It is an indicator of the extent to which an aggregation rule can yield well-defined choices.
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.
[1] 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.
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.
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).
(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
; that is, the set of maximal elements of the social preference.
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.
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.
be a (finite or infinite) nonempty set of individuals.
A veto player (vetoer) is an individual that belongs to all winning coalitions.
is acyclic (does not contain cycles) if for any finite number of alternatives
Note that acyclic relations are asymmetric, hence preferences.
A simple game with ordinal preferences is a pair
is the size (cardinal number) of the smallest collection of winning coalitions with empty intersection:[9] if
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.
Remarks 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.)
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.