Is there an idiom for making the same mistake repeatedly?
A Russian acquaintance of mine asked me whether there's an English equivalent of "They keep treading on the same rake": someone walking or running across the backyard or garden steps on the bow rake's teeth, making the handle swing rapidly upward and whack the blunderer in the face: a comic device used frequently in old cartoons (such as Tom and Jerry) or slapstick comedy.
Now, this only happens once or twice in a cartoon, and can be written off as an accident. That's not what the idiom is about; rather, it describes the behavior ... uh ... modus operandi, if you will ... of someone (or maybe a bunch of people) foolish enough to keep treading on the same rake (sitting in the same spot) over and over again. Or delusional, perhaps - you'll recall that Einstein once described crazy behavior as "doing the same thing in the exact same way while expecting different results."
Thus the idiom seems to apply to -
bankers (or banks, for that matter) who scheme, swindle and cheat, and get burned, yet resume their murky activities once the dust has settled
politicians (or political parties, for that matter) making an attempt, again, to make a system of ideas work that has failed many times in the past
people arguing with their spouses of many years even though neither party can recall a single instance of their fighting yielding even remotely satisfactory results in the past
folks purchasing $100 worth of lottery tickets every week for years on end
Etc, etc.
There's a bunch of English idioms that come pretty close ("fool's errand," "a fool repeats his folly," and something about stepping into something twice, etc, etc). None of them captures the humorous aspect of the behavior described above.
Here's a variation on the same theme:
Any suggestions?
"Treading" sounds odd because "to tread on" implies intentional stomping which isn't as comic/funny. In English I think we'd just say "They keep stepping on the same rake."
Other idiomatic images might be "They keep shooting themselves in the foot." or "...walking into the same wall."
Or for failed political systems one might say "Building castles out of sand."
More common is to simply say "Some fools never learn."
Also, the quote is actually: "Insanity is doing the same thing over and over and expecting a different result."
I'm aware of several idioms with different flavors, each pertaining to some but not all of the situations you listed:
bankers (or banks, for that matter) who scheme, swindle and cheat, and get burned, yet resume their murky activities once the dust has settled can be described as dyed-in-the-wool cheats.
politicians (or political parties, for that matter) making an attempt, again, to make a system of ideas work that has failed many times in the past may be flogging a dead horse, as may people arguing with their spouses of many years even though neither party can recall a single instance of their fighting yielding even remotely satisfactory results in the past, especially if they're fighting about the same issue. Or if it's the action of arguing or running for office and not the issues at hand, the problem could be that you can't teach an old dog new tricks and these people don't know how else to deal with their problems.
folks purchasing $100 worth of lottery tickets every week for years on end: this may be an example of the sunk cost fallacy, or throwing good money after bad, depending on why they're buying the tickets.
The slapstick in Tom and Jerry, as far as I can recall, is often predicated on repetition of different kinds. There's rhythmic repetition of the sort depicted in the Garfield comic, where a character suffers the same injury repeatedly but in a very short timespan and while they're still reeling from previous blows; I don't think the character flaws implied by the idioms and words in this and other answers apply, as the character isn't necessarily able to learn from the last mistake in time to avoid the next. However, another kind of repetition draws humor from the idea that the character is a slow learner (or too slow a learner to outwit the adversary/environment), which brings it back to quietsign's answer.
It's not uncommon to hear someone disapprovingly say, "this is like groundhog day," if they witness someone else repeat a mistake frequently.
groundhog day: a situation in which a series of unwelcome or tedious events appear to be recurring in exactly the same way.