Fair Coalition: Concepts & Mechanisms

• Fair Coalition is defined as coalition formation models where cooperation is guided by explicit fairness conditions in payoff allocation, stability, or domination structures.
• It involves a variety of allocation methods, including equal, proportional, dual, and Shapley value rules, applicable to wireless cooperation and cost-sharing settings.
• In graph theory, fair coalition characterizes pairs of non-fair dominating sets whose union forms a fair dominating set, reflecting structural regularity.

“Fair coalition” denotes a family of coalition concepts in which admissible cooperation is constrained by an explicit fairness condition. In transferable-utility cooperative games, fairness usually refers to how coalition gains or shared costs are allocated among participants while maintaining stability; in hedonic coalition formation, it is tied to the absence of profitable group deviations; in graph theory, it denotes a pair of non-fair dominating sets whose union is a fair dominating set (0802.2159, Chau et al., 2020, Woeginger, 2012, Alikhani et al., 20 Jul 2025). The unifying theme is that coalition formation is evaluated not only by aggregate surplus, but also by distributional regularity, incentive compatibility, or structural symmetry.

1. Conceptual scope

In cooperative wireless and networked systems, fair coalition models are typically formulated as coalitional games with transferable utility, where a characteristic function $v(S)$ maps each coalition $S$ to an aggregate payoff that can then be divided among members. In the wireless mesh network setting, a coalition is “fair” when the total payoff is maximized and the aggregated payoff is allocated so that each service provider gets at least what it could earn alone and no subset has an incentive to break away; the paper studies dual payoff and Shapley value as the relevant allocation concepts (Lu et al., 2014). In cost-sharing hedonic games, the primitive object is instead a payment rule $p_i(G)$, with utility defined by

$u_i(G)=C_i-p_i(G),$

so coalition preferences come from cost reductions relative to singleton operation (Chau et al., 2020).

A different but related tradition studies stability directly in terms of partitions. In hedonic coalition formation, a partition $\pi$ is core stable if there is no coalition $S$ such that every player in $S$ strictly prefers $S$ to $\pi(i)$, the coalition containing $i$ in $S$0 (Woeginger, 2012). This notion does not prescribe a particular allocation rule; instead it characterizes fairness through coalition-wise immunity to blocking deviations.

Graph theory uses the term in a sharply different sense. If $S$1 is a $S$2-fair dominating set when every $S$3 satisfies $S$4, then a fair coalition is a pair of disjoint sets $S$5 such that neither $S$6 nor $S$7 is a fair dominating set, but $S$8 is (Alikhani et al., 20 Jul 2025). This suggests that “fair coalition” is not a single universal solution concept, but a domain-specific label for coalition structures constrained by a fairness regularity.

2. Allocation principles inside coalitions

The most explicit payoff-level notion appears in wireless cooperation. In virtual-MIMO coalition formation, the coalition utility $S$9 is the total achievable rate over a frame after subtracting the power cost of intra-coalition information exchange, and two intra-coalition allocation rules are proposed. Equal-share fairness assigns

$p_i(G)$0

while proportional fairness assigns

$p_i(G)$1

The first rule splits the cooperation surplus symmetrically; the second makes the surplus contribution-aware through stand-alone utilities (0802.2159).

Wireless mesh networking uses a broader menu of fairness mechanisms. For a grand coalition of service providers, the paper derives a dual-payoff allocation from the LP dual variables and also computes the Shapley value. The dual payoff is explicitly constructed to lie in the core, while the Shapley value is justified by efficiency, symmetry, dummy, and additivity; in the numerical example, both allocations lie in the core region (Lu et al., 2014). This is a clear instance where fairness is not identified with equal splitting: one rule is usage-based through shadow prices, the other contribution-based through marginal contributions.

Cost-sharing coalition formation generalizes these ideas. Four “fair” mechanisms are studied: $p_i(G)$2 and the bargaining-based rules

$p_i(G)$3

with egalitarian-split and Nash bargaining coinciding in the model. Equal-split is symmetric in payments, proportional-split scales by outside options $p_i(G)$4, and egalitarian/Nash equalize utility gains rather than payments (Chau et al., 2020). A related sharing-economy analysis studies the same mechanisms together with usage-based cost sharing, where each facility cost is split among the users who actually use it (Chau et al., 2015).

Recent work extends fairness triggers into coalition dynamics themselves. A 2026 split-merge model defines fairness through coalition-restricted Shapley values $p_i(G)$5 and treats negative Shapley values as fairness violations; the aggregate fairness deficit is

$p_i(G)$6

so fairness is internalized as a state variable of the coalition-formation process rather than as a post hoc sharing rule (Zhu et al., 17 Mar 2026).

3. Stability and coalition-formation mechanisms

Different literatures operationalize fair coalitions through different stability notions. In hedonic games, the central benchmark is core stability: no coalition can block the current partition by making all its members strictly better off (Woeginger, 2012). In cost-sharing hedonic games, a stable coalition structure is defined by the absence of a blocking coalition $p_i(G)$7 such that every $p_i(G)$8 strictly prefers $p_i(G)$9 to its current coalition, which the paper identifies with strong Nash equilibrium in the induced hedonic game (Chau et al., 2020).

Wireless coalition formation uses local structural moves instead of direct core constraints. A family of coalitions $u_i(G)=C_i-p_i(G),$0 merges when

$u_i(G)=C_i-p_i(G),$1

and a coalition splits when the inequality is reversed. The resulting partition is always $u_i(G)=C_i-p_i(G),$2-stable, and if internal strict superadditivity together with an external incompatibility condition hold, the partition is strictly $u_i(G)=C_i-p_i(G),$3-stable, unique, and socially optimal (0802.2159).

Several papers connect fairness axioms to guaranteed stability. In “Solidarity to achieve stability,” a sharing rule satisfies solidarity when, for $u_i(G)=C_i-p_i(G),$4, if one incumbent agent is worse off in $u_i(G)=C_i-p_i(G),$5 than in $u_i(G)=C_i-p_i(G),$6, then no other incumbent agent can be strictly better off. The paper shows that solidarity is equivalent to endowment monotonicity plus consistency, and that a sharing rule satisfies solidarity if and only if every induced coalition formation problem is non-circular; hence the induced core is always non-empty (Alcalde-Unzu et al., 2023). This is a direct theorem linking a distributional fairness axiom to coalition stability.

Algorithmic dynamics also differ. The decentralized cost-sharing algorithm “Coln-Form” uses deferred-acceptance-style proposals over coalitions of size at most $u_i(G)=C_i-p_i(G),$7, and convergence follows from the absence of cyclic preferences under equal-split, proportional-split, egalitarian, and Nash bargaining (Chau et al., 2020). A task-driven multi-UAV coalition mechanism uses Shapley values as utilities inside coalitions and a marginal-utility preference order; the resulting coalition-formation game is an exact potential game with a Nash equilibrium solution (Lu et al., 2024). The 2026 Shapley-fair split-merge dynamics converge in finite time to Shapley-Fair and Merge-Stable partitions, or to value-preserving cycles in the invariant set identified by a vector Lyapunov function and discrete-time LaSalle analysis (Zhu et al., 17 Mar 2026).

4. Efficiency, welfare, and the fairness–performance trade-off

The literature does not support a single efficiency implication of fairness. In some settings, fairness and efficiency align through the grand coalition; in others, fairness requires smaller coalitions. In virtual-MIMO wireless transmission, cooperation is costly because users must first exchange data, and the coalition value is

$u_i(G)=C_i-p_i(G),$8

Because cooperation cost may exceed the slot power constraint, the game is generally non-superadditive and the core is often empty; the grand coalition is typically infeasible or inefficient. The merge-and-split algorithm therefore yields smaller stable coalitions, and simulations show that for $u_i(G)=C_i-p_i(G),$9 users the average individual utility improves by about $\pi$0 relative to non-cooperation (0802.2159).

The opposite conclusion appears in wireless mesh networking, where service providers pool nodes and spectrum. There the game is super-additive, the grand coalition maximizes aggregated payoff, and dual payoff yields a core allocation. In the three-provider example, the grand coalition has aggregated payoff $\pi$1, and both the dual payoff and the Shapley value lie in the core region (Lu et al., 2014). A plausible implication is that whether fair coalitions are small or large depends primarily on the structure of coalition costs: explicit cooperation cost can destroy superadditivity, whereas pure pooling of complementary resources may reinforce it.

For cost-sharing coalition formation, the principal efficiency metric is the strong price of anarchy (SPoA), comparing the worst stable coalition structure with the social optimum. One analysis proves a lower bound $\pi$2 and establishes that equal-split, proportional-split, egalitarian-split, and Nash bargaining all achieve $\pi$3 SPoA under cost monotonicity; in the P2P energy application the empirical SPoA is observed within $\pi$4 of the social optimal cost for coalition sizes 2 and 3, with egalitarian-split closest to optimal (Chau et al., 2020). An earlier sharing-economy analysis gave $\pi$5 for equal-split, proportional-split, and usage-based mechanisms under certain conditions, and $\pi$6 for egalitarian and Nash bargaining (Chau et al., 2015). The comparison shows that fairness rules can have materially different worst-case efficiency profiles even when all produce stable coalitions.

Coalition size also mediates a market-power versus uncertainty trade-off. In electricity markets with renewable producers, larger groups reduce forecast uncertainty through spatial diversification, but large groups also acquire market power and strategically lower aggregate output. The paper shows a “sweet spot”: groups large enough to achieve the uncertainty reduction of the grand coalition, but small enough that they have no significant market power; in the independent-error setting, asymptotically efficient structures have $\pi$7 and $\pi$8, with optimal scaling $\pi$9 and coalition size $S$0 (Zhang et al., 2015). In public-goods governance, a related conclusion appears at the institutional level: multi-coalition or polycentric governance allows uninformed actors to recognize marginal gains from cooperation better than a single inclusive coalition, thereby sustaining higher cooperation and participation (Vasconcelos et al., 2019).

5. Graph-theoretic fair coalitions

In graph theory, fair coalition is a combinatorial notion built on fair domination. A set $S$1 is a $S$2-fair dominating set if it is dominating and every $S$3 satisfies $S$4. A fair coalition in $S$5 is then a pair of disjoint sets $S$6 such that neither $S$7 nor $S$8 is a fair dominating set, but $S$9 is. A fair coalition partition $S$0 is a partition of $S$1 in which every part is either a singleton fair dominating set or a non-fair set that forms a fair coalition with another part; the fair coalition number $S$2 is the maximum size of such a partition (Alikhani et al., 20 Jul 2025).

This notion comes with structural bounds. If $S$3 has order $S$4 and no full vertex, then

$S$5

where $S$6 is the fair domatic number. If $S$7 is the fair domination number, then

$S$8

and for connected graphs of order $S$9,

$S$0

Exact values are obtained for several families: $S$1

$S$2

and for corona trees $S$3,

$S$4

Among small cubic graphs, the Petersen graph has $S$5 (Alikhani et al., 20 Jul 2025).

A later generalization fixes the fairness level $S$6. Two disjoint sets $S$7 form a $S$8-fair coalition if neither is a $S$9-fair dominating set and $\pi(i)$0 is one; a $\pi(i)$1-fair coalition partition requires each part either to be a $\pi(i)$2-fair dominating set with exactly $\pi(i)$3 vertices or to partner with another part in a $\pi(i)$4-fair coalition, and $\pi(i)$5 denotes the maximum number of parts (Jafari et al., 14 Sep 2025). This refinement yields exact formulas such as

$\pi(i)$6

$\pi(i)$7

and, for $\pi(i)$8-regular graphs,

$\pi(i)$9

This graph-theoretic strand is conceptually orthogonal to payoff allocation: fairness means equal domination multiplicity, not equitable sharing of coalition surplus.

6. Extensions, robustness, and common misconceptions

Several adjacent literatures generalize the idea of fair coalition beyond standard coalition-formation games. In abstract argumentation, coalition formability is studied for conflict-eliminable sets of arguments rather than conflict-free sets. Profitability $i$0 requires a larger set, a weakly better state, and no increase in unresolved attackers from the perspective of $i$1; mutual profitability $i$2 requires both sides to benefit, while $i$3 adds a maximal-profitability condition so that future coalition opportunities are not unduly sacrificed (Arisaka et al., 2016). In this setting, fairness is mutual non-worsening under internal compromise.

In blockchain protocol analysis, coalition fairness is formulated through equilibrium with virtual payoffs (EVP). A protocol is coalition-safe if no coalition can deviate so as to increase its utility, measured from the view of at least one honest participant, beyond small multiplicative and additive slack. The paper proves that weak fairness of reward allocation implies $i$4-EVP under relative rewards, shows that Fruitchain is $i$5-EVP for relative rewards and $i$6-EVP for absolute rewards minus cost, and shows that Bitcoin is not EVP under relative rewards because selfish mining can increase a coalition’s relative reward (Kiayias et al., 2019).

Coalitional robustness has also been quantified in fair division. For coalition size $i$7, the strong group incentive ratio $i$8 and group incentive ratio $i$9 measure the maximum and minimum multiplicative gains available to colluders when every corrupted agent is weakly better off. The tightly characterized values are: $S$00

$S$01

$S$02

These results reveal that fair division mechanisms that are individually manipulation-resistant can differ sharply in their vulnerability to collusive manipulation (Huang et al., 2 Oct 2025).

Finally, fairness can attach to the computational labor that precedes coalition formation. The Necklace-based Distributed Coalition Algorithm (N-DCA) addresses the problem of assigning coalition-value calculations in distributed characteristic-function games. It is described as a communication-free algorithm with provable no inter-agent communication, equitable allocation, no redundancy, balanced load, and self-interest, meaning that agents compute only coalition values for coalitions that contain them (Payne et al., 18 Apr 2026). Taken together, these extensions indicate that “fair coalition” may refer to fair payoff division, fair cost sharing, fair resistance to group manipulation, fair computation assignment, or fair structural domination, depending on the formal setting.

A recurring misconception is that fair coalition necessarily means equal splitting or the grand coalition. The literature contradicts both identifications. Equal splitting is only one among several fairness rules, alongside proportional, dual, Shapley, egalitarian, Nash, usage-based, and solidarity-based rules (0802.2159, Lu et al., 2014, Alcalde-Unzu et al., 2023). Likewise, some games are super-additive and sustain the grand coalition, while others are non-superadditive because coalition costs or market power make smaller coalitions more stable or more efficient (0802.2159, Lu et al., 2014, Zhang et al., 2015).