## Leonhard Euler

## Cooperatives, Even In Mathematics!!!

##

Emrullah Yiğit

Mehmet Efe Zengin

A mathematician from RUDN University developed a matrix system representation of set functions which makes the calculations way easier and more vivid. Among other things, the new development can be applied to cooperative game theory. Experts in cooperative theory examine the varying method of complex-decision making in multiple situations. In such a study, the performers or group members are awaited to come up with the most profitable decision regarding their benefits.

Set functions are one of the tools used to work with cooperative game theory. In these functions, the input data are sets of elements that can have different values. Simple explicit questions are quite rare in real life; therefore, the data on different elements can support or neutralize each other. Combinations of elements called coalitions can assume their own values. To work with this apparatus, scientists require an intuitive mathematical language.

Prof. Gleb Beliakov, a Candidate of Physics and Mathematics from RUDN University explained and suggested the new approach by stating:

"Our contribution to the mathematical language of cooperative game theory is based on the familiar notions of matrices and vectors. We have developed a formal approach to manipulations with set functions based on linear algebra. Our results can be practically applied to multicriteria decision analysis, group decision-making, operations with dependent goals, economic theories based on cooperative games, and aggregate functions theory."

The scientist obtained matrix expressions by transforming a derived set function expression. A derived function shows how a function transforms when its variables change. Having calculated a derived function, a specialist can give an accurate analysis of a certain situation. In linear algebra, treating an exponential set this way can simplify calculation methods and support effective implementation of many formulae in software. Prof. Beliakov also suggested new formulae for finding the Shapley vector—a version of 'fair distribution' in which the profit of each player is equal to their average contribution to respective coalitions. The new method makes it easier to obtain the Shapley vector in practical applications.

(Dec. 1, 2020)

References:

PhysOrg, RUDN University, 1 Dec. 2020, staff, Science X. “Mathematician Suggests New Approach to Cooperative Game.” Phys.org, Phys.org, 1 Dec. 2020, phys.org/news/2020-12-mathematician-approach-cooperative-game.html.

staff, Science X. “Mathematician Suggests New Approach to Cooperative Game.” Phys.org, Phys.org, 1 Dec. 2020, phys.org/news/2020-12-mathematician-approach-cooperative-game.html.