一次报价

Single Bid

专题: Probability / 概率
难度: L4
来源: Brainstellar

题目详情

你只能对一个物品报一次价。物品对卖家的价值（你所知）是 $0$ 到 $100$ 的均匀随机整数。

你很擅长使用该物品，因此该物品对你的价值比对他高 80%（即对你价值为对他价值的 1.8 倍）。

如果你的出价高于它对卖家的价值，卖家就会卖给你。

问：你应该出价多少才能最大化期望利润？

You have an opportunity to make one bid on an object, whose value to its owner is, as far as you know, uniformly random integer between $0 and $100. What you do know is that you are so much better at operating the widget than he is, that its value to you is 80% greater than its value to him. If you offer more than the widget is worth to the owner, he will sell it. But you get only one shot. How much should you bid? For example, if its actual value is $10, you bid & win at $11, and sell it for $18, making profit. But if you bid more than $18, you make lose! But since you don't know how much its actual price is, how do you bid in order to make some profit?

Hint

Consider the case he already won the bid at $x. What happens next?

解析

最优策略：不出价（出 0）。

如果你出价 $x$ 并买到，说明卖家价值 $y$ 在 $[0,x]$ 上均匀。你的期望价值为 $1.8\,E[y]=1.8\cdot x/2=0.9x$ ，但你支付了 $x$ 。

因此在“买到”的情形下期望利润为 $0.9x-x=-0.1x<0$ ；买不到时利润为 0。

总体期望利润非正，最优是不上场。

Original Explanation

Solution

CSEBLOG: We should bet only 0! (dont bet!) Suppose I bet $x and get the widget. So, the value of it for the owner would be $y, uniformly distributed between 0 and x. So, its value for me is $1.8y. Expected value for me is 1.8* Expected value of y = 1.8*x/2= 0.9x

So, if I get, expected value of the widget for me is 0.9x $ paying x $.

If x is less, i.e I am not getting it, I did not gain/lose anything.

So, overall I am losing. So, I should not bid.