Social Influence as a Voting System: a Complexity Analysis of Parameters and Properties

We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple games. Then we characterize the computational complexity of various problems on influence games, including measures (length and width), values (Shapley-Shubik and Banzhaf) and properties (of teams and players). Finally, we analyze those problems for some particular extremal cases, with respect to the propagation of influence, showing tighter complexity characterizations.

💡 Research Summary

The paper introduces “influence games,” a cooperative simple‑game model that captures how agents in a social network decide to participate in a collective task (e.g., vote) based on the willingness of other influential agents. The underlying diffusion mechanism is the linear threshold model: each player has a threshold, and when the total influence received from already active neighbors exceeds this threshold, the player becomes active and joins the coalition. A coalition succeeds if the number of active agents reaches a predefined quota (for instance, a simple majority).

The authors first prove that influence games are expressive enough to represent any simple game. By carefully constructing the influence graph and assigning thresholds, any collection of winning and losing coalitions can be reproduced, showing that the class of influence games coincides with the class of simple games.

The core of the work is a systematic complexity analysis of a wide range of decision and optimization problems defined on influence games. The problems fall into three broad categories: (i) structural measures (length and width), (ii) power indices (Shapley‑Shubik and Banzhaf values), and (iii) structural properties of coalitions and players (core membership, veto players, symmetric players, etc.).

1. Length and Width – Length is the size of a smallest winning coalition; width is the size of a largest losing coalition. Determining both quantities is NP‑hard. The hardness proof for length reduces from Minimum Vertex Cover, exploiting the fact that a vertex cover corresponds to a minimal set of influencers that can activate the whole graph. Width is shown NP‑hard by a complementary reduction.

2. Power Indices – Computing exact Shapley‑Shubik or Banzhaf values for a given player is #P‑complete. The reduction uses the counting version of the influence spread problem: each possible order of activation (or each subset of influencers) contributes to the marginal contribution of a player, leading to an exponential number of terms that must be summed. Consequently, even approximating these indices within a reasonable factor remains computationally challenging.

3. Coalition and Player Properties – The paper investigates decision problems such as: “Does a core exist?”, “Is a player a veto (or dictator) player?”, “Are two players symmetric?”, “Is a given coalition a minimal winning coalition?”, etc. Most of these problems are PSPACE‑complete or coNP‑complete. For example, checking core non‑emptiness is PSPACE‑complete because it can be expressed as a quantified Boolean formula over activation sequences; veto‑player detection reduces to complement of a NP‑hard reachability problem, landing in coNP‑complete.

The authors also study two extremal propagation regimes to obtain tighter characterizations:

• Complete Graph (unrestricted propagation) – When every player influences every other, the diffusion step is instantaneous. In this setting, length and width can be computed in polynomial time (they reduce to simple counting), but the power‑index problems remain #P‑complete, and core‑related questions stay PSPACE‑complete.

• Independent Set (no propagation) – When the influence graph contains no edges, activation never spreads beyond the initially chosen influencers. Here length and width are trivial (they equal the quota and its complement), yet the existence of a core, veto players, and symmetry still inherit the high complexity of the general case.

Parameter analysis shows that restricting thresholds (e.g., all thresholds equal to one) can lower the complexity of some measure problems but does not affect the hardness of power‑index computation. The paper emphasizes that these complexity results have practical implications: designing algorithms to identify key influencers, to compute fair power allocations, or to verify stability properties in real‑world social platforms will inevitably face #P‑hard or PSPACE‑hard obstacles, motivating the development of approximation schemes, fixed‑parameter algorithms, or heuristics tailored to specific network topologies.

In summary, the work bridges social influence diffusion, cooperative game theory, and computational complexity. It demonstrates that influence games are a universal representation for simple games, provides a comprehensive map of the computational difficulty of fundamental questions about them, and highlights how extreme diffusion scenarios affect these difficulties. This contributes both to the theoretical understanding of influence‑driven decision making and to the algorithmic challenges faced by designers of participatory systems in online social networks.