As of August 22, 2010, the results of the federal election are far from certain. In order to form a government, a party must obtain at least 76 seats (an absolute majority) of the 150 seats in the lower house: the House of Representatives. As yet, neither the two major parties: Labor, and the Coalition (Liberal + National) have been able to do so. Expected results are:

- 72 seats for Labor
- 73 seats for the Coalition
- 1 seat for the Australian Greens
- 4 independents

Of the independents, three of them are expected to form a block in discussion with either of the two major parties; the other independent is “his own man”. So we have the current voting blocks:

- 72 seats for Labor
- 73 seats for the Coalition
- 1 seat for the Australian Greens
- 1 independent
- 3 independent block

In what ways can these blocks form a majority? Given that Labor and the Coalition will never do so (they are what is known in voting theory as “quarrelling parties”), we have the following possibilities:

- Labor + Green + independent block (76 seats)
- Labor + independent + independent block (76 seats)
- Labor + Green + independent + independent block (77 seats)
- Coalition + independent block (76 seats)
- Coalition + Green + independent block (77 seats)
- Coalition + independent + independent block (77 seats)
- Coalition + Green + independent + independent block (78 seats)

But not every voting set is critical to each grouping above. For example, in the final grouping, neither the Green nor the independent are necessary to obtain 76 seats. So let’s list the above groupings again, with the critical votes:

- Labor + Green + independent block: all critical
- Labor + independent + independent block: all critical
- Labor + Green + independent + independent block: Only Labor and independent block are critical (if either the Green or the independent drop out, there are still 76 seats)
- Coalition + independent block: all critical
- Coalition + Green + independent block: only Coalition and independent block are critical
- Coalition + independent + independent block: only Coalition and independent block are critical
- Coalition + Green + independent + independent block: only Coalition and independent block are critical

Of the 7 groupings, each block is critical to a certain number of them:

- Labor: critical for 3 groupings
- Coalition: 4 groupings
- Greens: 1 grouping only (the first one)
- independent: 1
- independent block: 7

We can calculate the Banzhaf power indices by dividing each number by the sum:

- Labor: 3/16 = 0.1875
- Coalition: 4/16 = 0.25
- Greens: 1/16 = 0.0625
- Independent: 1/16 = 0.0625
- Independent block: 7/16 = 0.4375

We see that the independent block has by far the greatest power in establishing a government, and even though the Coalition has only one more seat in the lower house, they have 1/3 again as much power as Labor. Let’s hope the independent block use their power wisely!

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