The paper discusses the problem of roles distribution among participants of organizational systems (OS). Networking cooperation of unequal economic entities is a basis for modern systems organization with every entity taking responsibilities for results, establishing rules of interaction and decision-making by others. In mathematical studies, network organization principles to the fullest extent possible can be implemented in a game theoretical approach. An introduction of an appropriate classification system is necessary due to multiple variants of various game-theoretic models (GTM) describing roles distribution in OS. Roles distribution in OS with fixed number of participants in the proposed GTM classification system is described with a right of the first move (move order) and / or distribution of variables (controls). The following classes are presented: 1) models with a fixed right of the first move and with separated variables; 2) models with a nonfixed right of the first move and with separated variables; 3) models with a fixed right of the first move and with non-separated variables; 4) models with a non-fixed right of the first move and with non-separated variables. Overviews of basic GTM are presented for each class.