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A mathematical theory of team optimization

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If some is good, more must be better.

Mustn’t it?

Of course not, although figuring out where to draw the line isn’t always obvious.

Take, for example, the laudable principle that inclusion and collaboration usually yield better results than isolation and authoritarian decision-making.

Just don’t take it off a cliff.

Which appears to be exactly what’s happening in your average corporation — at least it is if the evidence and conclusions presented in “The Hard Evidence: Business is Slowing Down,” (Tom Monahan, Fortune, 1/28/2016) are to be believed. Among Monahan’s findings: The number of collaborators people have to work with in a typical corporate effort is at least 10, and often 20 or more.

Before we go on: I doubt this is really the number of collaborators people have to work with. I’d bet as much as a quarter Monahan counted the number of colleagues people include and involve in an effort in any capacity.

Not that I’m criticizing. Inclusion and involvement provide numerous benefits. No matter what problem you’re trying to solve you get more eyes on it, more different perspectives, more creativity, and when you’re finished, greater acceptance of whatever solution you and your collaborators have developed.

But (is anyone surprised this was the next word to appear?).

But, inclusion and collaboration don’t scale all that well, which is why there’s such a huge gulf separating consensus from voting as decision-making techniques.

All voting requires is an opinion and a willingness to invest the time needed to cast a ballot. This willingness increases in direct proportion to the level of distrust each voter has in their fellow voters.

Consensus, in contrast, depends on trust among those involved in a decision. Trust doesn’t require everyone involved in a decision to have the same goals, perspectives, assumptions  (conscious or otherwise), experiential background, or time zone of residence.

It does require the assumption on the part of everyone involved that they’re dealing with social peers who have no hidden agendas, who are open to new evidence and logic, and who honestly want to participate in creating a positive outcome.

Which leads us to a pair of mathematical curves.

The first curve is a polynomial, the output of the formula that computes the number of relationships between pairs of individuals on a team. The formula is K = P(P-1)/2.

K, that is, is the number of relationships that need maintaining in a team. So a ten-person team consists of 45 interpersonal relationships, while a 20-person team consists of 190. Maintaining trust among 45 pairs of individuals is difficult but not impossible. Maintaining it among 190 pairs is, at best, unlikely, even if one person has handpicked each team member with team compatibility their first selection criterion.

This is why inclusion and involvement don’t scale.

Which brings us to curve #2. It’s a variant of the logistics equation. In this case, dI/dP= rP*(1-P/K), where dI/dP represents how the number of good ideas (I) increases with team size (P).

In this formula, r is a constant — the raw idea generation power of an average team member. The logistics equation says that as team size increases, the outcome when it comes to more and better ideas will be an s-shaped curve, with the limiting factor (K) being the number of trust-based relationships required due to increasing team size.

This is nifty, don’t you think? It gives us a mathematical theory of team size optimization.

Sure, it isn’t all that surprising that the result is a point of diminishing returns as teams get bigger.

But reducing it to an equation is still pretty cool.

Comments (7)

  • Do you have any citations or references for the second equation?

  • Thank you! I was a Math major, and I think this is the first time I have seen calculus used in my professional career! And I love calculus! Now I want to see the curves.

  • First, consider Brook’s Law: Adding more workers to a late project will make it later – Fred Brooks in his 1975 book The Mythical Man-Month.

    I have always advocated a small working team: 3 people, unless the work is large, then subdivide so that each major task has 3 people. Add a 4th, and someone needs to manage – productivity declines to less than one person working alone.

    • My experience in local government and my parallel experience as a gentlemen farmer raising pigs tells me that “three” is a not a good number for optimum collaborations. I learned, quickly, that it is possible to raise 1 pig, 2 pigs, four pigs, – any number of pigs except three pigs. This is because three pigs always yields 2 hogs and a runt.
      You may “Find pigs” and “Replace With Collaborators”. To those who might object to the idea that collaborators and pigs have some relationship I would point out that at least pigs stick with being pigs.


  • I was also a math major and I would like to see the curves. But, on first glance it does seem reasonable and I think it’s a very cool of math. However, when you mentioned the word “trust”, I think you add some factor for the probability for the number of “jerks” likely to be included. From teaching, I’d say the probability of potential jerks in the group is naturally about 1 in 4.

    I don’t think these folks intend to be dysfunctional elements, but I sense that they sometimes don’t trust anyone who doesn’t exactly see things as they do. I suspect dealing with 2 out of 10 folks like that, is a lot easier than dealing with 5 out of 20. I don’t think the group toxicity increase is a linear function.

    Still, your equations can give managers a good place to start. Good stuff.

  • Consensus and collaboration are good to a point. Sooner or later, however, someone has to have enough courage to make a decision and provide some leadership! Some years back I worked at a company and one of the “rising stars” was sent to Stanford for management training. The general consensus was they destroyed him because when he returned he was no longer willing/able to make a decision. I also worked with a guy who used to say “Do something even if it is wrong.” In other words consensus and collaboration are good until they become an excuse not to make a decision.

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