Smart Cats Have 25 Fifes, .... in Expectation

You’re a contestant on the hit new game show, “You Bet Your Fife.” On the show, a random real number (i.e., decimals are allowed) is chosen between $0$ and $100.$ Your job is to guess a value that is less than this randomly chosen number. Your reward for winning is a novelty fife that is valued precisely at your guess.

What number should you guess to maximize the average value of your fifing winnings?

If you guess $x \in [0,100],$ then the payoff function in terms of $x$ and the randomly selected $Y \sim U(0,100)$ is $$v(x|Y) = \begin{cases} x, &\text{if $Y > x$;}\\ 0, &\text{if $Y \leq x.$}\end{cases}$$ So in expectation, we have $$V(x) = \mathbb{E}_Y[v(x|Y)] = x \mathbb{P}[ Y > x ] + 0 \mathbb{P}[ Y \leq x ] = x \left(1-\frac{x}{100}\right).$$

Since $V(x)$ is continuously differentiable and concave, it is maximized exactly at $\hat{x}$ which solves $$V^\prime(\hat{x}) = 1 - \frac{\hat{x}}{50} = 0.$$ That is, the optimal solution is to choose $\hat{x} = 50,$ at which point $$V(\hat{x}) = 50 \left( 1 - \frac{50}{100} \right) = 25.$$