A model-based approach to football strategy.
|May 1, 2004|
In recent years proposals have been made to begin overtime with an auction, in which teams bid for starting field position, rather than with the traditional coin toss and kickoff. In this article we will summarize some of the proposals, characterize their equilibria, and show that they are all essentially equivalent.
The motivation for the proposals to alter the format of overtime is that the coin-toss winner seems to have a substantial advantage. Since 1994, when kickoffs were moved to the 30-yard line, the team receiving the kickoff has won 59% of the games that were decided in overtime. This can be thought of as an estimate of the advantage of winning the coin toss in average wind conditions. Indoors, or outdoors in light wind, the advantage will be larger.
A modest change in the spot of the overtime kickoff wouldn't eliminate this advantage. In fact, because of the touchback rule and the relative lack of distance control on place kicks, the receiving team might retain an advantage regardless of the spot of the kickoff. Replacing the kickoff with a free kick from near midfield might do the trick, but this would be as drastic a rule change as an auction. And of course, it wouldn't take the wind conditions into account.
An alternative is to eliminate the overtime kickoff, and have one of the teams -- selected through a coin flip -- begin on offense at a pre-specified yard line. The difficulty with this approach is that one has to determine the starting point that is fair. One way to try to do so would be to look at a large number of NFL games in which, early in the game, a team begins a drive deep in its own end. For each such situation, we would record the yard line at which the drive began, and also which team was the next to score. (In essence, we would be making believe that the start of the drive is the beginning of overtime.) With enough data, we could estimate how the probability of being the next team to score varies as a function of the starting point for the drive.
This kind of analysis would certainly not be definitive, because overtime strategy is different. For example, early in a game a team might go for it on fourth down rather than kick a chip-shot field goal, but would never do so in overtime. In addition, the analysis just described doesn't take into account the factors that are specific to a particular game, such as the wind.
Another way to estimate the fair starting yard line is to use a model. Our own overtime model suggests that with evenly matched teams and no wind, the fair starting point would be just inside the 15 yard line. (Note: In the steady-state model of David Romer, the 15 yard line is a "zero-value state." This has led some people to conclude that the 15 yard line is a fair place for a team to begin on offense in overtime. However, a simple example shows that this is not really an implication of Romer's results.)
The auction proposals also eliminate the kickoff. However, rather than pre-specify the spot at which one of the teams takes possession, the auctions have the teams compete for possession. We will look at four approaches. The first is identical, and the second almost identical, to methods Andrew Quanbeck proposed to the NFL.
Method 1 ("sealed bids"): Each coach writes down on paper the yard line at which he is willing to begin on offense. The team with the lower bid begins on offense at that yardline. For example, if Team A bids 12 and Team B bids 10.5, then overtime starts with Team B on offense at the 10.5 yard line. To avoid ties, Quanbeck requires one of the coaches to bid an integer yard line (such as 10), and requires the other coach to bid a half-yard line (such as 10.5).
Method 2 ("Dutch auction"): The referee begins counting in half-yard increments ("one-half, one, one-and-a-half, two,...") and continues until stopped by the coach of one of the teams. The team whose coach stops the count begins the overtime on offense, at the yard line (or half-yard line) corresponding to the last number the referee called out. For example, if neither coach has spoken until the referee's count reaches 10.5, at which point the coach of team B announces "stop," then overtime begins with Team B on offense at the 10.5 yard line. As in Method 1, to avoid ties, we allow the coach of Team A to stop the count only at integer yard lines, and allow the coach of Team B to stop the count only at half-yard lines.
Method 3 ("open outcry"): Each coach can bid as often as he likes. A bid must be lower than any previous bid, and must be an integer yard line or a half-yard line. The auction continues until there are no more bids, at which point the last bidder's team starts on offense at the yard line (or half-yard line) of his last bid.
Methods 1 and 2 are superficially different, but they are "strategically equivalent." By this we mean that these two methods present the coaches with identical decision problems, and therefore ought to yield identical outcomes. To see this, notice that in Method 2, a coach's strategy is simply a choice of a yard line at which to stop the count, if the opposing coach hasn't already done so. Each coach might as well write that yard line down on paper before the referee begins counting. But then there is no need to have the referee count: He can just look at the two slips of paper, reducing Method 2 to Method 1. One's preference between these two auctions must therefore derive from aesthetics rather than substance.
Our own favorite method is analogous to the familiar way in which two people can divide a piece of cake fairly: One cuts, and the other chooses.
Method 4 ("cut-and-choose"): There is a coin flip. The loser chooses the yard line at which overtime will begin, and then the winner chooses whether to be on offense or defense. As an example suppose that Team B loses the coin toss and selects the 10.5 yard line, and then Team A chooses to defend. The overtime will begin with Team B on offense at the 10.5 yard line.
We like Method 4 because it features a coin flip followed by a decision by each team -- just as in the existing format. All that's missing is the kickoff. So perhaps this method would be more palatable to the NFL.
We will determine the outcomes of these four methods in a Nash equilibrium. A Nash Equilibrium is a pair of strategies, one for each team, with the following properties: Team A's strategy is its best possible strategy, relative to the strategy chosen by Team B; and similarly, the Team B's strategy is its best possible strategy, relative to the strategy chosen by Team A. So, in an equilibrium, each coach knows the other's strategy, but neither coach has any incentive to change his own strategy.
The situation will be roughly as depicted in the graph at left. The function f is upward sloping, since we are more likely to win the game if we start our possession farther upfield. Similarly, g is downward sloping, since if the opponents begin the overtime on offense, we are less likely to win the farther upfield they start. We will assume that the teams agree on the various probabilities, in other words they agree on the functions f and g.
As shown on the graph, the functions f and g will intersect at some yard line y*. In equilibrium, under Method 4 ("cut-and-choose") the overtime will begin at y*. To see this, suppose we lose the coin toss. If we select y*, then our probability of winning the game will be
If instead we select a yard line y that is larger than y*, the opponents will choose to start on offense, and our probability of winning will be
Next consider the Dutch auction (Method 2). Notice that at any yard line (or half-yard line) that comes at least a half yard before y*, it can never make sense to stop the count: You are better off remaining silent regardless of what the oposing coach intends to do at his next turn. In addition, in equilibrium neither coach will wait until some yard line significantly past y*. We prove this by backward induction. Notice that if the referee's count somehow reaches 99 (i.e., the 1 yard line at the other end), the coach of Team A will certainly stop the count and take possession; there can be no further benefit to waiting. If the count reaches 98.5, Team B's coach can either stop the count and take possession at that point, or remain silent, knowing that Team A will take possession at the 99. So of course, if the count reaches 98.5, Team B will take possession. Knowing this, Team A will stop the count if it reaches 98, and so on. This argument can be repeated for every yard line or half-yard line greater than or equal to y*, and shows that the referee's count will be stopped before it gets a half yard past y*.
These arguments taken together show that in a Dutch auction, the team that takes possession will do so within a half yard of y*. (The exact equilibrium outcome of the Dutch auction depends on additional aspects of the functions f and g; details are in the mathematical appendix.) By strategic equivalence, the same result holds for the sealed-bid auction (Method 1).
For the open outcry auction (Method 3), the argument is similar. It could never make sense to let the opponents win the auction at a yard line y that exceeds y*, if we could make a lower bid that still exceeds y*. And although it could in principle make sense to bid below y* (if you think the opponents will subsequently bid even lower), this can't happen in equilibrium. To see this, notice that neither coach would ever bid the one-half yard line: his team would have to start there on offense, since the opposing coach couldn't outbid him even if he wanted to. But knowing this, neither team will bid the one yard line, and so on. These arguments show that for the open outcry auction, the team that takes possession will do so within a half yard of y*.
In summary, with cut-and-choose the overtime will start at y*, the yard line at which each both teams are indifferent between being on offense or defense. (This result actually requires only that both coaches be rational, and that the coach of the coin-toss loser believes that the other coach is rational.) Under any of the auctions (Methods 1, 2, and 3), in equilibrium overtime will begin within a half yard of y*.
All of these methods are fair in the limited sense that, at the starting field position, neither the team on offense nor the team on defense would prefer to switch. These methods certainly don't make the teams equally likely to win, nor should they. The better team should be more likely to win. (As the functions f and g are drawn in the graph shown above, we have about a 52% chance of winning the game.)
The current format for overtime in the NFL gives far too big an advantage to the winner of the coin toss. Since there are fairer alternatives that involve minimal modifications to the existing rules, the NFL should make a change.
Copyright © 2004 by William S. Krasker