Imagine you have been set the following trolley problem by a villain. There is a central track, called CONTINUE. If you do nothing, the trolley will continue down this track, and kill whomever is at the end of it, then stop.
Part way along the line, there is a junction, with a lever. If you pull that lever, then the trolley will go down one of two tracks—STOP and LOOP. If it goes down STOP, then it stops, killing whoever is at the end of the line (if anyone).
If it goes down LOOP, it returns to the start of the track, killing whoever is on LOOP, and leading to the trolley returning to the junction. The lever determining which way the trolley will go is probabilistic, and the villain controls the probabilities.
The villain also controls how many people are tied to the tracks, and which tracks they are tied to. Importantly, if the trolley goes down LOOP, killing whoever is on there, then the villain will replace those victims with fresh ones (for the moment we’ll assume that he does so with the same number).
This lecture animated by a concern that deontological theorists have trouble accommodating ignorance and uncertainty into our theories, developed a broadly deontological approach to iterated, probabilistic decision problems like this one.