Solution to Garbled Games

This challenge was to find the optimal strategy for player 5 of a certain game. This must be solved with probability density. Let the PDF before the guess of player n be fn, and that player’s best guess, x0. The derivative of the expected value must be zero, which can be simplified to:

x_0 \operatorname{f_n}x_0 = \int_0^{x_0}\operatorname{f_n}x \delta x

Solving for player 1, we get 3x0 = 2, and so x0 = 3/2. Using a the same process for players two through 5, we can find the answer. Shown is a table of the best guess, chance of victory, and expected value for players one two five, as well as the probability density function for the weight of the prize before each guess. The answer is thus a guess of exactly 0.1 kilograms, and you will win it exactly half the time.

\begin{array}{|r|rcl|} \hline \\ \text{Player} & \text{Guess} & \operatorname P {win} & \text{Value} \\[4pt] \hline \\ 1 & 3\over2 & 3\over4 & 9\over8 \\[4pt] 2 & 1 & 1\over2 & 1\over2 \\[4pt] 3 & 1\over3 & 1\over2 & 1\over6 \\[4pt] 4 & 5\over24 & 23\over48 & \approx.1 \\[4pt] 5 & 1\over10 & 1\over2 & 1\over20 \\[4pt] \hline \end{array}

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