  This paper has a twofold scope. The first one is to clarify and put in
evidence the isomorphic character of two theories developed in quite different
fields: on one side, threshold logic, on the other side, simple games. One of
the main purposes in both theories is to determine when a simple game is
representable as a weighted game, which allows a very compact and easily
comprehensible representation. Deep results were found in threshold logic in
the sixties and seventies for this problem. However, game theory has taken the
lead and some new results have been obtained for the problem in the last two
decades. The second and main goal of this paper is to provide some new results
on this problem and propose several open questions and conjectures for future
research. The results we obtain depend on two significant parameters of the
game: the number of types of equivalent players and the number of types of
shift-minimal winning coalitions.
