Arrow, India and elections: Impossibility of solving the Impossibility theorem: Financial Express 13th March 2017

Having all sections participate in governing the country leading to policy formulation sounds fair enough, though at the end of the way there are other factors at work which make governments pay more obeisance to these forces like globalisation, fiscal prudence, etc

Kenneth Arrow’s famous impossibility theorem is compelling not just from the point of view of working out the math involved, in the lecture theatre of Delhi School of Economics, but because the contemporary developments in the world today make this theory relevant. It has a lot of applicability in our framework of political choices and economic decisions, where we always feel that the final result of any system of selection is never close to what one can call a collective view.

Arrow’s theorem basically says that when we have a system of voting to arrive at a collective social choice, we can rarely have one which satisfies five basic tenets. These are first, a situation where everyone has the chance to rank all the preferences that are offered. The second is transitivity in the ranking where if A>B and B>C, then A>C always holds. Third, the solution is Pareto efficient in so far as if every citizen prefers an option to another, then it is socially the best one. Fourth, there is definitely the absence of a dictator who overrules and decides what is good for the citizens, and last, decisions taken are independent of irrelevant alternatives.

One of the conditions put here is that we must have at least three alternatives rather than a referendum variety where the choice is between a ‘yes’ and ‘no’. Curiously, the Brexit had its share of critics. It was argued that a decision taken by a majority vote which was defined as being marginally over 50% though reflecting the ‘majority’ could still not be ideal. It was argued that higher ratios like 67% or 75% must be used. But if the 51% rule is not used and a higher cutoff is pitched for, then there will rarely be any mCloser to home the UP elections can be a good illustration of Arrow’s impossibility theorem. How about the other axioms? People surely did have their choices or preferences between the BJP, Congress-SP and BSP. If these parties are denoted by N,C and B, respectively, in general it could be argued that N>C based on what happened in the elections at the centre. Further, it may be assumed for the sake of argument that if C>B, then N>B must hold. But this will not be so as people in society have different ways of reacting to parties and persons as well as the perceived dogmas. Therefore, transitivity cannot be taken for granted in this case.

This conclusion will come out sharply in case one of the parties does not field a candidate. Let C not be in a constituency. Will all their votes go to N which should be the case if rational behaviour held? Unlikely again, and it is just possible that all the votes for C could go to B. This is why when multi parties contest the elections those which win may not need to get more than 30-35% of the votes and only have to ensure that the rest get a lower share of the votes. This is the condition of ‘irrelevant alternatives’ skewing the show while N is preferred to C which is better than B, when C steps out, B could emerge the winner. This has invariably been the case when it is a three or four cornered contest.

The same can be taken into the world of economy policy making where we assume that since there is a government that has been elected by the majority rule–which as discussed above could flout the logical way of social choices being expressed, now has the power over policy formulation. A look at the budgetary options can illustrate this point.

Let us assume that there are three options for allocation of funds which address the concerns of different segments of society. These are infra spending, subsidies and salaries. The rich talk of infrastructure as they would like to have swankier airports and highways where they can drive their BMWs. The poor will be happier with more subsidies as this affects them the most and this class which is not bothered about the wastages or leakages as long as they get something. The middle class is happier with salaries being increased because at the end of the day they need to run their households with a decent level of comfort.

In this situation too one can never reach a transitive state, as the rich will prefer infra to everything else with subsidies being the third preference. The poor will have an order of subsidies, infra and salaries while the middle class would have salary at the top of the list and could toss between the two. However, if the subsidy part is deemed irrelevant when going for a vote, then salaries may be preferred to infra creating an anomaly. Hence, if social choice is to be decided by logical ranking of all preferences based on numbers, the scales may tilt to subsidies as most people would like it and infra could be the last.

This is where the dictator plays the part which could be the PM or FM or FS where it is decided that society needs infra the most followed by salaries and subsidies and hence a decision is reached. Hence, the final outcome of any policy will never be the view of the majority and is invariably that of the elites which have been elected by the people to serve them.

This deduction is interesting because it can take one back to the theory of social contract where Jean Jacques Rousseau questioned whether there can be a legitimate political authority. By having such systems which Arrow spoke about, it may appear that at times people may be better off in isolation. When people in a democracy elect a government, which as can be seen may not be consistent, then individuals surrender their freedom to this government which then decides what is good for all. The policies which go beyond the Budget as was explained earlier would again tend to wean towards the elites and may not lead to an optimal social decision.

Is there a solution to this conundrum? Not really as there can never be a system where all ends meet. Having all sections participate in governing the country leading to policy formulation sounds fair enough, though at the end of the way there are other factors at work which make governments pay more obeisance to these forces like globalisation, fiscal prudence, etc. But quite ironically people still get swayed by slogans and not express themselves differently the next time at the ballot box.ajority decision which can be arrived at.