Lexicographic Ranking based on Minimal Winning Coalitions

In this paper, we consider the consistency of the desirability relation with the ranking of the players in a simple game provided by some well-known solutions, in particular the Public Good Index [14] and the criticality-based ranking [1]. We define a new ranking solution, the Lexicographic Ranking based on Minimal winning coalitions (LRM), strongly related to the Public Good Index being rooted in the minimal winning coalitions of the simple game, proving that it is monotonic with respect to the desirability relation [17], when it holds. A suitable characterization of the LRM solution is provided. Finally, we investigate the relation among the LRM solution and the criticality-based ranking, referring to the dual game.

💡 Research Summary

The paper introduces a novel ranking solution for simple cooperative games called the Lexicographic Ranking based on Minimal winning coalitions (LRM). Traditional power indices such as the Shapley‑Shubik, Banzhaf, and nucleolus assign numerical scores to players and satisfy a monotonicity property with respect to the desirability relation—a preorder that captures whether one player can replace another in any winning coalition without harming the outcome. However, indices that rely exclusively on minimal winning coalitions, notably the Public Good Index (PGI) and the Deegan‑Packel Index (DPI), often violate this monotonicity, leading to rankings that contradict the desirability relation.

To address this gap, the authors define for each player i a vector θ_v(i) = (i₁,…,i_n), where i_k counts the number of minimal winning coalitions of size k that contain i. The LRM solution orders players by the lexicographic comparison of these vectors: i is ranked at least as high as j if θ_v(i) ≥_L θ_v(j). This construction captures two intuitive ideas: (1) smaller minimal winning coalitions confer more power, and (2) among coalitions of the same size, the number of such coalitions matters.

The paper proves that LRM satisfies Desirable Monotonicity (DM): if two players are equivalent under the desirability relation (i ∼ j), then their vectors are identical and LRM treats them as indifferent; if i is strictly more desirable than j (i ≻ j), then there exists a smallest coalition size k where i appears more often than j, guaranteeing i P j in the lexicographic order. Consequently, LRM aligns perfectly with the desirability preorder whenever it is total, and it inherits weight‑based monotonicity for weighted majority games.

Beyond DM, the authors introduce two additional axioms inspired by earlier work on blocking coalitions: Anonymity of Minimal Winning Coalitions (AMWC), which requires that rankings be invariant under any bijective relabeling of the non‑i,j players that preserves the structure of minimal winning coalitions involving i and j; and Independence of Larger Minimal Winning Coalitions (ILMWC), which states that adding new minimal winning coalitions of larger size cannot overturn an existing ranking between two players. They show that the only ranking solution satisfying DM, AMWC, and ILMWC simultaneously is LRM, establishing a clean axiomatic characterization.

The authors also explore the relationship between LRM and the criticality‑based ranking introduced in earlier literature. By considering the dual (blocking) game, minimal winning coalitions correspond to blocking coalitions, and the criticality ranking—based on how often a player belongs to blocking coalitions—mirrors the structure of LRM. Thus LRM can be viewed as a refined, lexicographically ordered version of the criticality approach.

Illustrative examples throughout the paper demonstrate that PGI and DPI can rank players contrary to the desirability relation, while LRM yields rankings that coincide with it. In weighted majority games, LRM respects the natural ordering induced by player weights, confirming its practical relevance.

In conclusion, the paper provides a theoretically robust and practically useful ranking method that leverages minimal winning coalitions while guaranteeing consistency with the desirability relation. The LRM solution fills a notable gap between numerical power indices and ordinal ranking needs in applications such as computational biology, network centrality analysis, and political science. The authors suggest future work on algorithmic scalability, extensions to dynamic games, and integration with multi‑criteria decision frameworks.