I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. Specifically, one player has the opportunity to choose any value $\eta$ from the interval $[0, h]$ to transfer to Player 2. $h$, the maximum he may transfer, is drawn from the uniform distribution $U(0,1)$. Player 1 knows $h$, but Player 2 only knows its distribution, and rather than observe it directly, receives a noisy signal $S_h \sim U(h - \epsilon, h + \epsilon)$ about its value. Last, Player 2 also receives a noisy signal $S_\eta \sim U(\eta - \epsilon, \eta + \epsilon)$ about Player 1's choice.
$a$ is some (known) multiplier on the interval $[0,1]$. Both players know what $a$ is. $a$ defines the threshold that Player 2 will use to decide how to behave. For example, if $a = .75$, then Player 2 will be cooperative if Player 1 chooses $\eta$ of at least 75% of $h$. How can I find the CDF:
$$P(S_\eta \geq a \times S_h \mid \eta, h, a)$$
This is the probability that Player 2's signal about $\eta$ will be greater than what he believes the threshold is.
I've done some simulations to calculate the CDF empirically, so I know that it looks roughly like that of the normal distribution:
(Edit: for parameters $h = .8, a = .75, \epsilon = .01$):
But I'd really like to solve for it analytically. Does anyone have an idea about how I could do that?