On the face of it, the game was the Congress’ to lose. Four short of a simple majority with 17 members of the legislative assembly (MLAs), it was significantly ahead of the second largest party, the Bharatiya Janata Party (BJP), that had 13. The Goa Forward Party (GFP) and the Maharashtrawadi Gomantak Party (MGP) had three MLAs each, the Nationalist Congress Party (NCP) had one and there were three independents. And yet, the BJP, led by Manohar Parrikar, formed the government.

A little game theoretic analysis will show that the result is not as surprising as it seems. The power of parties in a legislature is clearly not just a matter of the proportion of legislators they control. For instance, in the Karnataka assembly with a strength of 225 MLAs, if the Congress controls 110 MLAs, the BJP controls 90, and the Janata Dal (Secular), or the JD(S), controls 25—the power of the JD(S) is at least equal to that of the two bigger parties.

The Shapley-Shubik power index developed by Nobel prize winner Lloyd Shapley and Martin Shubik assigns a precise measure of power to a party based on the probability that it is decisive, i.e. it converts a randomly chosen coalition from a losing one to a winning one by virtue of its inclusion. Robert Aumann and Roger Myerson, both Nobel prize winners in economics, apply the index to analyse coalition formation in hung legislatures.

Recall, the aim of both the leading parties is to stitch up a coalition that will achieve a majority and thereby enjoy 100% power. The decision on whom to invite and the decision of the invitees on whether to join, is based on assessments of how power would be divided between the different parties that constitute the government. For instance, the Congress could form a government with the support of the GFP and the NCP. But it would be foolish to do so because then the GFP and the NCP would both be able to topple the government at will.

The more small parties join the government, the less the large party depends on a particular one, and so the greater its power. This increase in power continues uptil the point that there are so many small parties that they could achieve a majority on their own. The power of the large party now comes down as it has lost its veto power.

The significant feature of the Goa assembly is that neither the Congress nor the BJP would join a coalition led by the other and the remaining parties and independents combined control a total of 10 seats only. Hence, the Congress and BJP never lose veto power with the addition of other parties and each potential coalition partner must keep in mind that whichever leading party it joins will attempt to stitch as wide a coalition as possible, excluding the other leading party. Therefore, it should assess its power in the resulting rainbow coalitions, one led by the Congress and one by the BJP, in order to decide which faction to join.

It stands to reason that the power of a small party in a coalition led by the smaller of the two leading parties would be greater. Foreseeing this, each small party and independent should reject overtures of the Congress and ally with the BJP.

In this model, it does not matter whether the governor first invites the single largest party to form the government or not. With perfect information about offers and rejections, all parties are aware of the course of events and things will turn out in favour of the second largest party. However, if there is imperfect information about the offers and rejections, then things change.

In an atmosphere of mutual suspicion, a small party would be inclined to accept the first offer to join the government rather than wait for a better offer and risk being left out in the cold. The natural advantage of the second largest party may continue to prevail when the governor is passive but if the governor invites the single largest party to form the government first, then a government led by that party becomes a natural outcome.

Such a government is likely to be more stable than one led by the second largest party. The government is now composed of six entities—BJP, MGP, GFP, NCP, and two independents. In the present configuration in Goa (and, indeed, in Manipur), despite the much larger size of the BJP, the Shapley-Shubik power index assigns equal power to the top three entities in the government, 26% each. The reason is that despite their differing sizes, the three entities are perfectly equivalent in terms of the number of cases in which they are the key to enabling a subset of the ruling coalition to cross the halfway mark.

Each of the singletons, on the other hand, enjoys a power of just 7% each. In a government led by the Congress, its power index would range from 33% to 65%, a higher figure, making for greater stability.

The BJP could increase its power by adding the remaining independent, who will also gain, and is probably keen to do so. With power evenly shared between a large number of parties, providing good governance would be a challenge, not good news for Goa’s corruption-infested natural resources sector.

Game theory tells us that with the governor on its side, the BJP needed Parrikar not to form the government but to steer the boat through the choppy waters that lie ahead.

The author wants to thank Deep Desai and Vinay Singh for the research assistance.

Rohit Prasad is a professor at MDI, Gurgaon and author of Blood Red River. Game Sutra is a fortnightly column based on game theory.

Comments are welcome at views@livemint.com

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