Brexit and Arrow’s impossibility theorem

Politics in the UK has been on a rollercoaster ride in the last few weeks over Brexit. There have been many high profile cabinet resignations by ministers who are unhappy with the proposed Chequers plan.  Brexiteers and  Remainers, who are bitterly opposed each other, have managed to find one thing that they do actually agree on. They both vehemently reject the proposed compromise in Theresa May’s Chequers plan for the transition period. Note that this is just the transitional agreement, not the final agreement that will govern the ongoing relationship between the UK and Europe in the future.  For that, we can look forward to another two years of negotiation, political fighting and uncertainty. 

Arrow’s Impossibility Theorem

This turmoil supports the truth of Arrow’s Impossibility Theorem. This states that it is actually not possible to fairly combine peoples wishes into the right outcome. The view of the people can not be forged into a single coherent view. In other words, what the British people want will always be different from what Britain wants.

Kenneth Arrow, a Nobel Laureate, set up three criteria for ‘fairness’ in an electoral system and then demonstrated that no system can be designed that will satisfy all three of these. His criteria were :

•             If every voter prefers X over Y, then the group prefers X over Y. 
•             If every voter’s preference between X and Y remains unchanged, then the group’s preference remains unchanged 
•           There is no ‘dictator’: no single voter possesses the power to set the group’s choice. 

I won’t go into the maths – you can look it up for yourself here on Wikipedia. It all hinges on the fact that there is a pivotal voter who tips the balance and therefore determines the outcome. So that pivotal voter is, in effect, a dictator: what he says goes.  

This mismatch between the voting public’s wishes and the higher level group outcome is a good example of catataxis. We can sum it up in the catataxic maxim :

“As above, not so below”

What happens at the group level is often the opposite of what is happening at the individual level. 

Plus ça change, plus c’est la même chose

We can draw another example by considering the differences between the American Revolution (1775) and the French Revolution (1789). Both happened within 15 year of each other but they had very different outcomes. The Founding Fathers in America made sure they built in checks and balances to the constitution to prevent any dictator emerging. In other words, conflict and argument were baked in at a lower level to guarantee stability at a higher level. 

In contrast, the French Revolution decapitated the state but did not redistribute its powers to other lower entities. So, in effect, the state was still a monolithic structure. This then led to the rise of Napoleon as a dictator followed by the Bourbon restoration which took them back to where they started. 

So maybe the optimistic conclusion to all the Brexit furore is that this vehement disagreement is  necessary. It is the messy process of democracy that guarantees a stable, unified state at the end.    At a higher level, it certainly seems to have pulled the EU together into a more coherent whole, judging by the unanimity from their side of the negotiating table.