A note on the growth of the dimension in complete simple games

The remoteness from a simple game to a weighted game can be measured by the concept of the dimension or the more general Boolean dimension. It is known that both notions can be exponential in the number of voters. For complete simple games it was only recently shown that the dimension can also be exponential. Here we show that this is also the case for complete simple games with two types of voters and for the Boolean dimension of general complete simple games, which was posed as an open problem.

💡 Research Summary

The paper investigates the representation complexity of complete simple games, focusing on two related notions: the dimension (the smallest number of weighted games whose intersection yields the given simple game) and the Boolean dimension (the smallest number of weighted games combined by logical ∧ and ∨ to represent the game). While it is known that both measures can be exponential in the number of voters for arbitrary simple games, recent work showed that the dimension can also be exponential for complete simple games, but the construction required an unbounded number of voter types.

The author first proves that even when the number of voter types is restricted to two, there exist families of complete simple games whose dimension grows exponentially with the number of voters. This result closes the gap left by earlier constructions and demonstrates that the complexity does not stem merely from a large variety of voter classes. The proof builds on a careful encoding of winning and losing coalitions using shift‑minimal winning vectors and shows that any representation as an intersection of weighted games must involve exponentially many components.

Next, the paper addresses an open problem posed in the earlier work: can the Boolean dimension of a complete simple game also be exponential? By counting arguments, the author notes that the total number of complete simple games on n voters is at least 2^{c·n} for a constant c, while the number of distinct Boolean formulas built from s weighted games is at most 2^{O(sn² log(sn))}. Since the former outpaces the latter for s polynomial in n, most complete simple games inevitably require a Boolean dimension that is exponential in n. This establishes that the Boolean dimension, like the ordinary dimension, can be super‑polynomial for complete simple games.

The paper also contributes several structural results. Lemma 3.1 shows that for any complete simple game one can construct a representation as an intersection of weighted games where all voters belonging to a chosen equivalence class receive the same weight. Proposition 3.2 demonstrates that, in general, it is impossible to enforce the strict ordering of voter strengths across different equivalence classes in every weighted component of a minimal representation; a concrete example with four classes is provided. Lemma 3.4 gives an upper bound on the Boolean dimension: if a game has r shift‑minimal winning vectors and t voter types, then its Boolean dimension is at most r·t. The proof constructs, for each shift‑minimal winning vector, a conjunction of t weighted games that isolates that vector, and then takes the disjunction over the r vectors.

Finally, the author discusses implications for voting system design. Many real‑world decision bodies (e.g., the European Union Council) are modeled by complete simple games that are not weighted. The results imply that any attempt to simplify such systems by assigning weights will either require an exponential number of weighted components or accept a loss of fidelity. Moreover, the impossibility of preserving the natural ordering of voter strengths across all components limits the usefulness of uniform weight assignments in practice.

Overall, the paper settles two previously open questions about the growth of dimension and Boolean dimension in complete simple games, extends the known exponential lower bounds to the case of only two voter types, and provides both lower‑ and upper‑bound constructions that deepen our understanding of the combinatorial complexity inherent in voting games.