Guest Column - Building Coalitions: The Banzhaf Index

By Brian Easton.

One of the advantages of MMP is it enables us to think more systematically about the political process (although given much of the nonsense that is being written at the moment, it does not appear to force us to). What this note sets out is a a mathematical procedure which enables us to think systematically about coalitions (although, and as I shall explain, like most mathematical models it has imitations).

However, first a word about its progenitor, John F. Banshaf III, a remarkable professor of law at George Washington University who is known as the 'Ralph Nader of the Tobacco Industry,' 'the Ralph Nader of Junk Food,' 'The Man Who Is Taking Fat to Court' and 'Mr. Anti-Smoking'. (He is also Faculty Advisor for the GWU Volleyball Team.) Earlier he had been an electrical engineer and obtained the first copyright ever registered on a computer program. He also developed a method for measuring the power of parties negotiating coalitions – the Banzhaf Index.

The Banzhaf Index

I use the Banzhaf Index in the current New Zealand political situation. I take the current allocation of seats. The specials may change the numbers but not the principles illustrated here.

There are eight parties vying for power, with a total of 122 seats. Each may be in or out of a coalition, so in total there is the possibility of 2⁸ or 256 coalitions (including the zero when nobody joins). As it happens there are 124 of these in which the coalition has the required 62 votes or more to govern (slightly less than half because 61 votes doesnt give a majority).

Of course not all parties belong to all the possible successful coalitions. Labour with 50 votes is in 68 (or 55%) of the coalitions. The figures for all the parties are shown in column 4 (and column 5) of Table I.

However, this does not discriminate between the importance of a party in a coalition. For instance in the coalition of all parties with 122 seats, if Progressives with their one seat walk out the coalition still has 121 seats, more than enough to govern.

So the Banzhaf index involves counting the number of times of a party to walk out of a coalition with the result that it loses its majority. In the current situation, any party can walk out of the ‘All’ coalition and it would still have a majority. (Were Labour to walk out, there would still have 72 (122-50) votes).

As it happens Labour is required in 68 of the 124 coalitions, while the Progressives are required in only 4 of them. (Column 6). The Banzhaf index adds the number of times this happens for all parties (238) calculates each party’s total as a proprtion of the grand total, and calls the level of power of each party relative to the rest. (So Labour’s relative power is 68/238 = 28.6%).

Table I: Illustrating the Banzhaf Index

Party	Seats		Coalitions		Vetos
Labour	50	41.0%	96	77.4%	68	28.6%
National	49	40.2%	92	74.2%	60	25.2%
NZ First	7	5.7%	80	64.5%	36	15.1%
Greens	6	4.9%	76	61.3%	28	11.8%
Maori P.	4	3.3%	72	58.1%	20	8.4%
U Future	3	2.5%	68	54.8%	12	5.0%
ACT	2	1.6%	67	54.0%	10	4.2%
Progressive	1	0.8%	64	51.6%	4	1.7%
Total	122	100.0%	124	100.0%	238	100.0%

The final column in bold shows the Banzhaf index of the power of each party in coalition forming.

We have the well known phenomenon that smaller parties seem to have power out of proportion to their votes.

Incompatibles

The previous section was highly idealised, assuming that the parties are only interested in power and have no principles (or backers, which is often the same thing). Sometimes parties are incompatible and wont go into coalition together.

So lets add some of those principles as follows:

1. Progressives always go with Labour
2. ACT and the Greens never go with each other
3. The Maori Party wont go with National because it wont entrench the Maori Seat provision.

(I can easily add some more, but this illustration suffices.).

Now there are only 46 viable coalitions, and as Table II shows Labour is in 91.3% of them.

Table II: Banzhaf Index with Some Political Restrictions

Party	Seats		Coalitions		Vetos
Labour + Progressives	51	41.8%	21	91.3%	19	36.5%
National	49	40.2%	14	60.9%	11	21.2%
NZ First	7	5.7%	16	69.6%	10	19.2%
Greens	6	4.9%	10	43.5%	5	9.6%
Maori P.	4	3.3%	6	26.1%	4	7.7%
U Future	3	2.5%	12	52.2%	2	3.8%
ACT	2	1.6%	7	30.4%	1	1.9%
Total	122	100.0%	46	100.0%	52	100.0%

The final column in bold shows the Banzhaf index of the power of each party in coalition forming.

Not surprisingly, those parties which do not rule out partners as a matter of principle generally have a higher proportion of vetos and more power relative to the unrestricted case. Conversely those that rule out some options have a lower proportion of the vetos and less power.

The exception is United Future. Through a quirk of the numbers, 3 seats is not a good number to have in this set of voting patterns. (Peter Dunne would agree. He would rather have 13 seats.)

Minority Government

Banzhaf designed his index for one-off situations. It is usually illustrated in the American literature, by the Electoral College for the US President, which comes together, votes on the sole matter of who is to be the next president, and then dissolves.

The New Zealand situation is different because the parties meet again after every election and, as we shall see, the prospect of that affects how they behave now. Moreover, the coalition process is an ongoing one in the intervening three years, particularly when there is a minority government, as there has been in eight of the last ten years and there is likely to be over the next three (and probably after).

We can adapt the index as follows. Let’s assume that Labour remains a minority government with the Progressives. It has to seek six coalitions of the remaining parties in parliament. There are coalitions available to it for this purpose (that is the government has to raise at least another 11 votes). Table III shows the calculations for the minority parties.

The Labour and Progressive row is deleted. The theory is not robust enough to measure the power of an incumbent minority government, which has a whole range of institutional instruments which enhance the power from their seats. However the Banzhaf Index can be used to measure the relative power of those outside government, as Table III shows.

Table III: Banzhaf Index for Relative Strengths of Outside Parties (assuming Labour and Progressives form a minority government)

Party	Seats		Coalitions		Vetos
National	49	69.0%	32	19.0%	15	35.7%
NZ First	7	9.9%	29	17.3%	9	21.4%
Greens	6	8.5%	28	16.7%	7	16.7%
Maori P.	4	5.6%	27	16.1%	5	11.9%
U Future	3	4.2%	26	15.5%	3	7.1%
ACT	2	2.8%	26	15.5%	3	7.1%
Total	71	100.0%	168	100.0%	78	100.0%

The final column in bold shows the Banzhaf index of the power of each party to influence the minority government.

Table III suggests that while National has more than two thirds of the seats outside the minority government, it has only just over one third of the power to form a coalition to influence the minority government. All the other parties have correspondingly more power.

National Remains Outside

However, this requires cooperating with the government which, for reasons good or bad, National has not done so in the past. Suppose they refuse to join in. Table IV shows the relative power of the remaining parties outside parliament (assuming that ACT is willing to cooperate).

Table IV: Banzhaf Index for Relative Strengths of Outside Parties (assuming Labour and Progressives form a minority government and National is not willing to cooperate).

Party	Seats		Coalitions		Vetos
NZ First	7	31.8%	26	23.2%	18	34.0%
Greens	6	27.3%	24	21.4%	14	26.4%
Maori P.	4	18.2%	22	19.6%	10	18.9%
U Future	3	13.6%	20	17.9%	6	11.3%
ACT	2	9.1%	20	17.9%	5	9.4%
Total	22	100.0%	112	100.0%	53	100.0%

The final column in bold shows the Banzhaf index of the power of each party in coalition forming.

The outcome is that the power of the remaining parties is close to their numbers of seats. Moreover, their power is higher than if National was a player. In effect National not joining in gives the others more power, for three or so years anyway.

Conclusion

The above has tried to clarify the current state of the coalition discussions using the Banzhaf index of power. It shows that if there is a minority Labour led government, and National does not join in the coalition making on a one policy basis, the role of the minor parties is strengthened.

There are at least two further caveats in this assessment. The theory is really about a series of one night stands. Coalitions, even those between those inside and outside government, often have more of a marriage element, insofar as one party may compromise against its immediate interests in order to get overall gains in the long run.

Second, while National will not join in the public glare of the House, it is well known that Select Committees are considerably more cooperative. No doubt the coalition principles explored here are relevant, although the caution about the ‘one night stand’ assumption applies here too.

Notes

More about the the theory of the Banzhaf index at http://www.cs.unc.edu/~livingst/Banzhaf/.

A computational algorithm is available at http://www.math.temple.edu/~cow/bpi.html. This makes some assumptions which results in estimates not quite as as precise as those given here, which are derived from a spreadsheet. This more tedious procedure gives the user a better feel of the underlying theory, and also allows the introduction of the incompatibility restrictions.

You can read more about John F. Banzhaf III at http://banzhaf.net/. Its enough to make someone need a hamburger.