Bayesian Persuasion Posterior Splitting

limitations

The sender's optimal strategy involves finding the concave envelope of their expected payoff as a function of the posterior belief. By splitting the prior p into a low posterior l and a high posterior h, the sender achieves a payoff on the line segment connecting (l, u(l)) and (h, u(h)). This assumes the sender can fully commit to an information structure and that the receiver has no other source of information. The model is also limited to binary states and finite outcomes.

worked-example

Suppose the prior is p = 0.25. Without any signal, the receiver's belief is 0.25 < 0.5, so they do not adopt (payoff 0). If the sender uses a signal that induces posteriors l = 0 and h = 0.5, we can compute the weights. The probability of the high signal (which leads to adoption) is 1 - w = 0.5. Thus, the adoption probability increases from 0 to 0.5.

1. Weight w = (p - l) / (h - l)
2. w = (0.25 - 0) / (0.5 - 0) = 0.5

experiment

Use the interactive module to adjust the prior probability and the chosen high and low posteriors. Observe how the weights are calculated (when Bayes plausible) and the resulting expected adoption probability. Check the interactive model here: Bayesian Persuasion Posterior Splitting.

1. Change the prior good probability to observe its effect.
2. Adjust the low and high posterior bounds to see the updated weights and payoffs.

references

For more details on Bayesian Persuasion and the concavification approach, see the original paper which introduced this concept:

• Kamenica, E., & Gentzkow, M. (2011). Bayesian Persuasion. The American Economic Review, 101(6), 2590-2615.