Coming soon to a television near you is the Poll Bowl, a postseason meeting of the purportedly best and second best college football teams. Every fourth year, in the evening of January 3 or 4, the Rose Bowl will be the Poll Bowl. In all other years, the Rose Bowl will forego, and the Poll Bowl will claim, any Big Ten or Pac-10 champion judged the nation's best or second best.

Shortly after the Soviet Union splintered, an official opined that to map the region would now require at least five colors. One mathematics journal quoted this remark under the headline, "Have We Got a Theorem for You." The editors referred to a famous result establishing that, if one wishes to map a plane of disjoint regions such that no adjacent regions appear in the same color, at most four colors are needed. Things quantitative also hold allure for sports enthusiasts. From broadcasters' competition in statistics du jour, soon we may learn of third down efficiency against opponents whose mascots hibernate. Neglected in this quantitative flurry is more fundamental mathematics, mathematics that punctures the Poll Bowl bubble.

THE PROBLEM. It is tempting to assume that every set of things esteemed contains a best member. A plausible claim to identify the best or nth-best of a set implies that one has imposed an ordering on the set. We express an ordering of the natural numbers when we write "0, 1, 2. . . ." For many sets (e.g., the set of paintings), there is no obvious ordering. For football teams, the ordering sought is some "better than" relation. Perhaps one could pronounce a team better than another if it has defeated or can be predicted to defeat the other. The premise that pairwise victory is decisive undergirds every national championship scheme. Yet pollsters who pledge allegiance to that premise often conclude, obliviously to the self-contradiction, that a game's loser is better than its winner. Such is often their verdict even as to season-ending games (e.g., Michigan victories over OSU in 1993, 1995 and 1996). Their theory, it seems, is that x is better than y if x has "achieved more." But what does that mean?

THE INDETERMINACY. For two reasons, we possess no rational method of ordering college football teams. [1] There is no reliable measure of team strength. (a) How good is a team that averages 4.9 seconds in the 40-yard dash and includes a placekicker with 75% accuracy up to 60 yards? Better or worse than a bigger team with a more experienced passer and a defense averaging twelve points allowed? While there is no lack of game performance data, the data are all opponent-relative. No team's sequence of opponents and playing conditions is the same as another's, to say nothing of chance. As there exists no uniform standard of comparison, the data are incommensurable. (b) Striving to overcome this, some imagine that won-lost records or victory margins can be rendered commensurable by adjustment for "the strength of opponents." To reckon an opponent's strength, the only data available are its opponent-relative performance data. Consequently any such adjustment would require that one first ascertain the strength of the opponents' opponents, which would require first ascertaining the strength of the opponents' opponents' opponents, and so on in an infinite regress. A purported measure or ordering of opponent strength is nothing other than a measure or ordering of teams, which it was the purpose of this exercise to construct. To define team strength by reference to opponent team strength is circular, a case of smuggling into the definition that which is to be defined. (c) In comparing teams by game outcomes, even consistency is unattainable. In a 1997 cycle, Wisconsin beat Iowa which beat Purdue which beat Wisconsin. Victory is not transitive. Without transitivity, one could scarcely hope to complete a total ordering of 111 Division I-A teams who play but a dozen games apiece, and in all events, transitivity is a necessary condition of an ordering. Presented annually with numerous cycles, pundits venture that pairwise competition is inconclusive, again contradicting a premise of a national championship event.

[2] Suppose that, conceding inability to measure team strengths, one elects to impanel voters and to accept as the conclusive ordering of teams an ordering derived from voters' orderings such as they may be. Published in 1951 and recognized with the Nobel Prize in economics, Arrow's impossibility theorem establishes that for two or more voters declaring their positionings of three or more candidates, no function exists that, by nondictatorial pairwise comparisons respecting unanimity, compiles any such positionings into one. By compiling ballots, one will sometimes generate a cycle—in graphical form, Floyd of Rosedale's tail. (Awarded to the winner of the Iowa-Minnesota game, Floyd is a pig.)

Given the circumstances of college football, we possess no rational method to determine, by measurement or voting, the participants in a championship event. This conclusion is general, as applicable to a computer algorithm as to a scheme executed with pencil and paper.

THE IMPOSTORS. [1] Football polls propound orderings of teams at the cost of breaching one of Arrow's rationality conditions. The polls operate the Borda count, a tally of points awarded candidates as follows: the most (or fewest) for a first place vote, the second most (or second fewest) for a second place vote, and so forth. Proposed as a means for electing members to the Academie Royale des Sciences by Jean Charles de Borda, French investigator of fluid mechanics and naval captain in the American Revolutionary War, the Borda count today is otherwise practiced only in society elections. It breaches the rationality condition that says this: in a positioning of candidates derived from ballots, the order of x and y may depend on the positions in which x and y appear on the ballots, but not on the position of any z other than x and y. Debatable in other contexts, this condition is difficult to dispute here. When a voter lists team x above y, one cannot plausibly associate precision with how many teams, if any, the voter puts in between. Coaches and reporters never see many teams for which they vote or decline to vote. They seem obligingly to list teams by won-lost records, a method insufficient even to avoid ties. Imagine how shaky would be the discriminations of a teacher asked to list 25 students in order of merit. One would have difficulty believing, for example, that students whom the teacher lists fifth and seventh should be treated as differing more in merit than those listed first and second.

The manipulability of a Borda count is notorious. To use an example recently alleged to be actual, suppose that a coach considers team x best and y second. In a Borda count poll, the coach can boost x's ranking by voting y third or lower.

[2] Poll Bowl participants will be chosen (in the "Bowl Championship Series") by a Borda count of seven weighted ballots. The ballots will be the positionings respectively produced by the following: (a) two extant Borda count polls (one of coaches, another of sportswriters), (b) three newspapers' "computer ratings," (c) an attempted measure of strength of opponents, and (d) each team's number of losses. This scheme like all others fails to accomplish what Arrow's theorem shows to be impossible. As for its components, the algorithms used to generate (b) manipulate a modicum of opponent-relative data—for example, The New York Times relies on game outcomes, including scoring margins—in the vain attempt to measure teams. Last year one algorithm yielded Tennessee as best after the regular season; after the Rose Bowl, the three declared Michigan second, third, and fourth, respectively. The method of (c) is merely to add won-lost records of opponents and opponents' opponents while arbitrarily assigning the latter half the weight of the former. In abandoning after two steps what is necessarily an infinite regress, this yields only incommensurables.

When three or more impressive teams next finish a season with identical records—as did five in 1996—the bright light of public attention may turn on this hocus-pocus with incommensurables. Sportswriters have already adopted the mantra that indeterminacy about the top two teams commends a playoff of the top eight. But that overlooks the main point. What the foregoing shows is the inability to determine the top n football teams for any n.

CONTRASTS. Why is the NCAA basketball championship feasible if a football championship is not? The former includes fully one-fifth of eligible teams (64 of 306) and spans but 19 calendar days. Arbitrariness in selection and seeding is tolerated in a tournament conducted in the absence of an ordering, if—as in the medieval jousts from which a tournament takes its name—the sport permits a highly inclusive draw played in a flash. What is feasible in football, an incomplete round robin tournament, already occurs as the "regular season."

The point of bursting the playoff bubble is not to suggest a misplaced scientific precision for sports. Rather the point is to ask, given that a reckoning of a best team is unattainable, whether the traditional, determinate Rose Bowl competition should go by the board. For C. L. Dodgson—Oxford mathematical don, observer of voting anomalies, and gifted illustrator of illogic better known as Lewis Carroll—fascination with No. 1 would have presented an easy mark. Upon the Messenger's return from a trip in Through the Looking Glass, the White King asks the Messenger whom he passed on the road. "Nobody," the Messenger replies. "So of course," responds the King, "nobody walks slower than you." Startled, the Messenger pauses, then ventures, "I do my best. I'm sure nobody walks much faster than I do." To this the King replies, "He can't do that, or else he'd have been here first."

Louis M. Guenin '72, lecturer on ethics, Harvard Medical School, is the third Michigan generation of a family including Yost's players Clement P. Quinn '13 and Cyril J. Quinn '14.

In "The Myth of the Best" (Michigan Quarterly Review [MQR], Winter, 1998) he describes from the academy's perspective recent negotiations concerning a college football championship, mathematics that assures indeterminancy, and financial considerations overlooked. (MQR is available through its editorial offices at 3032 Rackham Building, University of Michigan, Ann Arbor, MI 48109-1070; phone (734) 764-9265; e-mail michiganquarterlyreview@umich.edu.)

[See also a related article by Guenin.]