Friday, September 23, 2005



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

PartySeatsCoalitionsVetos
Labour5041.0%9677.4%6828.6%
National4940.2%9274.2%6025.2%
NZ First75.7%8064.5%3615.1%
Greens64.9%7661.3%2811.8%
Maori P.43.3%7258.1%208.4%
U Future32.5%6854.8%125.0%
ACT21.6%6754.0%104.2%
Progressive10.8%6451.6%41.7%
Total122100.0%124100.0%238100.0%

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

PartySeatsCoalitionsVetos
Labour + Progressives5141.8%2191.3%1936.5%
National4940.2%1460.9%1121.2%
NZ First75.7%1669.6%1019.2%
Greens64.9%1043.5%59.6%
Maori P.43.3%626.1%47.7%
U Future32.5%1252.2%23.8%
ACT21.6%730.4%11.9%
Total122100.0%46100.0%52100.0%

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)

PartySeatsCoalitionsVetos
National4969.0%3219.0%1535.7%
NZ First79.9%2917.3%921.4%
Greens68.5%2816.7%716.7%
Maori P.45.6%2716.1%511.9%
U Future34.2%2615.5%37.1%
ACT22.8%2615.5%37.1%
Total71100.0%168100.0%78100.0%

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).

PartySeatsCoalitionsVetos
NZ First731.8%2623.2%1834.0%
Greens627.3%2421.4%1426.4%
Maori P.418.2%2219.6%1018.9%
U Future313.6%2017.9%611.3%
ACT29.1%2017.9%59.4%
Total22100.0%112100.0%53100.0%

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.

10 comments:

One of the things that strikes me about the first table is how weak Labour appears - in fact how weak Labour and National both appear. My read would've been that Labour is in a very strong position, that level of weakness would only have been true if there was a single party shaping up as kingmaker (where Winston wanted to be).

I wonder how much that is because I start off with a very FPP mentality - after the votes are in we will have either a Labour or a National govt - with an assortment of support parties.

I guess the reality is that only Labour or National can lead the government, but the rest have the ability to prevent them doing it. There is a sense in which the big two have no power, all they can do is work to convince the others to allow them to govern.

Finally, I think it's fascinating how badly incompatibility hurts both parties. Both UF and Winston have tried to veto the Greens, and have had to back off because it hurts them too badly. It looks, for this analysis, that the loss is to their power-of-numbers (as well as political and public perception, which we've already seen)

I think both UF and Winston have worked out the best way to hurt the Greens will be to play "we'll go with the biggest party" and watch the Greens bleed to Labour. That way they reduce (or wipe out) the Greens power-of-numbers, without hurting themselves by taking the two-way hit of incompatibility.

Posted by Anita : 9/23/2005 08:13:00 PM

OK. Labour and National seem weak because they steal power from each other, and leech it to the center parties. That was rather the idea behind MMP, to stop the big boys doing as they pleased. Hmmm...

Compare the last election: 2002.
(I'll remove the 5% minimum and let a few more minnows in, as that actually de-powers the centre few).

61: {49,25,13,9,8,7,2,2,2,2,1}

votes power
49 60.43 %
25 7.57 %
13 7.57 %
9 6.51 %
8 5.44 %
7 4.12 %
2 1.86 %
1 0.93 %

And the same for the 1999 election.

61: {48,37,9,8,6,5,3,2,1,1,1}

votes power
48 42.54 %
37 14.03 %
9 11.16 %
8 9.83 %
6 6.96 %
5 6.3 %
3 3.43 %
2 2.1 %
1 1.22 %

Once you push the Labour/Alliance and Labour/Progressive blocks into one they continue to secure most of the power, quickly leaving National with no more power than the small center parties (as any one of them can act to pass a bill).


That's fairly intuative really, with a 59-59-2 block all three groups have equal say.
There's an awesome reason to get rid of the threshold, more minnows reduces the power of the minnows, giving it back to the big voting blocks.

It's also very cool that a willingness to cooperate with a broad range of others is the real path to power. Also quite interesting that 49 seat National will ultimately have no more power than the 7 seat NZF (assuming a L-G-P voting block at 5 short of a majority).

Posted by tussock : 9/23/2005 09:46:00 PM

Of course they're weak - you only have to look at the minimum (and likely) numbers of actors each needs to form a government or pass legislation to see that. Labour's apparent strength stems from the promises that have been made for parties to talk to them first, and Clark's perceived strength as a coalition-builder, not from the actual numbers; National's weakness from the fact that they have no friends, and Brash doesn't seem likely to make any.

Convincing others to let you govern is fundamentally what politics is all about. It's just that normally, those others are outside Parliament, rather than in it.

Posted by Idiot/Savant : 9/23/2005 09:54:00 PM

Tussock: I've been arguing that small parties would balance one another for quite some time, based on the experience of this term where Labour was able to get the support of the Greens, United or NZ First depending on what they wanted to pass. It's nice to see a mathematical basis for it.

But while willingness to cooperate improves your coalition maths and therefore your power, the blunt fact is that parties don't just want power, they want power to do certain things. And it's there, rather than in the raw maths, where the real problems are.

Posted by Idiot/Savant : 9/23/2005 10:02:00 PM

i/s: True, that's another real advantage of the big blocks having a lot of choice available when seeking the majority of votes on a peice of legislation: no one can demand too much in exchange, making it very hard to get fringe policies through, especially when the other minnows might give support to save legislation from moving further away from their own position.
Ideally National would always save the left having to make poor comprimises, but opposition parties don't seem to like being constructive, yet.

Posted by tussock : 9/24/2005 12:15:00 AM

Tussock,

When you add in the minnows do they take power from the majors or the minors? My rough calculations say they would take it from the majors and the minors would be as strong as ever, but I might be having a mathematicaly challenged day :)

Posted by Anita : 9/24/2005 11:01:00 AM

I think the majors are stronger than they appear because the public won't stomach the minors throwing their weight around. You just have to look at the loss of power for NZ First between the 1996 and 1996 elections (17->5 seats = 23.08% ->9.91%).

The media (and the other minors) love to play up the stroppy/unreasonable/power_hungry-ness of the minors, which handicaps them. The electoral math may say that NZ First have 34% of the power (given the last scenario - LP in minority government) but if they start behaving like it, their opponents and the media will make a huge fuss, and they'll be punished at the next election.

None of the minors can afford to be punished for tantrums and/or parliamentary blackmail in 2008 - so their power is necessarily limited.

Interestingly I'd say that the Greens are in a particularly odd position - if they seem unreasonable (or like the sandal-wearing lunatic fringe as a colleague likes to describe them) the electorate will punish Labour and thus the left as a whole. Those of us already committed to the Greens will happily accept that the Greens do (and should) have bottom lines, it's the centreist Labour voters who could be frightened into backing away from Labour if the Greens looked too stroppy.

Posted by Anita : 9/24/2005 11:24:00 AM

I think the majors are stronger than they appear because the public won't stomach the minors throwing their weight around. You just have to look at the loss of power for NZ First between the 1996 and 1996 elections (17->5 seats = 23.08% ->9.91%).

The media (and the other minors) love to play up the stroppy/unreasonable/power_hungry-ness of the minors, which handicaps them. The electoral math may say that NZ First have 34% of the power (given the last scenario - LP in minority government) but if they start behaving like it, their opponents and the media will make a huge fuss, and they'll be punished at the next election.

None of the minors can afford to be punished for tantrums and/or parliamentary blackmail in 2008 - so their power is necessarily limited.

Interestingly I'd say that the Greens are in a particularly odd position - if they seem unreasonable (or like the sandal-wearing lunatic fringe as a colleague likes to describe them) the electorate will punish Labour and thus the left as a whole. Those of us already committed to the Greens will happily accept that the Greens do (and should) have bottom lines, it's the centreist Labour voters who could be frightened into backing away from Labour if the Greens looked too stroppy.

Posted by Anita : 9/24/2005 01:49:00 PM

You expressed puzzlement over the low power of Labour.

1. Labour and National are going to have roughly equal power.
2. Each requires at least to other parties to form a government.

So the result is not that surprising.

Nor is it surprising that Labour's power fell from about 70% in 2002 to 30% this time around.

There are a couple of assumptions which make the index misleading.

1. They really are about a single game, not a repeated gain.
2. They make no allowance for when one gains the Treasury Benches the advantages of initiative that is built into that.

(They may be the same assumption.)

Basically we are going to get stable government, I think, but a more constrained Labour.

I ponder on how it can still be progressive in such circumstances. The Greens should ponder even more.

Posted by Anonymous : 9/24/2005 09:26:00 PM

Nice article, but there's an error in paragraph 5 that makes my head hurt.

This sentence is wrong:

"Labour with 50 votes is in 68 (or 55%) of the coalitions."

It should read:

"Labour with 50 votes is in 96 (or 77%) of the coalitions."

This is particularly confusing since you talk about column numbers but the columns aren't numbered, and is a 68 in the sixth column, and also in the forth column on another row.

Has anyone tried to repeat the second table without the third incompatible (Maori-National) but with the obvious one: that Labour and National wont form a grand coalition?

Thanks again for the article.

Gordon

Posted by Gordon Paynter : 9/25/2005 09:32:00 AM