Collective Agency and Coalitional Power in Games

Abstract

This paper explores how insights from the philosophy of collective agency can inform the development of coalition logic, focusing particularly on the conceptual distinctions among intentionality, preference, and coalitional power as foundational elements. While the interdisciplinary discussion mainly adopts a philosophical perspective, we also propose specific directions for broadening and refining coalition logic through philosophical theories. This expansion sheds light on phenomena often overlooked by logicians, including unstable joint actions, exogenous power, and the role of coalitional structures.

1. Introduction

Collective agency pervades our daily lives, given that institutional groups with stable structures or procedures possess the capacity for coordinated action as rational individual entities (cf. [1]). Nevertheless, empirical instances often elude straightforward explanations, as illustrated by the following case study.

OPEC+ Oil Production Negotiations (2020): During the COVID-19 pandemic, the collapse in global oil demand led the OPEC+ alliance to confront a strategic dilemma as it negotiated production cuts to stabilize plummeting oil prices. Despite agreements mandating collective output reduction, each member state faced strong incentives to cheat and overproduce in hopes of maximizing its revenue while benefiting from others’ compliance. The tension was further exacerbated by the United States’ annual increase in shale oil production, as the U.S., not a member of OPEC+, was not bound by any production cut agreements. Ultimately, these dynamics led Russia to conclude that production cuts were futile, refuse to cooperate fully, and eventually triggered a price war with Saudi Arabia. As a result, oil prices collapsed to historic lows, even briefly turning negative, hurting all OPEC+ members, as well as the United States.

This scenario instantiates a paradox of collective agency: despite OPEC+’s institutionalized structure (supported by international legal contracts and binding agreements), individual incentives eventually subverted collective welfare. This mirrors the Prisoner’s Dilemma in game theory: players prioritizing self-interest leads to a worse collective outcome. The logical principles for reasoning about individual and collective ability are captured in coalition logic [2], which prioritizes so-called $\alpha$-effectivity in modeling such coalitional power1. The similarity here reveals a conceptual interplay between philosophical accounts of collective agency and game-theoretic representations of coalitional dynamics. Furthermore, strategic games are insufficient to explain the outcomes of many collective action problems since many games lack a state that simultaneously satisfies Nash equilibria and Pareto optimality2. The structural aspect of a collective is often ignored in the game-theoretical context. These limitations suggest the necessity of interdisciplinary research integrating philosophical perspectives on agency with game-theoretical models.

On the other hand, exploring coalitional power within the framework of game theory and multi-agent systems presents a fertile ground for academic inquiry (cf. [2,5]). However, it remains mainly anchored in game-theoretical presuppositions, neglecting the philosophical dimensions of agency. Existing philosophical theories addressing concepts such as intentionality, moral responsibility, and collective ontology have yet to attract much attention from logicians. One may contend that the development of coalition logic requires no philosophical elaboration, yet this paper posits that agency philosophy offers universal analytical tools, inherently presupposed by game theory’s and coalition logic’s conceptual foundations. Therefore, promoting coalition logic requires reciprocal insights from both philosophy and game theory, rather than unidirectionally importing results from the latter.

By bridging these domains, this paper identifies several novel directions for coalition logic to enhance its explanatory power, particularly in unstable joint actions, exogenous power, and coalitional structure.

Structure of the paper. We first articulate the general distinction between methodologies from philosophy and game theory in Section 2. Then, in Section 3, we explain the foundational conceptual difference between desire and preference, followed by an argument for which concept is fundamental in Section 4. In Section 5, we explore the connections between intentionality, effectivity function, and coalitional power. Based on this analysis, Section 6.1 focuses on formal methods and discusses several potential ways to revise the existing coalitional logic.

2. Path Dispute: Agency Philosophy and Game Theory

As mentioned in the Introduction, philosophy and game theory share a deep connection in addressing collective problems, though this connection does not extend to methodological integration. Derived from an interest in game-theoretic problems, coalition logic focuses more on explicating collective dynamics through a game approach. Let S be a set of states and N be a finite non-empty set of agents, while a coalition is a set $C\subseteq N$ of agents. Then the main type of expression in coalition logic, $\left[C\right]\phi$, means that the coalition C can cooperate to guarantee that $\phi$ will be the case. Its semantics is given by an effectivity function $E:2^N\to 2^{2^S}$, which tells us that a coalition C can guarantee $\phi$ if it can choose a particular set of states $S^{\prime }\in E\left(C\right)$ that precisely guarantees $\phi$. The basic axiomatization of coalition logic (cf. [2]) is as follows in

Table 1. Axioms of Coalition Logic.

Axioms	Inference Rules
(C1) $\neg \left[C\right]\perp$ (Liveness)	(MP) ${⊢}_{CL}\phi ,{⊢}_{CL}\phi \to \psi \Rightarrow {⊢}_{CL}\psi$
(C2) $\left[C\right]\top$ (Safety)	(RG) ${⊢}_{CL}\phi \leftrightarrow \psi \Rightarrow {⊢}_{CL}\left[C\right]\phi \leftrightarrow \left[C\right]\psi$
(C3) $\neg [\varnothing ]\neg \phi \to \left[N\right]\phi$ (N-maximality)
(C4) $\left[C\right](\phi \wedge \psi )\to \left[C\right]\psi$ (Outcome Monotonicity)
(C5) $(\left[C_1\right]\phi \wedge \left[C_2\right]\psi )\to [C_1\cup C_2](\phi \wedge \psi )$ where $C_1\cap C_2=\varnothing$ (Supperadditivity)

:

Here ∅ is the empty coalition, that is, no one participates, and N is the grand coalition composed of all the agents. Then (C3) indicates that if the empty coalition cannot guarantee $\neg \phi$, then the grand coalition ensures $\phi$. (C4) represents logical monotonicity. (C5) signifies that non-overlapping coalitions can combine and have more abilities, a property also known as superadditivity. Stemming from Pauly’s foundational work, subsequent advancements in coalition logic have inherited key properties, including those defining playable effectivity functions, such as liveness, safety, N-maximality, monotonicity, and superadditivity3. Other logics emphasizing coalitional agency, such as Alternating-time Temporal Logic [6] and Stit Logic [7], are also closely related to coalition logic, especially since [8] demonstrated that coalition logic constitutes a fragment of Alternating-time Temporal Logic.

However, coalition logic remains narrowly focused on characterizing goals attainable through cooperative strategies among agent coalitions. Scenarios analogous to the Prisoner’s Dilemma cannot be resolved directly by the coalitional force modeled by coalition logic. For instance, below is a formalization of the previous OPEC+ scenario (

Table 2. OPEC+ Dilemma.

US
underprod.				overprod.
		RU				RU
		reduct.	overprod.			reduct.	overprod.
SA	reduct.	$\boxed{4,4,3}$	1, 4, 1	SA	reduct.	$\boxed{2,2,4}$	$\underline{1,3,3}$
	overprod.	4, 1, 1	$\underline{2,2,1}$		overprod.	$\underline{3,1,3}$	0, 0, 0

):

The above metric represents a strategic game illustrating the preferences of three nations, Saudi Arabia, Russia, and the United States, in terms of oil production levels. Specifically, Saudi Arabia (rows) and Russia (columns) choose between two actions: $reduction$ (complying with production limits) and $overproduction$ (exceeding agreed quotas). Meanwhile, the United States (left and right sides of the table) chooses between $underproduction$ (indicating a limited market share) and $overproduction$ (indicating an unrestricted expansion of shale oil production). A strategy profile, i.e., the combination of actions selected by all players, results in payoffs represented by triples, corresponding sequentially to Saudi Arabia, Russia, and the United States4.

The left side of the table represents a market scenario in which the United States holds a relatively small market share, while the right side shows an expanded U.S. market share due to shale oil, thereby intensifying tensions. Notably, even when Saudi Arabia and Russia both choose to overproduce while the U.S. remains underproduced, their payoffs are still positive at 2 each. However, when the United States also chooses to overproduce, excess global production results in oil prices dropping below, leaving all countries unprofitable, which is expressed as (0, 0, 0).

Analyzing the entire game reveals three Nash equilibrium outcomes, indicated by underlined payoffs: (2, 2, 1), (3, 1, 3), and (1, 3, 3), and two Pareto optimal outcomes, indicated by boxed payoffs: (4, 4, 3) and (2, 2, 4). Given that the U.S. is not part of OPEC+, focusing solely on Saudi Arabia and Russia, the left part mirrors the classic Prisoner’s Dilemma. Here, their Pareto optimal state (4, 4, 3) differs from the Nash equilibrium state (2, 2, 1). Each country is incentivized to overproduce, aiming to exploit the other’s compliance for short-term gains, ultimately resulting in collective losses. This scenario underscores the strategic value of OPEC+ agreements in guiding member countries toward optimal outcomes. However, when the U.S. shifts the scenario by increasing market share (right side), the payoffs for Saudi Arabia and Russia resemble a game of Chicken, where yielding first results in disadvantages, yet mutual stubbornness leads to a lose–lose outcome. In this scenario, three outcomes become Pareto optimal for the OPEC+ members, illustrating why Russia may perceive production cuts as futile.

The game’s structure demonstrates how changes in external conditions can significantly impact a collective. Such external changes can drastically transform a stable group environment (one equilibrium on the left side) into a more volatile circumstance (two equilibria on the right), where behaviors that deviate from the group are incentivized. This example is sufficient to demonstrate the complexity of the group problem. Without coordinated intervention, this game setting indicates that the OPEC+ group cannot consistently achieve its optimal outcome (4, 4, 3).

This reveals that the ostensibly cooperative OPEC+ agreement lacks enforceability mechanisms to sustain coalitional cohesion. For such a coalition to stabilize its agency, i.e., always aiming for collective optimality, its contractual framework must incorporate both compelling incentives to comply and credible penalties for defection, administered by a supranational authority capable of imposing sanctions. In reality, however, the international system lacks a centralized coercive institution to fulfill this regulatory role; cooperation instead hinges precariously on cost–benefit calculations among nations. So, from the source, the variation of each player’s preferences and the dynamic interdependence among them constitute the analytical core of game-theoretic inquiry.

In this game, coalitional power can be allocated to Saudi Arabia and Russia, respectively, yet extending such power to the joint of both cannot be simply given. This disjuncture underscores a critical limitation of coalition logic, which assumes a priori the existence of enforceable cooperative mechanisms (e.g., binding agreements or inherent coalitional power). In reality, however, the barrier to collective agency here arises precisely from the absence of such mechanisms. Stated more formally, only adding the credibility of cooperation commitment and coordination mechanisms may permit such a priori coalitional power. Nevertheless, real-world applications reveal that a guaranteed coalitional power to every group remains an essentially normative endeavor. This tension leads to a pivotal ontological question: What constitutes the source of coalitional power? The answer is more related to philosophical theories of collective agency.

What coalition logic elides is precisely the field interrogated by philosophies of collective agency—namely, the ontological and normative foundations enabling groups to act as agents. Whereas game-theoretic models abstract agency into utility functions and strategy profiles, philosophical inquiry probes deeper into the preconditions for collective intentionality. In philosophical accounts, agency entails multifaceted psychological elements (e.g., shared intentions). It extends beyond mere strategic dynamics (e.g., preference alignment) to grapple with epistemic constraints, moral commitments, and the social ontology of collectives.

From a philosophical perspective, agency primarily concerns the capacity of an entity or a system to act autonomously and rationally and make free choices. Collective agency, specifically, refers to the capacity of a group to exhibit shared autonomous, rational actions that surpass the mere sum of individual contributions (cf. [9]). The so-called ’orthodox theory’ of collective agency prefers individualism (probably inherited from modern sociology, cf. [10]) and attempts to reduce the very concept of a collective back to the individual level (cf. [11,12,13]). However, some recent work insists that there is an essence of collective agency that cannot be reduced (cf. [1,9,14]). The resulting framework, named relational ontology, suggests that collective agency is not simply an aggregation of individual entities but is intrinsically linked to a stable pattern of relations. This perspective shifts the emphasis from individual components to the interconnectedness of these components, suggesting that the identity and agency of a collective stem from these relational dynamics and can be analyzed further by formal study.

While interdisciplinary research exists that bridges game theory and agency philosophy, exemplified by frameworks like team reasoning in non-cooperative game contexts, these syntheses remain constrained by a degree of abstraction, with preferences as the underlying concept. The game-theoretical approach sidelines richer philosophical notions, inviting a critical question: If philosophical theories of agency (e.g., shared intentions) and game-theoretic models (e.g., preference alignment) operate on divergent conceptual planes, does this justify developing coalition logic in isolation from philosophical insights? To reconcile these domains, a conceptual groundwork for philosophy and game theory must first disentangle, namely, intentionality and preference.

3. From Preference to Desire

Game theory defines an economic agent as an entity with preferences quantified via utility. According to the traditional interpretation, utility is tied to subjective welfare rankings among options. These rankings, driven by factors such as income, natural resources, moral pursuits, or other desiderata, reflect internal psychological processes, echoing Benthamite utilitarianism. By the 1930s, behaviorism spurred economists to pursue objectivity and refuse to hold unobserved and unmeasurable psychological processes as the basis of utility. As a result, the revealed preference theory proposed by [15] redefines utility as a purely technical construct. Under the new explanation, an economic agent maximizing the utility of their action simply means consistently taking those actions to improve the possibility of its realization. This new definition sounds like a tautology, and it actually is. It fixes the explanation of preferences at the level of behavior. From then on, the economic agent is the entity that takes action to maximize utility; action is the process of maximizing utility; and utility is the object that the economic agent tries to maximize. These concepts interlock at the basic level and are not explained further, excluding any mental elements.

While philosophically contentious—given philosophy’s pursuit of universal explanations—this abstraction aligns with formal theory’s aims, prioritizing methodological tractability over ontological depth. However, when analyzing collectives in reality, preference-centric models falter. Two gaps emerge: (1) the need to interrogate preference origins beyond tautological assumptions and (2) the inevitable mental phenomena when formal theories encounter concrete situations. Such limitations foreground the analysis of desire—it is a concept, on the one hand, entangled with preference yet irreducible to it; on the other hand, it is a mental type that manifests intentionality.

It is worth noting that a parallel idea has emerged in recent decades within the field of logic, where scholars have formally addressed preferences and desires dynamically (cf. [16,17]). Furthermore, these logical approaches link the concept of desire to various nuanced aspects, such as conditional desires, desire strength, and causal inference (see [18,19,20]).

In philosophical discourse, while desire is a familiar concept, a comprehensive and reasonable account remains challenging due to its diverse daily manifestations, e.g., wanting to drink, preferring tea, or desiring familial love; however, philosophical accounts aim to address its nature, causes, and variations in a unified manner. The following are several attempts in the previous literature.

Action-based theories propose desires as dispositions to act; as Anscombe mentioned, “the primitive sign of wanting is trying to get.” [21]. When we say someone desires something, we mean that she is disposed to act to bring it about. Whether such an act will succeed and such a desired object will become true does not hamper her desire for it. This relatively naive formulation “desiring p = being disposed to act so as to bring about p” fails to account for counterexamples, e.g., a gymnast prone to challenge difficult routines and is more likely to make mistakes. Then, it follows from the definition that she desires to miss. A more sophisticated version links desire to goal-directed actions—being disposed to act on beliefs likely to bring about p—but still faces issues, e.g., musicians attempting difficult pieces despite error risks, but we cannot say she desires to make mistakes. General critics question whether desire can be reduced to action dispositions.

On the contrary, pleasure-based theories emphasize hedonic aspects of desire: desiring p entails dispositional pleasure at its realization and displeasure at its negation [22]. Strawson argues that exercising desire is a prerequisite for consciousness, and consciousness inherently links desire to pleasure/displeasure, even in non-acting beings. However, others distinguish desire from pleasure ontologically, since it is more likely that the satisfaction or dissatisfaction of desire causes pleasure or displeasure. They are like causes and effects [23].

Alternative theories include good-based (desire as a pursuit of perceived goodness [24]), attention-based (desire as focus on valued outcomes [25]), and learning-based accounts (desire as reward-system adaptation [26]). Each theory easily faces counterexamples, and combining them risks contradictions. For instance, a worker motivated solely by survival lacks pleasure/goodness features, complicating desire attribution under pluralistic criteria; divergent theoretical accounts may classify this as either a “desire” (e.g., action-based theories) or exclude it (e.g., pleasure- or good-based theories).

Holistic approaches resolve these tensions by treating desire as a Wittgensteinian family resemblance concept. Following this line of thinking, we can adopt two primary forms: (1) Functionalism: Desire is an internal state type fulfilling multiple causal roles suggested by its various similarities, such as action, pleasure, attention, etc. [27]. (2) Interpretationism: An agent’s actions that align with these similarities can be rationally interpreted by observers as evidence of desire [28]. These approaches avoid requiring universal features, instead allowing contextual judgments. By focusing on sufficient (not necessary) conditions, they bypass counterexamples while maintaining conceptual coherence. This flexibility addresses measurement challenges and observer-dependent interpretations, offering a unified yet adaptable account of desire.

4. Preference or Intentionality: Which Is Fundamental?

Desires, understood through holistic accounts, allow intensity comparisons in three measurement approaches. The first is from the agent-first perspective, equating desire strength with the agent’s conscious awareness of their mental states, which is a psychologically grounded process. The second is the interpretationist–verification path: observers infer desires via backward induction of behavioral patterns and seek agent confirmation. This approach reflects reality as much as possible but is limited in formal theories due to extensive communication demands. The last is the interpretationist–non-verification method: observers rationalize desire hierarchies solely from agents’ action selection. Under ceteris paribus conditions, stronger desires deterministically govern actions when mutually exclusive options conflict. Nevertheless, this approach prioritizes formal applicability, avoiding psychological speculation by anchoring desire metrics to observable choice dispositions. By doing so, it also throws away dimensions.

Game theory posits preference as foundational, reflecting stable choice hierarchies (cf. [29]). As mentioned above, the expected utility is the expected pleasure or relief from suffering from the object, according to the older approach, or it is the degree to which the agent would be inclined to choose it, according to the modern approach. The preference-based account [30,31,32] argues, for example, “wanting something is preferring it to certain relevant alternatives, the relevant alternatives being those possibilities that the agent believes will be realized if he does not get what he wants” [30]5. This implies a preorder between possible worlds that include or exclude the desideratum. However, recent philosophical and cognitive science research challenges the primacy of preference-based approaches. Specifically, we propose the following counterargument:

• Proposition 1—Ontological Exclusion: Human cognition tends to treat either desire or preference as the more basic conative state, with one concept derived from the other. This is reflected in theoretical construction, as many existing theories posit that desires are fundamental and preferences are constructed from them, or vice versa, supporting this dichotomy (cf. [33,34]).
• Proposition 2—Ability Necessity: The grounding of preference or desire necessarily depends on agents possessing fundamental psychological capacities, particularly the capacity to experience or compare desire intensity. Neuroscientific findings [35] emphasize that emotional faculties (crucial for experiencing and gauging desires) are essential for forming advantageous preferences, which is also reflected in philosophical debate (cf. [36]).
• Proposition 3—Naturalistic Constraint: Constructing or storing a complete, transitive preference ordering over numerous desires or options is cognitively infeasible for human agents. Encoding even 300 desire-based facts into a total preference structure generates a combinatorial explosion, on the order of a billion billions at the least [37], thus exceeding natural cognitive limitations. By contrast, desires operate as individually weighted attractors whose motivational intensities fluctuate with context. Humans attend to only a small subset of these salient desires at any moment, steering decisions through simple heuristics rather than exhaustive pairwise comparisons. Consequently, a desire-based explanatory framework aligns far more plausibly with human cognitive constraints. Similar views are echoed in theories of bounded rationality [38] and the contextual construction of preference [39].
• Conclusion—supported by the three Propositions: From a realistic perspective, it is difficult for the human brain to understand and calculate pairwise preferences as primitives. Thus, desire emerges as a more plausible foundational element for understanding motivation and decision-making than abstract preference relations, resonating with broader neuroscientific and psychological research (cf. [40]).

Moreover, it is important to recognize that desire itself is inherently intentional [41], embodying intentionality through its purposive and directive nature. However, desire alone is neither uniquely sufficient nor entirely exhaustive as a driver of human action. Intentionality encompasses a much broader spectrum of mental states beyond mere “intentions,” each playing distinct yet interrelated roles in shaping behavior. Cognitive states such as beliefs and perceptions represent reality with a mind-to-world direction of fit, thereby informing agents about how the world is. Conversely, conative states, including desires and intentions, seek to change reality through actions aimed at achieving specific outcomes to make a world-to-mind fit [42]. Affective states, such as emotions and moods, serve evaluative and motivational functions, often prompting immediate behavioral responses. Sensory perceptions and mental images contribute essential experiential input and imaginative rehearsal, respectively, further shaping and informing intentions [43]. While desires significantly motivate behavior by establishing goals, intentions transform these desires into committed plans of action. Thus, rather than narrowly focusing solely on desire, our analysis adopts a broader conceptualization of intentionality, highlighting its comprehensive role in facilitating rational decision-making.

Thus, desire, or more broadly, intentionality, can serve as the underlying concept of a unified theory, as it captures the subjective aim or teleological drive behind actions, e.g., a “thirst” that motivates water-seeking behavior. By contrast, “preference”, on which game theory relies, is derivative; it represents structured expressions of desire, e.g., ranking water over wine when thirsty. Nevertheless, specific scenarios exist where adopting preferences as the primary construct significantly streamlines formal analysis.

In formal modeling, desire (or intentionality) faces challenges in the following respects. First is non-comparability: Although philosophical thought experiments may posit the comparability of desires, such comparisons remain confined to a single agent’s brain. When utility functions are explicitly employed to enable cross-agent comparisons—such as in welfare economics or collective decision-making—they inevitably confront epistemological obstacles. For example, it becomes problematic to compare one agent’s “thirst” directly with another agent’s “hunger.” While it is acknowledged that some approaches use utility functions strictly without cross-agent comparisons (e.g., ordinal utility frameworks), the critical challenges highlighted here pertain explicitly to contexts where such comparisons are made. The second challenge is dynamics: Desires are contextually contingent, meaning that environmental changes induce desire shifts; e.g., thirst will subside after drinking. This variability complicates the abstraction of stable payoff structures in strategic game models. Third is opacity: Internal states are always more complex to formalize than observable choices; this is also why revealed preference theory has become the basis of modern game theory.

Unlike desire, preference provides distinct advantages for formalization in coalitional frameworks. First is structural clarity: Representing preferences through transitive orderings (e.g., $A\prec C$ derived from $A\prec B$ and $B\prec C$) reduces modeling complexity by inducing a linear hierarchy over agents’ choices, thereby systematizing their decision-making processes. Second is stability: Preferences inherently incorporate a layer of abstraction, which distills fleeting desires into stable, rank-ordered priorities that persist across contexts, which is critical for strategic game analyses. Third is intersubjectivity: Compared with desires, members sharing their own preferences is more conducive to forming collective preferences (e.g., we all prefer policy X to Y)6. These shared preferences enable coalitions to coordinate strategies and describe sophisticated endogenous concepts, such as Nash equilibria.

Therefore, formal systems prioritize simplicity and stability, abstracting fluid psychological factors like desire, which are considered acceptable. Preference as a simple and definite starting point is more attractive for these purposes. Like one of the directions in Section 6, where we reintroduce preference into the standard coalition logic system. In contrast, philosophical theories of intentionality can serve as a background for the formal theory, and these intentional states enter into the discussion only when we try to question further the source of given preferences and when we try to apply formal theories to concrete situations. As von Neumann and Morgenstern put it, “every measurement—or rather every claim of measurability—must ultimately be based on some immediate sensation, which possibly cannot and certainly need not be analyzed any further. In the case of utility the immediate sensation of preference—of one object or aggregate of objects as against another—provides this basis” ([29], p. 16). The interplay reflects a division of labor: formal models operationalize preferences for predictive coherence, while philosophical theories preserve intentionality’s richness for interpretative depth.

5. Intentionality, Effectivity, and Coalitional Power

As Heyting pointed out, the formal study of logic does not need to concern itself with the significance of the truth or the falsehood of propositions. However, once logic is employed for specific problem-solving, a corresponding ontology is necessitated to underpin its reasonable interpretation [44]. The intrinsic motivation behind coalition logic lies in using modal logic to characterize key properties of coalitional ability in games, which naturally compels logicians to engage with ontological considerations alongside the development of formal systems. However, the emergence of coalition logic is not rooted in preference-based reasoning but in the formalization of coalitional power through the mathematical construct of effectivity functions [2]. These functions, defined as $E:2^N\to 2^{2^S}$ where N is the set of agents and S the outcome space, assign to each coalition $C\subseteq N$ the sets of outcomes $X\subseteq S$ that C can be enforced independently of others. In the formal semantics of coalition logic, an effectivity function7 $E\left(s\right)$ is associated with every state in a model, and $\left[C\right]\phi$ holds if the extension of $\phi$ is in $E\left(s\right)\left(C\right)$. Here, the ontology resides in the very definition of effectivity: a coalition has the power to constrain outcomes, irrespective of individual preferences.

To further illustrate, consider two distinct forms of effectivity: $\alpha$-effectivity and $\beta$-effectivity categorized by [45]. $\alpha$-effectivity captures a coalition C’s capacity to guarantee an outcome within a predefined set X, regardless of external players’ strategies. Formally, a strategic game form consists of a set of players N, a set of outcome states S, a set of strategies $A_i$ for each player, and an outcome function o mapping any combination of strategies for each player to an outcome state. Crucially, the strategies $A_C$ available to a coalition $C\subseteq N$, construed merely as a subset of all players, are assumed to be all combinations of the individual strategies of the members of C—all possible joint actions. Then $\alpha$-effectivity is defined as follows. Coalition C is effective for the set of outcomes X iff

$$\exists s\in A_C\forall s^{\prime }\in A_{-C}:o(s,s^{\prime })\in X$$

where $-C=N\setminus C$ is the complement of C, indicating that C can unilaterally enforce X through a fixed, proactive strategy. As in daily life, a legislative supermajority capable of passing a bill despite any opposition instantiates a coalition with $\alpha$-effectivity.

$\beta$-effectivity, weaker than $\alpha$, describes C’s ability to reactively ensure X by responding to the opponent’s strategies. Put formally:

$$\forall s\in A_{-C}\exists s^{\prime }\in A_C:o(s,s^{\prime })\in X$$

In such cases, C’s strategy is contingent on others’ actions, allowing flexibility and conditional enforcement. For instance, a climate coalition adjusts its policy preferences to counteract opposition tactics, thereby achieving its objectives, demonstrating $\beta$-effectivity.

As the above example illustrates, coalitions may exhibit $\alpha -$or $\beta -$effectively in practice, depending on their strategic posture. The core difference between these two kinds of effectiveness lies in the different ordering of existential and universal quantifiers, which distinguishes proactive enforcement of outcomes from reactive strategic adaptability. However, in both cases, $E\left(C\right)$ is defined through strategic quantifiers, not ordinal rankings. These functions encode what coalitions can strategically guarantee, not their motivations or rationales. Unlike utility functions, effectivity functions abstract away from agents’ individual inputs. That is to say, coalitional power, as captured by $E\left(C\right)$, is treated as a primitive input in coalition logic; the framework itself remains agnostic to the sources of such power. Similar to the preference case, intentionality becomes indispensable when addressing questions of why and how a coalition acts.

Consider $\alpha$-effectivity: for a coalition to unilaterally enforce outcomes, its members must intentionally coordinate their strategies. However, from a philosophical perspective, mere aggregation of individual actions does not suffice to constitute a coalition—it lacks shared intentional commitment as well as interdependency between their sub-goals [13]. Similarly, $\beta$-effectivity presupposes the ability to dynamically adjust strategies in response to opponents’ moves. This actually presumes agents can anticipate intentions and revise their own intentional states iteratively. These capacities are irreducible to effectivity functions; instead, for further explanations, they demand intentionality’s flexibility.

The preceding analysis reveals two distinct formal pathways for modeling strategic agency: one centered on effectivity functions, which captures what coalitions can enforce through $\alpha /\beta$-power, and another on utility functions, which encode individual behaviors as preference orders. While these frameworks differ in scope—coalitional power in cooperative contexts versus individual rationality in non-cooperative settings—they share a common limitation: neither explicates their primitive concepts. Yet in practice, both formalisms implicitly presuppose intentional states; they are abstractions serving different modeling purposes rather than fundamentally incompatible views. Consider the OPEC+ scenario in Table 2 again: while Russia and Saudi Arabia’s oil production decisions are enforced by their respective coalitional roles, their relationship remains fundamentally competitive, as revealed by diverging preferences in the matrix. In such cases, OPEC+ cooperation is driven not by a stable, overarching coalition force, but rather by adjustments contingent on market dynamics, which is a more accurate reflection of strategic reality. Crucially, intentionality remains the foundational layer unifying these formalisms: both coalitional power and individual preferences ultimately stem from agents’ desires, commitments, and goal-directed reasoning.

The relationship between these concepts can be visualized as a two-layer model, as shown in Figure 1.

The above conceptual delineation shows the close relationship among philosophy, cooperative and non-cooperative game theory, and coalition logic. Since intentionality grounds both individual preferences and coalitional power, integrating agency-theoretical insights into the framework of coalition logic is both inevitable and essential for its advancement. This sets the stage for Section 6, where we extend coalition logic with various operators, continuing the previous trajectory.

6. Not Merely the Sum of the Parts

6.1. Coalition Logic and Superadditivity

Following the philosophical line of thought, we delve more into a formal analysis tailored to multi-agent systems, specifically within a game-theoretical framework that has been the subject of extensive formal research (cf. [46]).

The classical work on reasoning about coalitional power in games is coalition logic (cf. [2]), as mentioned in Section 2. Recall that its main operator is $\left[C\right]\phi$, expressing that at the current state, the set of agents C can force $\phi$ to be the case. One of the characteristic axioms of coalition logic is superadditivity:

$$(\left[C_1\right]{\phi }_1\wedge \left[C_2\right]{\phi }_2)\to [C_1\cup C_2]({\phi }_1\wedge {\phi }_2),\mathrm{where}{C}_1\cap C_2=\varnothing$$

— two disjoint coalitions can always form a new coalition with the combined power of the two coalitions in the specific sense that if the latter can enforce ${\phi }_1$ and ${\phi }_2$, respectively, then the combined coalition can enforce ${\phi }_1\wedge {\phi }_2$ to be true simultaneously. Superadditivity in coalition logic can be explained by the representation theorem [2,47]: there is a one-to-one relationship between the effectivity functions (see the previous section) making up the logical (neighborhood) models of coalition logic and non-cooperative strategic game forms, in the sense that coalitional power corresponds to joint actions, i.e., combinations of individual actions, and that the effectivity functions captures $\alpha$-effectivity. In more detail, the effectivity functions associated in the models of coalition logic are required to have certain properties called the true playability properties8, and it can be shown [2,47] that an effectivity function E has the true playability properties if and only if $E\left(C\right)$ is exactly the set of sets X of outcomes for which C is $\alpha$-effective. There is no restriction on such combinations of joint actions. This means, for example, that coalition logic cannot characterize any new endogenous stability that arises from the mutual constraining relations between individuals within the new group.

On “the other side” of game theory, we find cooperative (or coalitional) games, where coalitional power is divorced from individual power. Reasoning about cooperative games has also been formalized using coalition-logic-like logics. Reference [5] develops two logical systems for reasoning about coalitional games, namely Coalitional Game Logic (CGL) and a modal variation, Modal Coalitional Game Logic (MCGL). It shows that superadditivity does not hold and shows that MCGL can express the core, the stable set, and the bargaining set in cooperative games.

From another perspective, [48] extends the Logic of Functional Dependence (LFD, cf. [49]) to the Logic of Preference and Functional Dependence (LPFD) and its hybrid extension (HLPFD) to explicate concepts such as Pareto optimality, Nash equilibrium, and core, differentiating various levels of stable relational patterns. It shows that superadditivity does not also hold in LPFD and HLPFD. This highlights the importance of relational patterns in defining collective agency, asserting that such agency can only be instantiated no lower than the effectiveness level of the core, which shares the same spirit with [1].

These two studies significantly expand the characterization of coalitional power in games beyond the confines of traditional coalition logic, offering a more nuanced portrayal of concepts essential for collective effectiveness, like the core. The primary distinction between them lies in their distinct approaches to intersecting various preorders and the varied definitions they employ for the cooperative effectiveness function.

6.2. Logical Directions

Building upon these foundational works, we look closely at formalizing certain relational patterns of collective agency. On the one hand, the notion of power in coalition logic is too weak, not requiring, e.g., stability in interactions. In a sense, coalitions have “too much” power; merely the existence of a joint action that the agents could happen to do is enough, even if it is not rational. A weak notion of power gives a strong logic, e.g., with the superadditivity axiom. On the other hand, it is at the same time too strong: in a precise sense (see the representation theorem), a coalition cannot have more power than the aggregate power of its members. Cooperative games solve both these problems, but only by removing any structure on the relationship between the power of individuals and coalitions (and between different coalitions). Here we are interested in logical systems in-between: with partial (“rational”) superadditivity, and where coalitions can have exogenous powers.

More specifically, we propose three logical directions.

6.2.1. Unstable Joint Actions

First, we focus on concepts that encapsulate the motivation and inherent stability of collective choices, thereby enriching our understanding of the dynamics at play in these strategic interactions. Technically, we propose stronger coalition modalities ${\left[C\right]}^s$, where ${\left[C\right]}^s\phi$ means that C has a stable joint action that will ensure $\phi$: there is a joint action by C that is stable, and no matter which actions the other agents chose, $\phi$ will be true. To this end, we extend the models of coalition logic with preference relations. For the notion of stability, here we draw from both non-cooperative and cooperative game theory. On the one hand, the notion of Nash equilibrium gives us a notion of stability for a particular type of joint action—a joint action by the grand coalition (the coalition consisting of all agents). In the non-cooperative game setting of coalition logic, such a joint action uniquely identifies the outcome state. In our definition, we are, however, interested in the stability of joint action by a (possibly proper) subset of the grand coalition, effectively identifying a set of outcomes (corresponding to the outcomes resulting from the quantification “no matter which actions the other agents chose”). The notion of the core (and related notions such as stable sets and the bargaining set) identifies stable joint actions by sub-coalitions, but assumes that those joint actions pick out single outcomes rather than sets of outcomes. We need something “in-between”. To this end, we identify stability for two types of agents. First, assume that cautious agents only deviate from a joint group action picking out a set of outcomes X if they are guaranteed to be better off, i.e., if every outcome in the new set of outcomes is better than every outcome in X. Second, we assume that bold agents only deviate if they can possibly be better off, i.e., if there is at least one outcome in the new set of outcomes in which the agent is better off. Note that we are still considering $\alpha$-effectivity: the preference relations are only used to define stability, not to compare two different stable joint actions. Also note that under either of these assumptions, stability for the grand coalition consisting of all agents corresponds to a Nash equilibrium.

Thus, ${\left[C\right]}^s\phi$ expresses that C has a stable choice to enforce $\phi$; ${\left[N\right]}^s\phi$ (N is the grand coalition) that there is a Nash equilibrium $\phi$-outcome; ${\left[N\right]}^s\top$ that there is a Nash equilibrium outcome; and ${[\varnothing ]}^s\phi$ that all outcomes are $\phi$-outcomes (observe that the empty coalition cannot be unstable, so ${[\varnothing ]}^s\phi$ is equivalent to $[\varnothing ]\phi$).

As an example, consider the setting in Table 2 again. Assume weakly cautious agents. We have that $\left[\right\{SA,RU\left\}\right](443\vee 224)$ (here we use atomic propositions to represent the payoff combinations): SA and RU have a joint action, namely reduction–reduction, that will ensure that the outcome is either 443 or 224. However, this joint action is not stable: SA has an incentive to deviate and overproduce, which leads to the outcome set $\{411,313\}$. Thus, we have that $\neg {\left[\{SA,RU\}\right]}^s(443,224)$. Similarly, $\left[\right\{SA,RU,US\left\}\right]443$, but $\neg {\left[\{SA,RU,US\}\right]}^{s}443$—the joint action reduction–reduction–underproduction is not a Nash equilibrium.

Superadditivity is violated for both cautious and bold agents. From the example formulas just given, it can be seen that N-maximality ($\neg {[\varnothing ]}^s\neg \phi \to {\left[N\right]}^s\phi$) also fails—all outcomes might be $\phi$-outcomes, but not all of them might be Nash equilibria. For the same reason, ${\left[C\right]}^s\top$ fails. Finally, coalition-monotonicity (${\left[C\right]}^s\phi \to {\left[D\right]}^s\phi$ for $C\subseteq D$) fails as well: while a super-coalition can always perform the joint actions of one of its sub-coalitions, those actions might not be stable for the super-coalition, even if they are stable for the sub-coalition.

6.2.2. Exogenous Power

As argued above, coalitional power in coalition logic is intuitively also too strong: coalitional power is derived from individual power, ruling out modeling exogenous coalitional power, such as the powers of the university or the police department. A hiring committee, or the board of a company, has certain intrinsic powers that are not directly derived from the members of those groups.

Let us, first, discuss in which sense that intuition is correct, and, second, make a proposal for dealing with it.

As mentioned, the intuition comes from the combination of the semantics of the coalitional ability modalities in coalition logic and the representation theorem. The latter shows that coalitional ability in coalition logic corresponds exactly to $\alpha$-effectivity in games, defining the ability of a coalition in terms of joint actions composed of individual actions. So, on a semantic level at least, the intuition is true. But is it picked up on the syntactic level—is it manifested as theorems or axioms? In a certain precise sense, the answer is actually no. In fact, the following can be shown9:

If $D\subseteq C$ and $\left[C\right]\phi$ and $\psi$ are consistent, then $\left[C\right]\phi \wedge \left[D\right]\psi$ is consistent.

In other words, we can always extend the coalitional ability of a sub-coalition D without affecting the ability of a super-coalition C.

The extension of the ability of a coalition D might, however, be inconsistent with other groups disjoint with D. We have the following:

If $C\cap D=\varnothing$, then $\left[D\right]\phi \to \neg \left[C\right]\neg \phi$ is valid.

In other words, extending the coalitional ability of D might not be consistent with the ability of other, disjoint, coalitions. Why? It follows from the superadditivity axiom for standard coalition ability operators together with the liveness axiom $\neg \left[C\right]\perp$ (where ⊥ is a contradiction): if $\left[C\right]\neg \phi$ and $\left[D\right]\phi$ were the case, then $[C\cup D](\phi \wedge \neg \phi )$ would follow by superadditivity, which contradicts liveness.

Thus, superadditivity is also a barrier to exogenous power, and weakening it will allow for more coalitional abilities. One extreme option is to remove it altogether, like in [50], which generalizes the models of coalition logic and de-couples them from the representation theorem, in particular by removing the assumption that agents can make independent choices. As we have argued above, however, we do not want to completely divorce coalitional ability from individual ability. But we now have a tool for controlling restricted superadditivity, namely, the notion of stability introduced in the previous subsection. We propose using that also to weaken superadditivity to allow exogenous power.

6.2.3. Coalitional Structure

Third, we propose adding coalitional structure to coalition logic. Nearly all existing logical formalizations of coalitional power treat a coalition merely as a set of agents with no internal structure. We argue that formal logic is well-suited to also reason about such structures. Examples here range from a flat structure, where each individual makes their own decisions, to hierarchical structures where some individuals have power over others, and more abstractly, to social choice mechanisms such as voting. Let us explore two concrete variants.

Consider the case that any majority in a coalition can decide what the coalition does. This effectively extends the power of a coalition to the power of any coalition it is contained in, up to twice its size. Let us introduce a new operator ${\left[C\right]}^m$, capturing the power of the majority. In other words, ${\left[C\right]}^m\phi$ holds iff there is a D such that $C\subseteq D$, $\left|D\right|\le 2\ast \left|C\right|$, and $\left[D\right]\phi$ holds, where ${\left[C\right]}^m$ is the coalitional ability under the majority assumption and $\left[C\right]$ is the standard coalitional ability.

Observe, yet again, that superadditivity fails: let $C\cap C^{\prime }=\varnothing$ and ${\left[C\right]}^m\phi$ and ${\left[C^{\prime }\right]}^m\psi$. We thus have that $\left[D\right]\phi$ and $\left[D^{\prime }\right]\psi$ for $D\supseteq C$ and $D^{\prime }\supseteq C^{\prime }$, but we do not necessarily have that $D\cap D^{\prime }=\varnothing$ and thus not necessarily that $[D\cup D^{\prime }](\phi \wedge \psi )$ or ${[C\cup C^{\prime }]}^m(\phi \wedge \psi )$.

Hierarchical structures can be modeled in a very general way by a directed acyclic graph (DAG) like the one in Figure 2, where the idea is that an arrow from an agent a to agent b means that a has direct power over b. Hierarchical coalitional ability operators ${\left[C\right]}^h$ can be defined as follows: ${\left[C\right]}^h\phi$ holds iff $[C\ast ]\phi$ holds, where $C\ast$ is the set of all agents reachable from C by the reflexive transitive closure of the relation in the hierarchy graph, and $[C\ast ]$ is the standard coalitional ability operator $\left[D\right]$ for $D=C\ast$. By now, it should not be a surprise that superadditivity also fails in this case. In the example in Figure 2, a has the power to instruct both b and c, both b and c have power over d, and c also has power over e. We might have that ${\left[\left\{b\right\}\right]}^h\phi$ because $\left[\right\{b,d\left\}\right]\phi$ and ${\left[\left\{c\right\}\right]}^h\psi$ because $\left[\right\{c,d,e\left\}\right]\psi$, but not necessarily that ${\left[\{b,c\}\right]}^h(\phi \wedge \psi )$ since $\left\{b\right\}\ast \cap \left\{c\right\}\ast \ne \varnothing$. Said another way, there exist games such that $\left[\right\{b,d\left\}\right]\phi$, $\left[\right\{c,d,e\left\}\right]\psi$ and $\neg \left[\right\{b,c,d,e\left\}\right](\phi \wedge \psi )$ holds, and we thus have ${\left[\left\{b\right\}\right]}^h\phi \wedge {\left[\left\{c\right\}\right]}^h\psi$ but not necessarily ${\left[\{b,c\}\right]}^h(\phi \wedge \psi )$. In many—but far from all—cases, organizational charts are trees (DAGs where each node has at most one parent), and it can be shown that

When the hierarchical graph is a tree, then the superadditivity axiom for ${\left[C\right]}^h$ holds.

7. Final Remarks

In this paper, we bridge philosophical theories of agency with formal frameworks in game theory and coalition logic by examining grounded concepts such as intentionality, preference, and coalitional power. This analysis intertwines the complexity of multi-agent interactions, game theory, coalition formations, and agency. It demonstrates that the philosophical accounts of agency can not only underpin formal research but also navigate ways of formalizing logical systems.

In particular, we proposed three logical directions, tied together by a red thread: weakening of superadditivity.

The present work provides a long-term research program investigating the interplay between philosophy and logic. Instead of resolving existing debates, the purpose of this paper is to reframe questions and adopt an interdisciplinary perspective to engage both philosophical and technical audiences. The logical formalism introduced here is intentionally preliminary, serving as a conceptual scaffold for future investigations. Our forthcoming paper will expand the first methodological direction proposed in Section 6, integrating more technical details to rigorously formalize the dynamics between group agency and coalitional power.