Fit a logistic win curve to past quotes, multiply it by the margin each discount leaves, and solve for the peak; then see what a revenue commission does to it.
The optimal discount on a quote is the one that maximises expected contribution: the probability of winning at that discount times the contribution the deal leaves if it is won. On an illustrative 10,000-unit quote at a list price of 100.00 with a 28 per cent margin, where a fitted win curve gives 25 per cent at list and 65 per cent at a 10 per cent discount, the optimum is a 10.5 per cent discount, worth 117,124 of expected contribution against 70,000 at list. A salesperson paid on revenue would prefer 21.1 per cent, which gives away 45.6 per cent of that value.
Worked in full in Pricing Strategy by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Most discount debates run on one of two instincts. Finance wants the price held because every point comes off the margin; sales wants the deal won because a lost quote earns nothing. Both are half of the same product. Expected contribution multiplies the two, and once the win probability is written as a function of the discount, the argument becomes a number that can be solved for.
| Input | Value |
|---|---|
| List price per unit | 100.00 |
| Variable cost per unit | 72.00 |
| Unit margin at list (28.0%) | 28.00 |
| Units on the quote | 10,000 |
| Win probability at list | 25% |
| Win probability at a 10% discount | 65% |
| Commission on revenue | 3% |
Win rates rise with the discount but flatten at both ends: no discount makes a quote certain, and a token discount barely moves it. A logistic curve captures that shape with two parameters, and two observed points are enough to fix them. Here the intercept is −1.0986, the log-odds of a 25 per cent win rate, and the slope is 17.18 per unit of discount.
Win probability: P(d) = 1 / (1 + e−(a + b × d))
Expected contribution: E(d) = P(d) × units × (list × (1 − d) − variable cost)
Optimum where b × (1 − P) × (margin % − d) = 1
In Excel, with the discount in A2: =1/(1+EXP(-(a+b*A2))) for the win probability and =B2*Units*(List*(1-A2)-VarCost) for expected contribution. Fill down in steps of 0.5 per cent, or let Solver maximise the second column.
Step 1: fit the curve. At list, the log-odds of a 25 per cent win rate are ln(0.25 / 0.75) = −1.0986. At a 10 per cent discount, the log-odds of 65 per cent are 0.619. The slope is the difference over the discount: (0.619 + 1.0986) / 0.10 = 17.18.
Step 2: tabulate expected contribution. Each row multiplies the win probability by what the deal leaves if won.
| Discount | Win probability | Margin per unit | Contribution if won | Expected contribution |
|---|---|---|---|---|
| 0.0% | 25.0% | 28.00 | 280,000 | 70,000 |
| 2.5% | 33.9% | 25.50 | 255,000 | 86,362 |
| 5.0% | 44.0% | 23.00 | 230,000 | 101,278 |
| 7.5% | 54.7% | 20.50 | 205,000 | 112,189 |
| 10.0% | 65.0% | 18.00 | 180,000 | 117,000 |
| 12.5% | 74.0% | 15.50 | 155,000 | 114,774 |
| 15.0% | 81.4% | 13.00 | 130,000 | 105,852 |
| 20.0% | 91.2% | 8.00 | 80,000 | 72,950 |
Step 3: solve for the peak. The curve is flat near the top, so the exact optimum is worth solving rather than reading off the grid. Setting the derivative to zero gives the condition in the panel: (1 − P) × (0.28 − d) = 1 / 17.18 = 5.82 per cent. It holds at a discount of 10.5 per cent, where the win probability is 66.8 per cent and the margin per unit is 17.53.
The best discount on this quote is 10.5 per cent. It wins two times in three and leaves 175,343 of contribution when it does, for an expected 117,124. Quoting at list keeps the full 280,000 of margin but wins one time in four, which is worth 70,000. The discount is not a concession here: it adds 47,124 of expected contribution, because the win curve is steep in the range where it is given.
The condition reads simply. Each extra point of discount raises the win probability by b × P × (1 − P), which earns the margin still left on the extra deals won, and costs a point of price on every deal that would have been won anyway, P of them. Stop when the two are equal. A fatter margin pushes the optimum out; a curve that is already near its ceiling pulls it in, which is why a steeper curve does not always mean a deeper discount.
| Unit margin at list | Optimal discount | Win probability | Expected contribution |
|---|---|---|---|
| 20.00 | 7.3% | 54.0% | 68,421 |
| 28.00 | 10.5% | 66.8% | 117,124 |
| 35.00 | 12.5% | 74.1% | 166,608 |
| Win rate at 10% discount | Slope b | Optimal discount | Win probability | Expected contribution |
|---|---|---|---|---|
| 45% | 8.98 | 8.7% | 42.2% | 81,313 |
| 65% | 17.18 | 10.5% | 66.8% | 117,124 |
| 85% | 28.33 | 9.1% | 81.3% | 153,944 |
The second table holds the surprise. A more price-sensitive customer does not always justify a bigger discount. When the curve is very steep, a modest discount already carries the win rate to 81.3 per cent, and further points buy little extra probability while costing margin on every deal won. The optimum is highest in the middle, where the curve is steep and still has room to climb.
The common mistake is to optimise the wrong quantity. Maximise the win rate and the answer is always a larger discount: 19.2 per cent buys a 90 per cent win rate and an expected 79,308, less than the optimum by a third. Maximise revenue, which is what a commission on revenue asks a salesperson to do, and the best discount for the representative is 21.1 per cent: a win probability of 92.6 per cent, a margin of 6.88 a unit and expected contribution of 63,694. The company loses 53,430, or 45.6 per cent of the value of the quote, while the representative's expected commission at 3 per cent rises from 17,942 to 21,917. The arithmetic of that misalignment is worked in when a commission plan rewards a discount that loses money, and the authority levels that contain it in how to set discount approval levels.
The second mistake is to trust a win curve that was never fitted. Two points from a sales director's memory are a hypothesis, not a curve. Fit the logistic to the last few hundred won and lost quotes, segment it where customers behave differently, and refit it each year.
The book builds a deal table with the win-probability parameters as inputs, showing the company's optimum discount and the salesperson's under two commission plans; it is in the free workbook for this case on the Pricing Strategy companion page.
Fit a logistic regression to past quotes, with won or lost as the outcome and the discount as the driver, segmented where customers differ. Two points fix the curve in a pinch: a 25 per cent win rate at list and 65 per cent at a 10 per cent discount give an intercept of -1.0986 and a slope of 17.18. A curve from memory is a hypothesis to test against the quote history.
Because the representative maximises win probability times revenue, which ignores variable cost. On the illustrative quote that optimum is a 21.1 per cent discount, a 92.6 per cent win rate and only 6.88 of margin a unit, giving the company 63,694 of expected contribution against 117,124 at its own optimum of 10.5 per cent. Paying commission on margin aligns the two.
No. With a 28 per cent margin, raising the win rate at a 10 per cent discount from 65 to 85 per cent lowers the optimal discount from 10.5 to 9.1 per cent, because the steeper curve reaches a high win rate sooner and further points buy little probability. The optimum is largest where the win curve is steep and still well below its ceiling.
This article is one calculation from Pricing Strategy. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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