The Nakamura numbers for computable simple games

The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked.

💡 Research Summary

The paper investigates the Nakamura number—a pivotal measure in social choice theory that determines how many alternatives a collective decision rule can handle without generating cycles—in the context of computable simple (voting) games. A simple game ω is defined on a countably infinite set of players N = {0,1,2,…} as a collection of recursive coalitions (subsets of N whose membership can be decided by a Turing program). Each coalition is either winning (assigned 1) or losing (assigned 0). The authors adopt the standard four axioms used to classify simple games:

• Monotonicity (M) – supersets of winning coalitions are also winning.
• Properness (P) – no coalition and its complement are both winning.
• Strongness (S) – for every losing coalition, its complement is winning.
• Non‑weakness (W) – the intersection of all winning coalitions (the set of “veto players”) is non‑empty; a game that fails this condition is called weak.

In addition, they distinguish between finite games (those possessing a finite carrier—a coalition that determines the outcome of every other coalition) and infinite games (no such carrier). By assigning a plus sign to a satisfied axiom and a minus sign to a violated one, each game falls into one of 2⁴·2 = 32 possible “type” codes; the paper actually works with 25 distinct types because some combinations are impossible.

The Nakamura number ν(ω) is defined as the smallest cardinality of a collection of winning coalitions whose intersection is empty (if the game is non‑weak); otherwise ν(ω) = +∞. Nakamura’s classic theorem (1979) states that for any profile of individual strict preferences, the core (the set of undominated alternatives) is non‑empty for every finite set X of alternatives iff |X| < ν(ω). Kumabe and Mihara (2008) showed that for computable, non‑weak games ν(ω) is always finite, thereby linking computability to a bound on the number of alternatives that can be rationally aggregated.

The central contribution of the present paper is a complete taxonomy of attainable Nakamura numbers for each computable game type. The authors compile their findings in Table 1, which lists, for every type, the possible finite values of ν(ω) (and whether infinite values occur) for both finite‑carrier and infinite‑carrier games. The most striking pattern is:

1. A computable game can have a finite Nakamura number larger than 3 only if it is proper, non‑strong, and non‑weak. In the notation of the table this means only the types “+ + − +” (monotonic, proper, non‑strong, non‑weak) and “− + − +” (non‑monotonic, proper, non‑strong, non‑weak) admit ν ≥ 3. All other types either force ν = 2 (e.g., any strong game, any weak game, any improper game) or ν = +∞ (any weak game).

2. Strongness is the decisive blocker. If a game is strong, the smallest collection of winning coalitions with empty intersection can contain at most two coalitions (a coalition and its complement), so ν ≤ 2. Consequently, to achieve ν > 3 one must abandon strongness, unless the game is dictatorial (the unique strong‑and‑weak case, which still yields ν = +∞).

3. Monotonicity does not affect the upper bound. Both monotonic and non‑monotonic games can reach ν > 3 provided they satisfy the proper‑non‑strong‑non‑weak trio.

4. Finite versus infinite carriers do not change the attainable ν. For any given type (except the trivial dictatorial type), the same set of finite Nakamura numbers can be realized by a game with a finite carrier and by a game with no finite carrier. The authors construct explicit Turing programs (or recursive coalitions) to demonstrate both possibilities.

To substantiate these claims, the paper supplies constructive proofs for each entry of the table. For ν = k (k ≥ 3) in the “+ + − +” type, they design k mutually disjoint winning coalitions and ensure that none of their complements is winning, thereby violating strongness while preserving properness and non‑weakness. For ν = 2, they present games where a coalition and its complement are the only winning sets (strong, possibly improper). For ν = +∞, they give weak games (empty intersection of winning coalitions) or games that are strong but weak, showing that the core can be empty for arbitrarily many alternatives.

The authors also discuss strategic foundations for these constructions. By interpreting a simple game as the set of coalitions that can enforce a particular outcome in a strategic game form, they illustrate how non‑proper or non‑monotonic games arise naturally (e.g., unanimity voting, certain game forms with “effective” actions). This links the abstract combinatorial properties to concrete decision‑making environments.

A noteworthy comparative analysis is provided with non‑computable ultra‑filters. Ultra‑filters are non‑principal, non‑computable objects that are strong yet yield an infinite Nakamura number, thereby circumventing Arrow’s impossibility theorem. However, because they are not algorithmically describable, they are deemed impractical for real‑world social choice mechanisms. The paper thus emphasizes that computability imposes a normative “due process” constraint: only rules that can be implemented by a finite algorithm are acceptable, and under this constraint the trade‑off between strongness and the size of the admissible alternative set becomes unavoidable.

In summary, the paper delivers a complete characterization of how the classic axioms of simple games interact with computability to bound the Nakamura number. Its main theorem—finite ν > 3 ⇔ proper ∧ ¬strong ∧ non‑weak—provides a clear design guideline for anyone constructing voting rules or multi‑criterion aggregation procedures that must be both algorithmically implementable and capable of handling three or more alternatives without cycles. The work bridges recursion theory, cooperative game theory, and social choice, and opens avenues for future research on partial strongness, bounded rationality, and algorithmic implementations of the constructed games.