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2.1 Simple Voting Games We begin by defining the most general class of mathematical structures commonly used to model voting decision rules. This, then, is the basic definition of the theory. 2.1.1 Definition A simple voting game — briefly, SVG — is a collection W of subsets of a finite set N , satisfying the following three conditions: (1) N ∈ W; (2) ∅ ∈ W; (3) Monotonicity: whenever X ⊆ Y ⊆ N and X ∈ W then also Y ∈ W. W is said to be a proper SVG if, in addition, it satisfies the condition (4) Whenever X ∈ W and Y ∈ W then X ∩ Y = ∅. Otherwise, W is said to be improper. We shall refer to N , the largest set in W, as the latter’s assembly. The members of N are the voters of W. A set of voters (that is, a subset of N ) is called a coalition of W. A coalition S is said to be a winning or losing coalition, according as S ∈ W or S ∈ W. 2.1.2 Remarks (i) Apart from some inessential modifications, this definition is the same as that given by Shapley in [95] for what he calls ‘simple game’. He attributes the concept to von 11 12 2. Groundwork of the Theory Neumann and Morgenstern [108], but as a matter of fact his class of simple games is considerably wider than that admitted by [108]. To prevent confusion with the latter, we use the term ‘simple voting game’ for the broader concept. In...