Negative selection of personnel in the hierarchical structure of the enterprise
Introduction
We all know how important recruitment is in an organization. But often we have rather vague ideas about how exactly the structure of the organization affects the efficiency of this staff. Or, for example, how the distribution of employees with various parameters in the staffing table affects work efficiency.
When this question is raised, these are usually some common words, heuristics from boring textbooks with highquality descriptions , such as the hierarchical structure pushing forward and motivating daffodils, who are more interested in their own promotion, rather than the prosperity of the enterprise.
It turns out that some aspects of this issue are quite easy to mathematically model, which may open up opportunities for new options for managing companies. In the context of fairly simple assumptions about the properties of personnel, a huge number of important and interesting properties crawl out on their own, just when setting the structure of the organization. And then the question can be posed like this  Is there an opportunity to use this for the benefit of the organization?
Model description
Step by step, how a large organization develops $$ , in which each working person at each level of the hierarchy is described by two variables and two subsets:
$$
Where $$  a bunch of, $$  the set of all the bosses of this person, $$  the set of all subordinates of this person.$$  the degree of personal effectiveness (productivity) of how a person performs his tasks on $$ th level. At every level $$ where $$  uniform distribution. Thus, the average personal productivity of all together is 1. Here it is appropriate to recall the principle of Peter , which in the case of this task of effectiveness begins to sound like this: “In a hierarchical system, each individual tends to rise to the lower level of his incompetence.” Those. not the fact that a person is not effective even higher. It just rises only to where it has a failure in efficiency. The most effective collective farm chairman is not always at least not even a tolerable milkmaid, as we know. However, as we will see later, this principle also describes an absolutely idealistic situation (all the more so since it was justly criticized ).
$$  the degree of human egoism. Here, egoism means exclusively a certain “distilled” ability to do things that are not related to productivity or the performance of work functions, but according to a person should contribute to his career advancement.
The organization model is generated $$ a person with the characteristics described above, which are initially randomly distributed by $$ levels. The capacity of each level is set for the following reasons:
We all know a joke about mathematicians who order beer at a bar , each time asking for half of what the previous one requested:
$$
So for $$ level, the number of people at the level can be calculated by the following formula:
$$
Where $$  rounding down (floor). $$  normalization amount, so that in the final it turns out exactly $$ человек.Accordingly, fully organization $$ set as a dimension vector $$ where $$ level set by many employees $$
Each person has a set of bosses (except for the upper level) and a set of subordinates (except for the lower level)  $$ . To set this list for each person, we use the following algorithm:
Let an organization be given $$ . Then for every person $$ also owned $$ list $$ of $$ subordinates is set like this:
H.D=[]
if H.i != 0:
H.D = random.shuffle(O[i1])[:K]
for dependant in H.D:
dependant.C.append(H)
It turns out a graph expanding from the top of the hierarchy in the spirit of a tree, but not a tree, because there can be more than one path between two vertices.
At each iteration, each person’s performance $$ is calculated according to the following formula:
$$
Where $$  a lot of productivity of all subordinates. The essence of this formula is that the egoism of individual subordinates is leveled by their number $$ , while the overall system performance without taking into account selfishness is the performance of the weakest link in the subordination $$ .Further, at each iteration, the performance of all people is considered:
$$
As the main measure of the effectiveness of the structure, productivity normalized to the number of people is used: $$ . This indicator shows how much the result obtained is less than what would have happened if no one were selfish and each had a productivity of 1.At the end of each iteration, a frame advancement step takes place. Every turn $$ random people leave (retire or something else), after which their direct subordinates compete for vacated seats, and the very bottom of the pyramid is added $$ laid off with a hierarchy lowered to the lower level. This is done so as not to introduce new entities and avoid additional random walks of the result.
Three modifications of competition were considered:

A scenario with selfish promotion, when the promotion depends only on the employee’s selfishness,

A scenario with idealistic promotion, when promotion depends only on the employee’s productivity at the level of the displaced boss,

Mixed promotion scenario, a more complex promotion model described below.
The scenario with selfish promotion The
competition is that each direct subordinate of the dismissed person has the possibility of participating in the struggle for promotion with some probability. The winner is considered in accordance with the following formula:
$$
Where $$  the vector of all egoisms among subordinates $$ former boss $$ , *  elementwise multiplication, $$ Is a vector of random values from $$ dimensions $$ .
Сценарий с идеалистическим продвижением
The winner is considered in accordance with the following formula:
$$
Where $$ Is the vector of all potential productivity among subordinates $$ for $$ former boss $$ .
Mixed promotion scenario
It was proposed to take the competition model adopted in one of the developed countries. “The pyramid is divided into two halves. Bottom line the correlation with performance. From above, promotion is determined by the product of selfishness and total productivity throughout the chain down. The transition from the lower half to the upper is pure egoism. ”
It can be implemented like this:
prob = []
center = round(L/2)
for D in H.D:
if D.i<center:
prob.append(D.E + D.W)
elif D.i == center:
prob.append(D.E)
else:
prob.append(D.E*(D.P+Col(D)))
s = sum(prob)
r = random.random()
for p in prob:
r = r  p/s
if r<0:
winner = H.D[i]
The Col (D) function is a recursive function that summarizes the performance of all subordinates:
def Col(H):
result = 0
for D in H.D:
result += D.P + Col(D)
return result
What happens during the simulation?
Scenario with a selfish advance
Consider the result of the calculation according to a scenario with a selfish advance, where 60 iterations are “lived” for a population of 10,000 people. The first graph is how performance changes over the years. It is worth noting that a value of 1 would be only if all employees had perfect efficiency at all levels. And the second, this is how the distribution of egoists changes over the years (you can see that initially all egoists are at the level of 20% of the entire population of the level, which is rather optimistic).
Productivity drops more than twice. The number of egoists at the upper levels is growing, exponentially striving to completely fill the upper levels.
Well, the obvious conclusion is that the number of egoists at the upper levels is growing, and productivity is falling, because egoism is not always combined with good performance. Who would doubt that.
Scenario with idealistic promotion.
All the same parameters. 10,000 people, 60 iterations, 20% of egoists at any level in the beginning. The code has changed one line. The criterion for competition for places. Instead of selfishness, the level of potential productivity is now at the level of the displaced boss. In a particular case, it turns out like this.
Since the level of egoism is not correlated in any way with the level of productivity, the redistribution of productive people to higher levels does not affect the egoism of this level. But at the same time, the productivity of all is growing due to the promotion of talented ones.
Mixed Promotion Scenario The
parameters are the same. Just another rule. A complex scheme for separating the assessment of different levels of the hierarchy.
As you can see, in such an organization the number of egoists in the board is also growing, but not to such egregious values as in the selfish scenario. Moreover, the process does not converge to 1 at the highest level. Accordingly, the decline in productivity from the dominance of selfishness is significantly lower. Until 0.06, and not until 0.04, as in the selfish version.
Generalizations and confidence intervals
There may be a number of questions about how exactly these graphs can vary, depending on the different realizations of the random process. Maybe there is no such big difference between egoism and the mixed version? Just two as different implementations as possible.
For this, each of the scenarios was run 20 times with the same parameters. After that, graphs were constructed with confidence intervals for the average of each process. The results on the graph.
A number of interesting features of these dependencies can be noted.
Confidence intervals for a selfish scenario are narrower than in other cases. This is because the process after the second iteration is quite homogeneous. Some significant number of people with random performance characteristics will fall to a higher level than they will provide a decrease in productivity to all their subordinates. This is because, on the one hand, they are selfish enough to break even higher than they were. On the other hand, the graph has a fairly wide branching and coherence of the subordination grid, which will create a situation where egoists are more likely to get into some of the “competitions” to a new place higher in the hierarchy. And with each level, this probability is more and more until the upper level is filled with the most desperate egoists. В этот момент процесс начнет сходиться.
The reason for the wide confidence interval of the mixed variant is due (for the most part) to the fact that for this structure, the rotation of personnel at the highest levels quite quickly “throws off” the egoists who generate inefficient production chains. The man selfishly reached the top level and nothing disturbed him, and then suddenly (after another increase) his decisions made the work of a huge number of subordinates ineffective. And at the next iteration, this person does not win the promotion of a relatively more productive opponent. And when this happens, the improvement occurs rather abruptly, which can be seen on the chart with a special case (however, sometimes the decrease happens the same way). In principle, the scheme itself is not directly ideal, but we see a significant difference with egoism.
conclusions
Uncontrolled promotion of employees can cause significant harm to the enterprise and contribute to degradation and reduced efficiency. As shown above, the algorithm selected by the enterprise for staff rotation significantly affects the work of the entire organization.
In this article, we tried to give readers the opportunity to look and see the problem where no one had possibly noticed it before, and also to formulate it as an applied problem, which, possibly, has a number of adequate solutions.
But it is obvious that an idealistic model with the promotion of only effective employees at a new level is not too realistic. The problems begin already with the question, how to understand  will the employee be effective in the new position? What are his talents in her? Therefore, at this stage, we are not entitled to throw advice on how to optimize the situation. The transfer of model implementations to a specific real structure is always an extremely delicate thing and requires a fair amount of practical checks and evidence of the effectiveness of such a transfer.
However, in the end it will be appropriate to recall that in 2010 the Shnobel Prize was given to researchers from Italy, who, using the Peter principle, proved that the most effective promotion algorithm (if this principle is adequate to reality) would be to promote a random employee selected by a random generator numbers . It’s probably not worth the accident, but it would probably be more useful for everyone to choose a random one from the short list. But it is not exactly.
Автор статьи: Александр Беспалов, Data scientist, Maxilect
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