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# ‘Self-Love’ in the Time of Corona - A Game Theory Approach

This article is an attempt to show how self-centric decisions made by Individuals or States can affect the existence of everyone **(including them)** in their society. Here, I’ll use a common game theory called ‘Tragedy of the Commons’ (aka Common’s Dilemma) to depict the consequences of our decisions in these times of uncertainity.

### Game Theory

To put it simply, it is a hypothetical situation made by experts to study how people are likely to behave in a given strategic situation. This is applied in Economics, Political Science, Business Strategy Formulation, Law and many other fields to analyze how people or a group react to an existing dilemma. Just like chemistry explains how chemicals react with each other when exposed, game theory tries to show how people react when they are exposed to a particular given situation. Mathematics is useful in game theory as a tool to analyze players’ motivations and to predict outcomes.

### Tragedy of the Commons (aka Common’s Dilemma)

Many of us have heard about the Prisoner’s Dilemma. It’s about two prisoners having a dominant strategy to rat out the other while the jointly preferred outcome would be achieved if both of them cooperated. The ‘Tragedy of Commons’ is basically an n-person prisoner’s dilemma.

Here’s a beautiful illustration of this theory from **TED-Ed**:

Now that the concept is much more clear, let’s try to make some conclusions.

Since the beginning of time, men must have had the same urge to conquer and utilize the resources for their own benefit. With that said, we must have also formulated compatible strategies to cope up with every kind of social breakdown that threatens the existence of our race in common. To ensure the treatment of these threats promptly, we have:

- Formed Governments,
- Developed Law and Order,
- Implemented Judicial Systems,
- Formulated International Treaties and much more.

It’s logical to deduct that every social contract that ever made to share or control a particular resource, indirectly addressed this dilemma. So it’s clear that we, as humans, are bound to formulate such optimal strategies when the time asks for it.

### War Against COVID19

Since the global outbreak of this pandemic, we have been reading so much about how people and States are reacting to the shortage of resources. We see an inward-looking mentality of people and states that humans tend to show at the very first sign of danger. We see people stocking things(like hoarding toilet papers) and States closing borders, which results in a limited flow of resources among ourselves. This is expected and we need to be addressed.

So how to deal with this dilemma? Let’s do some math!

#### Crisis

The model am going to explain was originally discussed by Vivek Palaniappan, a Cambridge undergrad, in one of his medium posts. Imagine that there are ‘n’ people in this world, each able to choose how much toilet paper they want to consume and that the total toilet papers available is a constant K. Let the amount of toilet paper they each consume be k_i, for i from 1 to n. Also, each of them can consume from the remaining toilet papers that are left after consumption. For simplicity, we let each of the utility functions be logarithms. So the value function for each of these individuals, given some belief on what the others are gonna choose is:

The notation k_i denotes what each person believes the other people are going to choose. This belief can be founded on an informational advantage or some common knowledge of behavior. In most introductory game theory examples, we assume that all parties are rational and that information is also common knowledge ad infinitum.

Before we can proceed to find a solution, we need to introduce the concept of Nash equilibrium, discovered by mathematician John Nash. Nash equilibrium is informally defined as a solution where the individuals make a choice that maximizes their good for all strategies of the other individuals. Formally, it is the solution that maximizes the value function for all possible beliefs of all other parties. A simpler iterative way to understand is this:

- For A, find the choices that maximize his value function for every choice of B.
- For B, find the choices that maximize his value function for every choice of A.
- If there is a solution that is both optimal for A and B (from steps 1 and 2), then that solution is the Nash equilibrium.

This iterative procedure gives us some clue as to how to approach the problem at hand.

#### Solution

From our iterative procedure, we evaluated the best choices for a party for every other choice of other parties. This can also be done in the continuous analog by the following:

- Find the maximum point in the value function of A. (Note: it will be a function of the choices of the other parties)
- Repeat for all parties
- Now we will have n equations with n unknowns, which in most cases will give an intercept point (or plane, depending on the geometry of the value functions). This point is the Nash equilibrium.

Now we can attempt it for our problem. The stationary point of the value function is:

Now for n parties, this linear system becomes:

The solution to this linear equation is simple to derive and it gives:

So the optimal amount of toilet paper to be consumed by each individual is the total toilet paper available divided by number of people in the society **plus one**. This does make an intuitive sense: the plus one comes from the fact that your consumption also impacts you indirectly.

### Conclusion

As we saw from the above example, it is important to consider how your actions, as an individual, affect the resource you are consuming along with your fellow beings. Bear in mind the fact that any resource’s quality/quantity affects your own existence as well as others who are dependant on it! Use or consume what’s needed for your immediate necessities and don’t forget to share the source with those who are in need. To conclude, it’s logical to say that, caring for yourselves primarily includes caring for others.

Let’s fight this together!

#StayHomeStaySafe