Nakamura number
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 <math>a</math> socially preferred to alternative <math>b</math>, <math>b</math> to <math>c</math>, and <math>c</math> to <math>a</math>). 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 (1947–1979), a Japanese game theorist who proved the above fact that the rationality of collective choice critically depends on the number of alternatives.
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 [[cooperative-game-theory|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: <math>\{1,2,3\}</math>, <math>\{4,5,1\}</math>, <math>\{2,3,4\}</math>.
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 <math>x</math> such that there is no alternative <math>y</math> that every individual in a winning coalition prefers to <math>x</math>; 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 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 <math>x</math> is socially preferred to <math>y</math>, then increasing the support for <math>x</math> over <math>y</math> preserves this social preference) aggregation rules (Theorem 3.3), and obtain a theorem (Theorem 3.4) similar to Nakamua's.}} 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. 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.
Framework Let <math>N</math> be a (finite or infinite) nonempty set of individuals. The subsets of <math>N</math> are called coalitions. A simple game (voting game) is a collection <math>W</math> of coalitions. (Equivalently, it is a coalitional game that assigns either 1 or 0 to each coalition.) We assume that <math>W</math> is nonempty and does not contain an empty set. The coalitions belonging to <math>W</math> are winning; the others are losing. A simple game <math>W</math> is monotonic if <math>S \in W</math> and <math>S\subseteq T</math> imply <math>T \in W</math>. It is proper if <math>S \in W</math> implies <math>N\setminus S \notin W</math>. It is strong if <math>S \notin W</math> implies <math>N\setminus S \in W</math>. 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) <math>T \subseteq N</math> such that for all coalitions <math>S</math>, we have <math>S \in W</math> iff <math>S\cap T \in W</math>.
Let <math>X</math> be a (finite or infinite) set of alternatives, whose cardinal number (the number of elements) <math>\# X</math> is at least two. A (strict) preference is an asymmetric relation <math>\succ</math> on <math>X</math>: if <math>x \succ y</math> (read "<math>x</math> is preferred to <math>y</math>"), then <math>y\not \succ x</math>. We say that a preference <math>\succ</math> is acyclic (does not contain cycles) if for any finite number of alternatives <math>x_1, \ldots, x_m</math>, whenever <math>x_1 \succ x_2</math>, <math>x_2 \succ x_3</math>,…, <math>x_{m-1} \succ x_m</math>, we have <math>x_m \not\succ x_1</math>. Note that acyclic relations are asymmetric, hence preferences.
A profile is a list <math>p=(\succ_i^p)_{i \in N}</math> of individual preferences <math>\succ_i^p</math>. Here <math>x \succ_i^p y</math> means that individual <math>i</math> prefers alternative <math>x</math> to <math>y</math> at profile <math>p</math>.
A simple game with ordinal preferences is a pair <math>(W, p)</math> consisting of a simple game <math>W</math> and a profile <math>p</math>. Given <math>(W, p)</math>, a dominance (social preference) relation <math>\succ^p_W</math> is defined on <math>X</math> by <math>x \succ^p_W y</math> if and only if there is a winning coalition <math>S \in W</math> satisfying <math>x \succ_i^p y</math> for all <math>i \in S</math>. The core <math>C(W,p)</math> of <math>(W, p)</math> is the set of alternatives undominated by <math>\succ^p_W</math> (the set of maximal elements of <math>X</math> with respect to <math>\succ^p_W</math>): :<math>x \in C(W,p)</math> if and only if there is no <math>y\in X</math> such that <math>y \succ^p_W x</math>.
Definition and examples The Nakamura number <math>\nu(W)</math> of a simple game <math>W</math> is the size (cardinal number) of the smallest collection of winning coalitions with empty intersection: :<math>\nu(W)=\min\{\# W': W'\subseteq W; \cap W'=\emptyset \}</math> if <math>\cap W = \cap_{S \in W} S = \emptyset</math> (no veto player); without a veto player has a Nakamura number greater than 3 only if it is proper and nonstrong. |- ! 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). Let <math>W</math> be a simple game. Then the following three statements are equivalent: #<math>\# X < \nu(W)</math>; #the core <math>C^+(W,p)</math> without majority dissatisfaction is nonempty for all profiles <math>p</math> of preferences that have a maximal element; #the core <math>C(W,p)</math> is nonempty for all profiles <math>p</math> of preferences that have a maximal element.
- Remarks**
- Unlike Nakamura's original theorem, <math>X</math> being finite is not a necessary condition for <math>C^+(W,p)</math> or <math>C(W,p)</math> to be nonempty for all profiles <math>p</math>. Even if an agenda <math>X</math> has infinitely many alternatives, there is an element in the cores for appropriate profiles, as long as the inequality <math>\# X < \nu(W)</math> is satisfied.
- The statement of the theorem remains valid if we replace "for all profiles <math>p</math> of preferences that have a maximal element" in statements 2 and 3 by "for all profiles <math>p</math> of preferences that have exactly one maximal element" or "for all profiles <math>p</math> 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 <math>\mathcal{B}</math>-simple games. The theorem can be extended even further (1 and 2 are equivalent; they imply 3) to collections <math>W' \subseteq \mathcal{B}'</math> of winning sets by extending the notion of the Nakamura number.{{#tag:ref | The framework distinguishes the algebra <math>\mathcal{B}</math> of coalitions from the larger collection <math>\mathcal{B}'</math> of the sets of individuals to which winning/losing status can be assigned. For example, <math>\mathcal{B}</math> is the algebra of recursive sets and <math>\mathcal{B}'</math> is the lattice of recursively enumerable sets (Kumabe and Mihara, 2011, Section 4.2).}}