Weighted and Roughly Weighted Simple Games

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 simple games with a small number of players.

[1]  P. Fishburn,et al.  The Mathematics Of Preference Choice And Order Essays In Honor Of Peter C Fishburn , 2009 .

[2]  Caroline J. Klivans.  Threshold graphs, shifted complexes, and graphical complexes , 2007, Discret. Math..

[3]  Steven Vajda,et al.  The Theory of Linear Economic Models , 1960 .

[4]  Josep Freixas,et al.  Complete simple games , 1996 .

[5]  L. Shapley.  Simple games: an outline of the descriptive theory. , 2007, Behavioral science.

[6]  Caroline J. Klivans.  Obstructions to Shiftedness , 2005, Discret. Comput. Geom..

[7]  Calvin C. Elgot,et al.  Truth functions realizable by single threshold organs , 1961, SWCT.

[8]  M. Richardson,et al.  On finite projective games , 1956 .

[9]  Saburo Muroga,et al.  Threshold logic and its applications , 1971 .

[10]  Behruz Tayfeh-Rezaie,et al.  A Hadamard matrix of order 428 , 2005 .

[11]  Peter C. Fishburn,et al.  Failure of cancellation conditions for additive linear orders , 1997 .

[12]  Peter C. Fishburn,et al.  Finite linear qualitative probability , 1996 .

[13]  Robert O. Winder,et al.  Threshold logic , 1971, IEEE Spectrum.

[14]  Yiwei Huang,et al.  Uniqueness of some cyclic projective planes , 2009, Des. Codes Cryptogr..

[15]  Fred S. Roberts,et al.  The Mathematics of Preference, Choice and Order , 2009, Studies in Choice and Welfare.

[16]  Josep Freixas,et al.  Simple games and weighted games: A theoretical and computational viewpoint , 2009, Discret. Appl. Math..

[17]  Marston Conder,et al.  A counterexample to Fishburn's conjecture on finite linear qualitative probability , 2004 .

[18]  Simon Marshall.  On the Existence of Extremal Cones and Comparative Probability Orderings , 2005, ISIPTA.

[19]  J. Sylvester.  LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers , 1867 .

[20]  Alan D. Taylor,et al.  A characterization of weighted voting , 1992 .

[21]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[22]  Arkadii Slinko,et al.  Additive Representability of Finite Measurement Structures , 2009, The Mathematics of Preference, Choice and Order.

[23]  C. Kraft,et al.  Intuitive Probability on Finite Sets , 1959 .

[24]  William S. Zwicker,et al.  Simple Games and Magic Squares , 1995, J. Comb. Theory, Ser. A.