The mathematical study of social choice began in earnest in the late 1940s; see Riker [Rik61] for an early survey or the compilation [BGR09] for some modern results.
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The mathematical study of social choice began in earnest in the late 1940s; see Riker [Rik61] for an early survey or the compilation [BGR09] for some modern results. [...] [...] When there are just $2$ candidates, the majority function possesses all of the mathematical properties that seem desirable in a voting rule (e.g., May’s Theorem and Theorem 32). Unfortunately, as soon as there are $3$ (or more) candidates the problem of social choice becomes much more difficult. For example, suppose we have candidates $a$, [...] 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 \[ \mathbf{I}[f] = \sum_{i=1}^n \mathbf{Inf}_i[f]. \] [...] 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”. [...] In this chapter we introduce a number of important basic concepts including influences and noise stability. Many of these concepts are nicely motivated using the the language of social choice. The chapter is concluded with Kalai’s Fourierbased proof of Arrow’s Theorem. 

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