places means votes for minority parties have less weight.

The point of all this isn’t to say which system is right and which wrong – most of the serious psephological anoraks seem to favour either the system we have now or else STV. What does matter is for us to ask which sub-optimal outcome we prefer – bleating on about “fairness” is lazy argument! Saying you want a more proportional outcome is fine – this can be achieved in various different ways not all of which mean handing control over to party whips.

All choice systems based on voting – other than one between A and B – are subject to distortions and we will need (if we plan to debate change) to consider where we place the greatest value. Do we want strict (ceteris paribus) proportionality, representativeness or clarity of decision? Or do we require a combination of these factors? These are real and definable outcomes that can be designed into a system. What’s Arrow’s Theorem tell us is that choice systems – elections if you must – can never satisfy all the criteria that would make them “fair”.

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