Altruistic Hedonic Games: Computational Insights

• Altruistic hedonic games are coalition formation models where agents’ utilities integrate their own preferences with the welfare of their friends using friend-oriented valuation functions.
• Key results show that verifying core stability in these games is coNP-complete, with proofs leveraging gadget constructions that mirror clique problems.
• Extensions of the model incorporate alternative aggregation methods and broader coalition considerations, offering new insights into stability concepts in social choice.

Altruistic hedonic games (AHGs) are a class of coalition formation games that extend traditional hedonic games by embedding a form of social preference: an agent’s utility is influenced not only by her own preferences over coalitions but also by the welfare of her friends within those coalitions. This specification captures complex social dynamics where agents display varying degrees of altruism towards their direct contacts, leading to novel challenges and enriched theory in computational social choice and cooperative game theory. The formalization, central results, and computational properties of AHGs have been developed in a line of work initiated by Kerkmann et al. and Nguyen et al., and have recently been refined and extended to cover a broad range of stability notions and algorithmic questions (Rothe et al., 27 Nov 2025, Kerkmann et al., 2023).

1. Formal Model and Valuation Functions

A hedonic game is formally a pair $(N, \succeq)$, where $N$ is a finite set of players, and each player $i \in N$ is endowed with a complete, transitive preference relation $\succeq_i$ over all coalitions containing $i$. In the friend-oriented setting, the structure of interpersonal relationships is encoded by an undirected graph $G = (N, E)$, partitioning the set of non-self nodes into friends $F_i$ and enemies $E_i$ for each $i$, where $F_i = \{j \mid \{i, j\} \in E\}$ and $E_i = N \setminus (F_i \cup \{i\})$.

The canonical friend-oriented valuation for a player $i$ over a coalition $C \ni i$ is: $$\mathrm{val}_i(C) = n \cdot |F_i \cap C| - |E_i \cap C|,$$ where $n = |N|$ ensures friend-counts dominate. Standard (selfish) friend-oriented preferences order coalitions according to $\mathrm{val}_i(C)$.

In AHGs, each player further incorporates the coalition utilities of friends. The classic altruistic utility template for $C\ni i$ is: $$u_i(C) = (1-\alpha)\, p_i(C) + \alpha\; \frac{1}{|C_i|} \sum_{j \in C_i} p_j(C),$$ where $C_i = C \cap F_i$, $p_i$ is the base valuation, and $\alpha$ parameterizes the level of altruism ($0 \leq \alpha \leq 1$). In practice, three lexicographic treatment styles via a large weight $w \gg n$ are considered:

• Selfish-First (SF): $$u_i^{\mathrm{SF}}(C) = w\,\mathrm{val}_i(C) + \sum_{j\in C_i}\mathrm{val}_j(C)$$
• Equal-Treatment (EQ): $$u_i^{\mathrm{EQ}}(C) = \sum_{j\in C_i\cup\{i\}}\mathrm{val}_j(C)$$
• Altruistic-First (AL): $$u_i^{\mathrm{AL}}(C) = \mathrm{val}_i(C) + w\sum_{j\in C_i}\mathrm{val}_j(C)$$

These may be instantiated using alternative aggregation (average/minimum) of friends' values, leading to six utility variants: $$\begin{array}{ll} \text{Avg-based, SF:} & u_i^{\mathrm{avg-SF}}(C) = w\,\mathrm{val}_i(C) + \mathrm{avg}^i(C) \ \text{Min-based, SF:} & u_i^{\mathrm{min-SF}}(C) = w\,\mathrm{val}_i(C) + \mathrm{min}^i(C) \ \text{Avg-based, EQ:} & u_i^{\mathrm{avg-EQ}}(C) = \mathrm{avg}^i(C) \ \text{Min-based, EQ:} & u_i^{\mathrm{min-EQ}}(C) = \mathrm{min}^i(C) \ \text{Avg-based, AL:} & u_i^{\mathrm{avg-AL}}(C) = \mathrm{val}_i(C) + w\,\mathrm{avg}^i(C) \ \text{Min-based, AL:} & u_i^{\mathrm{min-AL}}(C) = \mathrm{val}_i(C) + w\,\mathrm{min}^i(C) \end{array}$$ with

$$\begin{aligned} \mathrm{avg}^i(C) &= \frac{1}{|(C \cap F_i) \cup \{i\}|} \sum_{j\in (C \cap F_i)\cup\{i\}} \mathrm{val}_j(C)\ \mathrm{min}^i(C) &= \min_{j\in (C \cap F_i)\cup\{i\}} \mathrm{val}_j(C) \end{aligned}$$

The treatment-style lexicographic structure (dominated by $w$) yields strict prioritization between self and altruistic terms (Rothe et al., 27 Nov 2025, Kerkmann et al., 2023).

2. Stability Concepts in Altruistic Hedonic Games

AHGs admit the full suite of stability notions from hedonic games, anchored by the core. For a coalition structure (partition) $\Gamma$, define $\Gamma(i)$ as the coalition containing $i$. A nonempty $C \subseteq N$ blocks $\Gamma$ if $\forall i \in C$ it holds that $C \succ_i \Gamma(i)$. The core is the set of partitions not blocked by any coalition. AHGs also support notions such as Nash stability (NS), individual stability (IS), contractual IS (CIS), total IS (TIS), strict core, popularity (POP), strict popularity (SPOP), and perfectness, each specified by canonical logical conditions over deviations of single players or coalitions (Kerkmann et al., 2023).

These stability concepts interact nontrivially with the altruistic assignment of utility, particularly in the presence of non-selfish or non-linear aggregation across friends' utilities. Example pathologies are observed in EQ and AL hedonic models where players may remain indifferent to friends’ improvements external to their own coalition, a limitation addressed when extending to global (partition-level) altruism (Kerkmann et al., 2023).

3. Computational Complexity of Core Stability Verification

A central body of work concerns the complexity of verifying core-stability in AHGs under different altruistic semantics. For the four “open” cases—avg-EQ, avg-AL, min-EQ, min-AL—defining the class of verification problems $\mathrm{VCS}^X$-$T$-AHG, where $X\in\{\mathrm{avg},\mathrm{min}\}$ and $T\in\{\mathrm{EQ},\mathrm{AL}\}$:

• Input: An AHG specified via $(N, G, w, \succeq)$ with corresponding utility and a coalition structure $\Gamma$.
• Question: Is $\Gamma$ core-stable?

The main theorem establishes that all four variants are coNP-complete (Rothe et al., 27 Nov 2025):

• Theorem 4.1: Verifying core-stability in min-based EQ or AL models is coNP-complete.
• Theorem 4.2: Verifying core-stability in avg-based EQ is coNP-complete.
• Theorem 4.3: Verifying core-stability in avg-based AL is coNP-complete.

Membership in coNP holds trivially via a blocking coalition certificate verifiable in polynomial time. The hardness results are by reductions from Clique, employing intricate gadget constructions in the underlying friend network. These constructions encode the combinatorial template of the clique problem into feasible blocking coalitions—requiring precise friend/enemy count relationships achievable only in the presence or absence of suitable cliques (Rothe et al., 27 Nov 2025).

4. Gadget Constructions and Proof Techniques

The complexity reductions rely on three major types of gadget structures representing vertices, edges, and incidences in the instance graph $H=(V, E)$ of the Clique problem. For min-based cases, circulant gadgets enforce constraints such that only one specially distinguished player in each gadget may participate in any deviation, effectively mirroring the selection of a clique subset. For avg-based cases, dome gadgets (and their pinched variants) create possible deviations constrained such that the average utility can only increase if exactly one distinguished top-node from each gadget is included.

These techniques ensure that any blocking coalition in the constructed AHG corresponds to a subgraph of $H$ with prescribed properties, thus mirroring the existence of a $k$-clique in the original problem. The gadget-based argumentation is crucial for enforcing the delicate degree and intersection requirements needed for reductions (Rothe et al., 27 Nov 2025).

5. Generalizations, Extensions, and Contrasts

Nguyen et al. and subsequent work (Kerkmann et al., 2023) identify the core limitation of classic AHGs: players only care about the welfare of friends within their own coalition. To address this, the framework is generalized to altruistic coalition formation games (ACFGs), wherein friend-oriented aggregation can span the entire partition $\mathcal{C}_N$. This yields more robust monotonicity and unanimity properties; for instance, in sum-based utilities, gaining a friend in any coalition or losing an enemy always improves a player’s welfare—a property not ensured in the hedonic EQ or AL models.

A suite of further stability notions is defined and analyzed in ACFGs, and it is shown that, for SF variants, certain stability properties are uniquely realized when coalition structures align with the clique-decomposition of the friend graph. Key results include characterizations of perfectness and the efficiency of verifying stability properties for the SF model, contrasted with the incomplete and often coNP-hard classification of such properties under EQ and AL models (Kerkmann et al., 2023).

6. Open Problems and Directions

Outstanding theoretical questions include classifying the complexity of core existence in AHGs/ACFGs with EQ and AL treatments, where the verification problem is proved coNP-complete but existence remains open. The complexity of verifying the strict core is in coNP but coNP-hardness is not yet established. Additional directions include seeking fixed-parameter tractable (FPT) or approximation algorithms parameterized by clique-size or the structural parameters (e.g., treewidth) of the friendship network. The exploration of alternative stability concepts, such as popularity or super-altruistic models, promises further generalization of social welfare properties and computational phenomena (Rothe et al., 27 Nov 2025, Kerkmann et al., 2023).

A plausible implication is that, given the coNP-completeness of verification across a broad landscape of AHGs, no efficient general algorithm exists for certifying core-stability in nontrivial instances, imposing a practical barrier for many applications of such coalition formation models. The richness of interaction between coalition valuation, friend-network topology, and algorithmic verification marks AHGs and their generalizations as a focal point in cooperative game theory and computational social choice.