Persuasion Ratio and the Optimal Signaling Structure

PERSUASION RATIO AND THE OPTIMAL SIGNALING STRUCTURE Introduction  In this section, we explore the concept of the persuasion ratio and its role in determining the optimal signaling structure for the sender.  The sender aims to maximize the total probability of action φ1, which involves comparing different sets of strategies formed by various pairs of induced strategies. Understanding the Persuasion Ratio  Definition 3: The persuasion ratio of a set of strategies S, denoted as PR(S, p), represents the maximum total probability of action φ1 taken (under a strategy within the set) divided by the prior probability p.  The persuasion ratio can be viewed as the relative utility gain a set of strategies offers to the sender at a specific prior value. Persuasion Ratio Curve  We can depict a persuasion ratio curve for each set of strategies as the prior varies in the range [0, 1].  Structural insights presented in Lemma 3 simplify the analysis by narrowing down the search space for the optimal signaling strategy.  Two key conditions are identified: 1. At least one IC (Incentive Compatibility) requirement of the constituent induced strategies is tight. 2. There is a signal that will be sent with probability 1 under the strategy. Simplifying the Analysis  Lemma 3: This lemma simplifies the problem-solving process by outlining two conditions that lead to an optimal strategy for the sender.  The proof demonstrates that when IC requirements are not tight for both induced strategies and both signals have non-zero probabilities, the sender can modify her strategies to increase her expected utility. Persuasion Ratio Curve for Different Strategy Types  Lemma 4: Given persuasion ratio curves for specific strategy types (e.g., (αA, γB) and (γA, dB)), this lemma helps derive the persuasion ratio curve for a set of incentive-compatible strategies (cA, dB).  It involves solving a maximization problem with linear equations, reducing the need for point-wise calculations. Optimal Signaling Strategy  Lemma 5: The persuasion ratio curve of the optimal signaling strategy PR∗(p) is the upper envelope of the persuasion curves of different strategy types.  The optimal signaling strategy is the one that reaches the frontier of the persuasion ratio curve at a specific prior value.  The sender chooses the strategy with the highest persuasion ratio, ensuring a higher expected utility for all prior probabilities. Corollary for Two-Phase Trials  Corollary 1: In the case of two-phase trials, if the optimal signaling strategy does not take action φ1 with probability 1, certain conditions about interim beliefs are met.  This simplifies the comparison of different types of strategies under varying interim beliefs. Comparison with Classical Bayesian Persuasion Strategies  After deriving the optimal signaling strategy, it's essential to quantify the sender's utility improvement compared to strategies similar to classical Bayesian persuasion.  Definition 4: Bayesian persuasion (BBP) strategies mix two possible states in one signal and reveal the true state on the other signal.  Comparison between the optimal signaling strategy and optimal BBP strategies is made based on expected utility. Conclusion  In this section, we introduced the concept of the persuasion ratio and its significance in determining the optimal signaling structure.  We explored key lemmas that simplify the analysis and enable the sender to make informed choices regarding signaling strategies.  The lecture provided insights into comparing optimal signaling with classical Bayesian persuasion strategies.