Chapter 9 exercises

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§9.6: Highlight: The Kahn–Kalai–Linial Theorem

Recalling the social choice setting of Chapter 2.5, consider a $2$-candidate, $n$-voter election using a monotone voting rule $f : \{-1,1\}^n \to \{-1,1\}$. We assume the impartial culture assumption (that the votes are independent and uniformly random), but with a twist: one of the candidates, say $b \in \{-1,1\}$, is able to secretly bribe $k$ [...]

§8.4: $p$-biased analysis

Perhaps the most common generalized domain in analysis of boolean functions is the case of the hypercube with “biased” bits.

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§8.2: Generalized Fourier formulas

In this section we will revisit a number of combinatorial/probabilistic notions and show that for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$, these notions have familiar Fourier formulas which don’t depend on the Fourier basis.

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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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