Elements of the Mathematical Theory of Human Systems, Part 3:

Mathematics of Cooperation and Confrontation of Human Systems



By Pavel Barseghyan, PhD

Yerevan, Armenia and

Plano, Texas, USA



The structural representation of people’s life and activity in the form of the equations of state has many applications, one of the most important of which is the quantitative description of the interaction and relationships between human systems.

To construct mathematical models of interactions and relationships between human systems, it is necessary to separate only the sub-flows of mutual actions from the common flow of actions of the parties and present these sub-flows by means of the equations of state.

Sub-flows of actions that represent interactions between human systems can easily be divided into two sub-sub-flows or sequences of actions that individually represent cooperation and confrontation between people.

In turn, the sub-flows of actions representing cooperation and confrontation between human systems can be presented in the form of a system of state equations, on the basis of which mathematical models of interaction between human systems are constructed.

These mathematical models can be used to manage cooperation and confrontation between human systems through purposeful changes in the values of the parameters of the parties.

These problems, which relate to mathematical modeling and control of the behavior of human systems, are discussed in the third part of the paper.

Key words: Human systems, mathematical theory, cooperation, confrontation, human interaction, state equations, equilibrium, non-equilibrium, human errors.


Relations between people and their interactions are the core of the functioning of any human system, and their quantitative representation can have many applications in various areas of organizational science.

Even superficial observations of the behavior and activities of human systems show that relations between people can be in very different states of consent or disagreement.

On the other hand, the same observations show that human relations are also the result of their mutual actions, which can create a balance or imbalance in the interaction between people.

This means that the universal mathematical model of human actions [1] can be used to quantify interactions between people as sequences of mutual actions.

The first step in this direction is the creation of mathematical models of balanced relations between people, for the implementation of which it is necessary to find quantitative equivalents for such a balance.

Since the balance in human relations also means consent between them, this circumstance indicates the nature of the quantitative equivalent of this consent.

On the other hand, since people having reached an agreement with each other, thereby guaranteeing each other the receipt of some equivalent benefit, the concretization and refinement of this statement can serve as a basis for describing such an equilibrium between people using mathematical equations.

With a more comprehensive examination of the problem, apart from mutual benefit, mutual losses can also be taken into account, since in addition to mutually beneficial stable relations between people there can be stable poor relations, for example between enemies or competitors.

For example, this is a very common case when different countries pursue a common policy directed against each other, but at the same time support a mutually beneficial economic policy.

This can be extended to different areas of the life and activities of human systems, where they simultaneously pursue a policy of cooperation and a policy of confrontation.

This circumstance makes it possible to use the method of state equations for the quantitative description of relations between people and various human systems.

In order to do that, the equations of state can quantitatively represent the benefits and losses of interconnected human systems that have functional links with the results obtained or created by the parties [1], and may also have the form of mutual positive or negative pressures [2].

Equilibrium and non-equilibrium of life and activity of human systems in light of their state equations

Let us consider the problem of quantitative description of the equilibrium and non-equilibrium states of human systems or associations of people of an arbitrary scale with the help of a parametric representation of their life activities.


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About the Author

Pavel Barseghyan
, PhD

Yerevan, Armenia
Plano, Texas, USA



Dr. Pavel Barseghyan is a consultant in the field of quantitative project management, project data mining and organizational science. Has over 45 years’ experience in academia, the electronics industry, the EDA industry and Project Management Research and tools development. During the period of 1999-2010 he was the Vice President of Research for Numetrics Management Systems. Prior to joining Numetrics, Dr. Barseghyan worked as an R&D manager at Infinite Technology Corp. in Texas. He was also a founder and the president of an EDA start-up company, DAN Technologies, Ltd. that focused on high-level chip design planning and RTL structural floor planning technologies. Before joining ITC, Dr. Barseghyan was head of the Electronic Design and CAD department at the State Engineering University of Armenia, focusing on development of the Theory of Massively Interconnected Systems and its applications to electronic design. During the period of 1975-1990, he was also a member of the University Educational Policy Commission for Electronic Design and CAD Direction in the Higher Education Ministry of the former USSR. Earlier in his career he was a senior researcher in Yerevan Research and Development Institute of Mathematical Machines (Armenia). He is an author of nine monographs and textbooks and more than 100 scientific articles in the area of quantitative project management, mathematical theory of human work, electronic design and EDA methodologies, and tools development. More than 10 Ph.D. degrees have been awarded under his supervision. Dr. Barseghyan holds an MS in Electrical Engineering (1967) and Ph.D. (1972) and Doctor of Technical Sciences (1990) in Computer Engineering from Yerevan Polytechnic Institute (Armenia). 

Pavel’s publications can be found here: http://www.scribd.com/pbarseghyan and here: http://pavelbarseghyan.wordpress.com/.  Pavel can be contacted at [email protected]