Explain epsilon in differential privacy to a product manager and budget a dashboard

A product manager wants a daily public dashboard of approval counts by region (40 regions) and by group (2 groups), "with differential privacy". They ask: "What is epsilon? What value should we pick? Will the numbers still be useful?" (a) Explain epsilon in plain language and with the formula. (b) For a single count, what noise does the Laplace mechanism add at epsilon 0.1 and 1? (c) The dashboard publishes daily for a year. How does privacy loss accumulate, and what do you propose? (d) What does the guarantee not cover? Use these results from a simulation: the error of a released group-gap was 0.0032 at epsilon 0.1 on 10,000 applicants but 0.076 on a 200-per-group slice.