Strategies in a (specific) Search Game

(This is pure curiosity for me.)

Hider's possible strategies are the (real-valued) random variables $X$ such that $\: E\big[|X|\big] \leq 1 \:\:$.

Define $\mathbf{S}$ to be the space of short maps $\: f : [0,\scriptsize+\normalsize\infty) \to (-\infty,\scriptsize+\normalsize\infty) \:$ such that $\: f(0) = 0 \:\:$.
Give $\mathbf{S}$ the topology of compact convergence. $\:$ The resulting topological space is a Polish Space.
Searcher's possible strategies are the $\mathbf{S}$-valued random variables $F$.

Hider wants to maximize, and Searcher wants to minimize, $\: E\big[\operatorname{inf}(\{t\in [0,\scriptsize+\normalsize\infty) : F(t) = X\})\big] \:\:$.

1. $\:$ Has this game been studied before?
2. $\:$ Is it known to have a value?
3. $\:$ What is known about the value (assuming it exists) and/or optimal (sequences of) strategies?

Obviously, Hider can force that expected value to be at least $2$ $($uniform sample from $\{-1,1\})$.
I also see that $\mathbf{S}$ is compact, although I don't know how one might use that.

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