"The Error Node" - A Policy Intervention...

I recently had a discussion with my brother on the following problem,

“Develop a function which returns a node from a binary tree with the following conditions

• Probability of the node returned being a root node or from a left sub-tree or a right sub-tree is equal
• Probability of any node from the tree being returned is equal”

At first it seems an impossible task, the probabilities don’t align but then a round-about solution was thought of – what if we introduce an error node?

For simplicity, let’s assume a binary tree with the root node, 6 nodes in the left sub-tree and 2 nodes in the right sub-tree.

Now, consider a 3 * 6 matrix (a matrix representation of the binary tree) as follows,

Where,

• Each non-zero number in the row represents either a root node or a node in the left or right sub-tree
• And most importantly, 0 represents the error node

And, finally, let’s define a function (as a solution to the above problem) such as

• Choose a random number from the set {1,2,3} with corresponds to each row of the matrix which in turn corresponds to a root node, a left or a right sub-tree
• Once the row is chosen, randomly choose a number from that row

In trying to achieve the objective, we have burdened the solution with a cost i.e. all the times we end up getting the error node.

If we were to quantify it, total cost can be represented as

C (cost incurred every time the function is called)

[We will reasonably assume it to be a constant]

*

P (probability of the error node being returned)

[= (1/3) * (5/6) + (1/3) * (0/6) + (1/3) * (4/6) => 1/2]      

*

N (number of times the function was called to return a node)

[We will reasonably assume it to be a fixed number]

i.e. 0.5CN which is 50% of the total cost incurred.

While the above example can easily be generalized, what is more important in it usefulness in understanding certain policy interventions.

Consider, instead of a binary tree, the society with its own sub-classifications (which can go down to the level of an individual) which are unequal along one of the many possible attributes.

How do we then ensure the equality enshrined in our constitution? (equal probability in the above problem)

The solution is simple (like above), introduce an "error node".

In a social context it means devising policies which abrupt the presumed normalcy. One such example (and a contentious one too) is reservation provided for the backward classes (including schedule tribes and scheduled castes).

Please note

• The word "error" is used only to highlight the fact that it’s abrupting the presumed normal and is supposed to be temporary (even the reservation was supposed to be temporary)
• The word "cost" is used only in reference to the word "error" being used
• Both of the above words cannot be taken to represent the meaning in the usual societal context
• Further, the example above highlights the need for that “error” (continuing in this case given that equality for them has not yet truly been been achieved)

However, doing so has a cost and in this case, Supreme Court felt that the acceptable cost was a cumulative of 50% of the total.

The above analogy is not to say that such policies are unwarranted but to the contrary why they are warranted and also get an understanding of the cost associated with it. The latter will give an understanding of to what an extent the policy should be abruptive and more importantly, highlight the fact that it’s supposed to be temporary (and in case of it continuing, the failure of society to rectify the underlying issue).