One bid for a widget of unknown value

Question

A widget is worth some amount V to its owner, and as far as you know V is uniformly random between 0 and 100. The widget is worth 1.8 times V to you, because you can use it better. You may make one sealed bid b. The owner sells if b is at least V; otherwise there is no deal.

What bid maximises your expected profit, and what is that profit?

For example, if V = 10 and you bid 11, you buy for 11 something worth 18 to you and profit 7.

Accepted Answer

Bid nothing. Every positive bid loses money on average. Condition on acceptance. If you bid b (at most 100) and the owner accepts, you learn that V is at most b. Given that, V is uniform on 0 to b, with expected value b/2. The widget is then worth 1.8 × b/2 = 0.9b to you, on average, and you paid b. Expected profit when the deal happens: 0.9b minus b = minus 0.1b. Unconditionally. The deal happens with probability b/100, so the expected profit of bidding b is (b/100) × (minus 0.1b) = minus b squared / 1,000, negative for every b above 0. The best you can do is bid 0 (or not bid) for an expect…

One bid for a widget of unknown value

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