Contemplating C. Ramanujacharyulu’s Work and the Bihar Assembly Poll Results

Voters stand in a queue to cast their votes in the first phase of Bihar assembly elections, at Alipur block, Kaimur district, October 28, 2020. Photo: PTI.

More than 50 years ago, an Indian mathematician named C. Ramanujacharyulu introduced a protocol that allows us to reconcile the results of a group of teams and their matches in a tournament, choose the winner and/or rank them in an order. Each team meets the other team (the paired comparison) once or several times and registers a result each time – win, tie or loss.

The recently concluded Bihar assembly polls were eagerly contested. As the results poured in on November 10, it was clear that there were five significant players in two broad coalitions: the NDA had the BJP and the JD(U), and the MGB (short for Mahagathbandhan) had the RJD, the Congress and what we can call the Left. On November 11, The Hindu carried the head-to-head results as these ‘teams’ paired off in various constituencies.

Table 1 (below) summarises this contest in the manner most mathematicians are used to, called a matrix. The zeros indicate that the teams don’t face off because they belong to the same coalition. We can think of this as a tournament in which these five teams faced off of each other. We can then use Ramanujacharyulu’s tournament metaphor to review the results instead of using what most psephologists call the strike rate.

At this stage, without invoking Ramanujacharyulu’s work, we can make a simplistic count of the strike rate. First, look at the rows. The totals are shown in the last column in table 1 for each party (or team). In the round-robin of just these five teams, the BJP with 72 wins/leads has the edge over the RJD, which has 70. The Congress and the Left are at the bottom. Next, look at the column-wise totals. These are a measure of losses/trails. The JD(U) has the worst record with 58, followed closely by the Congress with 46.

Since these totals – both wins/leads from rows and losses/trails from columns – depend on the ways the seats were shared by the fronts as they went into the polls, a better measure of success is the strike rate: this we define here as the ratio of wins to losses. Table 2 shows this list. The BJP has the best strike rate while the Congress has the worst.

We must now understand that the summing of rows and totals is predicated on a simplistic logic – that the strongest player (in this case the BJP, with strike rate 0.67) gets the same score for beating the next strongest rival (Left, 0.59) as it gets for a win against the weakest (Congress, 0.27). Is there a way to score wins based on the strength of the opponent?

Before Ramanujacharyulu, mathematicians had a neat answer to this dilemma: they used a recursive algorithm so that the points are weighted according to the strength of the loser. Let us call this the power score. The scores we see in table 1 row-wise are the power scores, and this can be complemented with the power scores after iteration and convergence has taken place. (The matrix theory behind all this is beyond the scope of this discussion.)

Table 3 (below) displays the power scores after the first and k-th rounds of recursive iteration using the tournament score matrix. We continue the iteration until the k-th round, when we have reached an acceptable degree of convergence. Note that the scores have been normalised to a total of 1.

In this reckoning, the RJD (0.333) has actually out-performed the BJP (0.319). However, this is a size-dependent score, as analysts will call it: it depends on the number of seats each party has contested. As a result, the lowest score goes to the Left. So what we need is the recursive equivalent to the strike rate. Ramanujacharyulu had an answer to this.

Ramanujacharyulu figured that the iterative methodology is only part of the story, based on a row-wise reading of the matrix. Another dimension appears when you read the matrix column-wise. We have already seen this as a prelude to computing the strike rate. Table 1 displays these scores as well, before the iterative computation has taken place. It is a measure of how you have lost.

Following the argument that losses must also be weighted proportionally, one can set up a recursive iteration to get a weakness score, where losing to a worse team is counted as being worse than losing to a stronger team. Table 4 (below) displays the weakness scores after the first and k-th rounds of recursive iterations. Now, the lower the score, the better it speaks of the team. But the size-dependency caveat applies again: the Left has the lowest score. This does not mean much, and we need a strike rate after the k-th round.

Ramanujacharyulu’s insight has important philosophical implications – particularly that there is no such thing as the ground truth when it comes to complex social systems. This is the same dilemma that rears its head in social choice theory as Arrow’s impossibility theorem (which describes the ‘paradox of voting’). Ramanujacharyulu revealed two ways to choose the best player, and explained how they may not always be convergent: the power method – a row-wise reading of the matrix – and the weakness method yielded slightly different rankings.

This said, Ramanujacharyulu also proposed an elegant solution to determine an ‘ultimate winner’, if one was necessary – to use the ratios of the scores, i.e. the power-weakness ratio (PWR) as the best approximation of the truth. This we can obtain by determining who can combine the greatest ability to win with the least susceptibility to lose. That is, “in tournaments one may be interested in locating the really talented man (sic) in the sense that he has won over the largest number of opponents but simultaneously he has been defeated by only a few opponents,” as Ramanujacharyulu wrote.

Table 5 (below) shows the ‘teams’ and their PWR scores after the first and k-th rounds of recursive iterations. The order is: BJP > Left > RJD > JD(U) > Congress. (If you are mathematically proficient, you may recognise that the strike rate and the PWR are related to each other: SR = PWR/{PWR + 1}.)

The Bihar mandate was mainly based on two groups or fronts. Table 6 (below) shows the consolidated tables from Ramanujacharyulu’s perspectives for the two groups. The NDA had the edge (1.058 to 0.947). However, it is anybody’s guess what would have happened if the Congress had refrained from participating, and conceded its share to the RJD and the Left, when they went to the polls.

Gangan Prathap is an aeronautical engineer and former scientist at the National Aeronautical Laboratory, Bangalore and former VC of Cochin University of Science and Technology. He is currently a professor at the A.P.J. Abdul Kalam Technological University, Thiruvananthapuram.