This chapter provides some basic background data. Basic set-theoretic notation is introduced in Section 1.1, and some combinatorics in Section 1.2. Formal descriptions of voting rules and related notation and terminology to be used throughout the book are given in Section 1.3. In Section 1.4 the basics of decision facing risk and expected utility theory are provided. Section 1.5 contains a short overview of a few basic concepts from game theory.

Basic set-theoretic notation

In general, sets are denoted by capitals (e.g. N, S, M, etc.), and when they are finite the same small case letter (n, s, m,…) denotes their number of elements or cardinality (sometimes denoted by #N, #S, #M,…). We write a ∈ S to express that an element a belongs to a set S, and a ∉ S otherwise. Given two sets A and B we write A ⊆ B to express that all elements in A are also in B, and we write A ⊂ B if A ⊆ B and A ≠ B. Symbols ‘∪’ and ‘∩’ denote the usual operations on sets of ‘union’ and ‘intersection’, while A\B denotes the set of those elements in A that do not belong to B. When B contains a single element i, i.e. B = {i}, we will often write A \ i instead of A \ {i} and A ∪ i instead of A ∪ {i}. The set consisting of all subsets of a set N is denoted by 2N (note that its cardinal is 2n).