Complexity of comparison of influence of players in simple games

Coalitional voting games appear in different forms in multi-agent systems, social choice and threshold logic. In this paper, the complexity of comparison of influence between players in coalitional voting games is characterized. The possible representations of simple games considered are simple games represented by winning coalitions, minimal winning coalitions, weighted voting game or a multiple weighted voting game. The influence of players is gauged from the viewpoint of basic player types, desirability relations and classical power indices such as Shapley-Shubik index, Banzhaf index, Holler index, Deegan-Packel index and Chow parameters. Among other results, it is shown that for a simple game represented by minimal winning coalitions, although it is easy to verify whether a player has zero or one voting power, computing the Banzhaf value of the player is #P-complete. Moreover, it is proved that multiple weighted voting games are the only representations for which it is NP-hard to verify whether the game is linear or not. For a simple game with a set W^m of minimal winning coalitions and n players, a O(n.|W^m|+(n^2)log(n)) algorithm is presented which returns `no’ if the game is non-linear and returns the strict desirability ordering otherwise. The complexity of transforming simple games into compact representations is also examined.

💡 Research Summary

The paper investigates the computational complexity of comparing the influence of players in simple cooperative voting games under several common representations and influence measures. Simple games are defined as monotone Boolean functions v : 2^N → {0,1} with v(∅)=0 and v(N)=1. The authors consider four principal representations: (i) the explicit list of all winning coalitions W, (ii) the list of minimal winning coalitions W^m, (iii) weighted voting games (WVG) specified by a quota q and integer weights w₁,…,wₙ, and (iv) multiple weighted voting games (MWVG), which are the logical conjunction of m WVGs. The latter captures games of higher dimension, i.e., those that cannot be expressed by a single WVG.

Influence is measured in three families. First, basic player types (dummy, passer, veto, dictator) are examined. The paper shows that dummy players can be identified in linear time when the game is given by W^m, and in polynomial time for the W representation, because a dummy never appears in any minimal winning coalition. Veto players, passers, and dictators are similarly easy to detect across all four representations.

Second, the desirability relation D (i ≽_D j) and its strict version are studied. A simple game is linear if D is a total preorder; equivalently, the game is swap‑robust or trade‑robust. For WVGs, a desirability ordering coincides with the ordering of the weights, which can be computed in polynomial time, but determining the strict ordering (i.e., testing whether two players are symmetric) is NP‑hard, as shown by Matsui and Matsui. For MWVGs the authors prove NP‑hardness of linearity testing by a reduction from the PARTITION problem: they construct two WVGs whose weight vectors differ only in the last four positions; a partition of the original numbers yields incomparable players, breaking linearity. Conversely, when the game is given by minimal winning coalitions, linearity can be decided in O(n·|W^m| + n²log n) time using Makino’s algorithm for testing regularity of monotone Boolean functions.

Third, classical power indices are considered: the Banzhaf index, the Shapley‑Shubik index, the Holler index, the Deegan‑Packel index, and the Chow parameters. The Banzhaf value η_i(v) (the number of coalitions where i is critical) is shown to be #P‑complete to compute even when the input is the set of minimal winning coalitions. This contrasts with the fact that checking whether a player’s Banzhaf power is 0 or 1 is trivial. The Shapley‑Shubik value and Chow parameters can be computed in polynomial time for any representation, because they are linear functions of the characteristic function and can be evaluated via known formulas or dynamic programming.

The paper also studies transformation problems. Deciding whether a given simple game is realizable as a WVG (WVG‑Realizable) is in P for games given by W or W^m, using linear programming techniques. However, for MWVG inputs the problem is NP‑hard, as it subsumes the dimension‑determination problem proved hard by Deinekó and Woeginger. Conversely, every simple game is MWVG‑Realizable (by definition of dimension), but constructing an explicit MWVG may require exponentially many weight vectors.

In summary, the authors provide a comprehensive map of the computational landscape for influence comparison in simple games. They identify easy cases (dummy/veto detection, linearity for WVGs, Shapley‑Shubik and Chow computation) and hard cases (#P‑completeness of Banzhaf, NP‑hardness of linearity for MWVGs, NP‑hardness of WVG‑realizability from MWVG). The results clarify which representations are suitable for algorithmic analysis of power and suggest directions for future work, such as approximation algorithms for #P‑hard indices, fixed‑parameter tractability with respect to the number of minimal winning coalitions, and efficient conversion between representations.