# Nakamura number
**Nakamura number** is a number associated with a simple game: it is the smallest number of winning coalitions whose intersection is empty, taken to be infinite if some individual belongs to every winning coalition (a vetoer).[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf) The concept is due to Kenjiro Nakamura, whose 1979 theorem gives a necessary and sufficient condition for a simple game with preferences to have a nonempty core for all profiles of individual preferences: the number of alternatives must be less than the game's Nakamura number.[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games) For a q-rule game with n voters, the Nakamura number is the lowest integer bound of n/(n − q), and the greater the number, the larger the set of alternatives from which the rule can always find a maximal element.[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf) • [2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games)
For a simple game, it is the smallest number of winning coalitions whose intersection is empty[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf). The concept is due to Kenjiro Nakamura, whose 1979 theorem gives a necessary and sufficient condition for a simple game with preferences to have a nonempty core for all profiles of individual preferences: the number of alternatives must be less than the game's Nakamura number[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games).
| Key fact | Detail |
|---|---|
| Definition | The smallest size of a collection of winning coalitions with an empty intersection; if some individual belongs to every winning coalition (a vetoer), the number is taken to be infinite[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf) |
| Nakamura's theorem (1979) | A simple game has a nonempty core for all preference profiles if and only if the number of alternatives is less than its Nakamura number, or the game is weak (has veto players)[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games) • [3](https://www.jstage.jst.go.jp/article/economics1950/29/3/29_3_259/_pdf) |
| q-rule value | For a q-rule game with n voters, the Nakamura number is the lowest integer bound of n/(n − q)[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf) |
| Meaning | The greater the Nakamura number, the larger the set of alternatives from which the rule can always find a maximal element[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games) |
| Computable games | A computable simple game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games) |
| Spatial variant | The spatial Nakamura number, based on convex hulls of connected winning coalitions, is never smaller than the ordinary Nakamura number[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf) |
## Definition
A simple game (voting game) is a pair G = (N, W), where N is a set of players and W is the set of winning coalitions, satisfying monotonicity: if S is winning and S is contained in T, then T is winning[3](https://www.jstage.jst.go.jp/article/economics1950/29/3/29_3_259/_pdf). The Nakamura number ν(G) is defined in two cases: if the intersection of all winning coalitions is nonempty, ν(G) = ∞; otherwise ν(G) is the minimum size of a subcollection σ of W whose intersection is empty[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf). In words, it is the minimal number of winning coalitions such that their intersection is empty. When an individual belongs to all winning coalitions, that individual is called a vetoer, and the Nakamura number is treated as tending to infinity[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf).
Two commonly studied classes of games are quota games, written G(n, q), in which a coalition is winning if and only if it has at least q members, and weighted games G(w₁, …, wₙ; q), in which player i has wᵢ votes and a coalition wins by gathering more than q votes[1](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf).
## Nakamura's theorem and the core
The core of a simple game with preferences is the set of alternatives that cannot be improved upon by any winning coalition; formally, alternative x is in the core if no alternative y dominates it via some winning coalition C, meaning every member of C strictly prefers y to x[5](https://www.researchgate.net/publication/24138070_On_the_Chacteristic_Numbers_of_Voting_Games). Nakamura's theorem (1979) states that the core is nonempty for every profile of preferences if and only if the number of alternatives is less than the Nakamura number of the game[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games). In his 1978 formulation for social choice functions, Nakamura showed that a binary social choice function is non-null if and only if either an individual with veto power exists or the number of alternatives in each feasible set is less than the Nakamura number; in the latter case there arise potential possibilities to avoid the application of "the voting paradox"[3](https://www.jstage.jst.go.jp/article/economics1950/29/3/29_3_259/_pdf).
The practical reading is a bound on the size of the alternative set. If the number of alternatives |A| is below ν, the core is nonempty for all profiles; if |A| exceeds ν, there exists a profile with which the social choice game produces a top cycle that can include Pareto-dominated alternatives[6](https://www.kier.kyoto-u.ac.jp/wp/wp-content/uploads/2021/03/DP936.pdf). Thus the greater the Nakamura number for a given game, the larger the set of alternatives from which the rule can always find a maximal element[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games).
## Values for q-rules
For q-rule games, in which q of n votes are needed to win, the Nakamura number is the lowest integer bound of n/(n − q)[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf). The reasoning is that each minimal winning coalition excludes exactly n − q voters, so a collection of winning coalitions with empty intersection must collectively exclude the entire electorate[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf). This bound connects directly to spatial voting: by a result of Greenberg (1979), in two dimensions, for a q-rule with quota greater than 2/3 of the voters, there must always be a core[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf).
## The spatial Nakamura number
In spatial voting, where alternatives are points in a multidimensional space, the **spatial Nakamura number** (SN) is the minimal number of minimally connected winning coalitions whose convex hull intersection is empty[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf). It can be no less than the ordinary Nakamura number, because any such collection of connected winning coalitions contains winning coalitions with an empty intersection[4](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf). In the continuous or smooth setting, a voting rule has a core if and only if the dimension of the alternative space is small relative to the rule's Nakamura number; for smooth rules an associated instability dimension classifies rules so that, for almost all smooth profiles, the core is empty when the policy-space dimension is at least the instability dimension (or at least one greater if the space has a boundary), in which case voting outcomes can depend on accidental details of the political process[7](https://www.sciencedirect.com/science/article/pii/0895717789904147).
## Restrictions and extensions
Kumabe and Mihara studied the restrictions that properties of a simple game place on its Nakamura number, in the setting of computable simple games. They found 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 has a finite carrier; the lack of strongness often results in alternatives that cannot be strictly ranked[2](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games). The Nakamura number also plays a role in preference aggregation and multi-criterion ranking generally: the number of alternatives that the players can always deal with rationally is less than this number[8](https://arxiv.org/abs/1107.0439).
Later work has placed the number in a broader picture. [Donald G. Saari](https://www.edgepedia.org/donald-g-saari), a mathematician known for work on voting theory, extended Sen's theorem on majority vote cycles to all q-rules and gave a new interpretation of the Nakamura number, subsuming Nakamura's number, Greenberg's theorem, and single-peaked preferences under a common explanation of paired-comparison voting difficulties[9](https://www.sciencedirect.com/science/article/pii/S016548961400002X). Research continues on bounds for the Nakamura number of different types of voting games and on its relations to other concepts of cooperative game theory[10](https://arxiv.org/abs/1711.06611).
## References
1. [On the Characteristic Numbers of Voting Games (CiteSeerX)](https://citeseerx.ist.psu.edu/document?doi=9ef67040783a69184687db67003abc9bf2482ab7&repid=rep1&type=pdf)
2. [Kumabe, M. and Mihara, H.R. (2008). The Nakamura numbers for computable simple games. Social Choice and Welfare 31:621–640](https://www.researchgate.net/publication/226720394_The_Nakamura_numbers_for_computable_simple_games)
3. [Nakamura, K. (1978). Necessary and Sufficient Conditions on the Existence of a Class of Social Choice Functions. The Economic Studies Quarterly 29(3):259](https://www.jstage.jst.go.jp/article/economics1950/29/3/29_3_259/_pdf)
4. [Grofman, B. and Feld, S. (1992). Group Decision Making. Organizational Behavior and Human Decision Processes](https://sites.socsci.uci.edu/~bgrofman/73%20Grofman%20and%20Feld.%201992.%20Group%20Decision%20Making...%20Org.%20Behav.%20and%20Hu.%20Per..pdf)
5. [On the Characteristic Numbers of Voting Games (ResearchGate)](https://www.researchgate.net/publication/24138070_On_the_Chacteristic_Numbers_of_Voting_Games)
6. [KIER Discussion Paper Series No. 936, Kyoto University](https://www.kier.kyoto-u.ac.jp/wp/wp-content/uploads/2021/03/DP936.pdf)
7. [Smooth social choice. Mathematical and Computer Modelling 12:417–435 (1989)](https://www.sciencedirect.com/science/article/pii/0895717789904147)
8. [The Nakamura numbers for computable simple games (arXiv)](https://arxiv.org/abs/1107.0439)
9. [Saari, D.G. (2014). Unifying voting theory from Nakamura's to Greenberg's theorems. Mathematical Social Sciences 69:1–11](https://www.sciencedirect.com/science/article/pii/S016548961400002X)
10. [Bounds for the Nakamura number (arXiv)](https://arxiv.org/abs/1711.06611)
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*Topic: Encyclopedia › Society and history › Politics and government › Elections and representation › Electoral systems and principles › Electoral theory and criteria › Voting paradoxes and impossibility results › Participation and consistency criteria results*
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