  This paper contributes to the program of numerical characterisation and
classification of simple games outlined in the classical monograph of von
Neumann and Morgenstern (1944). One of the most fundamental questions of this
program is what makes a simple game a weighted majority game. The necessary and
sufficient conditions that guarantee weightedness were obtained by Elgot (1961)
and refined by Taylor and Zwicker (1992). If a simple game does not have
weights, then rough weights may serve as a reasonable substitute (see their use
in Taylor and Zwicker, 1992). A simple game is roughly weighted if there exists
a system of weights and a threshold such that all coalitions whose combined
weight is above the threshold are winning and all coalitions whose combined
weight is below the threshold are losing and a tie-breaking is needed to
classify the coalitions whose combined weight is exactly the threshold. Not all
simple games are roughly weighted, and the class of projective games is a prime
example.
  In this paper we give necessary and sufficient conditions for a simple game
to have rough weights. We define two functions f(n) and g(n) that measure the
deviation of a simple game from a weighted majority game and roughly weighted
majority game, respectively. We formulate known results in terms of lower and
upper bounds for these functions and improve those bounds. We also investigate
rough weightedness of simle games with a small number of players.
