But for poor people $100 a month is the difference between eating or not eating some days, or having the heating on or going cold, or buying some books for the kids to read at home, or paying the bills on time vs paying them late.
Also, about books: I don't know if there's something like this in the US, but it seems like a good idea: https://literacytrust.org.uk/support-us/help-child-fall-love...
A meaningful amount, like $400~$500 a month could make a much more considerable impact. $100/month means "Maybe we can take the kids to a movie one a month or have a nice dinner to help distract ourselves."
$500/month could mean, "We can get all those extra school supplies or put our kids in extra school programs."
Seeing if people would make those decisions would be an equally important study.
These current programs see more designed to encourage people to use free programs with a small monitary incentive. I probably says more about those programs than the impact of the money itself.
If the money is not significant enough to meaningfully impact that assessment then a short term distraction is a rational way to spend the money.
It's analogous but no the same as the prisoners dilemma with a known number of games.