倒卖画作

若你出价为 $x$ 且赢得竞拍，则说明真实价值 $V\le x$，此时 $V\sim\mathrm{Unif}(0,x)$，所以 $\mathbb{E}[V\mid V\le x]=x/2$。

你卖出的期望收入为 $1.5\cdot x/2=0.75x$，小于支付的 $x$，期望利润为负。

因此最优是不参与，出价 0。

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Original Explanation

If your bid is lower than the value of the painting, you won’t lose anything, so no harm no foul. However, let's say that you do win the painting. This means that you bid higher than the actual value. Let $x$ be your bid. Given you won the bid, we know that the value is at most $x$. Therefore, the conditional distribution of the value of the painting is $\text{Unif}(0,x)$. On average, the value of the painting will be $\frac{x}{2}$. This means we will be able to sell it, on average, for $\frac{1.5 \cdot x}{2} = \frac{3}{4}x$. However, this is less than $x$, the amount we paid for the painting. So we shouldn’t bid on this painting, thus the answer is $0$.