A company is worth V to its owner, where V is uniform on [0, 100] and known only to him. Under your management it is worth 1.5V. You make one take it or leave it bid b, which the owner accepts exactly when b is at least V. What bid maximises your expected profit?

A company is worth V to its owner, where V is uniform on [0, 100] and known only to him. Under your management it is worth 1.5V. You make one take it or leave it bid b, which the owner accepts exactly when b is at least V. What bid maximises your expected profit?

Approach: Compute the expectation of V conditional on the bid being accepted rather than its unconditional mean, then write expected profit as a function of b.

0. Acceptance means V is at most b, so conditional on a trade V is uniform on [0, b] and E[V | accepted] = b/2. What you receive is worth 1.5 * b/2 = 0.75b against a price of b, so every accepted trade loses 0.25b. Expected profit is P(accept) * (0.75b - b) = (b/100) * (-0.25b), negative for every b > 0 and maximised at b = 0. The plausible wrong answer is about 67, from comparing b with 1.5 * E[V] = 75 using the unconditional mean. That ignores adverse selection: the owner sells only when V is low, which is the winner's curse in its purest form.

Follow-up: At what multiple of V under your management does a positive bid become profitable, and what is the optimal bid there?

Key concepts: adverse selection, conditional expectation, winner's curse.