Review Sheet for MATH 410 at UA
Review Sheet for MATH 410 at UA
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The Bowl Championship Series A Mathematical Review Thomas Callaghan Peter Mucha and Mason A Porter Introduction On February 29 2004 the college football Bowl Championship Series BCS announced a proposal to add a fifth game to the BCS bowls to improve access for midmajor teams ordinarily denied invi tations to these lucrative postseason games Al though still subject to final approval this agree ment is expected to be instituted with the new BCS contract just prior to the 2006 season There aren t too many ways that things could have gone worse this past college football season with the BCS Standings governing which teams play in the coveted BCS bowls The controversy over USC s absence from the BCS National Champi onship game despite being 1 in both polls gar nered most of the media attention 12 but it is the yearly treatment received by the nonBCS mid major schools that appears to have finally gener ated changes in the BCS system 15 Created from an abstruse combination of polls computer rankings schedule strength and qual ity wins the BCS Standings befuddle most fans and sportswriters as we repeatedly get national championship games between purported 1 and 2 teams in disagreement with the polls con Thomas Callaghan is an undergraduate majoring in ap plied mathematics Peter Mucha is assistant professor of mathematics and Mason Porter is a VIGRE Visiting assis tant professor all at Georgia Institute ofTechnology Peter Mucha s email address is much amath gatech eduThis work was partially supported by NSF VIGRE grant DMS 0135290 as aResearch Experiences for Undergraduate pro ject and by a Georgia Tech Presidential Undergraduate Research Award The simulated monkeys described herein do notknow that they live on Georgia Tech computers No actual monkeys were harmed in the course of this inves tigation SEPTEMBER 2004 sensus Meanwhile the top nonBCS squads have never been invited to a BCS bowl Predictably some have placed blame for such predicaments squarely on the computer nerds whose ranking algorithms form part of the BCS formula 7 14 Although we have no part in the BCS system and the moniker may be accurate in our personal cases we provide here a mathematically inclined review of the BCS We briefly discuss its individual components com pare it with a simple algorithm defined by ran dom walks on a biased graph attempt to predict whether the proposed changes will truly lead to in creased BCS bowl access for nonBCS schools and conclude by arguing that the true problem with the BCS Standings lies not in the computer algorithms but rather in misguided addition Motivation for the BCS The National Collegiate Athletic Association NCAA neither conducts a national championship in Di vision IA football nor is directly involved in the cur rent selection process For decades teams were se lected for major bowl games according to traditional conference pairings For example the Rose Bowl featured the conference champions from the Big Ten and Pac10 Consequently a match be tween the 1 and 2 teams in the nation rarely oc curred This frequently left multiple undefeated teams and cochampionsimost recently Michigan and Nebraska in 1997 It was also possible for a team with an easier schedule to go undefeated without having played a truly major opponent and be declared champion by the polls though the last two schools outside the current BCS agreement to do so were BYU in 1984 and Army in 1945 The BCS agreement forged between the six major BCS conferences the Pac10 Big 12 Big NOTICES or THE AMS 888 Ten ACC SEC and Big East plus Notre Dame as an independent was instituted in 1998 in an at tempt to fix such problems by matching the top two NCAA Division lA teams in an endofseason BCS National Championship game The BCS Standings tabulated by The National Football Foundation 18 selects the champions of the BCS conferences plus two atlarge teams to play in four endofseason BCS bowl games with the top two teams playing in a National Championship game that rotates among those bowls Those four bowl gamesiFiesta Orange Rose and Sugarigenerate more than 100 million annually for the six BCS conferences but less than 10 percent of this windfall trickles down to the other five nonBCS Division lA conferences 13 With the current system guaranteeing a BCS bowl bid to a nonBCS school only if that school fin ishes in the top 6 in the Standings those confer ences have complained that their barrier to ap pearing in a BCS bowl is unfairly high 2 0 Moreover the money directly generated by the BCS bowls is only one piece of the proverbial pie as the schools that appear in such highprofile games re ceive marked increases in both donations and ap plications Born from a desire to avoid controversy the short history of the BCS has been anything but un controversial In 2002 precisely two major teams Miami and Ohio State went undefeated during the regular season so it was natural for them to play each other for the championship In 2000 2001 and 2003 however three or four teams each year were arguably worthy of claiming one of the two invites to the championship game Meanwhile none of the nonBCS schools have ever been invited to play in a BCS bowl Tulane went undefeated in 1998 but finished 10th in the BCS Standings Sim ilarly Marshall went undefeated in 1999 but fin ished 12th in the BCS In 2003 with no undefeated teams and six oneloss teams the three BCS one loss teams Oklahoma LSU and USC finished 1st through 3rd respectively in the BCS Standings whereas the three nonBCS oneloss teams finished 11th Miami of Ohio 17th Boise State and 18th TCU The fundamental difficulty in accurately rank ing or even agreeing on a system of ranking the Di vision lA college football teams lies in two factors the paucity of games played by each team and the large disparities in the strength of individual sched ules With 117 Division lA football teams the 1013 regular season games including conference tournaments played by each team severely limits the quantity of information relative to for exam ple college and professional basketball and base ball schedules While the 32 teams in the profes sional National Football League NFL each play 16 regular season games against 13 distinct oppo nents the NFL subsequently uses regular season NOTICES OF THE AMS outcomes to seed a 12team playoff lndeed Division lA college football is one of the only lev els of any sport that does not currently determine its champion via a multigame playoff format1 Ranking teams is further complicated by the Divi sion lA conference structure as teams play most of their games within their own conferences which vary significantly in their level of play To make mat ters worse even the notion of top 2 teams is woefully nebulous Should these be the two teams who had the best aggregate season or those play ing best at the end of the season The BCS Formula and Its Components In the past national champions were selected by polls which have been absorbed as one component of the BCS formula However they have been ac cused of bias towards the traditional football pow ers and of making only conservative changes among teams that repeatedly win In attempts to provide unbiased rankings many different systems have been promoted by mathematically and statistically inclined fans A subset of these algorithms com prise the second component of the official BCS Standings Many of these schemes are sufficiently complicated mathematically that it is virtually im possible for lay sports enthusiasts to understand them Worse still the essential ingredients of some of the algorithms currently used by the BCS are not publicly declared This state of affairs has inspired the creation of software to develop one s own rank ings using a collection of polls and algorithms 21 and comical commentary on faking one s own mathematical algorithm 11 Let s break down the cause of all this confusion The BCS Standings are created from a sum of four numbers polls computer rankings a strength of schedule multiplier and the number of losses by each team Bonus points for quality wins are also awarded for victories against highly ranked teams The smaller the resulting sum for a given team the higher that team will be ranked in the BCS Stand ings The first number in the sum is the mean rank ing earned by a team in the AP Sportswriters Poll and the USA TodayESPN Coaches Poll The second factor is an average of computer rankings Seven sources currently provide the algorithms selected by the BCS The lowest computer ranking of each team is removed and the remaining six are averaged The sources of the participating ranking systems have changed over the short history of the system most recentlywhen the BCS mandated that the official computer ranking 1The absence ofd Division IA playoff is itself quite con troversial butwe do not intend to address this issue here Rather we are more immediately interested in possible so lu tions under the const rdintofthe NCAA mdnddte dgdinst playoffs VOLUME 51 NUMBER 8 Simple Random Walker Rankings Consider independent random walkers who each cast a single vote for the team they believe is the best Each walker occasionally considers changing its vote by examining the outcome of a single game selected randomly from those played by their favorite team recasting its vote for the winner of that game with probability p and for the loser with probability 1 7 p In selecting p E 12 1 to be the only parameter of this simple ranking system we explicitly ig nore margin of victory currently forbidden in official BCS systems and other potentially pertinent pieces of infor mation including the dates that games are played We denote the number of games team i played by 11 the number it won by w and the number it lost by If A tie not possible with the current NCAA overtime format is counted as both halfa win and half a loss so that n w If We denote the number of random walkers casting their single vote for team i as vi To avoid rewarding teams for the number of games played we set the rate at which a walker voting for team i decides to recast its vote to be proportional to n with those games then selected uniformly In other words the rate that a single game played by team i is considered by a walker at site ieg by a Poisson process is indepen dent of the other games played by team i Both because of this rate definition and to circumvent cycles that can arise in discretetime transition problems we find it convenient to consider the statistics of the random walkers in terms of differential equations for the expected populations For a game in which team ibeats team j the average rate at which a walker voting for j changes to i is propor tional to p gt as it is more likely that the winning team is actually the better team and the rate at which a walker already voting for i switches to j is proportional to 1 7 p The expected rates of change of the popula tions at each site are thus described by a homogeneous system of linear differential equations 1 1 D 1 where 1 is the T vector of the expected number 1 of votes cast for each of the T teams and D is the square matrix with components Dii pli 1 pwii 2 1 2p 7 1 DU NJ 2 Aij ii where Ni NJ is the number of headtohead games played between teams i and j and A 7A is the number of times team i beat team j minus the number of times team i lost to team j in those Ni games In particular if i and j played no more than a single headtohead game 3 AU 1 if team i beat team j A 71 if team i lost to team A 0 if team i tied or did not play team j If two teams play each other multiple times which can occur because of conference championships we sum the contribution to AU from each game This multiplicity also occurred in the calculations we performed because we treated all nonDivision l A teams as a single team which is naturally ranked lower than almost all of the 117 Di vision lA teams The matrix D encompasses all the winloss outcomes between teams The offdiagonal elements Di are nonneg ative vanishing only for teams i and j that did not play directly against one another because p lt 1 The steady state equilibrium 1 of 1 and 2 satisfies 4 D10 lying in the nullspace of D that is 1 is an eigenvector associated with a zero eigenvalue As long as the graph of teams connected by their games played comprises a single connected component then the matrix must have codi mension one for p lt 1 and 1 is unique up to a scalar multiple We therefore restrict the probability p of voting for the winner to the interval 6 1 the winning team is rewarded for winning but some uncertainty in voter behavior is maintained The distribution ofv is then joint binomial with expectation 1 and the expected populations of each site yield a rank ordering of the teams Although this random walker ranking system is grossly simplistic we have found 3 4 that this algorithm does a remarkably good job of ranking college football teams or at least arguably as good as the other available systems In the absence of sufficient detail to reproduce the official BCS computer rankings we use this simple random walker ranking scheme here to analyze the effects of possible changes to the BCS SEPTEMBER 2004 NOTICES OF THE AMS 889 algorithms were not allowed to use margin of vic tory starting with the 2002 season In the two sea sons since that change the seven official systems have been provided byAnderson amp Hester Billings ley Colley Massey The New York Times Sagarin and Wolfe None of these sources receive any com pensation for their time and effort indeed many of them appear to be motivated purely out of a com bined love of football and mathematics Never theless the creators of most of these systems guard their intellectual property closely An ex ception is Colley s ranking which is completely defined on his website 5 Billingsley 1 Massey 17 and Wolfe 2 3 provide significant information about the ingredients for their rankings but it is insufficient to reproduce their analysis Additional information about the BCS computer ranking al gorithms and numerous other ranking systems can be found on David Wilson s website 22 The third component of the BCS formula is a measurement of each team s schedule strength Specifically the BCS uses a variation of what is commonly known in sports as the Ratings Per centage lndex RPI which is employed in college basketball and college hockey to help seed their endofseason playoffs In the BCS the average winning percentage of each team s opponents is multiplied by 23 and added to 13 times the win ning percentage of its opponents opponents This schedule strength is used to assign a rank to each team with 1 assigned to that deemed most diffi cult That rank ordering is then divided by 25 to give the Schedule Rank the third additive com ponent of the BCS formula The fourth additive factor of the BCS sum is the total number of losses by each team Once these four numbers polls computers schedule strength and losses are summed a final quantity for quality wins is subtracted to account for victories against top teams The current re ward is 710 points for beating the 1 team de creasing in magnitude in steps of 01 down to 701 points for beating the 10 team It is not difficult to imagine that small changes in any of the above weightings have the potential to alter the BCS Standings dramatically However because of the large number of parameters in cluding unknown hidden parameters in the minds of poll voters and the algorithms of computers any attempt to exhaustively survey possible changes to the rankings is hopeless lnstead to demonstrate how weighting different factors can influence the rankings we discuss a simple ranking algorithm in terms of random walkers on a biased network Ranking Football Teams with Random Walkers Before introducing yet another ranking algorithm we emphasize that numerous schemes are available NOTICES OF THE AMS for ranking teams in all sports See for example 6 10 and 16 for reviews of different ranking methodologies and the listing and bibliography maintained online by David Wilson 22 Instead of attempting to incorporate every con ceivable factor that might determine a team s qual ity we took a minimalist approach questioning whether an exceptionally naive algorithm can pro vide reasonable rankings We consider a collection of random walkers who each cast a single vote for the team they believe is the best Their behavior is defined so simplisticallysee sidebar that it is rea sonable to think of them as a collection of trained monkeys Because the most natural arguments concerning the relative ranking of two teams arise from the outcome of headtohead competition each monkey routinely examines the outcome of a single game played by their favorite teamise lected at random from that team s scheduleiand determines its new vote based entirely on the out come of that game preferring but not absolutely certain to go with the winner 1n the simplest definition of this process the probability p of choosing the winner is the same for all voters and games played with p gt 12 be cause on average the winner should be the better team and p lt 1 to allow a simulated monkey to argue that the losing team is still the better team due perhaps to weather officiating injuries luck or the phase of the moon The behavior of each virtual monkey is driven by a simplified version of the but my team beat your team arguments one commonly hears For example much of the 2001 BCS controversy centered on the fact that BCS 2 Nebraska lost to BCS 3 Colorado and the 2000 BCS controversy was driven by BCS 4 Washington s de feat of BCS 3 Miami and Miami s win over BCS 2 Florida State The synthetic monkeys act as independent ran dom walkers on a graph with biased edges be tween teams that played headtohead games changing teams along an edge based on the win loss outcome of that game The random behavior of these individual voters is of course grossly simplistic lndeed under the specified range of p a given voter will never reach a certain conclusion about which team is the best rather it will forever change its allegiance from one team to another ul timately traversing the entire graph In practice however the macroscopic total of votes cast for each team by an aggregate of randomwalking vot ers quickly reaches a statistically steady ranking of the top teams according to the quality of their sea sons We propose this model on the strength of its simple interpretation of random walkers as a rea sonable way to rank the top college football teams or at least as reasonable as other available meth ods given the scarcity of games played relative to VOLUME 51 NUMBER 8 the number of teamsibut we warn that this naive random walker ranking does a poor job rank ing college basketball where the margin of victory and established homecourt advantage are signif icant 19 This simple scheme has the advantage of having only one explicit precisely defined parameter with a meaningful interpretation easily understood at the level of singlevoter behavior We have investigated the historical performance and mathematical properties of this ranking system elsewhere 3 4 At p close to 12 the ranking is dominated by an RPllike rank ing in terms ofa team s record opponent s records etc with little regard for individual game outcomes For p near 1 on the other hand the rank ing depends strongly on which teams won and lost against which other teams Our initial questions can now be rephrased play fully as follows Can a bunch of monkeys rank football teams as well as the systems currently in use Now that we have crossed over into the Year of the Monkey in the Chinese calender and the BCS has recently proposed changes to their nonBCS rules it seems reasonable to askwhether the mon keys can clarify the effects of these planned changes Impact of Proposed Changes on NonBCS Schools The complete details of the new agreement have not yet been released but indications are that the proposed rules would have given four atlarge BCS bids to nonBCS schools over the past six years 13 Based on the BCS Standings the best guesses at those four teams are 1998 Tulane 110 BCS 10 poll average 10 1999 Marshall 120 BCS 12 poll average 11 2000 TCU 101 BCS 14 poll av erage 145 and 2003 Miami of Ohio121 BCS1 1 poll average 145 However there are also indica tions that only nonBCS teams finishing in the BCS top 12 would automatically get bids 15 and each of the four schools above would have had to be given one of the atlarge bids over at least one team ahead of them in the BCS Standings 8 Given the perception that the polls unfairly favor BCS schools it is worth noting the contrary evidence from six seasons of BCS Standings In addition to the four schools listed above other notable non BCS campaigns were conducted this past season by Boise State 121 BCS 17 poll average 17 and TCU 111 BCS 18 poll average 19 Five of these six schools earned roughly the same ranking in the BCS standings and the polls The only significant ex ception was 2003 Miami of Ohio averaging 6th in the official BCS computer algorithms but only 145 in the polls While the new rules might indeed give BCS bowl bids to all nonBCS schools who finish in the top 12 it is worth inquiring how close nonBCS schools SEPTEMBER 2004 Okllhaml 12 1 Lsu 12 1 v soullnmcnl111 Mllml OH 124 mum 1 BoIIaSIUzt v 10 10 05 06 07 00 09 1 Figure l Randomwalking monkey rankings of selected teams for 2003 may have come to this or to a top 6 ranking that would have guaranteed them a bid during the past six years In particular 2003 was the first time in the BCS era that there were no undefeated teams remaining prior to the bowl games Given that there were six oneloss teams and no undefeateds what would have happened if one or more of the three nonBCS teams had instead gone undefeated While it is impossible to guess how the polls would have behaved and we are unable to reproduce most of the official computer rankings we can instead compute the resulting randomwalking monkey rankings for different values of the bias parame ter p As a baseline Figure 1 plots the endofsea son prebowlgame rankings of each of the six oneloss teams plus Michigan from the true 2003 season scaled logarithmically so that the top 2 top 6 and top 12 teams are clearly designated Now consider what would have transpired had Miami of Ohio TCU and Boise State all gone un defeated Figure 2 shows the resulting rankings of the same teams as Figure 1 under these alternative outcomes In the limit p a 1 going undefeated trumps any of the oneloss teams so each of these mythically undefeated schools ranks in the top 3 in this limit For TCU and Boise State however their range ofp in the top 6 is quite narrow If the new rules require only a top 12 finish for a non BCS team then the situation looks much brighter for an undefeated TCU which earned monkey rank ings in the top 11 at all p values However ac cording to the scenario plotted in Figure 2 an un defeated Boise State s claim on a BCS bid remains tenuous even under the proposed changes lndeed even had Boise State been the only undefeated team last season not shown the monkeys would have left them out of the top 10 and behind Miami of Ohio for p S 086 At the other extreme oneloss Miami of Ohio already has a legitimate claim to the top 12 according to both the monkeys and the real BCS Standings Note in particular the exalted ranking NOTICES OF THE AMS 1 oz Olanom 121 Lsu 121 Samm cal 111 Mimi OH 130 39rcu 120 BoiIISIUS o Midligln 1 o 10 05 06 07 08 09 1 Figure 2 Randomwalking monkey rankings of selected teams for an alternate universe 2003 in which the three nonBCS oneloss teams instead went undefeated the monkeys would have given Miami of Ohio had they won their season opener against Iowa their only loss in the actual 2003 season According to the monkeys they may have even had a reasonable argument to be placed in the championship game had they gone undefeated It was bad enough not being able to fit three teams onto the field for the BCS National Championship game but we might have been one Miami of Ohio victory over lowa away from wanting to crowd four squads into the mix As an example of how the effects of games prop agate into the rankings of other teams we also in clude Michigan s ranking in both figures even though their outcomes were not changed in the cal culations that produced the two plots Neverthe less because Michigan is a nextnearest neighbor of Miami of Ohio in the network both teams lost to Iowa in 2003 changing the outcome of the Iowa V Miami of Ohio game unsurprisingly affects Michigan s ranking detrimentally To conclude this section we stress that the above discussion is purely hypothetical as the monkeys provide only a standin for our inability to compute true BCS Standings under alternative outcomes The Problem at the Top and a Possible Solution While we focused above on nonBCS schools and the recent changes that improve their chances of playing in a BCS bowl game the larger BCS con troversy for many fans is the recurring inability of the BCS to generate a championship game between conclusive top 2 teams Each of the past four seasons the two polls agreed on the top two teams prior to the bowl games In three of those seasons however the top two spots in the BCS Standings included only one of the teams selected by the polls In 2000 and 2001 the 2 team in the polls ended up on the short end of the BCS stick whereas NOTICES OF THE AMS in 2003 it was USC the 1 team in both polls on the outside looking in Although it is easy to blame this situation on the computer rankings the true problem as we see it lies in the BCS formula of polls computers sched ule strength losses and quality wins Simply the polls and computers already account for schedule strength and quality wins or else the three non BCS oneloss teams Miami of Ohio TCU and Boise State would have placed in the top 6 in the 2003 BCS Standings Adding these factors again after the polls and computer rankings are determined dis astrously doublecounts these effects adversely degrading confidence in the BCS selections for the National Championship and the other BCS bowls One of the presumed motivations for including separate factors for schedule strength and quality wins was to reduce the assumed bias of the polls towards traditional football powers However as dis cussed above the top nonBCS teams over the past six years were ranked similarly in the polls and com puters Therefore one might rightly worry that the quality wins and schedule strength factors are mak ing it harder for nonBCS schools to do well in the standings as their schedules are typically ranked significantly lower and they have few opportunities for socalled quality wins USC was on the losing end of this double counting in 2003 having finished the regular sea son 1 in both polls and averaged 267 on the com puters LSU was 2 in both polls and averaged 193 on the computers and Oklahoma was 3 in both polls and averaged 117 on the computers One of the official computer systems even ranked nonBCS Miami of Ohio ahead of USC However al though the computers ranked Oklahoma ahead of the other teams it was Oklahoma s 11th place schedule strength and 705 quality win bonus for beating Texas that combined to give it an addi tional 155 BCSpoints edge compared to USC s 3 7th place schedule standings available from 18 With six oneloss teams in Division lA the rank ing algorithms predominantly favored Oklahoma be cause of its relatively difficult schedule and its vic tory over Texas Without those effects being included again in separate quality wins and sched ule strength factors a straightup averaging of the polls and the computers would rank USC first 1267367 LSU second 2193393 and Ok lahoma third 3117417 A reasonable kneejerk reaction to this proposal would be to reassert that schedule strength number of losses and socalled quality wins should matter Our point is that they are already incorporated in such a simple averaging scheme as the polls and the com puters necessarily consider such factors to produce reasonable rankings To explicitly add further BCS points for each of these considerations gives them VOLUME 51 NUMBER 8 more weight than the collective wisdom of the polls and computer rankings believe they should have Whatever solution is ultimately adopted we strongly advocate that modifications to the BCS re move such doublecounting and ideally provide a system that is more open to the community That the doublecounting problem is not widely appre ciated further supports our opinion that the BCS system needs to be more transparent The recently announced addition of a fifth BCS bowl does not address this problem2 College football fans should not have to accept computer rankings without a minimal explanation of their determining ingredients not only so that they have more confidence in these algorithms but also to open debate about what factors should be included and how much they should be weighted For example there is certainly a need to discuss how much losing a game late in the sea son or in a conference championship game as Oklahoma did in 2003 should matter compared to an earlier loss Even before the endofseason controversy in 2003 a survey conducted by New Media Strategies indicated that 75 percent of college football fans thought that the BCS system should be scrapped entirely 9 That number presumably increased after the new round of controversy Changes that lead to greater transparency and a simplified weighted averaging of the polls and computers are the only way anything resembling the current BCS system can maintain popular support Epilogue New information appearing after the original writ ing of this review claims that the doublecounting factors in the BCS formula may be scrapped in favor of an average of polls and computer rankings 2 We submitted advance copies of this article to BCS decision makers but we have no knowledge that any changes resulted directly from our input It was announced on July 15th that the new BCS Standings wil be determined by equally weighting the AP poll the USA TodayESPN Coaches poll and an average over the computer systems that is 23 polls 13 computers One might worry that this weighting effectively relegates the computers to tie breaking a posteriori yielding National Cham pionship pairings in agreement with the polls over each of the past six seasons and placing any pos sibility of a midmajor school getting a BCS bid al most wholly in the hands of poll voters Never theless such a change clearly simplifies the BCS Standings which we view as positive 2However it appears that even more recent changes may simplify the BCS formula by removing the double count ing see the Epilogue SEPTEMBER 2004 References 1R BILLINGSLEY College football research center http cfr39c com as downloaded in January 2004 2 T BARNHART More changes ahead for BCS The At lantafournalConstitution May 12 2004 3 T CALLAGHAN M A PORTER AND P J MUCHA Random walker ranking for NCAA Division lA football sub mitted 2003 Eprint arX39i v0r39gphys i cs0310148 4 T CALLAGHAN M A PORTER AND P J MUCHA How well can monkeys rank fOOtball teams http www math gatech eduN muchaBCS as downloaded in Janu ary 2004 S W N COLLEY COlleyMatrix COlley s bias free matrix rankings httpwwwco39l39leyr39ank39ingscom as downloaded in January 2004 6 G R CONNOR AND C P GRANT An extension Of Zermelo s model for ranking by paired comparisons European J Appl Math 11 2000 2257247 7 M FISH Stuff happens SIcom December 9 2003 8 P FIUTAK Cavalcade Of whimsy Fifth BCS game is silly Co 39I egeFootba39l 39INews com March 4 2004 9 T HYLAND Survey College football fans say BCS has tO go Baltimore Business December 3 2003 10 J P KEENER The PerronFrObenius theorem and the ranking Of football teams SIAMRev 3 S 1993 80793 1 1 B KIRLIN How tO fake having your own math formula rating system tO rank college football teams http wwwcaew isceduNdwi39lsonr39sfc h 39i storyk39i r39l 39i nfake htm39l as downloaded in 2003 12 T LAYDEN Pleading the fifth BCS powers don t go far enough in latest reform SIcom March 5 2004 13 M LONG BCS agrees tO add fifth bowl game AssO ciated Press March 1 2004 14 M LUPICA If this all adds up subtract C from BCS tO name bowl New York Daily News December 8 2003 15 S MANDEL Presidental seal Different decisionmak ers send BCS in unlikely direction SI com March 4 2004 16 J MARTINICH College football rankings DO the com puters know best Interfaces 32 2002 85794 17 K MASSEY Massey ratings htt p www masseyr at39ingscom as downloaded in January 2004 18 THE NATIONAL FOOTBALL FOUNDATION The bowl championship serie s overview h tt p www footba39l 39I foundat39i on combcs php as downloaded in January 2004 19 J SOKOL personal communication 20 W SUGGS Presidents see progress in Opening up big money bowls tO more colleges Chronicle oing herEa 50 November 28 2003 A38 2 1 WHITAKER SOFTWARE APPLICATIONS Football rankulator httpmmwvfootba3939 ranku39l ator39com as down loaded in 2003 22 D L WILSON American college fOOtballirankings httpwww caew i sc eduN dw i 39Isonr39sfc rate as downloaded in January 2004 23 P R WOLEE 2003 college football httpwww bo39l M a eduN prwo39l fecfootba39l39Ir39at39i ngs htmas downloaded in January 2004 NOTICES OF THE AMS
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