The theory of socially embedded games: The mathematics of social relationships, rule complexes, and action modalities

In their classic work, Von Neumann and Morgenstern defined a game as simply the totality of the rules which describe it. They did not, however, elabor ate a theory of rules. Such considerations lead to conceptualizing rules an d rule configurations as mathematical objects, specifying the principles fo r combining rules, developing the theory of revising, replacing, and, in ge neral transforming rules and rule complexes. The mathematics is based on co ntemporary developments at the interface of mathematics, logic, and compute r science. This article, drawing on the mathematical theory of rules and ru le complexes, extends and generalizes game theory (GGT). The theory of rule complexes is used to conceptualize and analyze diverse social relationship s, roles, and games as particular types of rule complexes. A social role, f or instance, is the major basis of an individual's action in a game. It con sists of at least four major components - which are mathematical objects - in the determination of action: value complex, model of reality (including beliefs and knowledge bases), a repertoire of acts, routines, programs, and strategies, and modalities, role-specific algorithms for determininig or g enerating action in game settings. The article focuses on three types of ac tion modality routine or habitual, normative, and instrumental modalities. The theory: (1) provides a cultural/institutional basis for a theory of gam es where games, social relationships, and roles are formalized in terms of rule complexes; (2) explains human action as a form of rule application or rule-following action, which underlies all modalities of action; (3) formul ates the theory that actors construct an action or make choices among alter native actions by making comparisons and judging similarity (or dissimilari ty) 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 so cial form and makes a distinction between open and closed games.