Optimizing Signaling Strategies with Sender-Designed Experiments in Multi-phase Trials

Table of Links

Abstract and 1. Introduction

2. Problem Formulation

2.1 Model of Binary-Outcome Experiments in Two-Phase Trials

3 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings

3.2 Assumptions and induced strategies

3.3 Constraints given by phase-II experiments

3.4 Persuasion ratio and the optimal signaling structure

3.5 Comparison with classical Bayesian persuasion strategies

4 Binary-outcome Experiments in Multi-phase trials and 4.1 Model of binary-outcome experiments in multi-phase trials

4.2 Determined versus sender-designed experiments

4.3 Multi-phase model and classical Bayesian persuasion and References

4.2 Determined versus sender-designed experiments

Given the model, the sender can manipulate the phase-K interim belief only when designing an experiment at phase K − 1. If an experiment at phase K − 1 is determined, then the phase-K interim belief is a function of interim belief at phase K − 1. Therefore, figuring out how these two types of experiments, determined and sender-designed experiments, will influence every given phase’s interim belief is the key to solving for the optimal signaling strategy. We start by noting that if the posterior belief at a leaf node is given, then the receiver’s action is determined - he will take the action with the highest posterior probability unless there is a tie, in which case he is indifferent and will follow the sender’s recommendation. Therefore, we can use backward iteration and the principle of optimality to determine the optimal signaling. We start by considering the last phase’s experiments when the sender can design them.

Experiments in phase N − 1 In the second-last phase, results in Section 3.4 describe a sender-designed experiment’s role in the optimal signaling strategy: pick the strategy on the frontier of all persuasion-ratio curves. However, if the experiment is determined in the second-last phase, an additional constraint on the interim belief between the second-last phase and the last phase is enforced. That is to say, the set of (feasible) strategies will shrink. Fortunately, after enforcing the constraints, the process of searching for the optimal signaling strategy under a determined experiment is the same as the sender-designed experiment, i.e., pick the strategy in the frontier of all persuasion ratio curves. Therefore, at each possible branch of phase N − 1, we can plot an optimal persuasion ratio curve capturing the sender’s optimal signaling strategy at phase N − 1 and phase N.

Non-binary Outcome Experiments When the experiments have non-binary outcomes, the same approach derived above works with an increased number of phases (if complexity is not an issue).

For general non-binary experiments, see the proof of Lemma 6 in [23] for a detailed construction from non-binary to binary experiments.