Explain the Four Quadrants With Numbers for a Retention Offer
A streaming service wants to send a retention offer (three months at a reduced price) to subscribers who are about to hit their renewal date. A past randomized test shows these segments among 8,000 at-risk subscribers. The offer costs $8 per subscriber who receives it (contact plus discount budget, paid whether or not they stay) and a retained subscriber is worth $60 of future margin.
| Segment | Size | P(\text{retain}) without offer | P(\text{retain}) with offer |
|---|---|---|---|
| A | 1,000 | 0.50 | 0.80 |
| B | 4,000 | 0.95 | 0.95 |
| C | 2,000 | 0.10 | 0.10 |
| D | 1,000 | 0.70 | 0.60 |
- Name each segment's quadrant and compute its uplift \tau.
- Compute the average treatment effect over the 8,000 subscribers and the profit of sending the offer to everyone.
- The team has a model of "probability to retain given the offer" and can afford 1,000 offers. Compute the incremental retained subscribers and profit of targeting by that model versus by uplift.
- A product manager says "B is our best audience, they respond to everything". Rebut this in two sentences.
1. Quadrants and uplift. \tau=P(\text{retain}\mid\text{offer})-P(\text{retain}\mid\text{no offer}).
| Segment | \tau | Quadrant |
|---|---|---|
| A | 0.80-0.50=+0.30 | persuadables (the offer changes the outcome) |
| B | 0.95-0.95=0 | sure things (retain anyway) |
| C | 0.10-0.10=0 | lost causes (leave regardless) |
| D | 0.60-0.70=-0.10 | sleeping dogs (the reminder or the plan change triggers a cancellation) |
2. ATE and treat-everyone profit. Incremental retained subscribers: 1{,}000(0.30)+0+0+1{,}000(-0.10)=200, so ATE =200/8{,}000=+0.025, a positive average. Profit per treated subscriber is v=60\tau-8: A =+10, B =-8, C =-8, D =60(-0.10)-8=-14. Total =1{,}000(10)+4{,}000(-8)+2{,}000(-8)+1{,}000(-14)=-\$52{,}000. A positive average effect, a loss from blanket treatment: the $8 is paid on 7,000 subscribers whose outcome does not improve (verified in Python).
3. Response model versus uplift. The response model ranks by P(\text{retain}\mid\text{offer}): B 0.95, A 0.80, D 0.60, C 0.10. With 1,000 offers it picks 1,000 of segment B: incremental retained =0, profit =1{,}000(-8)=-\$8{,}000. The uplift model ranks by \tau: A (+0.30) first; all 1,000 offers go to A: incremental retained =300, profit =1{,}000(+10)=+\$10{,}000. Same budget, an $18,000 swing. (Had the budget been larger, uplift would stop at A rather than spend on B or C because their value is negative.)
4. Rebuttal. B retains at 95 percent with or without the offer, so the high response rate is a property of the customers, not of the offer, and every dollar spent on B buys nothing. Judge an audience by the change in its outcome between treated and control, not by its outcome level.
What a strong answer adds. Individuals cannot be sorted into quadrants (that would need both potential outcomes), only segments with probabilities. Mention that in practice \tau is estimated, so state the uncertainty, and that D needs a guardrail metric (cancellations) because the harm is invisible in an average. Cost enters through v=(m)\tau-\text{cost}; the same ranking by \tau holds only when the cost is the same for every subscriber.
Share this question