The House-Selling Problem and Reservation Values

You're selling a house, or unwinding a large position by shopping it to buyers one at a time, or a trader is fielding one bid at a time for a block trade. Offers arrive sequentially, each drawn from some distribution you roughly know, and each round of waiting for the next offer costs you something — carrying cost, opportunity cost, the risk the market moves against you. Should you take the offer in front of you, or wait for a better one? Trying to plan the whole sequence of future offers seems hopeless since they're random. But the house-selling problem shows the optimal strategy collapses to one simple rule: pick a single number, and accept the first offer that clears it.

An analogy: fishing with a cost per cast

Imagine fishing where each cast costs a small fee, and each cast lands a fish of random size drawn from a distribution you know well from experience. You want to maximize (expected fish size kept) minus (total casting fees paid). It seems like you'd need to plan how many more casts to try based on everything you've caught so far — but the optimal rule turns out not to depend on history at all. There's one target fish size, and the rule is: keep casting until you land a fish at least that big, then stop. The target size doesn't change no matter how many small fish you've thrown back; each cast is a fresh independent draw, so past casts carry no information about future ones.

The idea, one symbol at a time

Let offers $X_1, X_2, \ldots$ arrive independently, each drawn from a known distribution $F$. Let $c$ be the cost incurred per round of waiting (search cost). The optimal strategy is a reservation value $x^*$: reject every offer below $x^*$, accept the first offer at or above $x^*$. The reservation value is defined by the equation:

$$x^* = \mathbb{E}[\max(X, x^*)] - c ,$$

or equivalently, using $\mathbb{E}[\max(X,x^*)] = x^* + \mathbb{E}[(X - x^*)^+]$ (where $(X-x^*)^+$ means "the amount by which $X$ exceeds $x^*$, or zero if it doesn't"):

$$c = \mathbb{E}[(X - x^*)^+] .$$

In plain English: $x^*$ is set so that the expected extra value gained by taking one more look — the average amount by which a fresh offer beats your current threshold, counting zero when it doesn't beat it — exactly equals the cost of taking that extra look. At the reservation value, you're indifferent between stopping now and taking one more draw; below it, one more draw is worth its cost; above it, it isn't. This is why the rule never depends on how many offers you've already seen and rejected: each new offer is a completely fresh, identical opportunity, so the calculus that determined $x^*$ never changes.

Worked example 1: uniform offers

Suppose offers are uniform on $[0, 100]$(in thousands) and the search cost is$c = 5$per round. We need$x^$solving$\mathbb{E}[(X - x^)^+] = 5$. For$X \sim \text{Uniform}[0,100]$,$\mathbb{E}[(X-x^)^+] = \frac{(100-x^)^2}{200}$. Setting this to 5:

$$\frac{(100 - x^*)^2}{200} = 5 \;\Rightarrow\; (100-x^*)^2 = 1000 \;\Rightarrow\; 100 - x^* \approx 31.6 \;\Rightarrow\; x^* \approx 68.4 .$$

So the rule: reject any offer below $68,400; accept the first one at or above it. Even though the average offer is only $50,000, it's worth holding out for something notably above average because the cost of waiting is small relative to the upside from a lucky high offer.

Worked example 2: what happens as search gets more expensive

Keep the same uniform offers, but suppose carrying costs rise to $c = 20$ per round (a much less patient seller). Solve $(100-x^*)^2/200 = 20 \Rightarrow (100-x^*)^2 = 4000 \Rightarrow 100 - x^* \approx 63.2 \Rightarrow x^* \approx 36.8$. The reservation value drops from $68,400 to $36,800 — a seller facing higher waiting costs should accept much lower offers, exactly matching the intuition that expensive patience isn't worth much.

What this means in practice

The reservation-value structure recurs anywhere someone is sampling opportunities sequentially at a cost: shopping a block trade to counterparties one at a time, a portfolio manager screening deal flow, or a job seeker fielding offers. Its practical payoff is that the optimal policy is memoryless and requires no complex lookahead — you compute one number from the offer distribution and the search cost, and then apply a simple threshold rule to every offer as it arrives, with no need to track history.