So, here is my solution to the problem of the last post. Let us call f(t) the expected win at the optimal strategy, if we get the value t. Then this function must have the following property.

\(f(t) = \max \{ t, \int_t^1 f(s) \, ds \} \)

The reason for this is that we can either keep our value t or continue. If we continue, we get the average win value of all larger numbers.

It is possible to prove that

\(f(t)=t\)

for all

\(t \ge a \),

and for \(t < a\)

\(f(t) = \int_t^1 f(s) \, ds \)

The point a must be chosen so that

\(a = \int_a^1 s \,ds \)

One gets

\(a=\sqrt{2}-1 \)

Differentiating the above equation for t < a, we see that

\(f(t) = \lambda e^{-t} , \qquad t<a. \)

with

\(\lambda = a / e^{-a} \).

So the optimal strategy is to keep anything larger than a, and the expected win is lambda.