The increasingly desperate shuffling protagonists start to long for something - anything - to put them out of their misery.
South Korea's Ban Ki-moon is the current incumbent who - whether masochist or martyr - put up his hand for a second term in 2011.
Tellingly, he was elected unopposed.
Somehow, I suspect he will be ticking off the remaining days and weeks of his tenure much as in the time-honoured tradition of the prisoner marking numerals on his cell wall.
I read a quote somewhere (can't find it now) to the effect that the various ethical calisthenics indulged in by the United Nations would constitute high comedy if it weren't for the fact it would be a tragedy if there was no UN at all.
An even bigger tragedy, though, lies in the UN failures in such killing fields as Rwanda, Srebrenica, Sri Lanka and Syria - the complete antithesis of everything it was meant to prevent.
Alas, the seeds of its ultimate ineffectiveness were sown in the formation of the realpolitik mechanisms inherent in the spectacularly misnamed Security Council itself.
The sterling work of various agencies within the UN - Helen Clark's Development Programme (UNDP) included - have rescued countless thousands and promoted health and education worldwide, but it is mainly in the nature of bottom-of-the-cliff ambulance work.
Rather like beleaguered CYFs workers depressingly patching up unending streams of casualties that should never have happened in the first place.
Overlaying these failings is an abiding public perception that the whole institution has turned into a very public trough in which the representatives and bloated entourages of many states least able to afford it are wallowing in obscene largesse.
Symptomatic of the whole cynical scenario surrounding selection of the new secretary-general is the politicking already going on - not on the basis of the merit or otherwise of potential candidates, but simply the geographical origin of the individuals concerned.
Russia has more or less demanded that it is time for an eastern European secretary-general, although, given the impotence of the post in the face of Security Council recalcitrance, the successful candidate's place of residence seems a tad academic.
Dag Hammarskjld, the second UN secretary-general, famously stated that the United Nations "was created not to lead mankind to heaven but to save humanity from hell".
The secretary-general's is a hell of a job, although of course nowhere near the hell endured by the legions of civilian casualties who bear testimony to UN inability to perform its primary function.
Helen is a control freak.
In her present position as UNDP head, as in her lengthy tenure as prime minister, she has a fairly firm hold on the helm.
As secretary-general, she would be at the mercies and whims of the permanent powers and their stultifying veto rights.
And she'd rather chew glass for breakfast than be subject to that particular regime.
