[Paper Review] The comparative statics of persuasion

Gregorio Curello, Ludvig Sinander|arXiv (Cornell University)|Apr 14, 2022

Opinion Dynamics and Social Influence4 citations

TL;DR

This paper establishes the first general comparative statics results for the persuasion model by identifying conditions under which shifts in the sender’s interim payoff lead to more informative signals. It introduces the 'crater property'—a geometric condition that characterizes when increased convexity in the sender’s payoff induces greater information provision, strictly generalizing the commonly used 'S-shaped' (convex–concave) assumption.

ABSTRACT

In the persuasion model, apart from a few special cases, 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 a coarse 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 convex-concave ('S') shape commonly imposed in the literature. We identify conditions under which increased alignment of interests between sender and receiver leads to comparative statics, and study a number of applications.

Motivation & Objective

• To resolve the long-open question of how changes in the sender’s interim payoff affect signal informativeness in the persuasion model.
• To characterize which shifts in the sender’s payoff lead to non-decreasing informativeness of optimal signals.
• To identify the necessary and sufficient condition on the sender’s payoff for such shifts to yield strictly more informative signals.
• To extend comparative statics results beyond the standard convex–concave ('S-shaped') payoff assumption.
• To examine when increased alignment of interests between sender and receiver leads to greater information provision.

Proposed method

• Introduces a coarse notion of 'less convex than' to characterize non-decreasing comparative statics in the persuasion model.
• Develops the 'crater property'—a geometric condition on the sender’s interim payoff that ensures shifts lead to more informative signals.
• Applies tools from monotone comparative statics, including Topkis’s theorem and interval dominance, to the persuasion framework.
• Uses weak regularity and convexity/concavity analysis of payoff functions over posterior beliefs.
• Employs a proof strategy based on contradiction, leveraging affine majorants and tangent line comparisons.
• Analyzes the role of the prior and the single-moment assumption, showing that prior shifts do not yield robust comparative statics.

Experimental results

Research questions

• RQ1 Which shifts in the sender’s interim payoff lead her to choose signals that are not less informative?
• RQ2 What additional condition on the sender’s interim payoff ensures that such shifts lead to strictly more informative signals?
• RQ3 How does increased alignment of interests between sender and receiver affect signal informativeness?
• RQ4 Can the standard 'S-shaped' (convex–concave) payoff assumption be strictly generalized for comparative statics?
• RQ5 Under what conditions do shifts in the prior lead to robust comparative statics in signal informativeness?

Key findings

• A coarse notion of 'less convex than' characterizes shifts in the sender’s interim payoff that lead to non-decreasing signal informativeness, regardless of the prior.
• The crater property is both necessary and sufficient for coarse-convexity shifts to induce weakly more informative signals, for any prior.
• The crater property generalizes the standard convex–concave ('S-shaped') payoff assumption, allowing for broader classes of payoff functions.
• The condition is satisfied in key applications, including persuasion with privately informed receivers, voter persuasion, risk warning design, and costly information acquisition.
• Shifts in the prior do not yield robust comparative statics, as they fail to preserve the monotonicity of signal informativeness.
• Increased alignment of interests between sender and receiver leads to a coarse-convexity shift, and thus potentially greater information provision, under a simple and nearly necessary condition.