First, Rhode Island Senator Jack Reed said ‘no’. He’s enjoying serving his constituents too much to consider leaving the Senate.

Then, former undersecretary of defence Michèle Flournoy announced she wasn’t interested either. She is committed to spending more time with her family.

Nobody, it would seem, wants to succeed the dismissed Chuck Hagel as Barack Obama’s defence secretary.

But who would want the job?

Given what Mr Hagel had to deal with, and what happened to him as a consequence, the downside of any such appointment is glaringly obvious.

Since the financial meltdown of 2008, it’s almost universally agreed that significant budget cuts are essential to restore the American economy. Cuts in military spending are an indispensable component of this because the overall share of the Armed Forces’ funding is at least 17 per cent, which is a significant part of annual government spending. Many believe that 17 per cent is a gross underestimate.

It is often said that the Pentagon is effectively the largest corporation in the United States. It is also one of America’s biggest employers, of both uniformed and civilian personnel. In such circumstances, one of the key tasks of any defence secretary is to preside over some pretty ruthless downsizing.

The pain is even greater because in this case the expenditures are public rather than private. Jobs are lost every time a base closes, a contract cancelled, or a deployment scrapped. Mr Hagel was initially brought in partly to oversee another phase in this politically damaging process.

His successor will be the fourth defence secretary to preside over the giant rollback.

As the process moves on, it becomes more difficult, and the cuts more controversial. Indeed, Mr Hagel was reportedly upset that Mr Obama did not fight harder in Congress to secure the Pentagon’s budget.

There is also the problem of Mr Obama’s reputation as a chief executive who doesn’t pay much attention to what members of his cabinet think, preferring instead the counsel of his hand-picked White House inner circle. This concern applies to every cabinet-level position but it becomes particularly acute for any defence secretary.

But the biggest disincentive for any prospective Pentagon boss must surely be the Obama administration’s policy on ISIL and the conflicts in Syria and Iraq. Mr Hagel didn’t exactly resign; he was dismissed. This was, in large part, because of a two-page memo to national security adviser Susan Rice, in which he was said to be “sharply critical” of the strategy for dealing with Islamist fanatics.

He was, apparently particularly worried that American ambiguity regarding the future of Syrian dictator Bashar Al Assad places the entire US plan in grave jeopardy. He has reportedly been strongly arguing that American policies and practices must clearly demonstrate that the campaign against ISIL will not benefit the Damascus regime.

But the administration, led by the president, has carefully avoided any suggestion that American intervention in Syria against ISIL will involve or seek regime change. Indeed, last week Mr Obama was asked point-blank if he was considering steps that might lead to Mr Assad’s removal. He curtly responded, “no”.

Yet, the week before, Mr Obama noted that to “make common cause” with Mr Assad against ISIL “would only turn more Sunnis in Syria in the direction of supporting ISIL and would weaken our coalition”.

He appears determined to leave it at that: there will be no open or tacit alliance with the Damascus dictatorship. But he will not say that the United States is committed to regime change in Syria as it pursues the mission to “degrade and ultimately destroy” ISIL.

Mr Hagel also reportedly found himself caught between administration demands for more air strikes in the Aleppo region, and Pentagon resistance to the idea based on alleged concerns about Syrian air defences.

His successor, whoever it is, will be handed primary responsibility for this frankly incoherent policy that, as it’s presently constructed, virtually insures its own inability to meet its broadest and most important goals. He or she will not be able to count on the president giving serious consideration to their perspective. Instead, if recent experience is any guide, especially as explained by another one of Mr Obama’s former defence secretaries Leon Panetta, the Pentagon’s new boss can expect their most difficult policies to be micromanaged by the White House.

Mr Panetta’s memoir was, of course, one of several angry accounts by former administration officials. Even so, someone willing to take the job will no doubt quickly be found. And they surely will be credible and qualified, and probably very distinguished. But anyone entering the administration as Mr Obama’s fourth defence secretary will have to wonder if two years isn’t enough time to add a fifth person to that list. Early departure seems to be built into the job description.