**Astract**

If an organisation has many levels, people are not bound by a common goal and simply “do their job”, then such a structure inevitably leads to chaos, absorption of all available resources and eventually self destruction.

—

The phenomenon described here is well known in Control Theory. It can be observed in a general setting but we are going to illustrate it using a simplest model first.

Let the outcome \(X(t)\) of some process be controlled by a manager who defines the speed of producing the outcome \(Y=dX/dt\)

The behaviour of the manager of first level is controlled by the manager of second level \(Z=dY/dt\) and so on up to the manager of the top level (level \(n\))

The manager of the top level in our model implements feedback. His decisions are not based on complying with the wishes of his managers but on producing the outcome. For instance he may want variable \(X\) to achieve a certain level \(x_{0}\) and he will push the managers of the level below him one way if this level is not reached and another level if it is exceeded.

The simplest model of this kind is as follows –

\(dX/dt = Y,dY/dt = Z,dZ/dt = -k(x_{0}-X)\)

Or equivalently

\(d^{n}X/dt^{n}=-k(x_{0}-X)\)

This model has an explicit solution. The stability of the required stationary solution \((x_{0}=X,y_{0}=z_{0}=\ldots=0)\) depends on whether the real parts of the roots of the characteristic equation \(1^{n}=-k\) are negative or not.

These roots are complex numbers located in the vertices of a regular \(n\)-gon. If \(n>2\) then some of the vertices will necessarily lay in the non-stable right half-plane \((Re>0)\). If \(n=1\) then the single root lies in the stable half-plane \((1=-k)\), if \(n=2\) the roots \((1=\pm i\sqrt{k})\) lie on stability boundary.

Conclusion: multi-level management described by this model is not stable if \(n>2\). It results in a butterfly-effect, whereby a small change in initial conditions results in unpredictable change in the outcome. Two-level management result in periodic oscillations, which do not lead to oscillation build-up, which happens in management hierarchies with three and more levels.

The true stability is possible only in one-level hierarchy, where the manager is interested in the outcome rather than wishes of his managers.

The conclusions derived from the simplified model stand the test of structured stability, where simplistic dependencies are replaced with arbitrary functions, which model organisational details, excluding the case of \(n=2\).

Two-level management can be either stable or not stable depending on organisational details, which were ignored in the simplistic model.

Real organisations often have multi-level management hierarchies. The long term survival of such organisations is explained by the fact that substantial parts of the organisation form flat structures where people voluntarily organise to work together on the common goals rather than focus on executing instructions of their managers. Without this multi-level management will ruin the organisation.

—

Vladimir Arnold – Soft and Hard Mathematical Models, publication of Moscow Center for Continuous Mathematical Education, 2004, page 17- The Danger of Multilevel Management.

Translated from original Russian by my friend Alex.

227