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## It's impossible

Kenneth Arrow's impossibility theorem states that there cannot be an ideal voting method. It has other interesting angles as well

The last week of February 2017 marked the passing away of Kenneth Joseph Arrow at the ripe age of 95. Arrow received a Nobel Prize in economics in 1972 at the age of 51 - the youngest person ever to receive the award. Among his numerous contributions to economic theory was the Arrow impossibility theorem - something that he proved in his doctoral thesis. This led to the creation of 'social -choice theory' - a new sub-discipline! This theorem has deep implications on election design in a democracy.

People vote in elections to express their preferences. These preferences must be aggregated across the entire voting population to make a joint decision. The question is if there is an 'ideal' vote-aggregating method. This is the problem that Arrow set out to ponder. But before that, we must take a look at a paradox that has been known for several centuries.

Suppose there are three options presented to a voter: X, Y, and Z. To illustrate in the context of the recent elections in UP, we could let X represent BJP, Y represent BSP and Z represent SP-Congress. A voter could have a preference of x>y>z. Other alternatives could be y>z>x or z>x>y. Remember, in this case, a voter preferring BJP over BSP and BSP over SP (x>y>z) is the same as preferring BJP over SP (x>z).

Now suppose that there are equal number of voters who preferred each of these options. Then two-third would prefer X over Y; equally two-third would prefer Y over Z. This implies that X would be preferred over Z (transitive ordering). However, we see that there are also 2/3 cases where Z wins over X. This is clearly paradoxical (circularity).

What is 'ideal'
In attempting to look for an 'ideal' method to aggregate individual votes, Arrow used a 'social-welfare function'. This function should ideally rank candidates in order of voter preferences and produce an outcome that is supposed to represent the joint 'will' of voters. Arrow postulated a set of properties that this method should have to make it reasonable:
1. There should be no single person whose ranking should dictate the final outcome. This is known as 'no dictator' (ND). After all, we are dealing with a democratic set up.
2. If every voter prefers X over Y, then the outcome for the group should reflect X over Y. This is also known as Pareto efficiency (PE).
3. The outcome should not change the relative rankings of X and Y if voters change rankings of other candidates but not their relative rankings of X and Y. This is known as independence of irrelevant alternatives (IIA)

Given these properties, Arrow reached the conclusion that for three or more candidates, there cannot be a social-welfare function that satisfies ND, PE and IIA all together. Alternatively, it can be stated that if you want a transitive group ordering that satisfies all of the above, you must have a dictator. This is known as the Arrow impossibility theorem - that there is no ideal voting method. The world of 'social choice' is not as simple as 'rational individual choice' that economists think they understand.

Other 'non-ideal' methods exist
Often people say that Arrow proved that there are no good/fair election methods. This is not true, since there are many methods that are not covered by the hypotheses of Arrow's theorem. In particular, Arrow's result applies only to the methods in which voters rank all candidates, a requirement not satisfied by many popular voting methods, e.g., approval voting or plurality voting.

Furthermore, for any given context, one may question whether the 'reasonable' criteria are truly reasonable in that context. For example, if there are only two candidates, it is easy to see that plurality voting (where a voter can vote for only one candidate) is a social-welfare function that satisfies ND, PE, and IIA (since there are no other candidates).

Plurality voting, too, suffers from tactical voting when there are more than two candidates. In this 'first past the post system', voters are pressured to vote for one of the two candidates they predict are most likely to win, even if their true preference is neither, because a vote for any other candidate will likely have no impact on the final result. Any other party will typically need to build up its votes and credibility over a series of elections before it is seen as electable. The difficulty is sometimes summed up, in an extreme form, as 'All votes for anyone other than the second place are votes for the winner,' because by voting for other candidates, they have denied those votes to the second-place candidate who could have won had they received them.

Arrow's realisation that there is no satisfactory way of arriving at collective preference on the basis of individual preferences remains a seminal contribution to understanding social choice.

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