Theory and Practice, Problems and Paradoxes
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  for what he calls ‘simple game’. He attributes the concept to von 11 12 2. Groundwork of the Theory Neumann and Morgenstern , but as a matter of fact his class of simple games is considerably wider than that admitted by . To prevent confusion with the latter, we use the term ‘simple voting game’ for the broader concept. In...
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