Would a customer churn without a promotion but remain subscribed with one? The probability of necessity and sufficiency (PNS) asks how often both counterfactual statements are true for the same customer. It measures the share for whom the promotion makes the difference, rather than merely the share who stay after receiving it. section 8.3.1
Let X=1 mean the cause is present and Y=1 mean the outcome occurs. Write Y₍X=1₎ and Y₍X=0₎ for the outcomes under those two conditions. Then PNS = P(Y₍X=1₎=1, Y₍X=0₎=0). Unlike the probability of necessity or the probability of sufficiency, PNS is not conditioned on which cause and outcome were observed. section 8.3.1 section 10.6.2
Promotion targeting
A subscriber who stays only when sent a promotion contributes to PNS. A subscriber who stays either way does not. If promotions cannot make anyone less likely to stay—a monotonicity assumption—PNS equals P(Y₍X=1₎=1) − P(Y₍X=0₎=1). The difference between the two intervention rates identifies the responsive share under that assumption; without it, harmful responses cannot be ruled out. section 10.6.2
A randomized experiment can estimate the intervention probabilities in that formula. With observational data, identifying them requires an appropriate strategy, such as backdoor or front-door adjustment. When PNS cannot be fully identified, assumption-dependent bounds may still be useful; bounds for binary in-game purchases and side-quest engagement illustrate this alternative. section 10.6.2 section 10.9
