Selling a gadget to three buyers

Question

You are selling a gadget to three interested buyers who bid one at a time. Each bid is an independent uniform random amount between $0 and $60 (in thousands). You must accept or decline each bid on the spot, and a declined buyer walks away for good. If you decline the first two, you must accept the third.

What is the optimal strategy, and what is the expected sale price?

Accepted Answer

Backward induction. For a uniform draw on [0,60] with acceptance bar t, a stage is worth E[\max(x, t)]: keep the bid if it beats t (a value above t averages 	frac{t+60}{2}), otherwise continue with value t. Third buyer (last). Value = 30 (the midpoint). Second buyer. Bar = 30: value = 	frac12(30) + 	frac12(45) = 37.5, since a bid above 30 averages 45. First buyer. Bar = 37.5. Acceptance chance 	frac{60 - 37.5}{60} = 0.375, kept average 	frac{37.5 + 60}{2} = 48.75: V1 = 0.625(37.5) + 0.375(48.75) = 23.4375 + 18.28125 \approx 41.72. Strategy: accept the first bid above \37.5k; otherwise ac…

Selling a gadget to three buyers

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