Persuasion with Non-Linear Preferences

The Economics of Partisan Gerrymandering

“…The next lemma restates some results from Kolotilin, Corrao, and Wolitzky (2023), which we use to prove Propositions 8 and 9.…”

“…Similar results were established by Friedman and Holden (2008) and, in the persuasion context, Kolotilin, Corrao, and Wolitzky (2023). 24 To see the intuition, suppose a districting plan creates two districts, 1 and 2, with the same threshold aggregate shock r * , but where District 1 consists entirely of moderates and District 2 consists of a mix of left-wing and right-wing extremists.…”

“…Proof of Proposition 6. The proposition follows from Theorem 3 in Kolotilin, Corrao, and Wolitzky (2023) for the state-independent sender case, where V (a, θ) = V (a). We illustrate the proof in the case where supp(F ) and supp(H) are finite.…”

“…Friedman and Holden assume a finite number of districts, and also assume that the median and mode of Q coincide. Kolotilin, Corrao, and Wolitzky (2023) give sufficient conditions for single-dippedness in a more general model that allows state-dependent designer preferences.…”

“…However, after conditioning on the pivotal value of the aggregate shock, district-level uncertainty in Gul and Pesendorfer plays the same role as aggregate uncertainty in our model. 15 Specifically, the designer's problem is equivalent to the state-independent sender case of the persuasion problem studied in Kolotilin, Corrao, and Wolitzky (2023), which specializes the general Bayesian persuasion problem of Kamenica and Gentzkow (2011) by assuming that the state and the receiver's action are one-dimensional, the receiver's utility is supermodular and concave in his action, and the sender's utility is independent of the state and increasing in the receiver's action. In the gerrymandering context, state-independent sender preferences reflect the fact that the designer cares only about how many districts he wins and not directly about the composition of these districts.…”