Polls indicating that Republican voters prefer this or that candidate for president are often as simplistic as they are hard to avoid. Many observers complain that media coverage of the campaign is too focused on the “horse race.” In some ways, though, the coverage isn’t even as nuanced as that of a horse race, where horses are picked to win, but also to place or to show.

One way to get a clearer picture of an electorate’s preferences is to ask prospective voters to rank the candidates and not merely say which one is their first choice. Who is their second choice, third, fourth, fifth? Doing this allows us to get a better overall view of their appeal or lack thereof. It also makes clear that “Who’s ahead?” is not by any means a question with a single, simple answer.

Let’s imagine that likely Republican voters were asked to rank Herman Cain, Newt Gingrich, Ron Paul, Rick Perry, and Mitt Romney (Michele Bachmann, Jon Huntsman and Rick Santorum, please accept my apologies). This is for illustration only, although it’s not that far off the mark, so let’s further imagine:

that 36.3% of them favored Romney to Gingrich to Paul to Cain to Perry; and 27.3% of them favored Cain to Paul to Gingrich to Perry to Romney; and 18.2% of them favored Perry to Paul to Gingrich to Cain to Romney; and 9.1% of them favored Gingrich to Perry to Cain to Paul to Romney; and 9.1% of them favored Paul to Gingrich to Perry to Romney to Cain.

Romney is clearly preferred by the highest percentage of voters so using the conventional method of plurality, Romney, the most conventional candidate, is the clear leader.

But impressed that the second highest percentage of voters prefer him (“Wow! 27.3% is almost exactly the sum of my 9-9-9 plan”), Cain might well argue that a runoff between him and Romney is appropriate. In such a runoff, the numbers above suggest that Cain would win since 54.6% all of the voters polled ranked him higher than Romney.

Poring over the preference rankings, Perry supporters might warm instead to the idea of what’s often called an instant runoff (a version of which was just used in the San Francisco mayoral election). This differs from a standard runoff in that the candidates with the fewest first place votes (Gingrich and Paul in this case) are summarily eliminated. Next the ranking for the others are adjusted (Perry would gain another 18.2% of the first place votes once Gingrich and Paul are gone). Then the method dictates that the candidate among the remaining three having the fewest first place votes (Cain in this case) is eliminated and the ranking for the two remaining candidates adjusted. After this only Romney and Perry are left, and Perry beats Romney handily.

Scowling, Gingrich insists that we should pay more attention to the overall rankings, not just to the voters’ most preferred candidates. He says it’s only fair that first place votes should each be accorded 5 points, second place votes 4 points, third place 3 points, fourth 2 points, and last place votes 1 point. Using this method each candidate amasses points from the entire preference ranking, which Gingrich argues will more fairly measure that candidate’s support. Needless to add, Gingrich wins if this method is adopted.

Ever the individualist, Paul contends that only mano-a-mano contests should count and notes that given the preference rankings he beats each of the other candidates in a head-to-head race. If they were the only two candidates, more voters would prefer Paul to Romney. Likewise, more would prefer him to Gingrich, to Perry, and to Cain. His claim is true and underlies his argument that he deserves to be overall winner.

So who’s ahead? Given the preference ranking above, each of the five candidates has a reasonable case for being called the front-runner. The numbers were, of course, cherry-picked to yield these different outcomes, but real races often have many of these same oddities, which together provide a more nuanced feel for the relative strengths of the various candidates.

Some pollsters have already made an effort to get beyond the traditional “Who would you vote for if the election were held today?” Gallup, for example, has been experimenting this year with a “positive intensity score,” which is intended to measure the ability of candidates to arouse enthusiasm among those voters who know them. And, in cities such as St. Paul, Minn., Portland, Me., and San Francisco, where instant run-offs are employed, pollsters have expended effort and will inevitably expend more to determine how the voters rank the candidates. Private polling by the candidates themselves is often more sophisticated than the polls the public gets to read about.

Going further and compiling a full preference ranking of the candidates would, no doubt, be difficult and costly. Many voters don’t know all the candidates, others don’t rank them rationally, preferences don’t always hold, and so on.

Nevertheless, to the usual questions about voters’ top choices, pollsters should add questions about their second or third choices and about those candidates they’d refuse to vote for under any circumstances. Elephant and donkey races deserve at least as much analysis of possible outcomes as horse races get.

John Allen Paulos, a professor of mathematics at Temple University, is the author of eight books, including “Innumeracy” and “A Mathematician Reads the Newspaper.”