No sooner had John McCain wrapped up the Republican presidential nomination than the inevitable speculation started over his running mate. Indeed, the Arizona senator has done nothing to quiet this quadrennial guessing game, noting he has a list of 20 prospects.

And while the Democratic race is proceeding at fever pitch, that has not prevented talk about whether Barack Obama and Hillary Clinton would combine on a “dream ticket” or, if not, who would be a logical political partner for each.

It’s been ever thus, especially since presidential candidates began conducting intensive background probes after the past slapdash selection process produced a Democrat who quit the ticket because of disclosures that he had undergone electric shock therapy for depression and a Republican forced to resign in a corruption probe of his previous job.

At this point, most discussion of specific names is sheer guesswork. It may be more useful to list the key factors likely to govern the choices.

Here are a few, in no particular order:

¢ Ready to serve on Day One. Vice-presidential candidates must be deemed qualified by pundits and politicians, whose instant assessments will fill airwaves and newspapers. Presidential nominees should avoid picking running mates whose skimpy resumes may cast doubt on their judgment.

¢ Been around. As the Democratic campaign shows, presidential politics is rough. It would be a gamble to pick someone with little experience in elective politics, like Condoleezza Rice or Wesley Clark.

¢ No skeletons. The VP candidate must be fully vetted to avoid unpleasant surprises about his or her background. After the 1972 Democratic fiasco over the late Tom Eagleton’s electric-shock history, candidates began to assign trusted aides to conduct intensive background checks. This process could favor someone who has sought the presidency itself and undergone the accompanying media scrutiny.

¢ Able to debate. The candidate must be able to do well in the vice-presidential debate, perhaps the only time except for selection day in which No. 2 gets the political spotlight. It’s important to avoid someone who will seem uninformed, inarticulate or weak.

¢ Helpful politically. Useful, but hard to calculate. Candidates sometimes use the choice to unite a divided party, as Ronald Reagan did with George H.W. Bush in 1980.

It helps for a candidate to bring a state, or region, that might otherwise be lost. Al Gore might have become president eight years ago had he picked Sen. Bob Graham of Florida – and had he known he would lose that key state and the presidency by 537 votes.

A candidate with blue-collar and union support like Dick Gephardt might have helped John Kerry win Ohio in 2004; John Edwards proved unhelpful in his home state of North Carolina.

Sometimes, the political help is less tangible: Bill Clinton’s choice of Gore in 1992 helped reinforce a sense of generational change. Dick Cheney helped offset George W. Bush’s lack of international experience, something Obama might want to do if he is the 2008 Democratic nominee.

¢ Political hit man. The ideal running mate is not only qualified for the presidency but able to perform the traditional vice-presidential role of taking the lead in political slugging. “Politics ain’t bean bag,” Finley Peter Dunne’s Mr. Dooley said, but the public seems to prefer having the top candidate avoid the tough stuff.

Despite its potential importance, there’s little statistical evidence to suggest that most vice-presidential choices have been decisive.

John F. Kennedy might not have won in 1960 without Lyndon B. Johnson to help carry Texas and keep conservative Democrats in line. On the other hand, the elder George Bush’s widely derided choice of Dan Quayle made no apparent difference in the 1988 outcome.