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The problem statement, all variables and given/known data

A philanthropist writes a positive number x on a piece of paper, shows it to an impartial observer, and then turns it face down on the table. The observer then flips a fair coin. If it shows heads, she writes the value 2x, and, if tails, the value x/2, on a piece of blue paper which she then turns face down on the table. Without knowing either the value x or the result of the coin flip, you have the option of turning over either red or the blue piece of paper. After doing so, and observing the number written on that paper, you may elect to receive as a reward either that amount or the (unknown) amount written on the other piece of paper. For instance, if you elect to turn over the blue paper and observe the value 100, then you can elect either to accept 100 as a reward or to take the amount (either 200 or 50) on the red paper. Suppose that you would like your expected reward to be large.

(a) Argue that there is no reason to turn over the red paper first because if you do so, then no matter what value you observe, it is always better to switch to the blue paper.

(b) Let y be a fixed nonnegative value, and consider the following strategy. Turn over the blue paper and if its value is at least y, then accept that amount. If it is less than y, then switch to the red paper. Let R_y(x) denote the reward obtained if the philanthropist writes the amount x and you employ this strategy. Find E[R_y(x)]. Note that E[R_0(x)] is the expected reward if the philanthropist writes the amount x when you employ the strategy of always choosing the blue paper.

The attempt at a solution

(a) If I look at the blue paper and observe 100, then either 200 or 50 is written on the red paper. If I look at the red paper and observe 100, then either 200 or 50 is written on the blue paper. It doesn't matter which paper I look at first then. If I decide to receive the amount in the unknown piece of paper, then I will obtain twice the value I saw 50% of the time right? But this means the value I observe will be twice that written on the other piece of paper 50% of the time. So it doesn't matter what decision I make.

Of course, if I just want to obtain twice what the philanthropist wrote, I will always pick the blue paper. So, under this criteria, there is no need to look at the red paper.

(b) E[R_0(x)] = 2x * 0.5 + x/2 * 0.5 = x + x/4 = 5/4 x right? For arbitrary y, if x < y, then R_y(x) = x. If x >= y, then R_y(x) = 2x or x/2. P{R_y(x) = x} = P{R_y(x) = x|x < y}P{x < y} + P{R_y(x) = x|x >= y}P{x >= y} which simplifies to P{x < y}. What is this though?

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# Homework Help: Which Paper Do I Pick?

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