Generalized Game Theory in Perspective: Foundations, Developments and Applications for Socio-Economic Decision Models

Abstract

Classical game theory provides powerful tools for modeling strategic interaction, but often overlooks the social, cultural, and institutional dimensions of human behavior. To address this gap, Tom Burns and collaborators developed generalized game theory (GGT) and later sociological game theory (SGT). These frameworks extend classical game theory by embedding rules, norms, values, beliefs, roles, and institutional structures into formal models of interaction. This review synthesizes thirty key contributions to this research program, organizing the literature into eight thematic areas and providing an integrated overview of the field. The originality of this work lies in its comprehensive approach, which advances conceptual and formal foundations while exploring practical applications and outlining directions for future research. GGT/SGT develops rule-based modeling, the analysis of norms and values, multiple modalities of action determination, and various equilibrium types, offering a rigorous framework for understanding strategic behavior in complex social contexts. In application, these approaches provide insights into organizational processes, negotiation, legitimacy, distributive justice, and institutionalized procedures, while integrating interactionist and group-theoretical perspectives. By linking formal modeling with normative and institutional analysis, GGT/SGT offers innovative socio-economic decision models that capture uncertainty, fairness, legitimacy, and institutional transformation. It extends classical game theory by bridging mathematics, economics, and sociology, providing a versatile theoretical tool for understanding complex socio-economic systems and improving strategic decision-making in contemporary society.

1. Introduction

Game theory has become one of the most influential analytical frameworks in the social sciences, mathematics, and economics [1,2,3]. Originating from John von Neumann and Oskar Morgenstern [4], and John Nash [5,6,7,8], classical game theory formalized strategic interaction, equilibrium concepts, and rational choice behavior. However, its underlying assumptions, such as perfect rationality, self-interest, complete information, and fixed game structures, have long been criticized. They fail to capture the social, cultural, and institutional dimensions of real-world human action.

In response to these limitations, Professor Tom R. Burns and his collaborators developed generalized game theory (GGT) [9,10,11] and later called sociological game theory (SGT) [12,13]. These frameworks extend the boundaries of classical models by embedding them within normative and institutional contexts. GGT formalized the role of rules, norms, and belief systems in shaping actors’ choices, while SGT further emphasized the interplay between strategic interaction and the social structures in which it occurs. Together, these approaches not only critique rational choice theory but also offer constructive alternatives, highlighting how legitimacy, distributive justice, and institutionalized procedures influence both individual strategies and collective outcomes.

This paper provides a systematic review of the research program developed by Tom Burns and his collaborators on GGT/SGT. Its purpose is twofold. First, to trace the conceptual development of GGT/SGT, highlighting the theoretical innovations introduced in response to the limitations of classical game theory. Second, to synthesize thirty key contributions in this research, organizing them into thematic clusters to clarify their core achievements and implications. This approach both traces the development of SGT and places its insights in the wider context of socio-economic discussions on rationality, rules, norms, and institutions.

Three dimensions stand out as central contributions of this work:

• Conceptual and formal foundations—the development of rule-based modeling, norms and values, modalities of action determination, and the introduction of different types of equilibria.
• Applications—analyses of organizational processes, negotiation, institutional change, legitimacy, distributive justice, and institutionalized procedures, including integration with group theory and interactionist perspectives.
• Future perspectives—the potential for further formal modeling, simulation, empirical operationalization, and exploration of new contexts such as digital platforms and networked systems.

To present this research systematically, the review surveys thirty studies authored by Burns and his collaborators, which can be grouped into eight interconnected research clusters. These clusters cover the foundations of GGT/SGT, such as socially embedded games, rule complexes, and the mathematical modeling of roles and norms. They also address the systematization of GGT/SGT, consolidating theoretical insights and contrasting them with classical game theory. Other research areas examine decision-making under risk, uncertainty, and multi-value contexts, extending the framework through fuzzy reasoning, multi-agent modeling, and multi-criteria evaluation. Burns and colleagues also analyzed the equilibria, the balance between legitimacy and effectiveness in collective decisions, applied SGT to questions of distributive justice, fairness, and institutionalized decision-making, and critically evaluated the boundaries of rational choice theory. Finally, the reformulation of game theory in SGT and interactionist game theory (IGT) integrates social rules, cultural institutions, and context-sensitive strategic behavior, linking game-theoretic modeling with group theory and broader interactionist perspectives.

The paper is organized as follows: Section 2 provides the historical and conceptual background, highlighting the limitations of classical game theory and the motivations for developing GGT/SGT. Section 3 explains the conceptual foundations of GGT and its expansion to SGT, focusing on normative, cultural, and institutional dimensions. It also discusses the modeling of game structures and processes, including modalities of action determination and equilibria. Section 4 synthesizes the literature on GGT/SGT, organized into thematic clusters. Section 5 demonstrates the application of SGT to the Prisoner’s Dilemma Game. It shows how social roles, norms, and value orientations reframe strategic choices and lead to different forms of equilibrium than those predicted by classical game theory. Section 6 explores the integration of group theory with SGT, examining the interplay between institutional dynamics and strategic behavior. Section 7 presents a model of the societal game, showing how social states, the concept of a social optimum, and legitimizing procedures interact to guide collective decision-making and produce socially accepted outcomes. Finally, Section 8 concludes by summarizing contributions, identifying limitations, and outlining directions for future research.

By tracing this conceptual evolution, the paper demonstrates how GGT/SGT transformed classical game theory into a richer, interdisciplinary framework capable of analyzing rule-governed, socially embedded, and normatively structured interactions. The review summarizes key theoretical contributions, demonstrates their practical relevance, and sets directions for further sociologically grounded game-theoretic research.

2. The Evolution of Game Theory: From Classical Foundations to Sociological and Computational Perspectives

This section presents a brief and selective review of several influential streams in game theory. The overview is not intended to be comprehensive, but rather highlights a series of milestones that have shaped the field and illustrate its diversity of ideas and applications. The examples discussed here reflect a particular emphasis on theoretical innovation and practical relevance, while many other important contributions necessarily remain beyond the scope of this survey.

Game theory, as a formal discipline, emerged in the mid-20th century as a mathematical and economic framework for analyzing strategic interactions among rational decision-makers. Its foundations were laid by John von Neumann and Oskar Morgenstern [4], whose 1944 book, “Theory of Games and Economic Behavior”, introduced the first comprehensive formal model for situations in which each participant’s outcomes depend not only on their own choices but also on the choices of others. In this classical conception, players are assumed to be self-interested, perfectly rational, and capable of logical deduction, with complete information about the rules of the game, available strategies, and payoff structures. The structure of the game itself is considered fixed, closed, and each player’s objective is to maximize expected utility, given the strategies chosen by others.

Within this framework, a game consists of a set of players, rules governing their interactions, strategies representing complete plans of action, and payoffs quantifying the desirability of outcomes. Utility functions are exogenous, typically one-dimensional, and stable over time. The central solution concept is equilibrium, particularly the Nash equilibrium, in which no player can improve their outcome unilaterally. Early milestones, such as von Neumann’s proof of the minimax theorem in 1928, demonstrated that in finite, two-player zero-sum games, optimal strategies exist that minimize potential losses. This result provided a rigorous method for analyzing adversarial scenarios.

Von Neumann and Morgenstern [4] formalized the distinction between cooperative and non-cooperative games, linking abstract mathematical reasoning to real-world economic and political behavior. The next major advancement came with John Nash [5,7], whose equilibrium concept generalized game theory beyond zero-sum contests, making it applicable to oligopoly pricing and environmental agreements. Nash’s [6] bargaining model offered a formal framework for negotiations, while Ariel Rubinstein [14] later extended this framework to incorporate the alternating-offers model, accounting for the dynamic effects of time, patience, and strategic advantage in negotiations.

At the same time, Leonid Hurwicz [15,16] developed mechanism design theory, which shifted attention from analyzing outcomes of given rules to designing rules that yield socially desirable outcomes even under self-interest. Likewise, Lloyd Shapley’s [17] value concept provided a method for fairly allocating cooperative gains, influencing political coalition building, joint ventures, and research funding allocation.

William Vickrey’s [18] developed auction theory, John Harsanyi’s [19,20] Bayesian games, and the modeling of interactions under uncertainty, establishing links with probability and statistical reasoning. Robert Aumann [21,22] introduced correlated equilibrium and repeated games, showing how cooperation can persist over time even among self-interested actors. Thomas Schelling [23,24] contributed to game theory by showing how strategic behavior and coordination can emerge even from simple individual choices, particularly through focal points and dynamic models of conflict and cooperation. John Maynard Smith [25] applied game-theoretic concepts to evolutionary biology, defining evolutionarily stable strategies that cannot be displaced once established in a population. The 1990s also brought computational advancements and refined mechanism design, enabling large-scale applications in telecommunications, market platforms, and auctions. In the 2000s, behavioral game theory incorporated insights from psychology, exploring bounded rationality, fairness, and reciprocity, challenging the assumption of perfect rationality.

Swedberg [26] examined the impact of game theory on sociology, showing how it had broadened the field’s analytical tools, strengthened mathematical approaches, and encouraged dialogue with other disciplines. While he acknowledged frequent criticisms that game theory was too artificial for an empirical science, he argued that it nonetheless possessed significant theoretical value and could at times be empirically applied. He further contended that sociology could develop its own version of game theory, one focused on counterfactuals and alternative strategies, thereby reintroducing choice, subjectivity, and agency into sociological analysis.

Tom Burns and collaborators introduced generalized game theory [10,27,28], shifting focus from purely rational, self-interested actors to players embedded in social, cultural, and institutional contexts. Burns emphasized that real-world interactions occur within systems of norms, laws, and role expectations that shape what strategies are legitimate or conceivable. This integration of social rules into formal models allowed game theory to address organizational behavior, policy compliance, and institutional change in ways classical frameworks could not.

From the 2010s onward, game theory has become deeply embedded in the digital economy and artificial intelligence. It underpins real-time ad auctions, ride-sharing surge pricing, online matching markets for organ donation and school placements, and negotiation protocols between autonomous agents in logistics and robotics. In blockchain systems, game-theoretic mechanisms ensure consensus and security [29,30].

Today, its applications span almost every domain. In economics and markets, it shapes auction design, antitrust policy, and matching markets. In politics and international relations, it informs voting systems, coalition-building, treaty design, and deterrence strategies. In military and security, it guides defense planning, cybersecurity, and counterterrorism resource allocation. In biology and ecology, it models evolutionary dynamics and cooperation among species. In computer science and AI, it coordinates multi-agent systems and resource allocation in networks. Finally, in business and management, it informs supply chain negotiations, contract structures, and competitive strategy.

The transformative impact of game theory is underscored by its recognition in the Nobel Prizes in Economics. John Nash, Reinhard Selten, and John Harsanyi (1994) were honored “for their pioneering analysis of equilibria in the theory of non-cooperative games”. William Vickrey and James Mirrlees (1996) received the prize “for their fundamental contributions to the economic theory of incentives under asymmetric information”. Robert Aumann and Tom Schelling (2005) “for having enhanced our understanding of conflict and cooperation through game-theory analysis”. Leonid Hurwicz, Eric Maskin, and Roger Myerson (2007) “for having laid the foundations of mechanism design theory”. Finally, Lloyd Shapley and Alvin Roth (2012), “for the theory of stable allocations and the practice of market design” (‘All Prizes in Economic Sciences’ [31]).

Taken together, this historical evolution illustrates how game theory has evolved from a purely mathematical tool into a multidisciplinary framework. Its extension into GGT/SGT highlights the importance of context, norms, and institutions in shaping human interaction, providing a bridge between formal modeling and the complexity of social life. This short review illustrates how classical foundations have continually adapted to incorporate behavioral, social, and computational dimensions, making game theory a versatile and enduring analytical instrument.

3. Generalized and Sociological Game Theory: From Formal Rules to Social Contexts

3.1. What Are GGT and SGT? Conceptual Foundations and Social Contexts

Generalized game theory (GGT), developed by Tom Burns and colleagues [10,11,12,28,32], initially emphasized the mathematical formalization of rules and strategies. While Burns’ work provided a systematic approach to modeling rules, it is important to recognize that the conceptual focus on rules resonates with broader theoretical traditions across multiple disciplines, including sociology, economics, cognitive science, and institutional theory. Rules serve both as constraints and enabling structures, shaping behavior, guiding interaction, and supporting coordination in social and organizational contexts. Over time, GGT evolved into sociological game theory (SGT), which integrates these sociological insights and emphasizes interactions embedded in norms, values, and institutional arrangements.

In new institutionalism, rules are treated as constitutive elements of institutions and as mechanisms guiding collective action [33,34,35,36,37]. Institutions cannot be fully explained in terms of individual actors and interests alone, but must also be analyzed through the rule systems that shape, constrain, and enable behavior [33,35]. Linguistic theory similarly treats rules as generative principles that make communication possible, acting both as constraints and resources for creativity [38]. Evolutionary sociology interprets rules as drivers of social adaptation and cultural transformation [37,39], highlighting that social evolution involves the transmission, variation, and transformation of rules across populations and over time. Philosophical perspectives, such as Wittgenstein’s [40], further emphasize that rules are intrinsic to meaning and social life, not merely external prescriptions.

This broader research context demonstrates that GGT and SGT are situated within an interdisciplinary dialogue about how rules structure social and cognitive processes, rather than being an isolated theoretical development centered solely on Burns’ work. Contemporary social science underscores that social rule systems, constituting cultural formations, normative frames, and institutional arrangements, are ubiquitous and partially determinative of social action. Actors introduce, interpret, adhere to, or sometimes selectively enforce rules for cognitive, instrumental, social, aesthetic, and normative reasons, reflecting the interplay between individual judgment and social context.

Recognizing the complexity and centrality of rules provides a crucial foundation for GGT, which formalizes rules and rule complexes [9]. This framework systematically classifies different categories of rules, such as value-oriented, normative, prescriptive, judgmental, and meta-rules. It also enables the study of higher-order social constructs, including roles, routines, models of reality, social relationships, and institutional arrangements. These considerations align with research on judgment and decision-making, particularly the work of Simon [41,42,43] and Tversky and Kahneman [44,45,46], who highlighted cognitive limits, framing effects, and context-dependent valuation. Simon’s [41] concept of bounded rationality emphasizes satisficing as an adaptive response to complexity and uncertainty, while Tversky and Kahneman [44,45,46] show that valuation is influenced by perception, framing, and context, challenging classical utility maximization.

Building on these foundations, SGT extends the framework by embedding judgment and rule complexes into socio-economic and institutional contexts. Behavior is described through rules and complexes of rules, with judgment based on norms and values rather than utility maximization, captured by a value complex reflecting institutions, identity, and social context. Decision-making encompasses multiple modes: instrumental, normative, emotional, habitual, and identity-based.

Rules and complexes of rules go beyond numerical representation, capturing structured, context-dependent patterns of behavior and institutional interactions. An illustrative analogy can be drawn to John Conway’s work on surreal numbers [47], illustrating the structured complexity of rules rather than providing a formal mathematical mapping, where numbers emerge from positions in complex games, encompassing both real and ordinal numbers. It is important to clarify that, unlike traditional mathematical game theory, SGT/GGT does not assign numerical values to emotions or social interactions in a conventional sense. Rules and complexes of rules are structured frameworks for judgment and decision-making, capturing relational, contextual, and institutional dimensions rather than simple numerical quantities. In this sense, the analogy to Conway’s surreal numbers illustrates how complex, structured representations can model general games beyond what real numbers or conventional utility functions allow. This approach captures the richness of social phenomena without reducing them to mere numerical values. Similarly, SGT/GGT’s rule complexes allow rich, dynamic modeling of judgment and decision-making in social contexts, reflecting both structural and contextual complexity. This perspective aligns with Simon’s [41] concept of bounded rationality, acknowledging imperfect information, limited knowledge, unequal power, and myopic foresight characteristic of social actors.

By embedding bounded rationality within social and institutional contexts, GGT sets the stage for SGT [48]. While cognitive constraints operate at the individual level, SGT emphasizes how judgments and decisions are shaped by social roles, normative expectations, and institutional arrangements. Games are thus understood as systems or complexes of rules that may be imprecise, partly inconsistent, or subject to change. Core social science categories such as norms, values, roles, beliefs, relationships, and institutions can be represented in terms of such rule structures [10,28]. Social games involve multiple actors whose strategies are influenced by existing institutional arrangements, including markets, governments, and organizations. SGT further highlights the potential for actors to revise or reconfigure rules, enabling meta-level judgment, innovation, and institutional transformation, dimensions largely absent in classical and behavioral decision theories.

As Stolz [49] has noted, the theory of social games assumes a model of the individual called “homo ludens”, based on six key points. “Homo ludens” speaks and understands language, has basic physical and social needs, recognizes social games and their rules, aligns personal goals with game goals, builds identity through participation and performance, and seeks to satisfy needs efficiently while balancing involvement in different games. Although not always perfectly rational, “homo ludens” generally aims to “play the games well.” This model combines norm-following (“homo sociologicus”) and rationality (“homo oeconomicus”), with flexible preferences shaped by the games being played, which can emphasize norms and social values. Central to the model is the symbolic nature of human action, which cannot always be fully captured or measured numerically, as social reality consists of symbolic games that must be interpreted to understand human behavior.

While the GGT/SGT framework has been developed over the past 20 years, this 2023 study [49] builds on that tradition, providing a contemporary contribution that aligns with and advances these long-standing research concepts. By connecting SGT to this broader interdisciplinary literature, the framework is positioned not only as a formal extension of classical game theory but also as a bridge linking sociology, institutional economics, and cognitive decision research. This situates SGT within a wider theoretical landscape. It demonstrates that these approaches provide robust tools for analyzing cooperation, conflict, and organizational processes in complex social and institutional contexts, while remaining responsive to cultural, normative, and institutional variation.

3.2. Mathematical Foundations: Rules and Rule Complexes

3.2.1. Rules as Formal Objects

In GGT, rules and rule configurations are formalized as abstract mathematical objects [10,27,28,32,50]. This formalization draws from modern mathematics, logic, and computer science. A rule is defined as a structured relation between premises, justifications, and a conclusion.

Definition 1 

([27,32,50]). Let $L$ be a language, and $FOR$ the set all formulae obtained according to some formation rule. Rule $r$ is a triary relation $r\subseteq \mathrm{\wp }\left(\right(FOR){)}^2\times FOR$. such that for any triple $(X,Y,\gamma )\in r$, $card\left(X\right) = card\left(Y\right)<{אּ}_0$. Here $X$ is a set of premises, $Y$ is a set of justifications, $\gamma$ is a conclusion of $r$.

Formally, $(X,Y,\gamma )\in r$ means: If all elements of $X$ hold and all elements of $Y$ may hold, then we conclude $\gamma$. Thus, $r$ is a default rule in the sense of Reiter [51].

Abstractly, this can be expressed as follows [32,50]:

$$r:{\displaystyle \frac{X:Y}{\mathsf{\gamma }}}$$

where $X$ is the set of premises or conditions, $Y$ is a set of justifications (default provisions or exception conditions), and $\mathsf{\gamma }$ is a “conclusion”.

Note that the notion of rule is more general than in traditional concept inference rule because exceptions are allowed.

If $Y\ne \mathrm{\varnothing }$, then $r$ is a default rule [51].

If $Y=\mathrm{\varnothing }$, r is an ordinary if-then rule.

If $X=Y=\mathrm{\varnothing }$, then $r$ is an axiomatic rule that can represent facts.

3.2.2. Rule Complexes

A rule complex generalizes a set of rules, emphasizing interdependencies, hierarchies, and functional relationships among rules [27,32,50]. It may include both individual rules and other rule complexes, enabling representation of multi-layered rule systems. This development was motivated by the need to capture the full repertoire of rules in their inherent complexity, where rules may overlap, interact, or even conflict.

Definition 2 

([27,32,50]). A rule complex is obtained according to the following formation rules:

(1) 

Any finite set of rules is a rule complex.

(2) 

If $C_1\mathit{,}{C}_2$ are rule complexes, then $C_1\cup C_2$ and $\mathit{\wp }\mathit{\left(}{C}_1\mathit{\right)}$ are rule complexes.

(3) 

If $C_1\mathit{\subseteq }{C}_2$ and $C_2$ is a rule complex, then $C_1$is a rule complex.

Thus, the class of rule complexes contains all finite sets of rules, is closed under set-theoretical union and power set, and preserves inclusion.

For any rule complexes $C_1$ and $C_2$, $C_1\cap C_2$ and $C_1-C_2$ are also rule complexes.

Definition 3 

([27,32,50]). A complex $B$ is a subcomplex of the complex $A$ if $B=A$, or $B$ may be obtained from $A$ by deleting some rules from $A$ and/or redundant parentheses.

3.3. GGT Model of Game Structure and Game Process

3.3.1. General Game Structure Model

Given a situation $S_t$ in context $t$ (time, space, social environment), a general game structure is represented as a rule complex $G\left(t\right)$ [10,11,27,28].

The $G\left(t\right)$ complex includes players’ roles, subcomplexes, and other relevant rules. For a group $I=\{1,\dots ,m\}$ of actors involved in a game $G(t$), let $ROLE(i,t,G)$ denote the role complex of actor $i$. $ROLE(I,t)$ denotes the role configuration of all actors in $I$ engaged in $G\left(t\right).$ Each individual role complex is a subcomplex of the collective role configuration, which in turn is a subcomplex of the game:

$$ROLE\left(i,t,G\right)\subseteq g ROLE\left(I,t,G\right)\subseteq gG\left(t\right).$$

The game structure consists of roles and general rules:

$$G\left(t\right) = \left[ROLE\right(1,t), ROLE(2,t),....,ROLE(k,t); R].$$

$R$ includes the rules of the game, general norms, practical rules (e.g., procedures), meta-rules, and rules specifying interpretation and adaptation to contexts.

An actor’s role is defined by four complexes:

• Value complex $VALUE\left(i,t\right)$: rules assigning value to objects, states, actions, and actors, forming preferences or meta-values.
• Model complex $MODEL(i,t)$: beliefs about self, others, and environment, including constraints, causal mechanisms, and possible scenarios. Actors can operate with incomplete, fuzzy, or even incorrect information.
• Action complex $ACT\left(i,t\right)$: available strategies and acts, specifying obligations, routines, programs, permissions, prohibitions, and principles for selecting strategies. Action determination may be instrumental, normative, habitual, ritualistic, or emotional, depending on context.
• Judgment complex $J(i,t)$: organizes decision-making and action selection, rules guiding evaluation of truth, value, and action choice, producing decisions or new rule complexes. Judgment operates on objects such as values, norms, beliefs, strategies, and outcomes to produce decisions, evaluations, or new rule complexes.

These complexes reflect social contexts, institutional arrangements, and relational interdependencies. Individuals emphasize specific dimensions depending on role expectations, routines, or cognitive limitations.

3.3.2. Game Process

The game process involves the application and interpretation of rule complexes [10,11,27,28]. Actors may adapt rules to situational constraints, bring values from other relationships, or deviate from norms intentionally or unintentionally. Conflicts may arise between rules, social pressures, and external constraints, making role behavior partially unpredictable.

The game structure $G\left(t\right)$ should be distinguished from the game process. While the structure defines the rules, the process concerns “playing the game”: gathering information, making judgments, performing roles, innovating, and exercising human agency. Actors continuously interpret, adapt, or deviate from rules, influenced by norms, values, and social roles.

Games can be closed or open. Closed games (classical game theory) with a fixed set of players, predetermined actions, and clearly specified outcomes. Participants primarily maximize their own results. Open games (real-world social systems), where actors can construct, adapt, and expand their repertoires of actions and strategies, and even modify their underlying value systems during interaction [12,28,52]. For example, in negotiation, participants may alter strategies, reconsider goals, or reassess preferences in response to emerging social cues. The structure of the social relationship—cooperative, competitive, or indifferent—shapes strategy formation, interaction patterns, and outcomes.

3.3.3. Action Determination Complexes and Equilibria

Action determination in social games can follow multiple modalities [53,54,55].

• Routine interactions follow habitual patterns, standard operating procedures, or interlocked algorithms, creating predictable process equilibria when conditions permit consistent execution.
• Consequentialist-oriented interactions emphasize outcomes, with actors selecting strategies that maximize payoffs or achieve collective or individual goals (“instrumental rationality”).
• Normativist-oriented interactions prioritize intrinsic qualities of actions, such as fairness, solidarity, or ethical correctness, evaluating behavior in terms of adherence to rules and norms rather than consequences (“duty theory”).
• Emotional, expressive, and symbolic interactions further diversify behavior, with actors often employing hybrid strategies that combine multiple modalities, reflecting the complex, multi-dimensional nature of human agency (“feel good theory,” “dramaturgy”).

Equilibria arise when actions align with rules, values, and social relationships, but they are context-dependent and often unstable. Hybrid modalities are common, reflecting the multi-dimensional and socially embedded nature of human action.

In practical terms, action determination complexes regulate behavior through rules, algorithms, and value judgments, shaping how actors select, adapt, and implement actions within their roles and social contexts [28,53,54]:

• Routine interactions: Predictable sub-algorithms generate process equilibrium, which can be disrupted by unexpected contingencies, conflicting rules, or mistakes.
• Consequentialist-oriented interactions: Strategic adaptation maximizes payoffs or goals, with equilibria emerging when outcomes meet thresholds, though they remain fragile under uncertainty.
• Normativist-oriented interactions: Actions are assessed according to shared norms and expectations; equilibria arise when norms are met, but instability may occur due to misunderstandings or conflicting rules.
• Hybrid modalities: By combining multiple orientations, actors produce adaptive, socially embedded behavior.

Equilibria are often fragile, especially in rivalry-based contexts or when rules are misinterpreted. However, patterns are constrained by rule complexes and institutional frameworks, offering insight into coordination, cooperation, conflict, and institutional dynamics.

Table 1. Classical game theory vs. generalized game theory—conceptual comparison.

Classical Game Theory (CGT)	Generalized Game Theory (GGT)
Game rules are fixed, predefined, and cannot be changed by players during the game.	Game rules are complex, dynamic, and embedded in physical, ecological, and social contexts; players may contribute to shaping or modifying the rules.
Players are universal, rational, utility-maximizing agents without creative or transformative capacities.	Players are diverse actors in different social roles, capable of interpretation, creativity, and transforming the game situation.
Games are treated as symmetrical, with players having similar strategic positions.	Games may be asymmetrical, involving differences in roles, status, power, and resources.
Game structures are static and fixed throughout the interaction.	Game structures may change through players’ innovative actions or external influences.
Games are closed with all parameters defined from the start.	Both open and closed games are possible.
The utility function is given exogenously and is stable; all preferences are negotiable and comparable.	$VALUE(i,t)$ complex: Values derive from social context (institutions, identity, roles). Some values are non-negotiable (“sacred core”).
Information is assumed to be perfect or nearly perfect; reasoning is clear, precise, and logically structured.	$MODEL(i,t$) complex: Cognitive models may be incomplete, fuzzy, or even incorrect; reasoning may not follow classical logic.
A set of strategies is predefined; communication is either disallowed (non-cooperative) or strictly formalized (cooperative).	$ACT(i,t)$ complex: Wide repertoire of actions including routines, strategies, varied communication forms, persuasion, or deception; communication may differ across roles.
Single mode of decision-making: instrumental rationality (maximizing expected utility).	$J(i,t)$ complex: Multiple modes of decision-making (instrumental, normative, emotional, habitual, identity-based).
Rationality is complete and coherent, based on fixed axioms; contradictions do not occur.	Rationality is bounded, context-dependent, and may involve internal contradictions or dilemmas.
The solution is defined as an equilibrium, usually the Nash equilibrium.	Solutions depend on players’ perspectives; disagreement about acceptable outcomes is expected.
Primarily one type: the classical Nash equilibrium.	Multiple types of equilibria: generalized Nash, social, and normative equilibria.

Source: based on [11,56].

presents a conceptual comparison between classical game theory (CGT) and generalized game theory (GGT), highlighting how the latter expands the concept of strategic interaction under fixed rules and rational utility maximization by incorporating social context, value systems, variables, the complex motivations of real actors, and the possibility of transforming game structures.

4. Generalized and Sociological Game Theory: Key Themes and Contributions

To provide a structured and accessible overview, the reviewed literature is organized into eight thematic clusters, corresponding to the main directions of theoretical and applied research. This review covers more than twenty years of research, tracing the evolution from generalized game theory to sociological game theory and interactionist game theory (IGT). The literature is discussed chronologically, highlighting foundational studies, methodological developments, and sociological reformulations that integrate social norms, roles, and institutions into game-theoretic frameworks. This section also synthesizes potential research opportunities emerging from the thematic clusters reviewed above, offering an integrated perspective on future studies in GGT and SGT.

4.1. Theory (GGT): Mathematical and Conceptual Foundations

Foundational studies on generalized game theory (GGT) focused on extending classical game theory by embedding it within social and institutional contexts, emphasizing norms, values, beliefs, and social relationships in shaping human action [10,27,28,32]. Burns and Gomolińska [10,27,32] introduced the formal mathematics of rules and rule complexes to model social roles and action modalities. Social action was conceptualized as rule-governed, with actors choosing actions based on value complexes, beliefs, repertoires of acts, and role-specific algorithms.

Building on this, Burns et al. [28] analyzed socially embedded games (both open and closed) using rule complexes to model social roles as composed of (1) value complex, (2) model of reality, (3) repertoire of acts, and (4) role-specific action algorithms. Applications included market bargaining games and the Prisoners’ Dilemma, illustrating how social embeddedness influences interaction patterns, outcomes, and social equilibria. In parallel research [9], belief revision in multi-agent systems, showing how trust, status, commitment, and collective sanctions affect how agents integrate or reject information, is examined. Later, Gomolińska [57] discusses the problem of deriving rules from rule complexes. It explores rule derivability, relative derivability, rule activation, and the (in)consistency of rule complexes. In another paper [50], this perspective is extended mathematically via granular computing, modeling rules and rule complexes as multilevel structures and exploring mereological relationships.

Furthermore, a previous study [58] describes the concept of rough rule-following, which enables modeling human behavior where individuals follow social norms imperfectly, adapting them flexibly to context. It highlights that this approach helps maintain social order without strict enforcement and highlights its importance for understanding how societies function.

Together, these studies laid the theoretical and mathematical foundations of GGT, creating a robust framework for analyzing social interactions as rule-governed processes and paving the way for research on decision-making under uncertainty and the integration of legitimacy and social context into collective action.

4.2. Systematization of GGT and Comparison with Classical Game Theory

The systematic consolidation of GGT was undertaken in some studies by Burns and Roszkowska [11,56], highlighting the contrasts between classical game theory and socially embedded approaches. Burns and Roszkowska [11] laid out the foundational GGT framework, representing games, norms, values, beliefs, roles, social relationships, and institutions through rules and rule complexes. This study emphasized the reconceptualization of games as socially embedded, introduced the idea of normative equilibria, and distinguished between open and closed games. It also highlighted the perspective-dependence of solutions and the interplay between bounded factual knowledge and extensive social competence. Lately [56], this framework has been extended to multi-value decision-making and the social–psychological complexity of interactions. The paper formalized the role of judgment in action determination and explored the concepts of instrumental, normative, and social equilibria. By integrating social context, norms, and values into the decision-making framework, a richer understanding of rationality compared to classical models is demonstrated.

Together, these studies provide a systematic comparison of GGT with classical game theory, illustrating how social embeddedness, context-dependent rationality, and multi-dimensional decision-making shape outcomes.

4.3. Decision-Making Under Risk, Uncertainty, and Multi-Value Contexts

Decision-making under risk, uncertainty, and multiple values has been a central theme in GGT and SGT. Subsequent studies progressively deepened the application of GGT to complex social and decision-making contexts. Burns et al. [55] focused on multi-agent modeling, demonstrating how social regulation and institutional mechanisms can be formalized as “social algorithms” that guide collective behavior. They highlighted the interplay between actors’ judgment, social norms, values, and roles, showing how these elements interact in multi-agent simulations to produce stable social outcomes. Their study [55] also presents a simulation model of a population of agents interacting locally in neighborhoods, choosing whether to comply with a social norm or contribute to a collective good. Agents differ in their commitment to norms, cognitive perception of neighbors’ behavior, and communication capabilities, and their decisions are influenced by local patterns and feedback over time. Case 1 results show that compliance rates depend on the initial proportion of committed “good citizens” and their spatial distribution, with clustered minorities significantly affecting outcomes. Case 2, which includes dynamic feedback on commitment, indicates that compliance levels are largely determined by the proportion of agent types, with relative commitment or communication rates having minimal effect.

Building on this, the study [56] extended the framework to incorporate multi-value decision-making, emphasizing the social and psychological complexity of actors’ choices. They modeled how bounded factual knowledge interacts with social and cultural competence, illustrating that real-world decisions are shaped not only by instrumental rationality but also by normative and social considerations.

Further work connected GGT with broader theories of judgment and choice. The study [48] highlighted the context-dependent nature of decision-making, showing that choices cannot be fully understood without considering the social and institutional environment in which they occur. It linked GGT to the work of Simon and Kahneman-Tversky, showing how bounded rationality, heuristics, and social context interact. The authors propose different types of models: deterministic models, partially ordered multi-criteria models, and non-quantified value-based models, demonstrating the diversity of human judgment processes and the scope of GGT.

Some studies [59,60] address the limitations of classical game theory in explaining negotiations and propose the concept of a “negotiation generalized game,” based on rule complexes. Negotiations are modeled as a decision-making process on two interdependent levels: interests and relationships between parties. The framework introduces four key components: $MODEL, ACT, VALUE$, and $JUDGMENT$, which together allow for the inclusion of psychological, social, cultural, and behavioral factors. Rules, which may be imprecise or modifiable, describe the situational context, communication, strategies, and decision-making methods. This representation enables the analysis of both distributive and integrative negotiations, negotiation styles (hard, soft, principled), verbal and non-verbal communication, and tactics, offering a more realistic and flexible approach that can also support computational analysis and decision-making.

In another study, Roszkowska and Burns [61] applied theoretical insights to practical decision-making under uncertainty, linking GGT/SGT constructs such as belief structures, value complexes, and action repertoires to multi-criteria evaluation methods like TOPSIS. This study illustrated how the theory can systematically support applied decision-making by evaluating alternatives based on proximity to ideal outcomes and distance from worst-case scenarios.

Collectively, these studies extend classical game theory by integrating fuzzy reasoning, multi-value criteria, social norms, and psychological complexity, demonstrating that both theoretical rigor and practical applicability are enhanced when social and normative dimensions are explicitly modeled.

4.4. Social and Economic Equilibria from the Perspective of GGT/SGT

The study of social and economic equilibria benefits from the GGT framework. Unlike traditional models that focus solely on payoff structures, GGT emphasizes players’ value systems, social relationships, and evaluative judgments, allowing for a more nuanced understanding of equilibrium formation in complex social contexts. A key innovation in GGT is the concept of fuzzy judgment, which enables players to make decisions under imprecise information and account for both individual preferences and shared norms.

A previous study [53], extending the general theory of games, introduces fuzzy judgment, decision-making, and game equilibria. It models how players use approximate reasoning and handle imprecise information in their decisions, interactions, and equilibrium formation. Fuzzy judgment is conceptualized as a two-step process: assessing similarity and dissimilarity through threshold functions, and evaluating fit or degree of membership as a fuzzy set ranging from 0 to 1. The study develops a theory of equilibria, formulating a generalized Nash equilibrium in fuzzy terms and distinguishing normative equilibria that satisfy shared norms or values from non-normative ones. This fuzzified GGT applies to both classical closed games, exemplified by the Prisoners’ Dilemma, and open games, illustrated with bilateral bargaining, showing how social embeddedness shapes players’ judgments, interaction patterns, and resulting equilibria. Overall, the work demonstrates how fuzzy reasoning enhances the analysis of strategic interaction under uncertainty and social complexity.

Expanding the 2004 work, another study [54] introduces generalized Nash equilibrium and normative social equilibrium concepts introduces the concepts of generalized Nash equilibrium and normative social equilibrium. The GGT framework is applied to four classical games, Coordination, Prisoners’ Dilemma, Chicken, and Battle of the Sexes, to illustrate how outcomes depend on social relationships. Interaction patterns and equilibria differ across relationship types such as solidarity, hierarchy, rivalry, or hostility, shaping whether players collaborate or defect. The analysis shows that social context, role expectations, and meta-rules influence outcomes beyond the nominal payoff structures of each game.

The authors of [56] emphasize and discuss the importance of the reconceptualization of game equilibrium, distinguishing between classical Nash equilibria and normative equilibria, where outcomes satisfy shared norms and values. By highlighting players’ evaluative judgments and collective commitments, this study shows how normative equilibria provide socially recognized, predictable focal points that support planning, strategy, and institutional order, unlike purely pragmatic or imposed solutions.

In another paper, Roszkowska and Burns [52] present the GGT approach to 2-person fuzzy bargaining games. Key concepts include players’ value structures, comprising ideal points and limits of acceptance, and fuzzy judgment functions that allow decision-making under imprecise information. These functions incorporate economic, socio-psychological, cultural, and institutional factors influencing the bargaining process. The model enables analysis of potential agreements, participants’ satisfaction, and whether the resulting “price” constitutes an equilibrium. The study further illustrates how six negotiation scenarios—ranging from no possible settlement to ideal agreements for both players—highlight the impact of social relationships on bargaining outcomes. Solidary, hierarchical, rival, or hostile relations shape transaction costs, patterns of satisfaction, and the stability of agreements. The model enables analysis of potential agreements, participants’ satisfaction, and whether the resulting “price” constitutes an equilibrium. Results show that asymmetries in expectations or satisfaction often create instability, while social norms and relational context guide adjustment processes and influence the likelihood of reaching mutually acceptable settlements. Overall, this work demonstrates that GGT provides a robust framework for formally analyzing negotiation processes while integrating relational, normative, and institutional dimensions.

Together, these studies illustrate how GGT extends classical game theory by embedding strategic interactions in social and normative contexts. Fuzzy judgment functions and generalized equilibria allow players to incorporate imprecise information, social norms, and institutional rules into decision-making. Normative equilibria emerge as socially recognized focal points that align individual strategies with collective values, providing a framework for analyzing social and economic outcomes.

4.5. Legitimacy Versus Effectiveness in Collective Decisions

A central line of research in generalized game theory has examined the tension between legitimacy and effectiveness in collective decision-making. The Pareto problematique has served as a key case for demonstrating how socially embedded procedures can resolve conflicts and achieve socially accepted outcomes, even when Pareto-optimality is unattainable.

A previous study [62] critiqued the traditional Pareto framework, highlighting its theoretical and empirical limitations in multi-value contexts where actors hold divergent and often contradictory values. Drawing on GGT, he proposed conceptualizing institutionalized decision procedures such as voting, negotiation, and administrative decision-making as “social algorithms” capable of generating social equilibria that are legitimate and stable within certain conditions.

An accompanying paper [63] further developed this approach by modeling these institutionalized procedures as regulatory meta-games. They showed how mechanisms such as adjudication, negotiation, and democratic decision-making can overcome stalemates, non-optimal outcomes, and collective action dilemmas. Importantly, this work also identified the limitations of such procedures, stressing that their legitimacy and efficiency depend on context-specific conditions.

Building on these contributions, the study [64] applied GGT to the Pareto multi-agent problem, modeling institutional mechanisms, adjudication, negotiation, and voting as tools for achieving socially accepted outcomes. This work emphasized that legitimacy can guide collective choice even when optimal solutions are unattainable.

A later study [65] examined multi-value governance, demonstrating that legitimate decisions accepted by society or key stakeholders do not always ensure effective outcomes. Authors analyzed how institutionalized governance mechanisms integrate technical, legal, and procedural knowledge to reconcile legitimacy with functionality, achieving societal equilibria.

Burns and Roszkowska [66] further elaborated on collective decision legitimation, highlighting that socially accepted choices may not be optimal in the classical sense. They identified governance structures and procedures that combine scientific, legal, technical, organizational, and cultural knowledge to support both effective and legitimate collective decision-making.

Together, these studies illustrate that achieving socially stable outcomes requires not only technical effectiveness but also adherence to institutionalized rules and legitimacy. GGT provides a comprehensive framework for understanding this interplay, showing that procedures which are socially accepted, even if not optimal in the classical sense, can nonetheless secure durable and efficient collective outcomes. They showed that even legitimized collective choices may not be optimal, highlighting the need for multi-governance systems integrating scientific, legal, and practical knowledge.

4.6. Distributive Justice and Social Order

A key line of research in SGT examines the interplay between distributive justice, legitimacy, and social order. Burns et al. [67] developed a theoretical framework for distributive justice grounded in SGT. They explored principles such as equality, performance-based differentiation, status, authority, and need, formalizing judgment procedures and algorithms for different social organizations. This work situated distributive decisions within broader moral and sociological contexts, drawing on Rawls, Elster, and the Warsaw School of Fair Division, providing systematic tools to compare distributive judgments across social settings.

Another study [68] examines group normative procedures and distributional rules that shape normative equilibria, the foundation of social order in groups and communities. A group is understood as an organizational structure with some division of roles, shared goals, a normative system, and recurring interaction patterns. The authors highlight three key mechanisms: (1) legitimation procedures for resolving conflicts and making collective decisions; (2) patterns of fair distribution aligned with principles of distributive justice; and (3) normative equilibria, meaning stable forms of interaction or decision-making that persist because they fulfill essential group norms.

The authors argue that combining legitimizing procedures with principles of distributive justice increases the chances of achieving normative equilibrium and social stability. If procedures are applied without considering justice, outcomes are likely to be challenged and lead to instability. Without legitimizing procedures, equilibrium is also unattainable. Such procedures can be combined with distributive justice in two ways: (1) in self-organizing groups, where members collectively establish and apply rules and distribution principles, or (2) in adjudication, where a leader, judge, or administrator applies relevant justice norms. The study highlighted how legitimizing procedures, just outcomes, and normative interaction patterns collectively contribute to stable collective decision-making. Institutional mechanisms such as voting and negotiation were emphasized as essential tools for conferring normative force to outcomes. The research also addressed the limits of legitimacy when core principles of distributive justice are violated, illustrating that fairness is a necessary, though not always sufficient, condition for social stability.

Together, these studies demonstrate that SGT integrates game-theoretic modeling with sociological theory. Both the structure of social institutions and shared understandings of legitimacy are critical for achieving fair and stable outcomes, showing that distributive justice is deeply intertwined with social order.

4.7. Rational Choice Theory and Its Limits

A previous study [69] critically evaluates rational choice theory (RCT), highlighting its limitations in explaining behaviors embedded in social contexts. The study emphasizes the role of bounded rationality, heuristics, and social norms, arguing that standard RCT models are insufficient for capturing the complexity of human decision-making. In particular, classical models often ignore social rules, institutional frameworks, and the legitimacy of procedures, focusing narrowly on instrumental rationality.

This critique provides the rationale for the development of sociologically informed game-theoretic models, which integrate social rules, norms, and institutional contexts into decision-making frameworks. By accounting for social structures, legitimacy, and norms, these approaches extend classical game theory toward a more realistic understanding of strategic interaction. Extending critiques of classical rationalist frameworks, research could explore how bounded rationality, heuristics, and social norms can be systematically integrated into GGT/SGT models. Empirical studies, experiments, and simulations could provide insights into the interplay between instrumental and normative considerations in complex decision environments.

4.8. Group Theory and Sociological Approaches to Game Theory

The evolution toward SGT and IGT represents the integration of social rules, norms, roles, and institutional arrangements into game-theoretic models.

Sociological game theory (SGT) and interactionist game theory (IGT) offer frameworks that explicitly incorporate social and institutional factors. Burns et al. [70] linked social group theory with SGT/IGT through rule system theory, introducing a cultural-institutional perspective for conceptualizing groups and their games. The study showed how distinct group subcultures, ranging from military units to business entities, shape “rules of the game” and interaction patterns, illustrating systematic differences in game dynamics across social contexts.

In a companion study [12], the authors contrasted SGT and Goffman’s IGT with classical game theory, highlighting the central role of social structures, norms, and institutions in cooperation, conflict, and negotiation. Further work by Burns et al. [13] develops sociologically grounded game theory by introducing SGT and IGT as alternatives to classical models. They argue that rule-following, values, and institutional arrangements are central to human action, not just rational calculation. The study provides detailed distinctions between classical and sociological theories, including rule complexes and modalities of action determination.

Finally, the authors of [71] synthesized these developments, offering a comparative perspective on SGT, IGT, and classical game theory. They emphasized the applicability of sociologically grounded frameworks in analyzing social games, interaction processes, and institutional dynamics, systematically demonstrating how social context, norms, and rules mediate behavior and outcomes in ways classical game theory cannot capture.

Collectively, these studies illustrate the evolution from classical rationalist models to sociologically informed game-theoretic frameworks. SGT and IGT integrate social rules, norms, and institutional arrangements into the analysis of strategic behavior, providing nuanced tools to study cooperation, conflict, and negotiation while maintaining strong connections to prior GGT research.

4.9. Summary and Future Research Directions

In summary, these works show that social norms and rule-following often shape behavior more than pure strategy, and that SGT and IGT offer nuanced tools for analyzing cooperation, conflict, and institutions, building on GGT.

Table 2. Key Clusters of research in generalized/sociological game theory.

Schematic Cluster	Key Studies	Main Contributions
Theory (GGT): Mathematical and Conceptual Foundations	[10,27,28,32,50,57,58]	Introduced socially embedded games, rules, and rule complexes, social roles, value systems, and action algorithms; formal mathematical foundations of GGT.
Systematization of GGT and Comparison with Classical Game Theory	[11,56]	Consolidated GGT concepts, compared with classical game theory, highlighted context-dependent rationality, multi-value decision-making, and normative equilibria.
Decision-Making under Risk, Uncertainty, and Multi-Value Contexts	[48,53,55,56,59,60,61]	Extended GGT to fuzzy reasoning, multi-criteria evaluation, social norms, and psychological complexity; integrated practical methods like TOPSIS for decision-making under uncertainty.
Social and Economic Equilibria from the Perspective of GGT/SGT	[52,53,54]	Integrated GGT with fuzzy judgment and social norms, introducing generalized and normative equilibria that account for values, relationships, and imprecise information in both classical games and bargaining contexts.
Legitimacy versus Effectiveness in Collective Decisions	[62,63,64,65,66]	Critiqued the Pareto framework and modeled institutionalized procedures (voting, negotiation, adjudication, administration) as social algorithms and regulatory meta-games; analyzed how legitimacy and functionality interact in collective choice, showing that socially accepted but non-optimal solutions can achieve stability and efficiency.
Distributive Justice and Social Order	[67,68]	Analyzed fairness, normative equilibria, and institutional mechanisms in maintaining social stability; formalized procedures for distributive justice within SGT.
Rational Choice Theory and Its Limits	[52,69]	Critically evaluated RCT, highlighting limitations in explaining socially embedded behavior; emphasized bounded rationality, heuristics, and norms; motivated sociologically informed game-theoretic frameworks.
Group Theory and Sociological Approaches to Game Theory	[12,13,70,71]	Developed SGT and IGT integrating social rules, norms, roles, and institutional arrangements; demonstrated how social context shapes cooperation, conflict, and negotiation.

summarizes key studies across eight thematic clusters.

The review of the presented research highlights a broad theoretical foundation and a wide range of applied analyses. However, several promising avenues for future research emerge across the thematic clusters. These directions concern the formal development of rule-based systems, empirical validation of equilibrium concepts, integration with computational simulation, and cross-cultural studies of institutional and distributive processes.

First, at the theoretical level, future research may further examine the formal properties of rules and rule complexes, including structural relations, derivability, and consistency. Investigating how these properties influence the stability and adaptability of social interactions would strengthen the mathematical foundations of SGT and facilitate its integration with computational models. Such analysis may also support the development of algorithms for deriving or modifying rule structures in dynamic social environments.

Second, comparative research is needed to assess how context-dependent rationality and normative equilibria differ from classical game-theoretic outcomes across institutional settings. Systematic comparison between GGT/SGT and classical models in multi-value decision contexts would clarify when socially embedded frameworks better predict behavior and when classical assumptions still suffice. This may include controlled experiments, case studies, and model-based scenario testing.

Third, a key direction concerns decision-making under uncertainty. Future work could integrate multi-criteria evaluation, fuzzy reasoning, and judgment models with agent-based and cognitive simulations. This would make it possible to analyze how bounded knowledge, social competence, and institutional constraints jointly influence decision outcomes. Such hybrid modeling approaches would also support the practical use of GGT/SGT in negotiation, governance, and organizational contexts.

Fourth, further work is required to empirically test generalized and normative equilibria in real-world social and economic environments. This may involve applying the GGT framework to bargaining situations, multi-stakeholder negotiations, and decentralized governance systems. AI simulations and large-scale behavioral data can examine how actors follow norms, rules, and institutional constraints, shaping collective outcomes. Dynamic models incorporating learning, evolving norms, and network structures would provide a more realistic representation of institutional change and social adaptation.

Fifth, research exploring the relationship between legitimacy and effectiveness in collective decisions could benefit from comparative and longitudinal studies. Such analyses would clarify when institutional procedures generate socially accepted outcomes and under what conditions legitimacy may conflict with functional efficiency. This direction is particularly relevant in multi-governance systems, public policy, and conflict resolution settings.

Sixth, the understanding of distributive justice within SGT would benefit from cross-cultural and multi-level studies. These could examine how norms of fairness, status, authority, and equality interact in diverse social environments, and how such norms shape social stability. Linking formal models with empirical data on distribution practices could significantly increase the explanatory power of SGT.

Seventh, there is considerable potential for integrating bounded rationality, heuristics, and rule-following behavior into formal GGT/SGT models. This may involve drawing on cognitive psychology, behavioral economics, and sociology to better capture how actors combine instrumental and normative reasoning in practice.

Finally, SGT and IGT offer strong foundations for analyzing multi-group and cross-institutional interactions. Future studies could use agent-based modeling, network simulation, and field research to investigate how subcultures, role expectations, and institutional arrangements shape cooperation and conflict across organizations and societies.

Taken together, these directions point toward a research agenda that is interdisciplinary, methodologically diverse, and attentive to both theoretical rigor and empirical relevance. Advancing SGT along these lines may deepen our understanding of how norms, institutions, and rule systems shape strategic action and collective outcomes in complex socio-economic environments.

5. Applications of SGT to the Prisoner’s Dilemma Game: Judgment, Interaction Patterns, and Social Equilibria

The central point in the sociological game approach is that actors do not enter games as abstract, self-contained utility maximizers, but as individuals embedded in social relationships that shape their values, expectations, and judgment patterns. Games in social life are shaped not only by strategic incentives but also by the relationships between players. These relationships influence how actors evaluate actions, make judgments, and select strategies. Common types of social relationships include rational egoists, solidary relationships, hierarchical (authority-based) relationships, competitive (rivalries), and adversary relationships. Whether individuals cooperate or defect in a game depends not only on payoffs but also on the norms and value orientations associated with their relationship. Each type produces distinct patterns of behavior and different kinds of equilibria in game situations.

In this section, we draw on [11,28,48] to show how the Prisoner’s Dilemma (PD) Game has been applied to the analysis of different social roles. The PD framework offers a clear way to illustrate how social relationships shape decision-making.

Table 3. Outcome matrix for a 2-actor Prisoner’s Dilemma.

Actor 1/Actor 2	Cooperate (C)	Not Cooperate (−C)
Cooperate ($C$)	5, 5	−10, 10
Not Cooperate ($-C$)	10, −10	−5, −5

shows a standard 2-actor PD payoff matrix.

In Table 3, the first number represents Actor 1’s payoff, and the second represents Actor 2’s payoff. Mutual cooperation $(C,C)$ produces moderate positive payoffs for both. If one defects while the other cooperates, the defector receives the highest payoff (10) while the cooperator suffers a large loss (−10). Mutual defection $(-C,-C)$ produces negative payoffs for both (−5). In classical game theory, the dominant strategy for each actor is to choose $-C$, resulting in the suboptimal Nash equilibrium $(-C,-C).$ However, sociological game theory shows that value orientations significantly alter this judgment. Actions can be judged cognitively by considering the potential consequences for both actors and the broader social context. These judgments depend on value orientations, norms, and expectations embedded in the relationship.

Rational egoists act according to instrumental self-interest without concern for relational or normative expectations. They focus purely on personal payoffs, where the judgments on options in time $t$ are ordered as follows:

$$J(1,t)(-C,C) > J(1,t)(C,C) >J(1,t)(-C,-C) >J(1,t)(C,-C)$$

$$J(2,t)(C,-C)>J(2,t)(C,C)>J(2,t)(-C,-C)>J(2,t)(-C,C)$$

In the PD game, mutual defection $(-C,-C)$ is the most rational choice, forming a situational equilibrium. Although stable, it is suboptimal for both actors, and rational egoists may attempt coordination mechanisms to achieve a better outcome.

In a solidary relationship, actors share a normative expectation to cooperate and place a high value on mutually satisfying outcomes. Shared norms emphasize joint gains, reciprocity, and relational stability. Actors evaluate possible actions not only in terms of immediate payoffs but also based on the relational consequences for their partner. In the PD game, actors rank cooperation $(C,C)$ highest as the most appropriate and meaningful, because it satisfies both material and relational objectives. Asymmetric outcomes $(-C,C)$ or $(C,-C)$ violate norms of fairness and are less desirable. Using a cognitive judgment framework, the options are ordered as follows:

$$J(1,t)(C,C) > J(1,t)(C,-C) = J(1,t)(-C,-C) = J(1,t)(-C,C)$$

$$J(2,t)(C,C)>J(2,t)(C,-C)=J(2,t)(-C,-C)=J(2,t)(-C,C)$$

The mutual expectation of cooperation, not merely the outcome, creates a normative equilibrium, where actions align with social expectations and reinforce relational trust.

Hierarchical relationships involve asymmetrical norms of authority. The superior has the right to lead, make decisions, and receive larger benefits, while subordinates are expected to comply and defer. Each expects asymmetry in the interaction process and the outcomes. In the PD game, this translates into asymmetric outcomes $(-C,C)$, where the superior defects and the subordinate cooperates. The judgments can be expressed as follows:

$$J(1,t)(-C,C) > J(1,t)(-C,-C) = J(1,t)(C,C) > J(1,t)(C,-C)$$

$$J(2,t)(-C,C)>J(2,t)(-C,-C)=J(2,t)(C,C)>J(2,t)(C,-C)$$

This satisfies the normative expectation of unequal outcomes, maintains the principle of asymmetric distributive justice, and creates a normative equilibrium. Hierarchical interactions are stable because the roles and norms are clearly understood and enforced.

Actors in rivalrous relationships prioritize outperforming the other, seeking outcomes where the difference between self and other is maximized. Both actors prefer asymmetric outcomes that favor themselves. They would rank order the options as follows:

$$J(1,t)(-C,C) > J(1,t)(-C,-C) = J(1,t)(C,C) > J(1,t)(C,-C)$$

$$J(2,t)(C,-C)>J(2,t)(-C,-C)=J(2,t)(C,C)>J(2,t)(-C,C)$$

In the PD game, since both attempt to dominate the outcome, the likely result is mutual non-cooperation $(-C,-C).$ The equilibrium is unstable because it fails to satisfy either party’s value orientation for superiority. Rival actors may attempt to exploit structural asymmetries or design strategies to improve relative outcomes, leading to possible changes in game structure and outcomes.

In adversary (antagonistic) relationships, players value causing harm or disadvantage to the other, even at some cost to themselves. Cooperation is rejected because it conflicts with the intention to inflict loss. The PD interaction leads both actors to defect $(-C,-C)$, because this action maximizes harm to the opponent according to their value orientation. Unlike rivalry, the goal is not relative advantage but the infliction of loss, making the mutual defection outcome a stable equilibrium based on shared adversarial values. The relationship is defined by hostility, resentment, or ongoing conflict. Reducing the other’s welfare is itself meaningful and rewarding. They would rank order the options as follows:

$$J(1,t)(-C,C) > J(1,t)(-C,-C) > J(1,t)(C,C) = J(1,t)(C,-C)$$

$$J(2,t)(C,-C)>J(2,t)(-C,-C)>J(2,t)(C,C)=J(2,t)(-C,C)$$

In the PD game, mutual defection $(-C,-C)$ aligns with these objectives. This outcome is stable as a value equilibrium, since it fulfills the mutual desire to cause harm and satisfies the shared adversarial goals of both actors. Cooperation is normatively rejected.

Table 4. Patterns of interaction and equilibria in PD games.

Type of Social Relationship	Value Orientation and Norms	Expected PD Outcome	Type of Equilibrium
Rational Egoists	Instrumental self-interest, no shared norms.	$(-C,-C)$	Situational equilibrium: stable but suboptimal; coordination may improve outcomes.
Solidary	Cooperation, equality, joint benefit, trust, and self-sacrifice.	$(C,C)$	Normative equilibrium: stable, reinforces relational bonds
Hierarchy	Authority-based, asymmetric norms; the superior leads, subordinate defers; asymmetric distributive justice applies.	$(-C,C)$	Normative equilibrium: stable, aligned with social norms.
Competitive	Maximize relative advantage; prefer asymmetric outcomes.	$(-C,-C)$	Situational equilibrium: unstable but suboptimal; may lead to strategic adjustments.
Adversary	Maximize harm to the other; hostility-driven.	$(-C,-C)$	Value-based (Harm-oriented) equilibrium: stable, aligned with adversarial goals.

Source: Own elaboration based on [11,48].

presents a summary of types of social relationships, their corresponding value orientations and norms, the expected outcome in the Prisoner’s Dilemma, and the resulting type of equilibrium.

In some studies [11,48], researchers also noticed that the same logic applies beyond the PD. Solidary actors in zero-sum settings attempt to minimize joint losses; rivals attempt to maximize advantage; adversaries may accept mutually destructive outcomes; and hierarchical actors expect stable asymmetry. Different relationships, therefore, produce different “games,” even when the formal payoff structure remains unchanged. The core insight is that social relationships shape game outcomes. Value orientations, norms, and role expectations structure how actors evaluate actions and interpret outcomes.

Finally, games are dynamic social processes shaped by shared meaning, identity, and relationships. Cooperation and conflict arise from the social context, not just payoffs. In open games, actors use values and norms to guide actions: solidary actors foster normative equilibria, while competitive actors create dynamic equilibria. Mediators or institutions can transform adversarial interactions into partial cooperation, establishing new relational equilibria.

6. Linking Group Theory and Sociological Approaches to Game Theory

6.1. Linking Group Theory to Sociological Game Theory

A previous study [70] proposes integrating group theory with SGT by analyzing groups as “group rule configurations”—structured sets of rules that shape behavior and outcomes. These configurations cover ten universal categories, including membership, identity, roles, norms, shared values, social organization, governance, activities, technologies, and temporal–spatial conditions. By formalizing these rule systems, the framework allows for systematic comparison of groups and offers a cognitive-normative basis for understanding and guiding collective behavior.

In the proposed model of group and organizational rule regimes, ten categories of rules are distinguished (see

Table 5. Key types of rule categories specifying group conditions, structures, and processes.

Rule Types	Definition
Type I	Identity rules—“Who are we?” “What symbolizes or defines us?”
Type II	Membership, Involvement, and Recruitment Rules—“Who belongs, who doesn’t?” “What characterizes members?” “How are they recruited?
Type III	Rules concerning shared value orientations and ideals—“What does the group consider good and bad?”
Type IV	Rules concerning shared beliefs and models—“What do we know and believe about ourselves, our group behavior, and our environment”.
Type V	Social relational, group structuring, and governance rules. “How do we relate to one another; what is our social structure?” “What are the authority and status differences characterizing the group?” “How do we interact and reciprocate with one another and with the leadership?” “What are the rules of internal governance and regulation?”
Type VI	Rules for dealing with environmental factors and agents (“external governance”). “How do we cope with, make gains in the environment, dominate, or avoid environmental threats?”
Type VII	Group production and activity rules. “What are our characteristic activities, practices, production programs, ceremonies, and rituals?” “How do we coordinate activities and make collective decisions?”
Type VIII	Rules and procedures for changing the rule regime, or for changing core group conditions and mechanisms. “How do we (or should we) go about changing group structures and processes, our goals, or our practices?”
Type IX	Technology and resource rules. “What are appropriate technologies and materials we should use in our activities (and possibly those that are excluded)?”
Type X	Time and Place Rules—“What are our appropriate places and times?”

Source: see Burns et al. [70].

). These rules address conditions of group agency, social structure, interaction, material resources, and the dimensions of time and space. Specifically, (A) four categories relate to agency, covering Identity (I); Group Membership (II); Shared Values, Ideals, and Goals (III); and Shared Knowledge and Beliefs (IV); (B) one category addresses group social relations and structure (V); (C) three categories concern group action and interaction patterns or orders (VI, VII, and VIII); (D) one category relates to the material and resource conditions of group action and interaction (IX); and (E) one category covers rules concerning temporal and spatial conditions for group meetings and interactions (X).

The authors of [70] highlight that a group’s rule regime constitutes the cognitive-normative framework that defines its identity, purpose, structure, roles, procedures, activities, and patterns of interaction. It has also been highlighted as functioning like a collective “codebook” of cultural tools and organizational principles, enabling members to coordinate, collaborate, and make sense of group life. These studies also emphasize that rules within a regime are generally known (often tacitly), applicable when resources allow, and regarded as legitimate. They guide behavior, judgments, and interactions, while also allowing for disagreements over their content or application, thus introducing a political dimension to social rules. Furthermore, it has been noted that not all ten universal rule categories are always fully specified. Institutionalization typically unfolds gradually, with long-established groups covering all categories. However, as highlighted in the literature, political, economic, technological, or social changes may disrupt regimes, reshaping roles, norms, and structures, for example, shifting from hierarchical to more egalitarian forms.

Their study [70] also emphasizes the dual nature of group interactions, distinguishing between routine and strategic games. Routine patterns of activity and interaction are often predictable, ritualized, or institutionalized, whereas non-routine situations involve collective decision-making, conflict resolution, or innovation. The concept of action determination is introduced to extend traditional notions of decision-making. Three ideal-type modalities are defined:

• (DET-I) following or implementing established rules and algorithms, resulting in routinized and predictable interactions;
• (DET-II) selecting among alternative actions based on normative similarity or instrumental value;
• (DET-III) generating or constructing new action alternatives, which are then evaluated and enacted through DET-I or DET-II processes.

These modalities demonstrate how social actors operate within the constraints of their rule regimes while exercising bounded choice, creativity, and strategic judgment in non-routine situations. Illustrative examples of ideal-type groups include military units, terrorist organizations, recreational or social groups, research teams, and business organizations. Each of these groups exhibits distinct rule configurations that determine interaction patterns, authority structures, production algorithms, and internal governance. For instance, military and terrorist groups enforce hierarchical and ritualized interactions to maintain internal order. In contrast, research groups or social clubs rely on negotiation, democratic decision-making, and cooperative problem-solving. Universal games such as collective problem-solving, conflict resolution, rule and policy changes, governance, and secrecy occur across all groups, but the specific rules, procedures, and norms vary according to the group’s subculture. Differences in group size, resources, membership capabilities, and external environments further influence how rules are enacted and how interactions unfold.

The study also highlights the dynamic and adaptive nature of group rule configurations. Groups may face incoherence or incompatibility within their rules, leading to performance failures, which generate internal pressures to increase coherence or adjust rules. Similarly, groups may confront external pressures (legal, economic, or social) that require modification or reinterpretation of rule regimes. By showing how rules shape group interactions and outcomes, the article connects group theory with SGT and argues that rule systems form the cognitive–normative basis for coordination, cooperation, innovation, accountability, and power. In sum, each group’s rule configuration functions as a “cultural logic” that defines its identity, differentiates it from other groups, and shapes the specific patterns and outcomes of its social games.

This study provides a solid theoretical basis for linking group theory and SGT, offering a systematic framework for analyzing groups as structured rule configurations. Its strength lies in formalizing universal categories of rules and demonstrating how they shape both routine and strategic interactions. By doing so, it provides a strong cognitive-normative foundation for understanding social games in diverse organizational and cultural contexts. It also offers a solid theoretical basis for comparing different types of groups and organizations.

6.2. Sociological and Interactionist Game Theory: A Comparative Perspective

In [12,13,71], two complementary approaches are elaborated—sociological game theory and interactionist game theory—which emphasize the role of social norms, rules, roles, and institutions in shaping human behavior.

Both perspectives “sociologize” game theory by embedding strategic interaction in broader social contexts. Comparing these approaches, the authors highlight the fundamental assumptions, analytical strengths, and explanatory potentials of each framework. Unlike classical game theory, which prioritizes strategic rationality, these approaches argue that rule-following and social embeddedness are more fundamental to action. The papers outline how actors rely on complex rule-based models, value systems, and repertoires of actions, often determined collectively.

Sociological game theory provides a systematic framework in which games are conceived as socially embedded rule systems [33,69]. In contrast to the hyper-rational actors of classical game theory, SGT conceives of players as socially situated beings whose decisions are shaped by institutional rules, normative commitments, and role expectations [10]. The framework also expands the concept of equilibria by introducing normative equilibria as foundations of social order [53]. Moreover, it distinguishes between open and closed games, where actors and institutions may themselves restructure or transform rules. This systematic approach enables the analysis of cooperation, conflict, negotiation, and policy processes under conditions of incomplete, ambiguous, or even false information.

Burns et al. [13] also highlight key differences between classical theory and sociological approaches across various dimensions, including human agency, social structure, norms, institutions, cultural forms, interaction patterns, and outcomes. They additionally distinguish how these approaches address conditions of cooperation and conflict, as well as processes of game restructuring, transformation, and empirical relevance. Sociologically oriented game theory provides a conceptual language and analytical tools that better capture the diversity, complexity, and dynamics of human interaction. As a result, it offers greater empirical relevance and explanatory power than classical theory.

Table 6. Classical game theory vs. sociological game theory—social embedding of games.

Category	Classical Game Theory (CGT)	Sociological Game Theory (SGT)
I. Games and Game Constraints	Games are defined as fixed systems of rules, mostly material/technical constraints. No theory of rules as social institutions. Game structures are closed and not transformable (except by theorists).	Games are understood as complexes of social, normative, and institutional rules, alongside material constraints. Games can be open, restructured, or transformed by actors and external agents.
II. Agency and Actors’ Capabilities	Players are abstract, role-less, rational utility-maximizers with perfect (or near-perfect) information. Limited action repertoires, mainly instrumental rationality.	Actors are social beings embedded in roles, groups, and institutions. They use diverse repertoires: strategic, ritual, normative, habitual, and creative actions. Information is often incomplete, fuzzy, or contested.
III. Social Relations and Structures	Social context largely absent. No explicit account of roles, power, trust, or communication. Cooperative vs. non-cooperative games are reduced to communication/no communication.	Games are socially embedded. Roles, status, authority, power, and trust shape interaction. Communication is structured by social rules and may involve cooperation, negotiation, or manipulation. Games can overlap with other games.
IV. Empirical Relevance	Criticized for limited connection to real social life. Often abstract and mathematically elegant but not empirically grounded.	Shown to explain real social interactions, institutions, and outcomes. Captures the transformation of games, normative conflicts, and the political dynamics of rule systems.

Source: own elaboration based on [13,33,69].

provides a concise summary of the key differences between classical and sociological approaches to game theory, emphasizing the shift from abstract, rational actors to socially embedded actors whose strategies are shaped by roles, norms, and institutional settings.

Interactionist game theory, on the other hand, draws inspiration from Erving Goffman’s interactionism [72] but develops its own analytical framework within the sociological game theory tradition. It focuses on the microdynamics of face-to-face encounters, highlighting rituals, impression management, deception, and trust as structuring mechanisms of interaction. Actors are not only decision-makers but also performers who manage impressions and strategically manipulate information, while sustaining social order through rituals and interactional rules [69]. Competence in interpreting cues, evaluating others’ trustworthiness, and navigating the rituals of everyday life becomes central. In doing so, IGT brings into the analysis the symbolic and ceremonial dimensions of interaction that classical and even sociological game theory often overlook, while also acknowledging the role of institutions and third parties in structuring encounters In this way, IGT extends game analysis to the symbolic and communicative dimensions of strategic interaction that remain invisible in classical or purely institutional models.

Both approaches, as the authors note, converge in rejecting the reductionist assumptions of classical game theory and instead stress the embeddedness of strategic interaction in social, cultural, and institutional contexts. They jointly stress the empirical relevance of game theory for understanding real social life, showing that strategic interaction is deeply embedded in cultural, institutional, and symbolic contexts. SGT offers a more systematic, rule-based framework, particularly suited to analyzing institutional and organizational processes, whereas IGT provides insight into the fragile dynamics of trust, ritual, and symbolic communication in everyday encounters.

The comparison of SGT and IGT made in the papers can be summarized in the following

Table 7. Comparison of SGT and IGT.

Aspect	Sociological Game Theory (SGT)	Interactionist Game Theory (IGT)
Focus	Social systems, institutions, rules, norms, values.	Face-to-face interactions, rituals, and impression management.
Actors	Socially situated, role-bound, creative agents with bounded rationality.	Performers and interpreters, managing impressions and trust.
Game Structure	Open or closed; rules and equilibria can be restructured or transformed.	Structured by communication, rituals, and situational framing.
Equilibria	Instrumental, normative, and social equilibria; normative equilibria are central to social order.	Less formalized; stability comes from rituals, trust, and interactional routines.
Key Contribution	Formal, systematic integration of institutions and norms into game theory.	Rich account of symbolic, communicative, and ritual aspects of interaction.
Applications	Organizational conflict, negotiation, policy games, and institutional analysis.	Everyday life interactions, gambling, secrecy, deception, and impression management.

Source: own elaboration based on [12,13,71].

[12,13,71].

These insights suggest that SGT and IGT extend the analytical core of game theory rather than replace it. While classical game theory provides a formal structure for strategic reasoning, sociological and interactionist approaches draw attention to the cultural, institutional, and symbolic environments within which strategic action becomes meaningful and possible.

Table 8. Classical game theory vs. sociological game approaches (SGT/IGT).

Aspect	Classical Game Theory	Sociological Game Theory & Interactionist Game Theory
Definition of the Game	A game is a fixed and closed structure defined by strategies and payoffs.	A game is a social system of rules, norms, roles, meanings, and institutional conditions; game rules may be modified or transformed.
Assumptions about Actors	Actors are rational, self-interested utility maximizers with stable preferences.	Actors are socially embedded, role-bound, value-oriented, and interpretive; motivations include norms, identity, morality, and loyalty.
Knowledge and Information	Assumes a clear and commonly known structure of the game, often with complete information.	Information is often incomplete, ambiguous, interpreted through norms, culture, symbols, and interactional context.
Interaction Context	Social context is external and usually not part of the model.	Social context is internal to the model: institutions, trust, norms, and identities influence strategies and outcomes.
Equilibrium Concept	Nash equilibrium based on mutual best responses.	Normative, social, or interactional equilibria based on legitimacy, shared expectations, ritual stability, and trust.
Types of Action Considered	Mainly instrumental–strategic action.	Both strategic and non-instrumental action: ritual, symbolic, moral, habitual, and expressive behavior.
Sources of Cooperation	Cooperation arises through strategic incentives (e.g., repeated games).	Cooperation is grounded in norms, relationships, trust, shared identities, and institutional arrangements.

Source: Own study based on [12,13,71].

summarizes this contrast by juxtaposing the foundational assumptions of classical game theory with sociological and interactionist perspectives. The comparison clarifies how the latter extends the analytical scope of game theory to include the cultural, normative, and symbolic dimensions of action.

7. Model of the Societal Game, Social Optimum, and Legitimizing Procedures in Socio-Economic Systems

This section presents a framework for understanding collective decision-making in socio-economic systems. It begins with the Pareto optimization problematique, highlighting its limitations, and introduces the GGT approach. Building on studies [54,62,65,66], the section summarizes and discusses a model for analyzing societal decision-making processes, with a focus on defining the societal game, the social optimum, and legitimizing procedures. The approach emphasizes the roles of agents, decision-making procedures, and multi-dimensional legitimacy in achieving socially accepted outcomes. It highlights not only the strategic behavior of agents but also the normative, cognitive, and institutional contexts in which decisions take place.

7.1. From Pareto Optimality to the GGT Procedural Approach

The concept of Pareto optimality analyzed in the literature defines a state as “efficient” if no change can improve the situation for one individual without making someone else worse off. While widely used in economics and decision theory, this criterion focuses exclusively on the absence of individual losses and is indifferent to fairness, justice, or broader social welfare. Consequently, the requirement of unanimity for any change renders Pareto optimality both empirically unrealistic and normatively problematic [62,63]. It can legitimize highly unequal or unjust social arrangements merely because a single actor can veto reform, even when most members of society would clearly benefit. Historical examples, such as the Polish liberum veto, illustrate the risks of requiring unanimity, which can paralyze governance, block reform, and hinder collective welfare. Contemporary applications would face analogous challenges: socially beneficial changes, including welfare reform, taxation, or institutional restructuring, are often blocked because some actors inevitably experience relative losses, despite net social gains.

Critics of Pareto efficiency have long highlighted these limitations. Hicks [73] noted that its dominance in economics is largely due to mathematical convenience rather than ethical adequacy, while Lockwood [74] argued that Pareto optimality discourages meaningful consideration of distributive justice. Sen [75] famously emphasized that a society can be Pareto-efficient yet ethically unacceptable. These critiques underscore that classical efficiency-focused models fail to account for social fairness, legitimacy, or normative acceptability, dimensions central to human and institutional decision-making.

To address these shortcomings, GGT introduces a procedural perspective, situating collective decisions within socially recognized and institutionalized frameworks. Societies can rely on formalized procedures such as democratic voting, adjudication, administrative decision-making, or negotiation to reconcile conflicting values and produce decisions that are normatively legitimate [52,54,66]. In this framework, outcomes are socially stable not because every participant benefits, but because the decision-making procedure is accepted as fair, authoritative, and normatively appropriate. Procedural legitimacy, rather than unanimous individual advantage, becomes the basis for stable social equilibria, enabling socially desirable transformations even when some actors incur relative losses.

By integrating critiques of Pareto optimality with the procedural and institutional focus of GGT, this approach demonstrates how generalized game-theoretic models provide both normative and practical guidance for complex socio-economic decision-making. It positions GGT as a framework that extends beyond classical efficiency models, linking formal modeling, institutional design, and considerations of fairness, legitimacy, and distributive justice, thus situating the theory within a broader interdisciplinary discourse rather than presenting it solely as a development of Tom Burns’ work.

7.2. Model of the Societal Game

A societal game is defined as an interaction process among agents who hold specific roles and relationships within a given social state. The societal game is defined using several formal elements [54]:

$Q$—the set of options, representing possible social states, such as different allocations of resources, institutional arrangements, or states of the world. This set forms the universe of choices available to society.

$I$—the set of all agents participating in the societal game.

$Iaut\subseteq I$—the subset of agents authorized to make collective decisions, whose actions guide the outcome of the game.

$G=\left[ROLE\right(I,G),R]$—the societal game itself, defined as a structured interaction process among agents. More specifically,

$G=\left[ROLE\right(Iaut,G),ROLE(I\backslash Iaut,G),R],$ distinguishing between authorized and non-authorized agents.

$ROLE(I,G)$—the role complex of agents in the game, which captures the functions, powers, and responsibilities of agents within the institutional context.

In GGT, the roles of agents are defined through four cognitive and normative components:

$MODEL(I,G$): The belief structures of agents, describing how they perceive social states, options, and consequences. This may include incomplete or fuzzy information and reasoning processes that do not follow standard logic.

$VALUE(I,G):$ The value-normative system shared by agents, defining norms and principles regarding what is allowed, obligatory, or forbidden.

$ACT(I,G)$: The repertoire of possible actions and strategies, including $PR$, the set of procedures available to agents.

$J\left(I,G\right):$ Judgment mechanisms, specifying how agents evaluate actions, outcomes, and compliance with norms. This includes mechanisms for collective judgments and “social algorithms.”

$Cpro\subset PR$: Measures that serve as social algorithms guiding collective action, such as formalized rules, rituals, or algorithms, which guide collective decision-making and ensure consistency, predictability, and legitimacy.

Thus, the societal game describes how agents in a society ($I$)—particularly those with decision-making authority ($Iaut$)—move between social states ($Q$) through interactions governed by role complexes, norms, and social algorithms. The model demonstrates that societal games are embedded in broader institutional and cultural structures, capturing the socially situated nature of decision-making by linking institutional arrangements, social relationships, and situational conditions.

The roles and subcomplexes described above are not independent but are integrated within wider institutional and social contexts (e.g., cultural patterns, political structures, economic relations). Accordingly, the societal game is conceptualized not as an abstract, isolated interaction, but as a socially embedded process through which agents navigate possible social states ($Q$), guided by norms, beliefs, procedures, and collective decision-making mechanisms.

7.3. Concept of Social Optimum

Building on the structure of the societal game, we define the concept of social optimum in terms of institutionalized procedures $Cpro\subseteq PR$ [54].

• A social procedure $Cpro$ from the set $PR$ is an institutionalized regulatory mechanism for addressing conflicts or suboptimality. It determines collectively whether to move from one option or state of the world $A\in Q$ to another $B\in Q$ (or to choose between them).
• A social regulatory procedure $Cpro$ from $PR$ is legitimate if agents involved in or affected by it recognize it as right and appropriate for deciding whether to move from $A$ to $B$, where $A,B\in Q$.
• A legitimate social improvement (or collective improvement) is a collective decision, based on a legitimate procedure $Cpro\in PR$, that results in movement from $A$ to $B$ and is accepted by those involved ($A,B\in Q$).
• An option from $Q$ (i.e., a state of the world such as an allocation of resources or institutional arrangement) is societally efficient or societally optimal if no further social improvement can be made.

A process of potential collective improvement can be described as follows [54]:

A norm or principle $r\in VALUE(I,G)$ legitimizes the use of a regulatory procedure $Cpro\in PR$ for collective judgment in situation $St$.

Agents with authority $(Iaut\subseteq I)$ apply $Cpro$ to decide whether moving from $A$ to $B$ (with $A,B\in Q$) constitutes an improvement.

The procedure $Cpro$ is implemented according to its rules.

All agents in $I$ accept the application of $Cpro$ by agents in $Iaut$, as well as its outcome, based on its legitimacy and the supporting sanctions and pressures.

In this way, players make both individual judgments based on their own $VALUE(i,G)$ and $J(i,G))$ and collective judgments (through the institutionalized procedure $Cpro$).

Societal improvement is primarily determined and deemed legitimate by the agents responsible for decision-making. Even those affected but not directly involved may prefer the status quo, yet they are likely to accept a transition from one social state to another if it is conducted through recognized and legitimate procedures. This highlights a central insight emphasized in prior research: adherence to legitimate procedures often outweighs dissatisfaction with the outcome, making procedural legitimacy a key mechanism for social acceptance and stability.

7.4. Procedural Legitimacy and Collective Decision-Making in Societal Improvement

Building on this foundation, the study [54] redefines societal improvement as more than the enhancement of welfare; it fundamentally involves the legitimate implementation of procedures. Within this framework, a social procedure regulates conflict and facilitates decision-making, a legitimized procedure is recognized as proper and authorized in context, societal improvement represents a legitimate transition from one state to another, and a social optimum occurs when no further improvement is possible through legitimate means. Negotiation, democratic voting, and legal adjudication exemplify how such procedures can produce socially accepted outcomes.

Table 9. Dimensions of conflict resolution procedures: a comparative overview.

Aspects of the Procedure	Administration/ Adjudication	Democratic Process	Negotiation
Actors involved	Judges; parties to the issue.	Parties to the issue (voters, proponents, opponents).	Parties to the issue (negotiators, proponents, opponents).
Roles and role relationships	Hierarchical relationship between judges and the disputing parties.	Horizontal relationships among voters, proponents, and opponents.	Horizontal relationships among negotiators and disputing parties.
Properties of the procedure	Initiative lies with the adjudicator, who follows designated phases and due process norms.	Mutual initiative in defining issues; proper voting procedure; aggregation of votes; application of a collective decision rule.	Mutual initiative; a give-and-take process aimed at reaching mutual agreement (or acknowledging lack thereof).
Norms/guidelines for the decision.	Decisions are guided by norms, rules, or precedents; specificity and strictness may vary.	Open-ended, though some legal or normative limits apply (certain issues cannot be decided by vote).	Open-ended, though certain forms of coercion or illegal acts are prohibited, and some issues cannot be negotiated (e.g., slavery, crimes).
Discursive properties	Legal reasoning is central.	Persuasion and rhetoric are central; legal reasoning may play a role.	Rhetoric, persuasive power, and strategic behavior (e.g., bluffing) are central; legal reasoning may play a role.
Risks and limitations	Risk of perceived bias or lack of competence; failure to follow procedure or norms.	Risk of improper voting procedure, improper aggregation of preferences, or flawed collective choice rule.	Risk of deceit or coercive pressure.

Source: based on [65].

summarizes and compares the properties of different conflict resolution procedures. Here, we focus on the formal procedural dimensions rather than the practical “game” dynamics or informal adaptations that often occur in real-world applications [65].

Later, Burns and Roszkowska [62] detailed models of the adjudication, negotiation, and democratic procedures and discussed their legitimacy bases, the limits of such societal procedures, and the accomplishment of societal efficiencies through the procedures.

Legitimacy is multi-dimensional, relying on the fulfillment of several interrelated criteria [54]. Decisions must be guided by normatively acceptable values, excluding illegitimate ones, and based on relevant, reliable knowledge. The process must clearly define whose voices count, ensuring procedural fairness, and adhere to formal rules regarding timing, location, and structure. When these conditions are met, collective decisions gain legitimacy and foster social equilibrium. When they are violated, legitimacy weakens, and conflict may arise or normative disequilibrium.

The studies also emphasize the importance of group decision-making in organizations and society [54]. Collective processes can generate more alternatives, foster deeper understanding, and strengthen commitment to implementation. However, they also pose risks such as slower decision-making, groupthink, polarization, and diffusion of responsibility. To address these, three main types of legitimizing procedures are highlighted. The authoritarian style involves a single individual making decisions efficiently, though it may generate discontent. Brainstorming is a creative and inclusive approach guided by a leader, allowing multiple options and consensus to emerge. Voting is a structured method that works well when alternatives are clearly defined, but it is less sensitive to individual preferences.

Finally, a previous study [54] stresses that societal improvement depends not only on procedural legitimacy but also on knowledge-based effectiveness. Legitimacy ensures social acceptance, while relevant expertise increases the likelihood that decisions achieve their intended outcomes. Integrating these dimensions produces socially accepted, stable, and impactful decisions. The framework thus captures the interplay between strategic, normative, and procedural aspects of social systems, providing a comprehensive model for understanding how societies achieve legitimate and effective collective outcomes.

8. Conclusions, Limitations, and Future Research

This paper examined generalized game theory (GGT) and its sociological extension, sociological game theory (SGT), developed by Tom Burns and collaborators. It highlighted their contribution to extending classical game theory by incorporating norms, values, and institutional rules. GGT situates actors within social contexts, moving beyond the narrow focus on rational choice. It models socio-economic interactions through formalized rule structures that account for ambiguity, exceptions, contradictions, uncertainty, and normative variability.

Key concepts of GGT include the distinction between game structures and dynamic processes, the centrality of rules and rule complexes, multiple action modalities (routine, consequentialist, normativist, and emotional), and the notion of open games, where actors may redefine roles, goals, and values. These insights are particularly relevant for understanding decision-making in complex and uncertain organizational and economic environments.

SGT further strengthens the connection between mathematical modeling and socio-economic theory, enabling the analysis of institutional change, cultural influence, and norm-driven behavior. It offers analytical tools for examining organizational processes, negotiation dynamics, regulatory frameworks, and social coordination, complementing approaches such as IGT that emphasize micro-level interaction rituals.

However, several limitations of GGT/SGT must be recognized. The high level of abstraction and formal complexity, while theoretically rigorous, poses challenges for empirical application and practical implementation. Modeling social games requires accounting for multiple layers of rules, heterogeneous interactions, and potential conflicts, complicating the development of parsimonious predictive models. The open-ended nature of games and the plurality of action modalities—including routine, consequentialist, normativist, and emotional behaviors—further limit the stability and clarity of outcomes. The methodological difficulty of translating complex rule structures, hybrid action modalities, and normative equilibria into empirically measurable variables also constrains systematic testing and comparison across cases.

Moreover, the flexibility of open games, in which actors may reinterpret or transform their roles, goals, and values, introduces inherent instability in predicted outcomes. Empirical evidence across diverse cultural and organizational contexts remains sparse, while the operationalization of normative and institutional configurations requires further refinement. As a result, practical applications of GGT/SGT demand careful contextual adaptation to capture institutional, cultural, and normative dimensions of strategic interaction.

Future research may investigate how complex rule configurations shape collective decision-making, how norms and institutions interact across different cultural and organizational settings, and how hybrid action modalities influence the stability of social equilibria. Further attention to power asymmetries, institutional pressures, and digitally mediated forms of interaction could deepen understanding of how social games evolve. Research on procedural legitimacy and knowledge-based effectiveness in real-world decision-making could also clarify the conditions under which social games achieve stable and socially desirable outcomes.

Overall, GGT/SGT offers a robust framework for understanding socio-economic behavior, linking formal modeling with socio-economic insight. Beyond its theoretical contributions, this approach enhances practical relevance. It provides guidance for analyzing cooperation, conflict, and institutional change in contemporary societies. It also aids in developing socio-economic decision models suited to complex institutional contexts. Ultimately, it points toward more effective and adaptive decision-making strategies in volatile and interconnected environments.

Importantly, similar ideas to those developed in SGT are found across the literature. For example, later studies [49] conceptualize social life as networks of interacting social games, emphasizing the role of rules, norms, and social roles in shaping behavior and outcomes. Stolz [49] argues that each game involves agents with resources who act according to rules, goals, and representations, producing specific outcomes. The theory integrates both instrumental and normative action across multiple social levels and links conceptual analysis with empirical research through descriptive–interpretive and explanatory heuristics, as well as agent-based modeling. While these works do not directly employ SGT tools, they resonate with Burns’ approach by integrating instrumental and normative action and exploring systematic modeling of social interactions. This demonstrates the broader relevance and influence of the SGT framework and highlights its conceptual alignment with ongoing research in sociology and organizational studies.