Appendix AToy example

In this section, we provide and expand upon a toy example. Recall that the inputs x and x0 need not correspond to real users but could instead represent hypothetical users. Example 5. Suppose that the regulatory guideline requires that users in the same geographical location receive similar weather forecasts. This can be written as "the weather forecasts that are selected by F should be similar for all users in the same geographical location", and S could be a randomly generated set of user pairs, where each pair corresponds to two (hypothetical) users in the same geographical location, and S could contain pairs across many locations. In the left-most panel, a filtering algorithm F takes in counterfactual inputs x and x0 and produces the content Z and Z0. Because a counterfactual regulation requires that F behave similarly under x and x0, the regulation is effectively requiring that content Z and Z0 are sufficiently similar (or, graphically, that they are close in Z). The question of how to quantify "similarity" is addressed in Section 2.1. The toy example in Example 5 is illustrated in the right-most panel.