Authors:
(1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu);
(2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu);
(3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). 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. This paper is available on arxiv under CC 4.0 license. Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Authors: Authors: (1) Shih-Tang Su, University of Michigan, Ann Arbor (shihtang@umich.edu); (2) Vijay G. Subramanian, University of Michigan, Ann Arbor and (vgsubram@umich.edu); (3) Grant Schoenebeck, University of Michigan, Ann Arbor (schoeneb@umich.edu). Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Problem Formulation 2. Problem Formulation 2.1 Model of Binary-Outcome Experiments in Two-Phase Trials 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 Binary-outcome Experiments in Two-phase Trials and 3.1 Experiments with screenings 3.2 Assumptions and induced strategies 3.2 Assumptions and induced strategies 3.3 Constraints given by phase-II experiments 3.3 Constraints given by phase-II experiments 3.4 Persuasion ratio and the optimal signaling structure 3.4 Persuasion ratio and the optimal signaling structure 3.5 Comparison with classical Bayesian persuasion strategies 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 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 4.2 Determined versus sender-designed experiments 4.3 Multi-phase model and classical Bayesian persuasion and References 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 . Experiments in phase N − 1 N N N 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). Non-binary Outcome Experiments For general non-binary experiments, see the proof of Lemma 6 in [23] for a detailed construction from non-binary to binary experiments. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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