Estimation of parameters of games with a hierarchical vector of interests

Many applied problems can be solved using the methods of the game theory. One of the issues studied in the game theory is finding of equilibrium situations, which presuppose a preliminary determination of the values of players' winnings. Among the various games there are games with a hierarchical interest vector. In such games it is assumed that a lot of players are distributed in hierarchically organized groups. Each player enters into several groups and allocates for each group a certain part of his resource, which allows him to receive a certain prize. In this case, the Nash equilibrium situation is a distribution of the resources of all players, in which each player will receive the maximum winnings in the game. The problem of finding the Nash equilibrium in games with a hierarchical vector of interests was solved by Germeyer and Vatel. To use this theorem, it is necessary to define certain conditions and parameters, which include, in particular, distribution of players in hierarchically ordered groups, the evaluation of the importance of groups for players, and the value of gains for players. In the work these problems are solved under the assumption that the distribution of players into groups is carried out on the basis of the coincidence of their goals. At the same time, to assess the importance of groups, the hierarchy analysis method was used, which makes it possible to give quantitative estimates based on qualitative comparisons of the players' goals. For the construction of the hierarchical structure of groups of players, coloured graphs were used, the vertices of which corresponded to the players, the edges reflected the coincidence of the goals of the players, and the colours of the ribs made it possible to distinguish goals. The groups of players in this case corresponded to the maximum single-colour clicks.