To bet or not to bet
I think the answer depends on the space $Y$ is coming from. Consider if $Y \in \{ 1, \frac{T+1}{T}+\epsilon \}$, where $\epsilon &gt; 0$ and the probability $p$ of the better result is 

$$p &gt; \frac{T^2}{\epsilon T + 1}$$

(Which just comes from solving:)

$$1\cdot (1-p) + \left(\frac{T+1}{T} + \epsilon \right)\cdot p &gt; T+1$$

Then no matter what utility function you have, risk averse or risk loving, you will always take the gamble (option B).

Did you mean to have some sort of restriction on $Y$ for this question? Otherwise we'd have to consider both $Y$ and $u$ to determine whether the player will take option B.