Cost-Based Thresholds Across Transaction Amounts

Your platform's cost parameters: chargeback fee F = $25, merchant margin m = 4 % of amount, false-decline churn cost C = $8 (flat, independent of amount). Use the expected-cost decision rule decline if p·(A + F) > (1-p)·(mA + C).

1. Derive the break-even fraud probability p* as a function of amount A.
2. Compute p* for A = $20, A = $200, and A = $2,000. What pattern do you see, and why does it make business sense?
3. Your model outputs one calibrated probability per transaction, but the threshold varies by amount. Is that a problem for the model, the policy, or neither? Explain the separation of concerns.