The Peter principle is a management concept that describes a common observation within hierarchical organizations. It says that individuals tend to be promoted to positions of increasing responsibility based on their success in previous roles. At some point, they may reach a level where they are no longer competent. Because the skills and abilities that led to success in one position do not automatically guarantee competence in a different role.
I want to justify this sound judgment mathematically. I think we can assume the following simplified setting:
Hierarchical organization: A countably infinite partially ordered set $S$ with the following properties.
- Every element has a unique immediate predecessor (Everyone has a manager)
- Join semi-lattice (Any two people have a common manager)
- Finite antichain condition (For any person, the number of incomparable individuals in the organization is finite)
Promotions: A sequence of $\{f_n\}_{n\in\mathbb{N}}$ of bijective non-decreasing (need not to be monotone) functions on $S$
Competency level: $\color{Red}{\text{???}}$ (I'm not sure how we can model this notion)
In the end, I suppose the Peter principle should follows from the fact that, for any $a\in S$ the set $\{a, f_1(a), f_2f_1(a), \cdots\}$ is finite.
Can somebody help me fill in the missing part of this model. Or, correct me if I am doing something nonsensical.