§2.3: Total influence

A very important quantity in the analysis of a boolean function is the sum of its influences.

Definition 26 The total influence of $f : \{-1,1\}^n \to {\mathbb R}$ is defined to be $\mathbf{I}[f] = \sum_{i=1}^n \mathbf{Inf}_i[f].$

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§2.2: Influences and derivatives

Given a voting rule $f : \{-1,1\}^n \to \{-1,1\}$ it’s natural to try to measure the “influence” or “power” of the $i$th voter. One can define this to be the “probability that the $i$th vote affects the outcome”.

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§2.1: Social choice functions

In this section we describe some rudiments of the mathematics of social choice, a topic studied by economists, political scientists, mathematicians, and computer scientists. The fundamental question in this area is how best to aggregate the opinions of many agents. Examples where this problem arises include citizens voting in an election, committees deciding on [...]

§1.6: Highlight: Almost linear functions and the BLR Test

In linear algebra there are two equivalent definitions of what it means for a function to be linear:

Definition 29 A function $f : {\mathbb F}_2^n \to {\mathbb F}_2$ is linear if either of the following equivalent conditions hold:

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