Binary-Outcome Experiments in Two-Phase

BINARY-OUTCOME EXPERIMENTS IN TWO-PHASE TRIALS Introduction  Objective: In this section, we delve into solving the sender's optimization problem in two-phase trials, specifically focusing on binary-outcome experiments.  We aim to understand how different types of experiments (determined vs. senderdesigned) influence the sender's optimal signaling strategy. Two-Phase Trials  We start with a scenario involving only two phases in the trial.  Analysis in this section provides insights into how determined and senderdesigned experiments impact signaling strategy.  A subset of two-phase trials closely resembles single-phase trials, where one phase-II experiment is trivial (non-informative). Experiments with Screenings  We begin by analyzing the sender's optimal strategy in a scenario with one nontrivial experiment in phase II.  We define a "trivial experiment" as one whose outcomes are independent of the state of the world.  A trivial experiment in phase II keeps the posterior belief unchanged, similar to interim belief in classical Bayesian persuasion. Impact of Trivial Experiments  Lemma 1 states that when a trivial experiment exists in phase II, the sender and receiver's expected utilities remain the same as in classical Bayesian persuasion.  Trivial experiments do not constrain the sender as they can always choose to reveal the true state when the non-trivial experiment is conducted. Assumptions and Induced Strategies  We make two assumptions for clarity in analysis. 1. Probability of passing a phase-II experiment under θ1 is greater than or equal to that under θ2. 2. When θ1 is true, the experiment conducted when outcome ωA occurs is more informative than when outcome ωB occurs.  These assumptions help categorize sender's induced strategies.  Induced strategies are choices available to the sender based on phase-I experiment outcomes. Incentive Compatibility  The sender's strategy includes choosing phase-I experiment parameters (p1, p2) and persuasion strategies in phase-II.  To ensure receiver's incentive compatibility, certain induced strategies are applicable based on phase-I experiment outcomes.  Induced strategies are categorized as (αX), (βX), and (γX) based on the sender's persuasion approach in phase II.  Claim: In certain cases, a strategy where the receiver takes action φ1 when the experiment fails and φ2 when it passes is not incentive-compatible for the receiver. Constraints Given by Phase-II Experiments  Sender's choice in phase I determines interim beliefs for phase-II experiments.  These interim beliefs lead to specific induced strategies that are incentivecompatible.  The sender's choice of (p1, p2) is constrained by the requirement to satisfy these incentive compatibility conditions.  Table 1 summarizes the impact of phase-II experiments on (p1, p2) based on the constraints imposed by induced strategies.  The sender's experiment design in phase I involves selecting combinations of induced strategies that meet the IC requirements for each strategy. Conclusion  In this section, we explored the impact of binary-outcome experiments in twophase trials on the sender's signaling strategy.  We discussed the concept of trivial experiments and their influence on signaling.  Additionally, we introduced induced strategies and examined their relationships with IC requirements and the sender's expected utility.