Nonanonymity and sensitivity of computable simple games

This paper investigates algorithmic computability of simple games (voting games). It shows that (i) games with a finite carrier are computable, (ii) computable games have both finite winning coalitions and cofinite losing coalitions, and (iii) computable games violate any conceivable notion of anonymity, including finite anonymity and measurebased anonymity. The paper argues that computable games are excluded from the intuitive class of gniceh infinite games, employing the notion of ginsensitivityh\-equal treatment of any two coalitions that differ only on a finite set.Voting games, infinitely many players, ultrafilters, recursion theory, Turing computability, finite carriers, finite winning coalitions, algorithms

Computability of simple games: A characterization and application to the core

The class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura's theorem about the nonemptyness of the core and shows that computable games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted.Comment: 35 pages; To appear in Journal of Mathematical Economics; Appendix added, Propositions, Remarks, etc. are renumbere

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

Computability of simple games: A characterization and application to the core

It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable simple games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted

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 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 infinitegames and noncomputable ones. This strongly suggests that computability is logically, as well as conceptually, unrelated to the conventional axioms