Forms of representation for simple games: Sizes, conversions and equivalences

•We survey the main forms of representation for simple games, regular games and weighted games in literature.•We study the complexity of the conversion problem, i.e., the problem of computing a representation of a simple game given another representation.•We prove that several changes of representation that require exponential time can be solved with polynomial-delay.

Simple games are cooperative games in which the benefit that a coalition may have is always binary, i.e., a coalition may either win or loose. This paper surveys different forms of representation of simple games, and those for some of their subfamilies like regular games and weighted games. We analyze the forms of representations that have been proposed in the literature based on different data structures for sets of sets. We provide bounds on the computational resources needed to transform a game from one form of representation to another one. This includes the study of the problem of enumerating the fundamental families of coalitions of a simple game. In particular we prove that several changes of representation that require exponential time can be solved with polynomial-delay and highlight some open problems.