If I told you that a fair democracy didn’t exist, what would you think? 96 of the world’s countries consider themselves a ‘democracy’, a voting system based on equality, where everyone has their say. In 1950, Kenneth Arrow published ‘Arrow’s Impossibility Theorem’, based on the controversial statement, ‘There is no such thing as a fair voting system’. So what is ‘Arrows Impossibility Theorem’ and why is it mathematically impossible to have a ‘fair’ democracy?
Firstly how democracy is defined? In Arrow’s theorem a voting system is fair if it meets five criteria:
- Voters are unrestricted in the preferences they can have, meaning the voting system must account for the preferences of each voter, also known as unrestricted domain
- Preferences of voters must be transitive, meaning if a voter chooses candidate A over B, and B over C, they would consequently choose A over C
- The ‘Unanimity criterion’ is satisfied. For example if all voters choose candidate A over B then the results must reflect this preference
- There must be no dictators – voters must not be influenced by the views of another, for example if one voter chooses candidate A, the other voters must not feel ‘forced’ to choose candidate A too
- There should be ‘independence from irrelevant alternatives’, meaning preferences for candidate A over B should not be influenced C.
These criteria may seem straightforward however when trying to ensure they occur simultaneously it gets more complex. To take a look at these criteria in practice we can analyse some existing democratic systems to evaluate their success.
The simplest form of democratic voting is the plurality method. In this system each voter indicates their preference for one candidate, the candidate with the most votes wins. Sounds fair right? Not quite. Here there is no independence of irrelevant alternatives. For example if candidate A gets 40% of votes, B gets 35% and C gets 25%, then A is the winner, however if C dropped out of the race and their supporters then voted for B, A would have 40% while B would have 60%, making B the winner. Furthermore, the votes may not be transitive as while some candidates will prefer A to B and B to C, some may prefer C to A.
Another voting system is the instant runoff method which is often used in local elections. Here instead of submitting one candidate of preference, voters are asked to rank their preferences for each candidate, then the candidate with the lowest preference is eliminated and the voters get their second choice if said candidate was their first, until one is chosen. This seems efficient,however issues are seen in the following example:
There are candidates A,B and C and nine voters. Four rank in order C-A-B, three with B-A-C and two for A-C-B. we can see that A has the least number of first preference votes, so they are eliminated. In the second round the preferences are 6 C-B and 3 B-C, making C the winner. However, in round one six of nine voters preferred A to C but the inclusion of B altered the outcome, meaning there is not independence from irrelevant alternatives.
The final example is the Borda count method. Here weighting is given to preferences, determining who would win based of their points (first preference would have the highest weighting and decrease with lower preference). This method allows for widely- liked but not favourite candidates to win, for example if there were 100 voters and A got a first preference of 60 voters (where 1st weighting is 3, 2nd is 2 etc) their total points would be 180, however if B got the second vote of 95 of the voters, they would get 190 points, meaning even if they are not the majority favourite, they would win.
Overall, we could look at all of the democratic systems but would reach the same conclusion – a ‘fair’ democracy is mathematically impossible. However something this theorem doesn’t take into account is weight of preference between candidates which could sway the vote to the majority. Finally, it also relies on the assumption that voter preferences are rational, but when has society truly behaved rationally?
