The right (or wrong) size for a committee: less than 20 but not equal to 8

New Scientist reporter Mark Buchanan has a fascinating article this week on "the curse of work." The title might be the least satisfactory thing about this examination of a new mathematical article that attempts to explain the inexplicable:

"Parkinson's law", first published in an article of 1955, states: work expands to fill the time available for its completion. Is it more than just a cynical slogan? (Image: OJO Images/Rex Features)

It is 1944, and there is a war on. In a joint army and air force headquarters somewhere in England, Major Parkinson must oil the administrative wheels of the fight against Nazi Germany. The stream of vital paperwork from on high is more like a flood, perpetually threatening to engulf him.

Then disaster strikes. The chief of the base, the air vice-marshal, goes on leave. His deputy, an army colonel, falls sick. The colonel's deputy, an air force wing commander, is called away on urgent business. Major Parkinson is left to soldier on alone.

At that point, an odd thing happens - nothing at all. The paper flood ceases; the war goes on regardless. As Major Parkinson later mused: "There had never been anything to do. We'd just been making work for each other." (Mark Buchanan, New Scientist)

I recommend you read this article. I'm only going to discuss a small part, but all of it is interesting. It discusses Parkinson's work and also a current paper by two physicists, Peter Klimek, Rudolf Hanel and Stefan Thurner of the Medical University of Vienna in Austria. They chose a "small world" topology for a network of nodes that interact. In such a network most nodes aren't neighbors but can reach any other node with just a small number of steps (the famous six degrees of separation is such a small world network). There are other kinds of networks (random and scale free being the best known), so their results depend to an unknown extent on their choice. But what they found is extremely interesting:

They grouped the nodes of the network - the committee members- in tightly knit clusters with a few further links between clusters tying the overall network together, reflecting the clumping tendencies of like-minded people known to exist in human interactions. To start off, each person in the network had one of two opposing opinions, represented as a 0 or a 1. At each time step in the model, each member would adopt the opinion held by the majority of their immediate neighbours.

Such a process can have two outcomes: either the network will reach a consensus, with 0s or 1s throughout, or it will get stuck at an entrenched disagreement between two factions. A striking transition between these two possibilities emerged as the number of participants grew - around Parkinson's magic number of 20. Groups with fewer than 20 members tend to reach agreement, whereas those larger than 20 generally splinter into subgroups that agree within themselves, but become frozen in permanent disagreement with each other. "With larger groups, there's a combinatorial explosion in the number of ways to form factions," says Thurner.

Qualitatively this is not surprising as these topologies readily form subnetworks. But the interesting quantitative results makes this extremely interesting:

Parkinson was also interested in other aspects of management dynamics, in particular the workings of committees. How many members can a committee have and still be effective? Parkinson's own guess was based on the 700-year history of England's highest council of state- in its modern incarnation, the UK cabinet. Five times in succession between 1257 and 1955, this council grew from small beginnings to a membership of just over 20. Each time it reached that point, it was replaced by a new, smaller body, which began growing again. This was no coincidence, Parkinson argued: beyond about 20 members, groups become structurally unable to come to consensus.

A look around the globe today, courtesy of data collected by the US Central Intelligence Agency, indicates that Parkinson might have been onto something. The highest executive bodies of most countries have between 13 and 20 members. "Cabinets are commonly constituted with memberships close to Parkinson's limit," says Thurner, "but not above it." And that is not all, says Klimek: the size of the executive is also inversely correlated to measures of life expectancy, adult literacy, economic purchasing power and political stability. "The more members there are, the more likely a country is to be less stable politically, and less developed," he says.

So less than 20 seems to work, more than 20 is a recipe for fractiousness. There is one exception. For some reason, 8 seems to be an unlucky number, leading significantly more often to deadlock:

But once again, Parkinson had anticipated it, noting in 1955 that no nation had a cabinet of eight members. Intriguingly, the same is true today, and other committees charged with making momentous decisions tend to fall either side of the bedevilled number: the Bank of England's monetary policy committee, for example, has nine; the US National Security Council has six.

I love papers like this. It's available free at the open access arXiv site.