# Bayesian persuasion

**Bayesian persuasion** is a kind of mechanism design setting. One person (called the "sender") wants to persuade another person (the "receiver") to do a certain action that is beneficial for the sender, by sending a certain signal about the state of the world. The receiver is assumed to be rational and to follow the Bayes' rule when revising his belief about the state of the world. The model was introduced by Kamenica and Gentzkow.^{[1]}

Bayesian persuasion is a special case of a principal–agent problem: the principal is the sender and the agent is the receiver.

## Example

Kamenica and Gentzkow^{[1]} use the following example. The sender is a medicine company, and the receiver is the regulator. The company produces a new medicine, and needs the approval of the regulator. There are two possible states of the world: the medicine can be either "good" or "bad". The company and the regulator do not know the true state. However, the company can run an experiment and report the results to the regulator. The question is what experiment should it run in order to get the best outcome? The assumptions are:

- The company gains utility if and only if the medicine is approved - regardless of its quality.
- The regulator gains utility if and only if it approves a good medicine, or rejects a bad medicine.
- Both company and regulator know the prior probability that the medicine is good.

For the sake of example, suppose the prior probability that the medicine is good is 1/3. Consider the following three options for the company:

- Do a thorough experiment, that always detects whether the medicine is good or bad, and truthfully report the results to the regulator. In this case, the regulator will approve the medicine with probability 1/3, so the expected utility of the company is 1/3.
- Don't do any experiment; always say "the medicine is good". In this case, the signal does not give any information to the regulator. As the regulator believes that the medicine is good with probability 1/3, he will always reject it, to maximize his expected utility. Therefore, the expected utility of the company is 0.
- Do an experiment that, if the medicine is good, it always reports "good", and if the medicine is bad, it reports "good" or "bad" with probability 1/2. Here, the regulator uses Bayes' rule: given a signal "good", the probability that the medicine is good is 1/2, so the regulator approves it. Given a signal "bad", the probability that the medicine is good is 0, so the regulator rejects it. All in all, the regulator approves the medicine in 2/3 of the cases, so the expected utility of the company is 2/3.

In this case, the optimal policy for the sender is policy 3. Using the Bayes rule, the sender has managed to persuade the receiver to act in a favorable way for the sender.

## Extensions

Camara, Hartline and Johnsen extend the model by removing the assumption that the sender and receiver have a common prior on the state of the world.^{[2]}

## References

- ↑
^{1.0}^{1.1}Kamenica, Emir; Gentzkow, Matthew (2011-10-01). "Bayesian Persuasion" (in en).*American Economic Review***101**(6): 2590–2615. doi:10.1257/aer.101.6.2590. ISSN 0002-8282. https://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590. - ↑ Camara, Modibo K.; Hartline, Jason D.; Johnsen, Aleck (2020-11-01). "Mechanisms for a No-Regret Agent: Beyond the Common Prior".
*2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)*(IEEE). doi:10.1109/focs46700.2020.00033. http://dx.doi.org/10.1109/focs46700.2020.00033.

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Original source: https://en.wikipedia.org/wiki/Bayesian persuasion.
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