Kenneth Arrow’s Impossible Theorem and India

Kenneth Arrow (died 21-Feb-2017) was an American economist, writer, and political theorist and winner of 1972 Nobel Prize in Economics. He will be best remembered for his impossible theorem.

When Nobel Laureate Kenneth Arrow explained his impossibility theorem it seemed a matter of theoretical learning but because of the recent developments in politics across the globe it appears to be a relevant theory and almost all the economic and political theories of the modern world rests on this theory.

Arrow’s ‘impossible theorem’

Arrow’s theorem states that when voters have three or more distinct options, the ranked preferences of individuals cannot be converted into a community-wide (complete and transitive) ranking meeting a pre-specified set of criteria by any ranked order voting system. The theorem basically says that there is no rank-order voting system that always satisfies these three criteria:

Assume that X, Y and Z are three distinct alternatives then:

• If every voter prefers X over Y, then the group prefers X over Y.
• If each and every voter’s preference between X and Y remains unchanged, then the group’s preference follows the same
• No single voter possesses the power to determine the group’s preference or in simple words there is no dictatorship of any voter.

Generally, it is impossible to implement these conditions without creating cycling group preferences.

This theorem is applicable to both economic and political arena especially if we apply this hypothesis to political situation of India.

Impossibility theorem and India

If we look at Arrow’s theorem, it seems mathematical and much complex to us. But if this hypothesis is applied to the biggest festival of democracy in India- elections, it will completely decipher the theorem especially the 2014 elections where BJP cleanly swiped the other parties.

Suppose the voters have to choose out of three distinct political faces that locked horns in 2014 polls- Modi, Rahul Gandhi and Arvind Kejriwal. Their political preferences would vary. Suppose Voter no. 1 would prefer Modi over Rahul Gandhi and Gandhi over Kejriwal, Voter no. 2 prefer Gandhi over Kejriwal and Kejriwal over Modi and Voter no. 3 would prefer Kejriwal over Modi and Modi over Rahul Gandhi. This means two voters prefer Modi to Rahul Gandhi, two voters prefer Gandhi over Kejriwal and two voters would prefer Kejriwal. There is a rule of transitivity in mathematics i.e. if x>y and y>z and therefore, x>z but this rule is not applicable in Arrow’s theorem as the result of this social choice is cyclical rather than transitive and hence will confuse non-Economists as there will be no clear winner and nobody will be left happy. But in real scenario, the individual would not influence the result alone but the collective choice does. The voting rule ought to replicate the preferences of voters that if each elector prefers one candidate to a different, then that candidate ought to be stratified higher by the choice rule. With many voting rules on provide, for certain there ought to be some that satisfy the cheap criteria of being decisive, free of unrelated alternatives and deferential to the consensus view. However, Arrow established that any social selection rule that satisfies all the aforementioned conditions might solely do so as a result of the existence of a dictator whose individual preference would be the social inclination no matter the rankings of others. This is the renowned Impossibility Theorem that has generated an entire branch of analysis referred to as social selection theory.

Similarly, if Indian economic scenario is concerned especially the budget for the different segments, the impossibility theorem can explain it well. In terms of allocation of funds for these different segments, there are three options: infrastructure, social security schemes and subsidies. The different classes such as the upper class would desire infrastructure spending and poor would ask for subsidies whereas middle class would only need strengthening of social security schemes.  Now it would be very difficult for the Finance Minister to allocate according to these preferences as there is no transitivity of preferences hence again a decision like dictator about what society needs will take place and for example the dictator would decide that social security schemes are preferable than the other two. Hence, the outcome of any policy will never reflect the majority and is not an optimal decision.

Conclusion

All the democratic countries in the world has presumptuous leaders who lock horns with each other in the elections; be it Al Gore and Bush; Hillary Clinton or Trump or in Indian context, Modi or Rahul Gandhi. The elections in these countries are a big affair in these countries with lot of money, time and respect at stake. Hence, voters play a significant role in the elections but not individually but as a part of group or community especially in India where we have pluralistic democracy where each person has different dogmas and beliefs and hence different communities exercise their beliefs and opinions in elections, for example a large section of people recently voted to BJP in UP, who believed in the developmental works of Modi whereas in Punjab people voted against the incumbent government.

Once elected, the leader has to formulate policies for the citizens and has to choose from different range of options available and if different segments are asked of this it would leave the leader confused of choices. Hence, the leader has to take bold decision which favours all the classes.

These scenarios can be well explained by Arrow’s prize winning theorem and made him the Albert Einstein of Economics.