4 research outputs found
Computability of simple games: A complete investigation of the sixty-four possibilities
Classify simple games into sixteen "types" in terms of the four conventional
axioms: monotonicity, properness, strongness, and nonweakness. Further classify
them into sixty-four classes in terms of finiteness (existence of a finite
carrier) and algorithmic computability. For each such class, we either show
that it is empty or give an example of a game belonging to it. We observe that
if a type contains an infinite game, then it contains both computable ones and
noncomputable ones. This strongly suggests that computability is logically, as
well as conceptually, unrelated to the conventional axioms.Comment: 25 page
Preference aggregation theory without acyclicity: The core without majority dissatisfaction
Acyclicity of individual preferences is a minimal assumption in social choice
theory. We replace that assumption by the direct assumption that preferences
have maximal elements on a fixed agenda. We show that the core of a simple game
is nonempty for all profiles of such preferences if and only if the number of
alternatives in the agenda is less than the Nakamura number of the game. The
same is true if we replace the core by the core without majority
dissatisfaction, obtained by deleting from the agenda all the alternatives that
are non-maximal for all players in a winning coalition. Unlike the core, the
core without majority dissatisfaction depends only on the players' sets of
maximal elements and is included in the union of such sets. A result for an
extended framework gives another sense in which the core without majority
dissatisfaction behaves better than the core.Comment: 27+3 page
The Nakamura numbers for computable simple games
The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find 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 whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked.Nakamura number; voting games; the core; Turing computability; axiomatic method; multi-criterion decision-making