In an National Football League overtime, a coin is tossed to determine which team will receive the kick off. In the sudden death format starting on offense has a significantly higher chance of winning. This makes coin tossing one of the most climatic moments and immediately confers an advantage to one team. Proposals to improve the ex post fairness by guaranteeing each team one possession reduce the efficiency and excitement of the sudden-death format. We propose a simple approach motivated by economics: the coaches of opposing teams bid on the yardage from its end zone at which his team would begin offense, with the low bidder winning the offense right. For instance, if coaches of teams A and B bid 18 and 21 yards, respectively, team A would win and begin its offense at 18 from its end zone. This note analyzes the equilibrium outcome of such an auction and discusses its properties

Che, Yeon-Koo

Hendershott, Terry

Columbia University Academic Commons

 Columbia University  Department of Economics Discussion Paper Series           Auctioning the NFL Overtime Possession  Yeon-Koo Che Terry Hendershott           Discussion Paper No.: 0506-25        Department of Economics Columbia University New York, NY 10027 May 2006  Auctioning the NFL Overtime PossessionYeon-Koo Che∗ Terry Hendershott†May 8, 2006Abstract: In an National Football League overtime, a coin is tossed to determinewhich team will receive the kick off. In the sudden death format starting on offensehas a significantly higher chance of winning. This makes coin tossing one of the mostclimatic moments and immediately confers an advantage to one team. Proposals toimprove the ex post fairness by guaranteeing each team one possession reduce theefficiency and excitement of the sudden-death format. We propose a simple approachmotivated by economics: the coaches of opposing teams bid on the yardage from its endzone at which his team would begin offense, with the low bidder winning the offenseright. For instance, if coaches of teams A and B bid 18 and 21 yards, respectively,team A would win and begin its offense at 18 from its end zone. This note analyzesthe equilibrium outcome of such an auction and discusses its properties.∗Department of Economics, Columbia University; email: yc2271@columbia.edu.†Haas School of Business, UC-Berkeley; email: hender@haas.berkeley.edu.11 IntroductionIn the National Football League (NFL), games ending in a tie are determined by sudden-deathovertime where the first team to score wins. Sudden death is an efficient means to decide a gamewhich is violent and exhausting. It is certainly preferable to the parody of the game used to breakties in college football.1 However, the sudden death nature confers a significant advantage on theteam who has the first possession. While the outcome of a coin flip to determine first possessionis ex ante fair, immediately after the toss it is no longer fair because the winning team has asignificant chance of scoring on its first possession. NFL Commissioner Paul Tagliabue has saidthat “There has been a trend in the last seven or eight seasons that the team winning the toss inovertime wins the game. That advantage of receiving the ball first is becoming unbalanced.” (NewYork Times, 2003a).2Suggested reform often includes guaranteeing both teams at least one possession (New YorkTimes, 2003a, 2003c). Unfortunately, this conflicts with the efficiency of using sudden death.It also only postpones the fairness issue. If the teams remain tied after their initial overtimepossession, the original problem reappears. Thus, both teams must be be guaranteed an equalnumber of possession indefinitely, as in the college overtime system. Economic theory with itsfocus on the fair and efficient allocation of scarce resources suggests a natural solution.We propose a system that is fair even after the initial possession is awarded and maintains thethe efficiency of sudden death. To begin overtime, the teams bid on the starting yard line of thefirst possession. The team that bids closest to its own goal line gets possession and starts at itsbid. Then sudden death commences with the team scoring first winning. This system is more expost fair because even after the first possession is decided, teams are indifferent to whether or notthey start with the ball. Below we construct a simple model to examine the equilibrium propertiesof our auction.2 Equilibrium of the auction game and its propertiesSuppose teams 1 and 2 tie in regulation, turning the game into an overtime. The continuationoutcome at the beginning of overtime is characterized by the probability that team 1 would win.Let f(x) and g(y) denote team 1’s probability of winning when it begins offense at x and whenteam 2 begins its offense at y, respectively, where yardage is measured from one’s own end zone.Obviously, 1−g(y) and 1−f(x) will then represent team 2’s probability of winning when it begins1When a college football game goes to overtime, each team is given one possession from its opponent’s twenty-fiveyard line. The leader after those possessions, if there is one, is declared the winner. If the teams remain tied, thiscontinues, switching the order of possessions for each overtime, until one team leads the other at the end of theovertime. To prevent too many overtimes/possessions, extra points do not count from the 3rd overtime on, makingit necessary for teams scoring touchdowns to attempt a two-point conversion.2Of the first 355 Nation Football League overtime games 100 were decided by the team winning the coin tossand then winning the game on its first possession (New York Times, 2003c). In 2002 of 25 overtime games, 10 weredecided on the first possession (New York Times, 2003b) with 17 games eventually won by the team winning thecoin flip.2offense at y and when team 1 begins its offense at x, respectively.3 These probabilities are assumedto be a common knowledge for the coaches. A coach representing its team is assumed to maximizethe probability of his team winning. We look for a Nash equilibrium of this game. It is reasonableto assume the following properties for f(·) and g(·):•Monotonicityf(·) is nondecreasing and g(·) is nonincreasing.•Continuity: f(·) and g(·) are continuous.• Interiority: f(0) ≤ g(0) and f(100) ≥ g(100).Figure 1 portrays f(·) and g(·) satisfying these properties.. ............. ..... .... ... ... ... .... ..... ....... .... .... ..... ..... .... ... .. .. ... .... ..... ....... ..... .... ... .. ........ ...... ..... ............................................. ... .... ...... ...... .... ... .. ....... ..... .... .... ... .. ... .... ..... ...... ....... ............................................... .......................................................... ....... ..................................... ....... .. .. ... ......... ............ .... ... ....... ..... ...... ....... ..... .... ..... ..... .... ... .............. .......... .. ... .... ..... ...... ....... ...........................................f(x)g(x)1winningchance1000yardage from the endzonex∗Figure 1Monotonicity means that the further away from its end zone a team begins its offense, thehigher is its chance to win. Continuity is self explanatory. Interiority means these f(·) and g(·)crosses at some x ∈ [0, 1], as illustrated in the figure: A team would rather defend at the opponentteam’s end zone than begin offense at its own end zone. All three conditions are quite reasonable.Given these three conditions, the equilibrium of this bidding game occurs at the crossing of f(·)and g(·), as will be shown in the next proposition.4Proposition 1 Given the three conditions, it is an equilibrium for each team to bid x∗ such thatf(x∗) = g(x∗). Team 1 wins with probability f(x∗) in any equilibrium.Proof. Given the three conditions, there exists x∗ such that f(x∗) = g(x∗). If coach 1 picks x3To analyze the case where one team kicks off to the other, x and y can be where the teams kick off from andf(x) and g(y) the associated expected probabilities of winning. The current NFL rule is that the kick off is fromthe 30 yard line. An similar mechanism can be constructed where the kick off position could be auctioned off ratherthan starting field position.4If interiority does not hold, equilibrium is at zero or 100, as one can clearly see from the logic of the argumentused in the proposition.3against coach 2 bidding x∗, the former’s payoff ispi1(x, x∗) =f(x) if x < x∗,f(x∗)+g(x∗)2 if x = x∗,g(x∗) if x > x∗.Since f(x∗)+g(x∗)2 = g(x∗) = f(x∗), pi1(x, x∗) is nondecreasing for any x ≤ x∗ and stays constantfor x > x∗. (In Figure 1, this payoff coincides with f(x) for x ≤ x∗ and with the dotted linefor x ≥ x∗.) So, the payoff is maximized at x = x∗. Hence, there is no profitable deviation forcoach 1. A symmetric argument works for coach 2, using the fact that 1− f(x∗) = 1− g(x∗). Theuniqueness in the probability of winning follows from the fact that the game is a two-person zerosum game.The equilibrium outcome has a couple of desirable properties, compared with the current ruleof selecting a team at random to kick off from its 30 yards. Indifference to the Assignment: Because the assignment of the offense right is exogenous to theperformance of the teams, arguably the particular assignment should not matter for the fortuneof each time. The current rule does not guarantee this. Auctioning ensures that the assignmentdoes not matter, for teams are indifferent to it in equilibrium. Fairness of the Assignment: It is not easy to delineate the precise requirements of “fairness” inan assignment rule when teams are asymmetric. After all, the entertainment value of a game maybe enhanced if a stronger team is handicapped to a certain extent.5 Even if a stronger team ishandicapped, however, it seems desirable for the handicapping not to be so extreme that a strongerteam will more likely lose after the assignment. This minimal notion of fairness seems not onlyuncontroversial but it also implies the obvious desideratum: teams with the equal capabilities musthave the equal chance of winning no matter which team is assigned the offense right.To describe our fairness notion formally, say team 1 is more capable than team 2 if f(x) ≥1− g(x) for all x, and strictly so if the inequality holds strictly. This condition simply means thatteam 1 is more likely to win than team 2, when starting its offense at the same offense yardageor when facing an offense by its opponent team at the same yardage. In the same vein, say twoteams are equally capable if f(x) = 1− g(x) for all x.It is easy to see that the current rule is unlikely to satisfy our fairness notion. For instance,even if team 1 is more capable than team 2, it is possible that, if team 2 is selected (by randomcoin tossing) to start on offense (i.e., receive a kickoff), team 1 has more than half chance oflosing. Two equally capable teams will not have the equal chance of winning after the assignment,except by a mere coincidence. Clearly, the equally capable teams have the same chance of winningprior to coin tossing, but what matters is the winning chance after the coin tossing, which is notnecessarily a half even when the teams are equally capable.It is easy to see that auctioning the offense right will guarantee the notion of fairness.5While handicapping may increase excitement it is rarely used in the sporting contest itself, presumably forfairness and incentive reasons. Handicapping to increase competitive parity occurs outside of the contests throughscheduling, draft rules, salary limits, or revenue sharing.4Corollary 1 Given the auctioning of the offense right, a (strictly) more capable team is (strictly)more likely to win, and equally capable teams have the equal chance of winning at the start of anovertime play in equilibrium.Proof. Let x∗ be an equilibrium bid. Without any loss, suppose team 1 is more capable thanteam 2. Then, f(x∗) ≥ 1 − g(x∗). By Proposition 1, f(x∗) = g(x∗). Combining the two facts,we have f(x∗) ≥ 12 . The inequality is strict if team 1 is strictly more capable than team 2, andbecomes an equality if two teams are equally capable.3 ConclusionHow to break ties arises in all sporting endeavors. Sports like baseball and basketball that havesufficient volatility of scoring relative to how physically draining they are can easily break tieswithout departing from the normal game process. Football, soccer, hockey, and tennis all even-tually adopt some bastardization of themselves to finally determine a winner. Tie breaking rulestrade off efficiency (quick ending) against making the tie breaking contest like the game itself.Because fairness conflicts with efficiency college and professional football have taken divergentapproaches to overtime. We propose auctioning off where to begin offense (or kick off) to eliminatefairness issues while ensuring efficiency. This tie breaking system may also increase excitement.A relatively better offensive team would bid aggressively, valuing first possession over better fieldposition. Similarly, a better defensive team would bid less aggressively, valuing the better fieldposition on the second possession if they initially stop their opponents.When the Wall Street Journal ran an online forum on this issue (Wall Street Journal, 2003),Tom Donahoe, then president and general manager of the Buffalo Bills, wrote “[w]e don’t like thecurrent system. It just seems that too much depends on the coin flip – who wins it and who losesit.” Donahoe wrote of the auction proposal being “way too revolutionary for the National FootballLeague. It sounds more like a videogame to me. I couldn’t see that happening.” However, theNFL’s steady increase of coaching decisions to enhance fairness, e.g., the challenge rule for instantreplay reviews of plays, are similar in spirit.References[1] New York Times, 2003a, N.F.L. Ponders Changing How Overtime Is Played, January 25,Section D, page 5.[2] New York Times, 2003b, N.F.L. Decides Not to Make Changes, for Now, March 27, Section S,page 5.[3] New York Times, 2003c, Heads or Tails Is No Way To Determine Overtime, November 4,Section D, page 1.[4] Wall Street Journal Online, 2003, Should the Outcome of a Coin Flip Mean So Much in NFLOvertime?, December 22.5

Auctioning the NFL overtime possession

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Auctioning the NFL overtime possession

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