Question
A used-camera dealer values one standard camera at £100 and posts a buying price (bid) \(b\) and a selling price (ask) \(a\). Both must be chosen from fixed grids: \(b\in\{\text{£90},\text{£94},\text{£98}\}\) and \(a\in\{\text{£102},\text{£106},\text{£110}\}\). Exactly one customer then arrives, drawn from the following three types:
| Type | Chance | Lowest bid at which they would sell a camera | Highest ask at which they would buy one |
|---|---|---|---|
| I | \(\frac{1}{2}\) | £90 | £102 |
| II | \(\frac{1}{3}\) | £94 | £106 |
| III | \(\frac{1}{6}\) | £98 | £110 |
A customer buys from the dealer if \(a\) is at most their buying limit. If not, they sell to the dealer if \(b\) is at least their selling threshold. Otherwise they leave. This buy-first rule applies even if both quotes would have been acceptable: one customer can make only one trade.
The dealer makes \(a-100\) pounds if a customer buys, \(100-b\) pounds if the customer sells, and zero otherwise. Which pair \((b,a)\) maximizes expected profit per arriving customer? Justify the optimum over the entire grid.
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Hint 1
For each quote pair, classify the three customer types as buy, sell or leave before calculating profit. Never count both a buy and a sell from one type.
Check your answer
Solution
For each type, apply the stipulated decision order: buy if \(a\) is within the type's buying limit; otherwise sell if \(b\) meets its selling threshold; otherwise leave. The dealer's profit from a purchase is \(100-b\) and from a sale is \(a-100\). Since there is exactly one customer, these actions are mutually exclusive.
Enumerating the three bids and three asks gives the following expected profits (pounds per customer), with rows \(b=90,94,98\) and columns \(a=102,106,110\):
| b \ a | £102 | £106 | £110 |
|---|---|---|---|
| £90 | 2 | 8 | \(\frac{20}{3}\) |
| £94 | 2 | 6 | \(\frac{20}{3}\) |
| £98 | 2 | 4 | \(\frac{10}{3}\) |
For example, at \(b=\text{£90}\) and \(a=\text{£106}\), type I sells to the dealer for £90, yielding £10; types II and III buy for £106, yielding £6 each. Their probabilities are \(\frac{1}{2}\), \(\frac{1}{3}\) and \(\frac{1}{6}\), so expected profit is \(\left(\frac{1}{2}\right)\times10+\left(\frac{1}{3}\right)\times6+\left(\frac{1}{6}\right)\times6=5+2+1=\text{£8}\).
At ask £102 every type buys, earning £2 regardless of the bid. At ask £110, type III buys for a £10 margin; types I and II may sell at the bid if their thresholds are met. Applying those possibilities gives the last column. Every other feasible pair earns at most £\(\frac{20}{3}\) or £6, so £8 is a strict global maximum and \((90,106)\) is the unique optimal quote pair.
A model that independently added probabilities of selling at the bid and buying at the ask could double count types for which both quotes are appealing (for instance, type II at bid £94 and ask £106). The customer's buy-first choice, not two independent order streams, determines the actual trade.
Assumptions
Each customer belongs to exactly one of the three types, with probabilities summing to one.
The listed selling and buying thresholds are fixed for each type; if both quote conditions hold, the customer buys rather than sells.
The dealer can execute at most one transaction with this customer, has a £100 reference value, and faces no other costs or inventory constraints.
Final Answer
Unique maximizing quote pair: \(b=\text{£90}\), \(a=\text{£106}\), with expected profit £8 per customer.