Information affects behavior by affecting beliefs. Information design

studies how to disclose information to a group of players to incentivize

them to behave in a desired way. This paper is a theoretical investigation

of information design, culminating with a representation theorem and a

fundamental application of it. We adopt a random posterior perspective,

viewing information design as belief manipulation rather than information

disclosure. The representation theorem shows that it is as if the designer

manipulated beliefs in a specific way, giving form to the random posterior

approach in games, as did Kamenica and Gentzkow (2011) in one-agent

problems. The representation theorem can also be implemented in specific

problems, for example in the beauty contest and multiple-agent problems.

We focus on an application that we call the Mother's Problem.