The Nakamura numbers for computable simple games

TL;DR

This paper investigates how properties of computable simple games influence their Nakamura number, revealing that certain conditions are necessary for the number to be finite and greater than three, impacting preference aggregation.

Contribution

It provides a comprehensive analysis of restrictions on the Nakamura number imposed by various properties of computable simple games, highlighting conditions for finiteness and size.

Findings

Finite Nakamura number > 3 requires proper, nonstrong, nonweak properties

Computable games with certain properties have limited ranking capabilities

Lack of strongness affects the strict ranking of alternatives

Abstract

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.