# Nakamura number

In cooperative game theory and social choice theory, the **Nakamura number** measures the degree of rationality
of preference aggregation rules (collective decision rules), such as voting rules.
It is an indicator of the extent to which an aggregation rule can yield well-defined choices.

- If the number of alternatives (candidates; options) to choose from is less than this number, then the rule in question will identify "best" alternatives without any problem.

In contrast,

- if the number of alternatives is greater than or equal to this number, the rule will fail to identify "best" alternatives for some pattern of voting (i.e., for some
**profile**(tuple) of individual preferences), because a voting paradox will arise (a*cycle*generated such as alternative socially preferred to alternative , to , and to ).

The larger the Nakamura number a rule has, the greater the number of alternatives the rule can rationally deal with.
For example, since (except in the case of four individuals (voters)) the Nakamura number of majority rule is three,
the rule can deal with up to two alternatives rationally (without causing a paradox).
The number is named after Kenjiro Nakamura (1947–1979), a Japanese game theorist who proved the above fact
that the rationality of collective choice critically depends on the number of alternatives.^{[1]}

## Overview

To introduce a precise definition of the Nakamura number, we give an example of a "game" (underlying the rule in question)
to which a Nakamura number will be assigned.
Suppose the set of individuals consists of individuals 1, 2, 3, 4, and 5.
Behind majority rule is the following collection of ("decisive") *coalitions* (subsets of individuals) having at least three members:

- { {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5} }

A Nakamura number can be assigned to such collections, which we call *simple games*.
More precisely, a **simple game** is just an arbitrary collection of coalitions;
the coalitions belonging to the collection are said to be *winning*; the others *losing*.
If all the (at least three, in the example above) members of a winning coalition prefer alternative x to alternative y,
then the society (of five individuals, in the example above) will adopt the same ranking (*social preference*).

The **Nakamura number** of a simple game is defined as the minimum number of winning coalitions with empty intersection.
(By intersecting this number of winning coalitions, one can sometimes obtain an empty set.
But by intersecting less than this number, one can never obtain an empty set.)
The Nakamura number of the simple game above is three, for example,
since the intersection of any two winning coalitions contains at least one individual
but the intersection of the following three winning coalitions is empty: , , .

**Nakamura's theorem** (1979^{[2]}) gives the following necessary (also sufficient if the set of alternatives is finite) condition for a simple game to have a nonempty "core" (the set of socially "best" alternatives) for all profiles of individual preferences:
the number of alternatives is less than the Nakamura number of the simple game.
Here, the **core** of a simple game with respect to the profile of preferences is the set of all alternatives
such that there is no alternative
that every individual in a winning coalition prefers to ; that is, the set of *maximal* elements of the social preference.
For the majority game example above, the theorem implies that the core will be empty (no alternative will be deemed "best") for some profile,
if there are three or more alternatives.

Variants of Nakamura's theorem exist that provide a condition for the core to be nonempty
(i) for all profiles of *acyclic* preferences;
(ii) for all profiles of *transitive* preferences; and
(iii) for all profiles of *linear orders*.
There is a different kind of variant (Kumabe and Mihara, 2011^{[3]}),
which dispenses with *acyclicity*, the weak requirement of rationality.
The variant gives a condition for the core to be nonempty for all profiles of preferences that have *maximal elements*.

For *ranking* alternatives, there is a very well known result called "Arrow's impossibility theorem" in social choice theory,
which points out the difficulty for a group of individuals in ranking three or more alternatives.
For *choosing* from a set of alternatives (instead of *ranking* them), Nakamura's theorem is more relevant.^{[5]}
An interesting question is how large the Nakamura number can be.
It has been shown that for a (finite or) algorithmically *computable* simple game that has no *veto player*
(an individual that belongs to every winning coalition)
to have a Nakamura number greater than three, the game has to be *non-strong*.^{[6]}
This means that there is a *losing* (i.e., not winning) coalition whose complement is also losing.
This in turn implies that nonemptyness of the core is assured for a set of three or more alternatives
only if the core may contain several alternatives that cannot be strictly ranked.^{[8]}

## Framework

Let be a (finite or infinite) nonempty set of *individuals*.
The subsets of are called **coalitions**.
A **simple game** (voting game) is a collection of coalitions.
(Equivalently, it is a coalitional game that assigns either 1 or 0 to each coalition.)
We assume that is nonempty and does not contain an empty set.
The coalitions belonging to are *winning*; the others are *losing*.
A simple game is **monotonic** if and
imply .
It is **proper** if implies .
It is **strong** if imples.
A **veto player** (vetoer) is an individual that belongs to all winning coalitions.
A simple game is **nonweak** if it has no veto player.
It is **finite** if there is a finite set (called a *carrier*) such that for all coalitions ,
we have iff .

Let be a (finite or infinite) set of *alternatives*, whose cardinal number (the number of elements)
is at least two.
A (strict) **preference** is an *asymmetric* relation on :
if (read " is preferred to "),
then .
We say that a preference is *acyclic* (does not contain *cycles*) if
for any finite number of alternatives ,
whenever , ,…, ,
we have . Note that acyclic relations are asymmetric, hence preferences.

A **profile** is a list of individual preferences .
Here means that individual prefers alternative
to at profile .

A *simple game with ordinal preferences* is a pair consisting
of a simple game and a profile .
Given , a *dominance* (social preference) relation is defined
on by if and only if there is a winning coalition
satisfying for all .
The **core** of is the set of alternatives undominated by
(the set of maximal elements of with respect to ):

## The Nakamura number: the definition and examples

The **Nakamura number** of a simple game is the size (cardinal number)
of the smallest collection of winning coalitions with empty intersection:^{[9]}

if (no veto player);^{[2]}
otherwise, (greater than any cardinal number).

it is easy to prove that if is a simple game without a veto player, then .

**Examples** for finitely many individuals () (see Austen-Smith and Banks (1999), Lemma 3.2^{[4]}).
Let be a simple game that is monotonic and proper.

- If is strong and without a veto player, then .
- If is the majority game (i.e., a coalition is winning if and only if it consists of more than half of individuals), then if ; if .
- If is a -rule (i.e., a coalition is winning if and only if it consists of at least individuals) with , then , where is the smallest integer greater than or equal to .

**Examples** for at most countably many individuals ().
Kumabe and Mihara (2008) comprehensively study the restrictions that various properties
(monotonicity, properness, strongness, nonweakness, and finiteness) for simple games
impose on their Nakamura number (the Table "Possible Nakamura Numbers" below summarizes the results).
In particular, they show that an algorithmically *computable* simple
game ^{[10]}
without a veto player has a Nakamura number greater than 3 only if it is proper and nonstrong.^{[6]}

Type | Finite games | Infinite games |
---|---|---|

1111 | 3 | 3 |

1110 | +∞ | none |

1101 | ≥3 | ≥3 |

1100 | +∞ | +∞ |

1011 | 2 | 2 |

1010 | none | none |

1001 | 2 | 2 |

1000 | none | none |

0111 | 2 | 2 |

0110 | none | none |

0101 | ≥2 | ≥2 |

0100 | +∞ | +∞ |

0011 | 2 | 2 |

0010 | none | none |

0001 | 2 | 2 |

0000 | none | none |

## Nakamura's theorem for acyclic preferences

**Nakamura's theorem** (Nakamura, 1979, Theorems 2.3 and 2.5^{[2]}).
Let be a simple game. Then the core is nonempty for all profiles of acyclic preferences if and only if is finite and .

**Remarks**

- Nakamura's theorem is often cited in the following form, without reference to the core (e.g., Austen-Smith and Banks, 1999, Theorem 3.2
^{[4]}): The dominance relation is acyclic for all profiles of acyclic preferences if and only if for all finite (Nakamura 1979, Theorem 3.1^{[2]}).

- The statement of the theorem remains valid if we replace "for all profiles of
*acyclic*preferences" by "for all profiles of*negatively transitive*preferences" or by "for all profiles of*linearly ordered*(i.e., transitive and total) preferences".^{[12]}

- The theorem can be extended to -simple games. Here, the collection of coalitions is an arbitrary Boolean algebra of subsets of , such as the -algebra of Lebesgue measurable sets. A -
*simple game*is a subcollection of . Profiles are suitably restricted to measurable ones: a profile is*measurable*if for all , we have .^{[3]}

## A variant of Nakamura's theorem for preferences that may contain cycles

In this section, we discard the usual assumption of acyclic preferences.
Instead, we restrict preferences to those having a maximal element on a given *agenda* (*opportunity set* that a group of individuals are confronted with),
a subset of some underlying set of alternatives.
(This weak restriction on preferences might be of some interest from the viewpoint of behavioral economics.)
Accordingly, it is appropriate to think of as an *agenda* here.
An alternative is a *maximal* element with respect to
(i.e., has a maximal element ) if there is no such that . If a preference is acyclic over the underlying set of alternatives, then it has a maximal element on every *finite* subset .

We introduce a strengthening of the core before stating the variant of Nakamura's theorem.
An alternative can be in the core even if there is a winning coalition of individuals that are "dissatisfied" with
(i.e., each prefers some to ).
The following solution excludes such an :^{[3]}

- An alternative is in the
**core****without majority dissatisfaction**if there is no winning coalition such that for all , is*non-maximal*(there exists some satisfying ).

It is easy to prove that depends only on the set of maximal elements of each individual and is included in the union of such sets. Moreover, for each profile , we have .

**A variant of Nakamura's theorem** (Kumabe and Mihara, 2011, Theorem 2^{[3]}).
Let be a simple game. Then the following three statements are equivalent:

- ;
- the core without majority dissatisfaction is nonempty for all profiles of preferences that have a maximal element;
- the core is nonempty for all profiles of preferences that have a maximal element.

**Remarks**

- Unlike Nakamura's original theorem, being finite is
*not a necessary condition*for or to be nonempty for all profiles . Even if an agenda has infinitely many alternatives, there is an element in the cores for appropriate profiles, as long as the inequality is satisfied.

- The statement of the theorem remains valid if we replace "for all profiles of preferences that have a maximal element" in statements 2 and 3 by "for all profiles of preferences that have
*exactly one*maximal element" or "for all profiles of*linearly ordered*preferences that have a maximal element" (Kumabe and Mihara, 2011, Proposition 1).

- Like Nakamura's theorem for acyclic preferences, this theorem can be extended to -simple games. The theorem can be extended even further (1 and 2 are equivalent; they imply 3) to
*collections**of winning sets*by extending the notion of the Nakamura number.^{[13]}

## See also

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }} Nakamura received Doctor's degree in Social Engineering in 1975 from Tokyo Institute of Technology.
- ↑
^{2.0}^{2.1}^{2.2}^{2.3}Template:Cite doi - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Template:Cite doi - ↑
^{4.0}^{4.1}^{4.2}^{4.3}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑
Nakamura's
*original*theorem is directly relevant to the class of*simple*preference aggregation rules, the rules completely described by their family of decisive (winning) coalitions. (Given an aggregation rule, a coalition is*decisive*if whenever every individual in prefers to , then so does the society.) Austen-Smith and Banks (1999),^{[4]}a textbook on social choice theory that emphasizes the role of the Nakamura number, extends the Nakamura number to the wider (and empirically important) class of*neutral*(i.e., the labeling of alternatives does not matter) and*monotonic*(if is socially preferred to , then increasing the support for over preserves this social preference) aggregation rules (Theorem 3.3), and obtain a theorem (Theorem 3.4) similar to Nakamua's. - ↑
^{6.0}^{6.1}Template:Cite doi - ↑ Template:Cite doi
- ↑
There exist monotonic, proper, strong simple games without a veto player
that have an infinite Nakamura number. A nonprincipal ultrafilter is an example, which can be used to
define an aggregation rule (social welfare function) satisfying Arrow's conditions
if there are infinitely many individuals.
^{[7]}A serious drawback of nonprincipal ultrafilters for this purpose is that they are not algorithmically computable. - ↑ The minimum element of the following set exists since every nonempty set of ordinal numbers has a least element.
- ↑ See a section for Rice's theorem
for the definition of a
*computable*simple game. In particular, all finite games are computable. - ↑ Possible Nakamura numbers for computable simple games are given in each entry, assuming that an empty coalition is losing. The sixteen types are deﬁned in terms of the four properties: monotonicity, properness, strongness, and nonweakness (lack of a veto player). For example, the row corresponding to type 1110 indicates that among the monotonic (1), proper (1), strong (1), weak (0, because not nonweak) computable simple games, ﬁnite ones have a Nakamura number equal to and inﬁnite ones do not exist. The row corresponding to type 1101 indicates that any (and no ) is the Nakamura number of some finite (alternatively, infinite) simple game of this type. Observe that among nonweak simple games, only types 1101 and 0101 attain a Nakamura number greater than 3.
- ↑ The "if" direction is obvious while the "only if" direction is stronger than the statement of the theorem given above
(the proof is essentially the same).
These results are often stated in terms of
*weak*preferences (e.g, Austen-Smith and Banks, 1999, Theorem 3.2^{[4]}). Define the weak preference by . Then is asymmetric iff is complete; is negatively transitive iff is transitive. is*total*if implies or . - ↑ The framework distinguishes the algebra of
*coalitions*from the larger collection of the sets of individuals to which winning/losing status can be assigned. For example, is the algebra of recursive sets and is the lattice of recursively enumerable sets (Kumabe and Mihara, 2011, Section 4.2).