I don't speak maths too well (engineer) so simple language preferred or could you describe it as a graph please?
This has probably been asked but I have no idea what to search...
Follow up question:
$$s = w_1 - w_2$$
What is $p(s \mid w_1,w_2)$?
Edit: my top guess if that both are just uniform distributions, as you have no other priors.
edit2: the wider question:
$p(y \mid w_1,w_2) = \int\int p(y \mid t) p(t \mid s) p(s \mid w_1,w_2) dsdt$
y is the outcome of a 'game'
$t \sim N(t \mid s, 1)$
$s = w_1 - w_2$
So the explanation i have jotted down is that $p(y \mid t)$ is a step function. I reckon $p(t \mid s)$ is a gaussian centred wherever s (the skill difference) is. this all makes sense. im just trying to get a handle on the last term...
Furthermore, the notes go on to day the above integral simplifies:
$p(y \mid w_1,w_2) = \int\int p(y \mid t) p(t \mid s) p(s \mid w_1,w_2) dsdt = \int p(y \mid t) p(t \mid w_1,w_2) dt$
Using the delta function explanation that was the accepted answer, this sort of makes sense in my head as "integrating over s sifts out a gaussian probability distribution of t that is dependent on w1 and w2". the whole variable vs known stuff that is eluded to in the answer comments goes over my head a bit but sort of explains the form above.