Quality & Quantity 34: 379–406, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
The Theory of Socially Embedded Games: The
Mathematics of Social Relationships, Rule
Complexes, and Action Modalities
TOM R. BURNS
and ANNA GOMOLI
Uppsala Theory Circle, Department of Sociology, University of Uppsala, Box 821, 75108 Uppsala,
Sweden, e-mail: email@example.com;
Department of Mathematics, University of Białystok,
Akademicka 2, 15267 Białystok, Poland, e-mail: firstname.lastname@example.org
Abstract. 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, elaborate a theory of rules. Such considerations
lead to conceptualizing rules and rule conﬁgurations as mathematical objects, specifying the prin-
ciples for combining rules, developing the theory of revising, replacing, and, in general transforming
rules and rule complexes. The mathematics is based on contemporary developments at the interface
of mathematics, logic, and computer science. This article, drawing on the mathematical theory of
rules and rule complexes, extends and generalizes game theory (GGT). The theory of rule complexes
is used to conceptualize and analyze diverse social relationships, roles, and games as particular types
of rule complexes. A social role, for instance, is the major basis of an individual’s action in a game.
It consists 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
generating action in game settings. The article focuses on three types of action modality: routine
or habitual, normative, and instrumental modalities. The theory: (1) provides a cultural/institutional
basis for a theory of games 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) formulates the theory that actors construct an action
or make choices among alternative actions by making comparisons and judging similarity (or dis-
similarity) 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.
Key words: rule, rule complex, application, role, action modality, value complex, judgement,
consequences, action determination, instrumental, normative, habitual, open game, closed game,
A part of this work was presented at the XXth World Congress of Sociology, Research Commit-
tee on Rational Choice (RC45), Session on New Developments in Rational Actor Theory, Montreal,
Canada, 26 July–1 August, 1998.