Alliance Instability in Intimate Organizations: A Cross-Disciplinary Analysis from Game Theory, Institutional Economics, and Anthropology

The family is one of humanity's oldest organizational forms, yet it is also one of the most unstable arenas for alliance structures. Why do intimate groups bound by blood and marriage find it harder to maintain stable cooperative relationships than commercial organizations? Why are alliances between certain members always brief and fragile? This article attempts to move beyond the framework of everyday language, employing coalition game theory, institutional economics, and anthropological analytical tools to deconstruct the deep structure of power dynamics within intimate organizations. This is not a narrative about any specific family, but rather an academic exploration of a universal social phenomenon.

1. Intimate Organizations as Arenas for Coalition Games

Traditional game theory primarily studies non-cooperative games, assuming each participant makes independent decisions to maximize individual utility. However, many real-world decision-making arenas -- including families -- are better analyzed using cooperative games or coalition games.^[1] In coalition games, participants can form various subsets (coalitions) to act collectively for greater total payoffs, but the core question remains: how should the value created by a coalition be distributed?

Nobel laureate Lloyd Shapley proposed the Shapley Value, which provides an axiomatic answer: each participant's share should equal the expected value of their marginal contribution across all possible orderings of joining.^[2] The mathematical expression is as follows:

φ_i(v) = Σ_S⊆N\{i} [ |S|! (n-|S|-1)! / n! ] × [ v(S∪{i}) - v(S) ]

Here, φ_i is the Shapley value of participant i, v(S) is the value that coalition S can create, and n is the total number of participants. The core insight of this formula is: what a person "deserves" depends on their marginal contribution to different coalitions, not on their identity or status.

However, the dilemma in intimate organizations is that "marginal contribution" is extremely difficult to measure objectively. Whose emotional labor is more important? Whose economic contribution is more critical? Who puts more effort into maintaining family harmony? These questions have no market prices for reference, leading each member to hold systematically biased perceptions of their own "deserved share" -- what psychologists call "self-serving bias."^[3] When everyone believes their Shapley value is being underestimated, the stability of the alliance is fundamentally threatened.

2. The Minimum Winning Coalition and Power Restructuring in Intimate Organizations

Political scientist William Riker's Minimum Winning Coalition theory posits that in zero-sum or near-zero-sum distributive situations, coalitions have an inherent tendency to shrink to "just enough to win."^[4] Why? Because the larger the coalition, the smaller each member's share. Rational participants tend to exclude "surplus" allies, retaining only the minimum number of members necessary to maintain their advantage.

Applying this logic to intimate organizations, we can observe a profound structural tension: when multiple potential "winning coalitions" exist within an organization, members continuously engage in implicit coalition restructuring.

Consider a three-person organization (denoted A, B, C), where any two-person coalition is sufficient to "win" (for example, forming a majority in family decisions). According to Riker's theory, this system is inherently unstable:

• If the A-B coalition forms and C is excluded, C has an incentive to offer A or B more favorable distribution terms
• Suppose C successfully recruits B to form a B-C coalition, then A is excluded
• A then offers terms to C, forming an A-C coalition...

This dynamic of "coalition rotation" is extremely common in intimate organizations, yet it is often understood by the parties involved as "interpersonal problems" or "personality conflicts" rather than as structurally determined.^[5]

What is even more noteworthy is that "winning" in intimate organizations is often not a clear voting outcome, but rather a contest over "attention," "emotional resources," or "decision-making influence." This makes the formation and dissolution of coalitions more covert, but the dynamic structure still follows Riker's logic.

3. Repeated Games, Reputation, and the Evolution of Cooperation

If interactions within intimate organizations were one-shot games, Riker's pessimistic prediction might hold: alliances would be perpetually unstable, and betrayal would be the rational choice. However, the hallmark of intimate organizations is precisely the repeated game -- members must coexist over the long term, and the shadow of the future profoundly influences present behavior.^[6]

Nobel laureate Robert Aumann's Perfect Folk Theorem proves that in infinitely repeated games, as long as participants are sufficiently patient (discount factor δ sufficiently close to 1), any "individually rational" and "feasible" payoff vector can be sustained as a Subgame Perfect Equilibrium.^[7] In other words, cooperation is theoretically sustainable, as long as the "future" matters enough.

This theorem has profound institutional implications. Under the condition where discount factor δ < 1, cooperative equilibrium requires:

π_cooperation / (1 - δ) > π_betrayal + δ × π_punishment / (1 - δ)

Here, π_cooperation is the per-period payoff from cooperation, π_betrayal is the one-time gain from betrayal, and π_punishment is the payoff during punishment. This inequality tells us: the stability of cooperation depends on three factors: (1) how much the future is valued (δ), (2) the one-time gain from betrayal, and (3) the severity of punishment.

In intimate organizations, all three factors have unique complexities:

1. Heterogeneity of discount factors: Different members value the "future" to different degrees. Older members may be more concerned with present dignity, while younger members may be more willing to invest in long-term relationships.
2. Ambiguity of betrayal payoffs: "Betrayal" in intimate organizations is often ambiguous -- is it an unacknowledged act of service? A careless remark? This ambiguity makes the very definition of "betrayal" a focal point of contention.^[8]
3. The punishment dilemma: The cost of implementing "punishment" in intimate organizations is extremely high, because punishment often simultaneously harms the punisher. This is why the Tit-for-Tat strategy has limited effectiveness in intimate organizations.^[9]

4. The Principal-Agent Problem and Information Asymmetry

The tensions within intimate organizations can also be understood through the lens of the Principal-Agent Problem.^[10] In traditional economic analysis, a principal hires an agent to perform tasks, but due to information asymmetry, the agent may pursue their own interests rather than the principal's objectives.

In intimate organizations, "principal" and "agent" relationships are often implicit and multifaceted. Take the extended family as an example:

• Elders may see themselves as "principals," believing younger members should conform to their values and expectations
• In in-law relationships, the birth family may view marriage as an "investment," expecting emotional or material "returns"
• The allocation of caregiving responsibilities involves defining who is the "principal" (the person needing care) and who is the "agent" (the caregiver)

The problem is: principal-agent relationships in intimate organizations have never been explicitly contracted. This leads to fundamental disagreements among parties regarding rights and obligations. Economist Oliver Hart's Incomplete Contract Theory points out that when contracts cannot cover all possible scenarios, the allocation of "residual control rights" becomes critical.^[11] In intimate organizations, the struggle over "residual control rights" -- who has the right to make decisions in scenarios not covered by the contract? -- is precisely the root of many conflicts.

Information asymmetry further exacerbates this problem. Different members' understanding of each other's contributions, motivations, and feelings is necessarily incomplete. When A believes their contributions are not seen by B, while B considers A's contributions to be taken for granted, both parties feel aggrieved. This is not a moral problem, but a product of the information structure.^[12]

5. The Family as a Common Pool Resource

Nobel laureate Elinor Ostrom is renowned for her research on the governance of Common Pool Resources (CPR).^[13] CPRs are characterized by: (1) difficulty of exclusion, and (2) rivalrousness -- one person's use reduces the quantity available to others. Classic examples include fisheries, irrigation systems, and forests.

Many "resources" within intimate organizations also exhibit CPR characteristics:

• Emotional attention: The attention and emotional energy of core members (such as elders or primary caregivers) is finite; when one person receives more, others receive less
• Economic resources: Family finances, inheritance, gifts, etc.
• Decision-making influence: The right to speak on major family decisions
• Symbolic capital: "Face," recognized contributions, status in the eyes of outsiders

Ostrom's research found that sustainable governance of CPRs requires meeting eight "design principles," including: clearly defined boundaries, rules adapted to local conditions, participation of users in rule-making, effective monitoring mechanisms, graduated sanctions, conflict resolution mechanisms, minimal external interference, and nested governance structures.^[14]

Examining intimate organizations through this framework, we can identify many structural reasons for "governance failure":

1. Ambiguous boundaries: Who constitutes "us"? Do in-laws count as "full members"? The answer to this question often varies by time and circumstance
2. Unclear rules: Intimate organizations rarely have an explicit "family constitution"; most rely on implicit expectations and traditions
3. Monitoring difficulties: Who determines whether a member has "violated the rules"? Without an external arbiter, monitoring itself becomes a source of conflict
4. High cost of sanctions: Implementing sanctions in intimate organizations is extremely costly, often "harming the enemy by a thousand while losing eight hundred oneself"

6. Role Conflict and the Structural-Functionalist Perspective

Structural-Functionalism in sociology provides another analytical perspective. Talcott Parsons argued that the stable operation of social systems requires clear differentiation and integration of different roles.^[15] Each role has its "function," and the relationships between roles constitute the social structure.

However, the core challenge facing modern intimate organizations is precisely Role Conflict -- when a person simultaneously assumes multiple roles whose expectations contradict each other, conflict becomes inevitable.^[16]

Consider the modern family structure: a woman may simultaneously be:

• Her husband's wife (expected to be loyal to the nuclear family)
• Her in-laws' daughter-in-law (expected to show respect and deference)
• Her own parents' daughter (expected to maintain ties to the birth family)
• Her children's mother (expected to prioritize her children's interests)
• A professional in the workplace (expected to commit to her career)

The expectations of these roles frequently conflict. When resources are limited (time, energy, emotion), fulfilling one role's expectations often means failing to meet another's. This is not a personal "failure," but structurally determined.^[17]

What is even more noteworthy is that role expectations themselves are products of history and culture, yet they are often treated as "natural" standards. When members from different generations and cultural backgrounds hold different understandings of role expectations, conflict becomes unavoidable.

7. The Anthropological Perspective on Kinship Systems: Levi-Strauss and Alliance Theory

French anthropologist Claude Levi-Strauss's kinship system theory provides a deeper structural perspective.^[18] He argued that the kinship systems of human societies are essentially "exchange systems" -- through marriage, groups exchange women (or men) to establish alliance relationships.

The "elementary structure of kinship" proposed by Levi-Strauss encompasses four basic roles:

Brother ↔ Sister ↔ Sister's Husband ↔ Sister's Son

This "atomic structure" reveals the core tension of kinship systems: every marriage simultaneously creates both alliance and division. When a person moves from the "birth family" to the "marital family," she simultaneously belongs to two systems whose interests are not necessarily aligned.^[19]

From this perspective, tensions within intimate organizations are not products of modern society but structural features of human kinship systems. Any attempt to integrate an "outsider" into an intimate group must grapple with the problem of "divided loyalty."

8. Historical Cases: Cross-Cultural Comparison of Kinship Structures

8.1 China's Traditional "Three Obediences and Four Virtues" and the Patrilineal System

The traditional Chinese kinship system is centered on patrilineal descent, where women are transformed from "outsiders" into members of the husband's family through marriage. The "Three Obediences" -- obey the father before marriage, the husband after marriage, and the son after the husband's death -- reflect precisely the transfer of a woman's "belonging" at different life stages.^[20]

However, this system also created inherent structural tensions. Anthropologist Margery Wolf proposed the concept of the "uterine family" in Women and the Family in Rural Taiwan: a woman's status in the husband's family depends on her bond with her son, not her relationship with her husband.^[21] This means:

• A woman's interest lies in strengthening the emotional bond with her son
• This may conflict with the interests of her son's spouse
• The tension between two generations of women is structurally determined, not a personal problem

From a game-theoretic perspective, this is an "intertemporal commitment problem": a young woman entering the husband's family starts in a weak position, but through raising her son, she expects to gain status and care in old age. However, once the son marries, whether this "commitment" can be honored depends on complex alliance dynamics.^[22]

8.2 Japan's "Marrying In" System and Primogeniture

Japan's traditional yomeiri (marrying in) system is likewise based on the patrilineal system but possesses distinctive structural features.^[23] The primogeniture principle means that the eldest son is obligated to live with his parents and inherit the family business, but also enjoys priority inheritance rights. This creates a clear allocation of "residual control rights": the eldest son (and his spouse) will inherit the family's core resources, while other children are expected to establish branch families (bunke).

In a sense, this system resolves the instability of the "minimum winning coalition" -- by pre-designating the "winner," it reduces the space for coalition restructuring. However, it also creates new tensions:

• The eldest son's spouse bears caregiving responsibilities but may diverge from the in-laws in values and lifestyle
• Other children may question the fairness of the distribution
• In modern society, women's economic independence undermines the exchange logic of "marrying in"^[24]

8.3 The Evolution of the Western Nuclear Family

The transition in Western societies from the extended family to the nuclear family is often viewed as a hallmark of "modernization."^[25] Historian Lawrence Stone traced this transformation in The Family, Sex and Marriage in England 1500-1800: from kinship networks as economic and political units to the "intimate family" based on emotional bonds.^[26]

The nuclear family model to some extent "resolves" the alliance instability of extended families -- by narrowing the family's boundaries, it reduces the number of participants requiring coordination. However, economist Gary Becker's research indicated that the nuclear family also has its vulnerabilities:^[27]

• Reduced risk-sharing capacity (no longer having extended kinship networks as a safety net)
• Outsourcing of caregiving functions (requiring market or state-provided services)
• Diminished intergenerational knowledge transmission

What is even more noteworthy is that the nuclear family has not eliminated alliance dynamics, but merely displaced them to different arenas -- for example, co-parenting after divorce, step-family dynamics in remarried families, and so on.^[28]

9. An Integrated Model: Alliance Dynamics in Intimate Organizations

Synthesizing the above analysis, we can construct an integrated model for understanding alliance dynamics within intimate organizations. Let N = {1, 2, ..., n} be the set of organizational members. For any coalition S ⊆ N, define its "characteristic function" v(S) as the total value that coalition can obtain (including material resources, decision-making influence, emotional satisfaction, etc.).

The stability of a coalition depends on whether a "core" exists -- that is, whether no sub-coalition has an incentive to break away and allocate independently. Mathematically, an allocation (x₁, x₂, ..., x_n) belongs to the core if and only if:

∀ S ⊆ N: Σ_i∈S x_i ≥ v(S)

However, the core of many games is empty, meaning no stable allocation exists.^[29] We hypothesize that the coalition game in intimate organizations possesses the following characteristics, making its core more likely to be empty:

1. Weakened superadditivity: In some cases, v(S∪T) < v(S) + v(T) -- the value of two coalitions merging is actually less than the sum of their separate values (for example, due to internal friction and coordination costs)
2. Divergent value assessments: Different members hold inconsistent perceptions of v(S)
3. Existence of outside options: In modern society, members can "exit" intimate organizations, which alters negotiation leverage

When the core is empty, the organization's functioning relies on other mechanisms: norms, traditions, emotional bonds, or iterative negotiation. These mechanisms may maintain surface-level stability, but the underlying structural tensions persist.

10. Implications for Institutional Design

If the tensions within intimate organizations are structurally determined, then efforts to "improve relationships" cannot remain at the individual level alone (for example, asking someone to "be more empathetic" or "be more willing to communicate") but must consider institutional design.^[30]

Based on the above analysis, I propose the following institutional design principles:

1. Make implicit contracts explicit: Many conflicts stem from divergent understandings of "who should do what." Making implicit expectations explicit -- even if the process is painful -- helps reduce misunderstandings. This corresponds to Ostrom's principle of "clear rules."^[31]
2. Establish credible external arbitration: Intimate organizations have difficulty effectively implementing monitoring and sanctions internally. Introducing credible external arbiters (whether trusted relatives, professional mediators, or cultural norms) can reduce the costs of self-enforcement.^[32]
3. Design exit options: When the costs of remaining in the organization are too high, members need dignified exit paths. This is not encouraging division, but ensuring that those who stay have genuinely chosen to stay rather than being trapped.^[33]
4. Recognize the role of structure: When members understand that tensions are "structural" rather than "personal," they are better able to externalize problems, reduce moral blame, and focus on solutions. This is one of the core contributions of this article.
5. Regular renegotiation: The environment of intimate organizations is constantly changing (members' ages, health, financial circumstances, external opportunities). Static "agreements" cannot adapt to dynamic realities, necessitating the establishment of mechanisms for periodic renegotiation.^[34]

11. Conclusion: Moving Beyond the "Whose Fault" Mindset

Tensions within intimate organizations are a universal phenomenon in human society, yet we often lack the appropriate vocabulary to discuss them. Everyday language tends to attribute problems to specific individuals ("she's too domineering," "he's not considerate enough"), while ignoring structural factors.

This article's analysis attempts to offer a different framework: the alliance instability within intimate organizations is largely a product of game structure, information asymmetry, role conflict, and historical institutions. This is not about "exonerating" anyone, but about moving beyond the "whose fault" mindset toward a more constructive question: how can we design better institutional arrangements to manage these structural tensions?

From coalition games to common pool resources, from the principal-agent problem to kinship system theory, analytical tools from different disciplines reveal the same deep structure: cooperation in intimate relationships is precious but fragile, requiring continuous maintenance, clear rules, and a sober awareness of structural constraints.^[35]

This is perhaps not a comforting conclusion. But as Ostrom discovered in her research on common pool resources: understanding the structure of a problem is the first step toward solving it. When we no longer view tensions in intimate organizations as someone's "fault" but instead see them as institutional challenges to be collectively addressed, we can finally begin a constructive dialogue.^[36]