n-person single games and n-component reliability structures

Abstract

<p>Characteristic functions of "(0,1) n-person simple games" and structure functions of "n-component reliability structure system" are studied as "diadic functions". The latter function may belong to a number of categories, which are characterized and interrelated. These categories include the class of Bolean diadic functions, monotone diadic functions, super-additive diadic functions, sub-additive diadic functions, and inessential diadic functions. It is helpful in this connection to introduce certain concepts and constructs new to both games and systems, and to interpret certain concepts in terms of the area other than that of their origin. Additional constructs are reexamined in the context of the entire domain of diadic functions. Among the latter are certain "generating vectors" and "veto vectors", for which appropriate generating algorithms are introduced;The "Shapley form of an n-person game" is studied over the entire domain of diadic functions, and a version of the Shapley form for monotone diadic functions, based on generating vectors, is shown to be equivalence to the "minimal path representation" of system theory. Further, the uniqueness of the "Shapley value", is considered for diadic and more general functions. To this end, a set of revised "Shapley axioms" are introduced. Finally, several methods for expressing and relating characteristics or structure functions are surveyed, as well as several indices for measuring the importance of a player or a component. Here, again, the theme is maintained, of interpreting a given index in terms of the area other than that of its origin.</p>