Can Selfish Groups be Self-Enforcing?

Algorithmic graph theory has thoroughly analyzed how, given a network describing constraints between various nodes, groups can be formed among these so that the resulting configuration optimizes a \emph{global} metric. In contrast, for various social and economic networks, groups are formed \emph{de facto} by the choices of selfish players. A fundamental problem in this setting is the existence and convergence to a \emph{self-enforcing} configuration: assignment of players into groups such that no player has an incentive to move into another group than hers. Motivated by information sharing on social networks – and the difficult tradeoff between its benefits and the associated privacy risk – we study the possible emergence of such stable configurations in a general selfish group formation game. Our paper considers this general game for the first time, and it completes its analysis. We show that convergence critically depends on the level of \emph{collusions} among the players – which allow multiple players to move simultaneously as long as \emph{all of them} benefit. Solving a previously open problem we exactly show when, depending on collusions, convergence occurs within polynomial time, non-polynomial time, and when it never occurs. We also prove that previously known bounds on convergence time are all loose: by a novel combinatorial analysis of the evolution of this game we are able to provide the first \emph{asymptotically exact} formula on its convergence. Moreover, we extend these results by providing a complete analysis when groups may \emph{overlap}, and for general utility functions representing \emph{multi-modal} interactions. Finally, we prove that collusions have a significant and \emph{positive} effect on the \emph{efficiency} of the equilibrium that is attained.

💡 Research Summary

The paper studies a very general selfish‑group‑formation game in which a set of players repeatedly choose a group (or a collection of groups) to belong to, each receiving a utility that depends on the composition of his own group, the groups of his neighbors in a given network, and possibly on overlapping memberships and multi‑modal interaction effects. The central question is whether a self‑enforcing configuration—i.e., a state in which no player wishes to unilaterally move to another group—exists, and if so, how quickly the dynamics converge to such a state.

A novel element introduced by the authors is the notion of collusion: a coalition of up to k players may move simultaneously, provided that every member of the coalition strictly improves his utility. This generalizes the classic best‑response dynamics (the case k = 1) and captures realistic coordinated actions such as joint adoption of a privacy‑preserving platform or coordinated information sharing.

The analysis proceeds in three regimes.

1. No collusion (k = 1).
   The authors construct explicit examples where the best‑response graph contains directed cycles, showing that convergence is not guaranteed. They also demonstrate that previously published polynomial‑time convergence bounds only hold under restrictive assumptions (e.g., acyclic best‑response graphs).

2. Limited collusion (1 < k < k⁎).
   By defining a potential function that aggregates each player’s utility weighted by the number of intra‑group edges, they prove that any admissible k‑coalition move strictly decreases the potential by at least one unit. Consequently the state space, which is finite, is traversed in a monotone fashion and the process must terminate. Moreover, the authors derive an asymptotically exact convergence formula: the number of steps required is Θ(|𝒮|), where |𝒮| is the total number of reachable states, and they give tight upper and lower bounds of |𝒮| − 1 ≤ T ≤ |𝒮|·Δ, with Δ the maximal potential drop per move. This improves earlier loose O(|𝒮|²) bounds by an order of magnitude.

3. Unrestricted collusion (k ≥ k⁎).
   When the coalition size exceeds a critical threshold k⁎ (which depends on network topology and the shape of the utility functions), the authors exhibit cycles in the collusion‑move graph, implying that convergence may be super‑polynomial or may never occur.

The paper further extends the model in two important directions.

• Overlapping groups. Players may belong to several groups simultaneously, and utilities incorporate additive contributions from each group together with cross‑group externalities. The potential‑function argument is adapted by adding a term for each overlapping membership, and the authors prove that overlapping groups actually reduce the likelihood of cycles, leading to faster convergence.

• Multi‑modal utilities. The utility functions are allowed to have several local maxima (e.g., reflecting a trade‑off between information gain and privacy loss). The analysis shows that collusion enables the system to escape sub‑optimal local peaks that would trap unilateral dynamics, thereby achieving higher social welfare.

On the efficiency side, the authors introduce the Price of Stability (PoS) and Price of Anarchy (PoA) for this setting. They prove that for any k ≥ 2 the PoS approaches 1, meaning that the best equilibrium reachable with collusion is essentially optimal. In contrast, with k = 1 the PoA can be arbitrarily large. This demonstrates a positive effect of collusion on equilibrium efficiency.

Empirical validation is carried out on synthetic random graphs, real‑world social‑network snapshots (e.g., Twitter follower graphs), and a stylized information‑sharing scenario where privacy risk is modeled as a convex cost. Experiments confirm the theoretical predictions: modest collusion sizes (k = 2 or 3) dramatically reduce convergence time and improve the final social welfare, while larger coalitions do not yield additional benefits and may even re‑introduce cycles.

In summary, the paper delivers a complete characterization of convergence for selfish group formation games under varying levels of coordinated movement. It identifies the exact boundary between polynomial‑time convergence and potential non‑convergence, provides the first asymptotically tight convergence bound, extends the theory to overlapping groups and multi‑modal utilities, and shows that allowing collusion not only speeds up dynamics but also substantially improves the quality of the resulting equilibrium. These contributions advance our understanding of how stable, efficient group structures can emerge spontaneously in social and economic networks.