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

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

Arrow's Theorem, countably many agents, and more visible invisible dictators

For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrowfs conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain goracleh resolves Miharafs impossibility result (1997) about computability of social welfare functions.Arrow impossibility theorem, Turing computability, recursion theory, oracle algorithms, free ultrafilters

Coalitionally strategyproof functions depend only on the most-preferred alternatives

In a framework allowing infinitely many individuals, I prove that coalitionally strategyproof social choice functions satisfy gtops only.h That is, they depend only on which alternative each individual prefers the most, not on which alternative she prefers the second most, the third, . . . , or the least. The functions are defined on the domain of profiles measurable with respect to a Boolean algebra of coalitions. The unrestricted domain of profiles is an example of such a domain. I also prove an extension theorem.Gibbard-Satterthwaite theorem, dominant strategy implementation, social choice functions, infinitely large societies, tops only