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.
A Characterization of Ideal Weighted Secret Sharing Schemes
Beimel，Tassa and Weinreb（2008）和Farras and Padro（2010）从合成运算的角度对理想加权门限秘密共享方案的访问结构进行了部分刻画。他们对不可分解的理想加权门限访问结构进行了分类，并证明了任何其他的理想加权门限访问结构都是不可分解的理想加权门限访问结构的合成。目前还不清楚哪些组成的不可分解的加权阈值访问结构的加权。在本文中，我们填补了空白。利用博弈论的技巧，我们确定了不可分解的理想访问结构的组合是加权的，并得到了理想加权门限秘密共享方案的一个充要条件刻画。
