# BJP-Shiv Sena Government, Game Theory and Shapley- Shubik Power Index

Shiv Sena on Thursday criticised the Bhartiya Janata Party over the contentious Land Acquisition Bill.In a scathing attack, the Shiv Sena’s mouthpiece Saamna said the party will not support any bill that stands against farmers.

– India Today

BJP and Shiv Sena have formed coalition government in Maharashtra, as neither of them can form government on their own. Though both claim to be pro- Hindu, there is no love lost between them. Given this situation BJP is not in a mood to spoil its relations with Sharad Pawar’s NCP, which in turn means it will have to go slow on corruption cases during NCP regime.

Question is how critical is Shiv Sena in coalition game. Answer to this lies in Game Theory.

Noble Prize winner Lloyd Shapley is expert in area of Game Theory. He along with Martin Shubik, came up with Power Index in 1954 to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface.

In case of power index, assumption is all are rational players and they join coalition in sequence i.e. one at a time.

Let us imaging that there are 4 players i.e 4 political parties – P1, P2, P3 and P4 and they have 7, 5, 3 and 2 seats respectively. To pass any bill in assembly, you need at least 10 votes. No party on its own can pass the bill, therefore need for coalition.

Suppose P1 proposes a bill, P2 first supports it, then P4 supports it and finally P3 support it, then sequence is P1, P2, P4, P3. There can be other sequences also ex. P3, P1, P2, P4.

Number of such sequences for 4 parties will be 4! = 4x3x2x1 = 24.

Now suppose P1 proposes a bill and it is supported by P3 then it gets required 7+3=10 votes, then P3 becomes pivotal player (since because of P3 you got required 10 votes), it could also be P2 proposes bill, P3 supports it first and next P4 supports it i.e. 5+3+2=10, here required votes were obtained when P4 joined coalition therefore P4 becomes pivotal player.

Now for all the 24 sequences you find a pivotal player in each sequence. Calculations will show that P1 is pivotal player in 10 sequences, P2 in 6, P3 in 6 and P4 in 2 sequences.

Now we calculate power index using formula given below

Power index for a party = No. of times a party has pivotal position/Total no. of sequences.

Therefore power index for P1= 10/24, P2=6/24, P3=6/24 and P4=2/24. Higher number indicates more power. It is interesting to note that P2 and P3 have same power though P2 has more seats.

In Maharashtra Assembly majority means 145 seats, BJP has 122, Shiv Sena has 63, NCP has 41 and Congress 42. This is very interesting combination, so let us wait and see how the game is played.