The comparative models of persuasion

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In the canonical persuasion model, comparative statics has been an open question. We answer it, delineating which shifts of the sender's interim payoff lead her optimally to choose a more informative signal. Our first theorem identifies an ordinal notion of 'increased convexity' that we show characterises those shifts of the sender's interim payoff that lead her optimally to choose no less informative signals. To strengthen this conclusion to 'more informative' requires further assumptions: our second theorem identifies the necessary and sufficient condition on the sender's interim payoff, which strictly generalises the 'S'-shape commonly imposed in the literature.