One price, unknown demand: pricing a perishable room block by expected value
1. Before you start
You have to set one price now, before you know how many people will show up, for something that cannot be stored: a hotel room-night, a restaurant seat, an event ticket. Sell it and it earns; leave it empty at midnight and it is gone for good. Expected value is the tool for this. It asks: across the different demand outcomes that could happen, each with its own chance, what is the probability-weighted profit of each price I might publish? A tiny example: at a given price you have a 50% chance of earning $100 and a 50% chance of earning $40. The single “most likely” number is not the point — the expected value is 0.5 × $100 + 0.5 × $40 = $70, and that $70, compared against the expected value of every other price, is what you decide on.
Three honest statements before you start:
- This is a Decide course. You read a situation, learn the ideas, and make a call. You do not write or run any code.
- The companies below, Cape Lorne Hotel and Solstice Table, are composite — invented firms built from ordinary figures so the arithmetic stays clean. No number is a claim about any real firm.
- This is not a certification. It proves, to you, that you can price a perishable capacity under uncertain demand and defend the call.
You need arithmetic, and the willingness to hold three possible futures in your head at once instead of one. That second habit is the whole difficulty here.
2. The Situation
Cape Lorne Hotel must publish one nightly rate for its 100-room block on a festival Saturday, weeks ahead, and cannot change it once it is out. The revenue manager wants $260 — “the festival always fills, charge for it.” The GM is nervous — “and if it doesn’t fill, we are staring at eighty empty $260 rooms that earn nothing.” Two things are unknown at once: how many rooms sell, and therefore which price actually earns the most — and the price you commit to changes the demand you get.
The trap is that a single best-guess forecast makes one price look obviously right, while the spread of outcomes around that guess quietly points at a different one.
3. What you’ll be able to do
After this course you will be able to:
- Build an expected-value table across demand scenarios and pick the price that maximises probability-weighted profit — not the price that wins under a single forecast.
- Name and size the two costs in tension — spoilage (empty perishable capacity) and underpricing (selling cheap what would have sold dear) — and say which one a given price is exposed to.
- Explain why the profit-maximising price under uncertainty can sit below the price a single best-guess forecast would choose, and state the one change that flips the call.
4. Prerequisites & time box
Prerequisites: arithmetic, comfort reading a small table, and the idea of a probability as a weight that sums to 1. Helpful but not required: contribution margin (price minus variable cost) — if that phrase is new, the composite figures below make it concrete. No spreadsheet needed, though one speeds the sums.
Time box: about 24 minutes of reading (measured), plus real thinking time on the call — this one rewards working the numbers yourself. That is under the 25-minute cap for a concept course.
Difficulty: 7 / 8 — a senior/director decision: the factors interact, the right call is genuinely contestable, and you’re judged on the quality of your reasoning.
Free-tier honesty: no signup, no GPU (requires_gpu: false), nothing to install.
5. The case & where the numbers come from
Cape Lorne Hotel is a composite hospitality company: its capacity, cost, price, and demand figures are in-course assumptions chosen for clean arithmetic, not drawn from or claimed about any real hotel. The methods — expected value across scenarios, and the overage/underage (newsvendor) trade-off for a perishable, fixed capacity — are standard and cited in section 11. Every dollar below is an assumption; every expected value is computed from these figures inside the course, so you can reproduce each one by hand.
The festival-Saturday decision:
| Item | Figure |
|---|---|
| Rooms in the block (fixed capacity) | 100 |
| Variable cost per occupied room (cleaning, amenities, utilities) | $40 |
| Cost of an empty room-night | $0 earned, and it cannot be re-sold — it perishes |
| Candidate nightly rates (pick one, publish it) | $180, $220, $260 |
Demand is uncertain. The team scopes three scenarios and assigns each a probability; the probabilities sum to 1. Because a higher rate deters some guests, the rooms wanted fall as the rate rises:
| Demand scenario | Probability | Rooms wanted at $180 | at $220 | at $260 |
|---|---|---|---|---|
| Soft | 0.45 | 90 | 55 | 25 |
| Mid | 0.35 | 120 | 95 | 85 |
| Strong | 0.20 | 150 | 120 | 110 |
Two things to hold onto. Rooms sold can never exceed the 100-room capacity, so demand above 100 is capped. And the Soft scenario — the one that hurts a high rate most — is the single most likely at 0.45, more likely than either other outcome. Keep that in view; a point-estimate forecast tends to forget it.
6. The Concepts
Expected value across demand scenarios
Contribution per occupied room is the rate minus the $40 variable cost: $140 at the $180 rate, $180 at $220, $220 at $260. Rooms sold in a scenario is the smaller of rooms-wanted and the 100-room capacity. Profit in a scenario is rooms sold × contribution. Expected value (EV) of a rate is the probability-weighted sum of its scenario profits.
Work the $260 rate all the way through, because it is the one everyone argues about. Contribution $220, capacity 100:
| Scenario | Prob | Wanted | Sold = min(wanted, 100) | Profit = sold × $220 |
|---|---|---|---|---|
| Soft | 0.45 | 25 | 25 | $5,500 |
| Mid | 0.35 | 85 | 85 | $18,700 |
| Strong | 0.20 | 110 | 100 | $22,000 |
EV at $260 = 0.45 × $5,500 + 0.35 × $18,700 + 0.20 × $22,000 = $2,475 + $6,545 + $4,400 = $13,420.
Now the $180 rate (contribution $140). Its Soft demand of 90 sells all 90; its Mid and Strong demand (120 and 150) both exceed the 100-room block, so rooms sold cap at 100:
| Rate | Soft profit | Mid profit | Strong profit | EV = 0.45·Soft + 0.35·Mid + 0.20·Strong |
|---|---|---|---|---|
| $180 | 90×140 = $12,600 | 100×140 = $14,000 | 100×140 = $14,000 | $5,670 + $4,900 + $2,800 = $13,370 |
The $220 rate (contribution $180) is where this sum most often goes wrong, so watch the capacity cap here. In the Strong scenario the rooms wanted are 120, but only 100 can be sold. Forget the cap and you compute the Strong row as 120 × $180 = $21,600, weight it (0.20 × $21,600 = $4,320), and land at a wrong EV of $4,455 + $5,985 + $4,320 = $14,760. Cap it properly — 100 × $180 = $18,000, weighted 0.20 × $18,000 = $3,600 — and the row falls into line:
| Rate | Soft profit | Mid profit | Strong profit (capped at 100) | EV = 0.45·Soft + 0.35·Mid + 0.20·Strong |
|---|---|---|---|---|
| $220 | 55×180 = $9,900 | 95×180 = $17,100 | 100×180 = $18,000 | $4,455 + $5,985 + $3,600 = $14,040 |
Check every “wanted” figure against the 100-room cap before you multiply — that one step is where these expected-value sums die most often.
So the three expected values are:
| Rate | Expected value |
|---|---|
| $180 | $13,370 |
| $220 | $14,040 |
| $260 | $13,420 |
The $220 rate has the highest expected profit. Not the top rate the revenue manager wanted, and not the floor rate — the middle one. Below is the interactive model of exactly this table.
(An interactive calculator sits here — enter the capacity, the per-room cost, the three candidate rates, the three scenario probabilities, and the rooms-wanted grid, and it returns the expected value of each rate and which one to publish. Change the probabilities or the per-room cost and watch the winning rate flip.)
Spoilage versus underpricing
Every price on a perishable, fixed capacity is exposed to two opposite costs, and the expected-value table is really pricing the balance between them.
- Spoilage (overage). Set the rate high and, if demand comes in Soft, rooms sit empty at midnight. At $260 in the Soft scenario, 25 rooms sell and 75 perish — each one a $220 contribution that vanishes and can never be re-earned. Spoilage is the cost of pricing above what the night will bear.
- Underpricing (underage). Set the rate low and, when demand is Strong, you sell out fast at a rate well under what guests would have paid. At $180 in the Strong scenario you fill all 100 rooms but at $140 contribution each; every one of them would have contributed $220 at the top rate — that gap is contribution left on the table. Underage is the cost of pricing below what the night will bear.
You cannot escape both, because you must commit before you know the night. The EV-maximising rate is the one whose expected spoilage-plus-underpricing cost is smallest. Raising the rate trades underpricing risk for spoilage risk; lowering it does the reverse. This is the newsvendor trade-off in a pricing skin: the size of each cost, weighted by how likely it is, decides the call.
A direct consequence, tested later: change the size of either cost and the winning rate moves. If empty rooms could be salvaged (sold late at a discount), spoilage would cost less, and the balance would tilt back toward a higher rate. That lever is the heart of section 7.
The point estimate, and why it can mislead
The revenue manager’s argument — “the festival always fills, charge $260” — is a point estimate: decide on a single best-guess scenario instead of the whole distribution. Take the Mid scenario as that single guess and read the profits straight off the table:
| Rate | Profit in the single Mid-case forecast |
|---|---|
| $180 | $14,000 |
| $220 | $17,100 |
| $260 | $18,700 |
On the point estimate, $260 wins outright — it is the obvious call, and it is the wrong one. The point estimate throws away two facts the EV table keeps: that Soft demand is the most likely single outcome (0.45), and that Soft demand is precisely where $260 collapses to $5,500. Averaging the Soft disaster in with the Mid and Strong cases is what drags $260’s expected value down to $13,420 — below $220’s $14,040. The point estimate is not a lie; it is a true statement about one future, sold as if it were the only one.
Why the uncertainty-aware price differs
Put the two views side by side: the single-forecast call is $260; the expected-value call is $220. They disagree, and the disagreement is the lesson.
The profit-maximising price under uncertainty sits below the point-estimate price whenever the high rate’s downside is both severe (empty premium rooms earn nothing) and likely (Soft demand at 0.45). Expected value prices that downside in; the point estimate cannot see it. The middle rate wins not because it is highest in any one scenario — it never is — but because its profit holds up across the scenarios: it does not crater in Soft the way $260 does, and it does not give rooms away in Strong the way $180 does. Choosing under uncertainty means choosing the rate with the best weighted outcome, and that is rarely the rate that looks best in the story you find easiest to imagine.
One caveat keeps this honest: the answer is only as good as the probabilities. Shift the odds toward Strong, or shrink the cost of an empty room, and the EV-optimal rate rises again. The next section makes you do exactly that on new ground.
7. Your Call
You have seen how expected value, spoilage versus underpricing, and the point-estimate trap decide Cape Lorne’s single published rate. Now a different decision lands on your desk.
Solstice Table is a composite fine-dining restaurant setting one prix-fixe price for its 40-cover Valentine’s night, committed on the booking platform weeks ahead and unchangeable after. The variable cost per cover (food and service) is $40, so contribution is $70 at a $110 price and $110 at a $150 price. An empty seat at close of service perishes — $0, gone. Demand is uncertain:
| Scenario | Probability | Covers wanted at $110 | Covers wanted at $150 |
|---|---|---|---|
| Quiet | 0.55 | 34 | 10 |
| Typical | 0.30 | 40 | 32 |
| Full | 0.15 | 40 | 40 |
Covers sold is the smaller of covers-wanted and the 40-seat capacity. The added wrinkle: 48 hours out, any unsold seat can be released to a last-minute dining app that reliably fills it at $65 — a $25 contribution per salvaged seat. So under that option an empty seat is no longer a total loss.
How this differs from the taught case (the transfer): Solstice Table is a different company in a different corner of hospitality — a restaurant’s single-night prix-fixe, not a hotel’s room block — and every figure is different, so the arithmetic must be redone from scratch. It also adds a constraint the hotel case did not have: a last-minute salvage channel that changes the size of the overage (spoilage) cost. The concept under test is unchanged — pricing a fixed, perishable capacity under uncertain demand by expected value.
8. Self-check
Before you write the memo, make sure you can say each of these in one line:
- What does an expected-value table compute, and why is it a better basis for the call than the single most-likely scenario?
- For a given published price, which cost are you exposed to — spoilage or underpricing — and what makes it large?
- Why can the profit-maximising price under uncertainty differ from the point-estimate price, and name the one change (a probability, or the cost of an empty unit) that would flip your call?
If any is fuzzy, reread section 6 — expected value, the spoilage/underpricing trade-off, and the point-estimate trap are the whole course.
9. Stretch
Push the decision further on your own:
- Back at Cape Lorne: find the Soft-scenario probability at which the $260 rate’s expected value overtakes $220’s. (Hold the Mid and Strong figures and re-solve for the weight on Soft — the genuinely hard one, because you are solving for the probability that makes two weighted sums equal, then asking whether a forecast that confident is defensible.)
- At Solstice, suppose the salvage app is not unlimited — it will take at most 8 late seats, and the rest still perish. Does salvage still flip the call to $150? At what salvage cap does the flip disappear?
- The whole method assumes you can name the scenarios and their probabilities. Where would those numbers actually come from for a real festival weekend, and what would you do if two experienced people handed you very different probability sets?
10. Ship it — your decision memo
Write a one-page memo to Cape Lorne’s leadership. State the call (publish $220 — highest expected profit at $14,040, versus $13,370 at $180 and $13,420 at $260 — not the $260 the single forecast favours). Show the reasoning in two or three lines (the Soft night is the most likely single outcome at 0.45, and it guts the $260 rate to $5,500; weighting all three scenarios, the middle rate holds up best). Name what you rejected (publishing $260 on the point estimate) and why (it prices one scenario and ignores the likely, punishing Soft night). Name the one thing that would change your mind (firmer odds tilted toward Strong demand, or a way to salvage empty rooms that cuts the spoilage cost — either would lift the winning rate). Keep it to a single page leadership grasps in two minutes. This memo is your own argued claim — not a credential.
11. Sources
Cape Lorne Hotel and Solstice Table, and every figure attached to them, are composite and illustrative — constructed for clean teaching arithmetic, not drawn from or claimed about any real company. The methods used to price them are standard; references below.
| Concept / claim | Source (publisher) | URL | Accessed |
|---|---|---|---|
| Expected value = probability-weighted sum of outcomes | Wikipedia — Expected value | https://en.wikipedia.org/wiki/Expected_value | 2026-07-19 |
| Overage/underage trade-off for a fixed, perishable capacity | Wikipedia — Newsvendor model | https://en.wikipedia.org/wiki/Newsvendor_model | 2026-07-19 |
| Pricing perishable capacity to demand (hotels, restaurants, events) | Wikipedia — Revenue management | https://en.wikipedia.org/wiki/Revenue_management | 2026-07-19 |
| Selling a fixed, time-limited capacity at the right price | Wikipedia — Yield management | https://en.wikipedia.org/wiki/Yield_management | 2026-07-19 |
| Choosing among options by expected value under uncertainty | Wikipedia — Decision theory | https://en.wikipedia.org/wiki/Decision_theory | 2026-07-19 |
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