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.

In contrast,

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 [ja] (1947–1979), a Japanese game theorist who proved the above fact that the rationality of collective choice critically depends on the number of alternatives.

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:

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: ${\displaystyle \{1,2,3\}}$, ${\displaystyle \{4,5,1\}}$, ${\displaystyle \{2,3,4\}}$.

Nakamura's theorem (1979) 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 ${\displaystyle x}$ such that there is no alternative ${\displaystyle y}$ that every individual in a winning coalition prefers to ${\displaystyle x}$; 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), 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.