--- title: 'Fair Coalition Partition: Models & Algorithms' url: https://www.emergentmind.com/topics/fair-coalition-partition type: topic --- # Fair Coalition Partition: Models & Algorithms Searching arXiv for the specified topic and cited papers to ground the article in the literature. Fair coalition partition denotes a family of partitioning problems in which a set of agents, voters, vertices, or tasks is divided into coalitions, districts, or blocks subject to a fairness or stability requirement on both the realized partition and the coalitions that could challenge it. In current arXiv literature, the phrase does not name a single formalism. It appears in locally fair redistricting, Shapley-based coalition stability, domination-based graph partitions, balanced grouping on friendship graphs, and task-driven multi-agent coalition formation [2112.06899], [2603.17153], [2507.15080], [2503.10830], [2403.05108]. ## 1. Scope of the concept Across these literatures, the recurring object is a partition $\Pi=\{S_1,\dots,S_k\}$ of a ground set, together with a criterion saying that the partition is acceptable only if some class of coalitional deviations is ruled out. In some models the constraint is geometric or combinatorial, as in contiguous intervals on a line; in others it is allocative, as in equal or Shapley-based payoff division; in graph-theoretic variants it is domination-based, so that a block is acceptable only if it is itself a special dominating set or can pair with another block to form one [2112.06899], [2304.10651], [2507.15080]. | Setting | Ground set | Defining condition | |---|---|---| | Locally fair partitioning | Ordered red/blue points | No allowable $\beta$-deviating interval | | SFMS coalition formation | Players in a TU game | Shapley fairness and merge stability | | Friendship partitioning | Agents on a friendship graph | Balanced partition with EF/EFX/EF1/PROP/MMS constraints | | Graph fair coalition partition | Vertex partition of a graph | Each block is a singleton fair dominating set or has a fair coalition partner | | Partition-form coalition games | Coalitions with externalities | No blocking coalition under the selected stability rule | A common theme is the replacement of purely global welfare criteria by coalition-stability criteria. In the redistricting model, a contiguous group of unhappy voters should not be able to carve out a better district. In TU coalition games, no coalition should contain a player with negative restricted-game Shapley value, and no pair of coalitions should have a strictly profitable merge. In domination-based graph models, blocks are validated through their coalition partners rather than through intrinsic optimality alone [2112.06899], [2603.17153], [2507.15080]. This suggests that “fair coalition partition” is best understood as an umbrella notion for partition problems in which fairness is imposed at the level of feasible coalitional deviations, not only at the level of aggregate outcomes. ## 2. Local fairness and contiguous deviations In "Locally Fair Partitioning" [2112.06899], the input is a 1D instance $X=x_1,\dots,x_n$ with each $x_i\in\{R,B\}$, and a parliament size $k$ induces the ideal district size $\sigma=n/k$. For $\epsilon\in[0,1/2]$, an interval $\pi_t$ is $\epsilon$-allowable if $$(1-\epsilon)\sigma \le |\pi_t| \le (1+\epsilon)\sigma,$$ and a partition is balanced if every interval is allowable. For an interval $I$, the majority color is $C(I)=R$ if $|R\cap I|>|B\cap I|$, and $C(I)=B$ otherwise, with ties breaking to blue. A point is unhappy if it belongs to the minority party in its interval. If $U(I,\Pi)$ is the set of unhappy points in $I$, then an allowable interval $D$ is a $\beta$-deviating group if $$\max\{|U(D,\Pi)\cap R|,\ |U(D,\Pi)\cap B|\}>\max\{\mu(D)/2,\beta\},$$ where $\mu(D)=|D|/\sigma$. A balanced partition is $(\epsilon,\beta)$-locally fair if there is no $\beta$-deviating group. The paper explicitly interprets this as a core-stability condition for contiguous feasible coalitions [2112.06899]. The adversarial existence theory is sharp. For any instance, there is a value $$ \bar\beta(\epsilon)= \begin{cases} \max\{1-\epsilon,(1+3\epsilon)/2\}+O(\delta), & \epsilon\in[0,1/3],\\ \max\{3(1-\epsilon)/2,2\epsilon\}+O(\delta), & \epsilon\in[1/3,1/2], \end{cases} $$ with $\delta\le 1/(n/\sigma-1)+1/\sigma$, such that any $\beta>\bar\beta(\epsilon)$ guarantees an $(\epsilon,\beta)$-locally fair partition. For $\epsilon\le 1/5$, the threshold is sharp at $\beta=1-\epsilon$: if $\beta\ge 1-\epsilon$, near-uniform partitions suffice, while if $\beta<1-\epsilon$ there are instances with no locally fair partition. The negative theorem gives such instances with $$|X| = O\!\left(\frac{\beta\sigma}{1-\epsilon-\beta}\right).$$ The impossibility is also robust across a set of target sizes $S=\{\sigma_1,\dots,\sigma_M\}$ by the multi-$\sigma$ corollary [2112.06899]. Beyond worst case, the paper studies clustered inputs represented as alternating runs $$X=R_1,B_1,R_2,\dots,R_\eta,B_\eta.$$ If all runs satisfy $\mu(R_j),\mu(B_j)\ge 2$, then there exists an $\alpha$-balanced $\epsilon$-locally fair partition with $$\alpha=\frac{(1-\epsilon)\sigma}{n}=\frac{1-\epsilon}{k}.$$ If only a $\gamma n$ mass of vertices lies in runs of measure $<2$, then $$\alpha=\frac{(1-\epsilon)\sigma}{n}+\gamma$$ suffices. Under stronger run-length conditions, exact balancedness is obtained. Algorithmically, existence is decided by a dynamic program with state $$LF((i,j],i_1,i_2,i_3),$$ which tracks the last three boundaries to detect deviating groups spanning at most four intervals. The running time is $O(n\sigma^3)$ for $\epsilon\in[0,1/3)$ and $O(n\sigma^4)$ for $\epsilon\in[1/3,1/2)$. The paper emphasizes that while the lower bounds extend to 2D, existence and efficient computation in 2D remain open, even for rectangles or hierarchical partitions [2112.06899]. ## 3. Cooperative-game formulations and stability A second major line of work treats fair coalition partition as a stability notion for TU cooperative games. In "Partition-based Stability of Coalitional Games" [2304.10651], the basic object is a partition-allocation pair $({\cal P},\vec x)$, where $\vec x$ is individually rational and coalition-wise efficient: $$ \mathscr{X}(N,\vec v,{\cal P})= \left\{\vec x\in\mathbb{R}^N: x(i)\ge v(\{i\})\ \forall i,\ \sum_{i\in S}x(i)=v(S)\ \forall S\in{\cal P} \right\}. $$ The paper defines strong, medium, and weak fission resistance, together with fusion resistance. Strong stability is core-compatible: for the grand-coalition partition $\{N\}$, the strongly stable allocations are exactly the core. Medium and weak stability are universal: every game has a mediumly stable and weakly stable partition-allocation pair. Medium stability is characterized by local maximality of the partition worth $$\tilde w(N,\vec v,{\cal P})=\sum_{S\in{\cal P}} v(S),$$ and a steepest ascent method moves through the partition lattice until no improving fission or fusion move remains [2304.10651]. "Split-Merge Dynamics for Shapley-Fair Coalition Formation" [2603.17153] imposes a more specific fairness criterion. For a coalition $S$, the restricted-game Shapley value is $$ \phi_i(S,v|_S)= \sum_{T\subseteq S\setminus\{i\}} \frac{|T|!(|S|-|T|-1)!}{|S|!}\,[v(T\cup\{i\})-v(T)]. $$ A partition is Shapley-fair if every player has nonnegative restricted-game Shapley value. The aggregate fairness deficit is $$ \Psi(\Pi)=\sum_{S\in\Pi}\sum_{i\in S}\max\{0,-\phi_i(S,v|_S)\}, $$ and system surplus is $$ \Phi(\Pi)=\sum_{S\in\Pi}v(S). $$ The split operator removes players with negative Shapley values, while the merge operator fuses coalitions only when $$ \Delta(A,B)=v(A\cup B)-[v(A)+v(B)]>0. $$ Fixed points are precisely Shapley-Fair and Merge-Stable (SFMS) partitions, and convergence follows from the lexicographic Lyapunov function $$ \mathcal V(\Pi)=(-\Psi(\Pi),\Phi(\Pi)). $$ The paper also notes that exact Shapley computation is NP-hard and discusses permutation sampling and Bayesian Monte Carlo approximations [2603.17153]. A related but distinct stability theory appears in generalized claims problems with a minimum coalition size $\theta$ [2311.03950]. If $E<\sum_i c_i$ and $p=E/\sum_i c_i$, then a coalition $S$ receives $$ E_S=p\sum_{i\in S} c_i $$ when $|S|\ge \theta$, and zero otherwise. Under continuous, resource monotonic, and consistent rules, stable partitions with mostly $\theta$-size coalitions emerge. For the constrained equal awards rule, $$ x_i^{CEA}(S)=\min\{c_i,\lambda_S\}, $$ with $\lambda_S$ chosen so that $\sum_{i\in S}\min\{c_i,\lambda_S\}=E_S$; for the constrained equal losses rule, $$ x_i^{CEL}(S)=\max\{c_i-\mu_S,0\}, $$ with $\mu_S$ chosen so that $\sum_{i\in S}\max\{c_i-\mu_S,0\}=E_S$. The paper proves that when $n$ is divisible by $\theta$, there is a stable partition entirely composed of $\theta$-size coalitions, and it gives explicit CEA and CEL algorithms to construct such partitions [2311.03950]. Several partition-form models place fairness inside coalitions while keeping the partition itself strategically endogenous. In lattice-theoretic TU sharing on the partition lattice, atoms are edges $[ij]$, and surplus is distributed to edges rather than directly to coalitions; the size-uniform rule $$ \phi^{SU}_a(f)=\sum_{x\ge a}\frac{\mu^f(x)}{s(x)} $$ has a fixed-point property, while the chain-uniform rule averages marginal contributions along maximal chains [1808.01563]. In cooperative resource sharing with an adamant player, coalition utilities are shared via the Shapley value, the all-alone partition is the unique Nash equilibrium for $n>4$, and the price of anarchy grows like $O(n)$ [2012.02960]. In coalition formation in congestion games, fair equal sharing within coalitions is stability-equivalent, and sufficiently large communication cost $\beta$ leaves only the all-alone partition stable [2410.06797]. ## 4. Graph-theoretic coalition partitions In graph theory, coalition partitions are traditionally domination-based. A coalition in a graph $G$ is a pair of disjoint sets $V_1,V_2\subseteq V(G)$ such that neither is dominating but $V_1\cup V_2$ is dominating. A coalition partition is a vertex partition in which every part is either a singleton dominating set or forms a coalition with another part. The coalition number $C(G)$ is the maximum cardinality of such a partition. For trees, the extremal cases are sharp: $C(T)=n$ iff $T=P_n$ with $n\le 4$, and $C(T)=n-1$ iff $T\in\{K_{1,3},P_5,P_6,S(2,1)\}$ [2111.08945]. For cubic graphs, $C(G)\le 9$, and "On cubic graphs having the maximal coalition number" constructs an infinite family of cubic graphs with $C(G)=9$, showing the bound is tight [2404.06245]. The minmin coalition number refines this by minimizing the order of a minimal coalition partition. It is defined as $$ c_{\min}(G)=\min\{|\mathcal X|:\ \mathcal X \text{ is a minimal } c\text{-partition of }G\}. $$ The paper proves $$ 2\le c_{\min}(G)\le n, $$ characterizes connected graphs with $c_{\min}(G)=n$ as exactly the graphs in the recursively defined family $\mathcal M$, gives a polynomial-time algorithm to decide whether $c_{\min}(G)=2$, proves that $c_{\min}(G)\ge 3$ iff every closed neighborhood $N[v]$ is dominating when $G$ has no universal vertex, and characterizes the graphs with $\delta(G)=2$ and $c_{\min}(G)=4$ as exactly the family $\mathcal F$ [2307.01222]. A further refinement is the coalition count $c(G)$, the maximum number of different coalitions in any coalition partition. This parameter is not comparable with $C(G)$: for $K_n$, one has $C(K_n)=n$ but $c(K_n)=0$, whereas for $C_4$, one has $c(C_4)=6>C(C_4)=4$. The paper proves that every graph is the coalition graph of some graph $H^*$ with a smaller order and size than the earlier realization construction, and it characterizes the graphs with $c(G)=1$ by $$ c(G)=1 \iff \alpha(G)=n-f \iff G\cong (K_f+pK_1)\cup qK_1, $$ where $f$ is the number of full vertices and $f+p+q=n$ [2511.21112]. The fair-domination variant changes the admissible coalition unions. A $k$-fair dominating set is a set $S$ such that every vertex outside $S$ has exactly $k$ neighbors in $S$. "Fair coalition in graphs" defines a fair coalition partition $\Upsilon$ by requiring each block either to be a singleton fair dominating set or to form a fair coalition with another block. The fair coalition number $\mathcal C_f(G)$ satisfies $\mathcal C_f(G)\ge 2d_f(G)$ when $G$ has no full vertices, and $$ \mathcal C_f(G)\le n-\gamma_f(G)+2, \qquad \mathcal C_f(G)\le n-\gamma_f(G)\ \text{for connected }G\text{ of order }n\ge 3. $$ Exact values are obtained for paths, cycles, corona trees, and small cubic graphs; for example, $\mathcal C_f(P_n)=4$ for $n\ge 2$, while for cycles $$ \mathcal C_f(C_{3k})=6,\quad \mathcal C_f(C_{3k+1})=5,\quad \mathcal C_f(C_{3k+2})=4 $$ [2507.15080]. The parameterized version is the $k$-fair coalition number $C_{kf}(G)$ [2509.11358]. A partition is $k$-fair if each block is either a $k$-fair dominating set with exactly $k$ vertices or has a $k$-fair coalition partner. The paper proves the lower bound $$ C_{kf}(G)\ge 2d_{kf}(G) $$ for connected graphs and $k\ge 2$, and the upper bound $$ C_{kf}(G)\le \Delta(G)-k+3 $$ for $k>\delta(G)$. Exact values are obtained for several families, including $$ C_{kf}(K_n)=n-k+2 \quad (2\le k\le n-1), $$ and for $k=2$ one has $C_{2f}(P_n)=3$ for $n\ge 4$ and $$ C_{2f}(C_n)= \begin{cases} 4, & n\text{ even},\\ 3, & n\text{ odd}. \end{cases} $$ This literature makes explicit that graph-theoretic fair coalition partition is not just a variant of domination; it is a partition problem in which domination, exact neighborhood counts, and coalition partnership all interact. ## 5. Balanced grouping, social networks, and task-driven coalitions In "Balanced and Fair Partitioning of Friends" [2503.10830], the ground set is a friendship graph $G=(V,E)$, and the goal is a balanced $k$-partition $$ \lfloor n/k\rfloor \le |P_j| \le \lceil n/k\rceil \qquad \forall j\in[k]. $$ An agent compares groups through replacement semantics: if $i$ compares its current group with another group $P_b$, it evaluates $u_i(P_b\cup\{i\}\setminus\{j\})$. This yields the coalition-style envy notions \[ \text{EF},\quad \text{EFX}_0,\quad \text{EFX},\quad \text{EF1}, \] together with proportionality $$ u_i(P_i)\ge \frac{u_i(F(i))}{k} $$ and balanced maximin share $$ \mu_i=\max_{Q\in \Pi_k}\min_{Q_r\in Q}u_i(Q_r). $$ The paper shows that EF is not guaranteed even on a 3-agent path, MMS may fail to exist in general, and deciding whether $\mu_i\ge \sigma$ is NP-hard even with $k=2$ and symmetric utilities. On the positive side, for forests and monotone utilities, MMS and EFX always exist and are computable by a constructive BFS budget algorithm; for additive utilities that algorithm runs in linear time. The paper also gives FPT results parameterized by vertex cover and XP algorithms parameterized by $(k,\mathrm{tw})$ [2503.10830]. In task-driven multi-UAV coalition formation, fairness is internal to each coalition and relevance is encoded by a task-specific threshold [2403.05108]. For task $i$, coalition work capacity is $$ e^i(C_i)=\sum_{j\in C_i} e_j^i, $$ completion time is $$ t_i(C_i)=\frac{Q_i}{e^i(C_i)}, $$ and the coalition value is $$ V_i(C_i)=R_i(e^i(C_i))-L_i(C_i). $$ The revenue function $R_i$ is increasing up to the coalition revenue threshold $\beta_i$ and decreasing for $\beta_i