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# How Valuable is our Prime Minister? A Mathematician’s Perspective

- June 25, 2020
- Authored by: Aniruddha Singhal
- Category: Mathematics Politics

In this interesting piece, a mathematician analyzes the worth of our politicians through Shapley value.

How Valuable is our Prime Minister? A Mathematician’s perspective

We are often faced with a situation where we have to evaluate how valuable something or someone is. For example, a creative director has a team which brainstorms for ideas when they have to create something new, but due to shortage of budget the director has to let some of the team members go. In such a situation, to make an informed decision, he needs to evaluate how valuable each team member is. Similarly, in a sports team, a captain has to determine the set of most valuable players which should be sent to an international tournament. Such situations are common and we have several systems to make decision in such situations. Sometimes people judge based on performance metrics, sometimes on intuition, sometimes on recent performance.

The people who are lucky and get selected have nothing to complain about, but the people who are left or criticized, always talk about the bias and unfairness involved in the selection process. Because of this, sometimes people who are capable and worthy do not get the opportunity and the praise that they deserve.

Mathematicians are not heartless and therefore they have
tried to deal with this problem. It is well known that a mathematician’s way to
deal with problems is highly objective. They have methods of calculating
exactly how valuable someone is. **Lloyd Shapley introduced a concept for
estimation of value for which he won ****the Nobel Prize in Economics in
2012.**

In a mathematical framework, competitive situations are modeled as cooperative game and the people involved are called players. In such a scenario, a player’s Shapley value equals her marginal contribution when she is added to a coalition averaged across all possible orderings in which the coalition of everyone forms.

Imagine three players, Arun, Balram, and Chirag who work in an Innovation Company. They have a task where they have to think about novel usage of cutting an edge distributed ledger technology called block chains. Arun and Chirag each think of six ideas and Balram thinks of seven. The group has a total of nine unique creative ideas. Now their manager is considering one of them for promotion and wants to know who is the most valuable team member. Someone advised him to use the Shapley’s method.

To compute the Shapley values, we could write down all six possible orders in which the group could form, give individuals credit only for unique ideas added to the group, and then average over all six cases. Essentially –

Anyone who proposes a unique idea always receives full credit. In the figure, we denote those ideas, such as Arun’s idea of art transactions, in **bold** font. If two people think of an idea, each has a one-in-two chance of joining the group first. Similarly, if all three people think of an idea, each has a one-in-three chance of joining first. It follows that allocating credit equally among people who thought of an idea produces Shapley values. Thus, it is the unique way to assign values that satisfies the four axioms. These values show that Arun, though he did not have the most ideas, adds the most value.

Group Creativity = Arun’s + Balram’s + Chirag’s creativity = 3.5 + 3 + 2.5 = 9

A very interesting application of this method can be in understanding value of political leaders. Specifically in India, it is common to see criticism of India’s PM Shree Narendra Modi, especially in comparison to governance of India’s first Prime Minister, Jawahar Lal Nehru. It is argued that Nehru and his successors, Indira Gandhi, Rajiv Gandhi etc were better at governance as compared to Shree Narendra Modi.

Let us do a Shapley analysis of J.L Nehru, Indira Gandhi and Shree Narendra Modi. The Figure below shows contribution of each of the PM. The bold items represent novel contributions and the rest are common with at least one of the PM.

(A different set of contributions can be chosen and a similar analysis can be done. Here each PM’s 3 major contribution from his era are taken. Based on differences and opportunities in space and time each contribution is accordingly associated with the PMs predecessors or successors.)

Let us analyze three major contributions of each Prime minister and argue how unique each of them is given the constraints of their times.

**J.L. Nehru:**

Establishment of educational and medical institution is not a ground breaking idea and all the countries who got independence in that time established such institutions. Therefore this contribution of Nehru could have been done by any of the other two PMs and thus we give these a value of each. The reorganization of states is supposedly a novel contribution because in a country like America, each state is independent and therefore it can be argued that it was not a necessity of time, but Nehru Government came up with it. This contributions deserves a values of 1. This makes total shapley value of Nehru to be 1.66.

**Indira Gandhi:**

Indira Gandhi brought green revolution to India in a time when people were dying of famine. It was a do or die situation where if no intervention was made, the country could not have survived. Therefore this contribution was not novel and any PM in that era would have done the same because of which it earns a Shapley value of . Privy Purse and Nationalization of banks was not extremely essential and was not motivated by politicians of that time. These may not have been done by other two prime ministers and thus should earn a value of 1 each. The total Shapley value of Indira Gandhi is 2.33

**Shree Narendra Modi:**

Shree Narendra Modi made Gas cylinders highly available which could have been done by Indira Gandhi as she had an opportunity to do it. Nehru never had a chance to do such a change in his time and would probably have done it if given a chance. Therefore the value associated with this is . Construction of Toilets and abrogation of article 370 and 35A was an opportunity available to both the previous governments upon which they never acted. The execution of these two initiatives is unique to Shree Narendra Modi’s Government and therefore earns a value of 1 each. This makes total Shaply value of Shree Narendra Modi equal to 2.5.

It can be seen that given this set of contributions –

**Shree Narendra Modi > Indira Gandhi > J.L. Nehru**

**Colloquially it can be said that
Shree Narendra Modi is the most valuable Prime Minister of India**. A more detailed and comprehensive
analysis will require a complete list of all the PMs of India and their
corresponding contributions. It will also require an unbiased historical perspective
to interpret the Shapley value of each PM.