Rule complex

Abstract: In their classic work, Von Neumann and Morgensterndefined a game as simply the totality of the rules which describe it. Theydid not, however, elaborate a theory of rules. Such considerations lead toconceptualizing rules and rule configurations as mathematical objects, specifyingthe principles for combining rules, developing the theory of revising,replacing, and, in general transforming rules and rule complexes. Themathematics is based on contemporary developments at the interface ofmathematics, logic, and computer science. This article, drawing on themathematical theory of rules and rule complexes, extends and generalizes gametheory (GGT). The theory of rule complexes is used to conceptualize andanalyze diverse social relationships, roles, and games as particulartypes of rule complexes. A social role, for instance, is the major basisof an individual's action in a game. It consists of at least four majorcomponents – which are mathematical objects – in the determinationof action: value complex, model of reality (including beliefs and knowledgebases), a repertoire of acts, routines, programs, and strategies, and modalities,role-specific algorithms for determininig or generating action in gamesettings. The article focuses on three types of action modality routineor habitual, normative, andinstrumental modalities. The theory: (1) provides a cultural/institutionalbasis for a theory of gameswhere games, social relationships, and rolesare formalized in terms of rule complexes; (2) explains human action as a formof rule application or rule-following action, which underlies allmodalities of action; (3) formulates the theory that actors construct an action or make choices amongalternative actions by making comparisons and judging similarity (ordissimilarity) between an option or options considered and their norms and values,and, in general, determine whether or not, and to what degree, a value,norm, or goal will be realized or satisfied; (4)reconceptualizes ``game'' as a social form and makes a distinction between open and closed games.

Abstract: In the paper we present the notion of rule complex as a promising tool to formalize social game systems. We also hope to arouse the interest of the computer science community in application of the rough-set and other current computing methods to the social game theory.

Abstract: The aim of this article is to present the key mathematical notions of the theory of socially embedded (or generalized) games (GGT) such as rule, rule application, and rule complex from the standpoint of granularity of information. In GGT, social actors as well as social games and, more generally, interactions, are represented by granules of information called rule complexes. Rule complexes are special cases of complexes of points of some space where the points are rules over a considered language. In many cases, rule complexes may be viewed as granules of information just because the rules constituting the rule complexes are drawn together on the base of similarity and/or functionality. On the other hand, rules represent granules of information. Therefore, rule complexes are multilevel structures representing granules of information as well. In this chapter, we also discuss mereological questions, and, in particular, we define three kinds of “crisp” ingredient-whole relationships with respect to complexes of points (viz., being of a gelement, being of a gsubset, and being of a subcomplex of a complex of points) and present some ideas on how to define the notion of a part (ingredient) to a degree in cases of complexes of points.