Suppose there is a pond with infinite number of fish. Weights of the fish are iid uniform $(0,1)$. We catch fish from this pond with the following rules:
- Each day we catch at most one fish from the pond.
- We need in total 3 fish and we are given 10 days.
- Each day (before catching), if we have already 3 fish, we can choose to release one of the fish and catch a new one.
- Our goal is to maximise the expected sum of the weight of the 3 fish we have on day 10.
What's the optimal strategy and what's the expected weight under this strategy?
I think the difficulty is that we only catch one per day. If we are allowed to release and catch any number of fish, the problem becomes that famous dice problem (see here).
Here's my attempt under the current setup. Let $Y_k$ be the expected total weight of the 3 fish by the end of day $k$. Then we have $Y_3=1.5$, and \begin{align} Y_4=&P\left(X_1+X_2+X_3\ge Y_3\right) E\left(X_1+X_2+X_3|X_1+X_2+X_3\ge Y_3\right)\\ &+P\left(X_1+X_2+X_3< Y_3\right)E\left(X_1+X_2+X_3+X_4-\min(X_1,X_2,X_3)\right). \end{align} Unfortunately this formula cannot be generalised to $Y_5$ and above.