Draft: Do Not Quote Without Author’s Permission
Exploring the Incentive Structure of NASCAR Events
John R. Crooker Department of Economics and Finance University of Central Missouri Warrensburg, MO 64093-5074 (660) 543-8601
[email protected] Christopher D. Azevedo Department of Economics and Finance University of Central Missouri Warrensburg, MO 64093-5074 (660) 543-4219
[email protected]
May 16, 2006
Introduction In many professional sport tournaments, the prizes are highly disproportionate with the top finishers earning multiples of the next best finisher’s monetary award. An empirical exception seems to be the NASCAR monetary award structure. NASCAR awards are largely uniform across race participants (Leeds and von Allmen, 2005), a stylized fact that Leeds and von Allmen refer to the “NASCAR exception.” NASCAR is different from many other sports in that competitors can engage in activities that can hinder other competitors from succeeding, and a catastrophic result may be experienced when the competitors are especially aggressive. Von Allmen (2001) argues that the empirical uniformity of payments across race finishers may reflect the desire of NASCAR administrators to ensure that drivers do not engage in “win at all costs” behavior to take home a top prize. In coming to this conclusion, he draws upon research by Lazear (1989) that shows that when sabotage is possible between workers a flat wage profile may be more efficient than a standard tournament theory rank-order payoff, as well as research by Becker and Huselid (1992) that explores the link between driver effort and the distribution of monetary awards. The purpose of our paper is to theoretically explore how award structures may impact the strategic behavior of drivers in a NASCAR race. We use game theoretic arguments to consider the likely strategic behavior of drivers. We begin by developing a stylized and simplistic formulation to identify some basic incentives facing drivers, and progress to incorporate more complicated NASCAR scoring procedures and contingency awards. In our analysis, we do not find it likely that the uniform payment scheme is designed to discourage risky driver behavior. Instead, we suggest NASCAR actively encourages a high degree of contention between drivers. Likely, this is motivated to enhance fan interest. We argue that another possibility suggested in
von Allmen (2001) that seems more likely is that the awards payment scheme is due to concerns for race parity, full field races, and perhaps the market power of track owners. It is important to point out that our results do not imply that NASCAR officials are uninterested in driver safety. We simply are stating that the monetary award structure is not the instrument used by NASCAR to increase driver safety. We begin by developing a theoretical model to explore the incentives that NASCAR drivers face. We then test the implications of the model using data drawn from NASCAR races run at the Daytona International Speedway, and finish by summarizing the results.
Explaining the NASCAR exception We begin with a simple theoretical model of the incentives faced by two drivers competing for the win on the final lap of a race. For simplicity, we will suppose that the players each have two available strategies: “Aggressive” and “Passive.” In the “Aggressive” strategy, the driver undertakes aggressive tactics to earn a victory. If the driver is currently the race leader, which we refer to as the incumbent, these aggressive tactics include things such as using the automobile to serve as a blocker or deviating from the preferred line around the track to prevent the challenger from making a run. In the “Passive” strategy, the driver behaves passively to ensure a high finish. The pay-off matrix is shown below with the row player being the challenger, or driver currently running in second place, and the column player being the incumbent, or driver currently running in first place. The possible rewards are designated as A 1 (first place monetary award), A2 (second place monetary award), and D (catastrophe). The return to the column player is given above the diagonal in each of the cells while the return to the row player is given beneath the diagonal for each cell.
Table 1: Last-Lap Pay-Off Matrix for Top Two Drivers Incumbent Aggressive
Passive
Aggressive
D
D
A2
A1
A1
A2
Challenger Passive A2
A1
If the Incumbent driver chooses the Passive strategy, the Challenger does better by contesting the Incumbent and driving aggressively. The result is that the Challenger would overcome the Incumbent and receive the top award, A1 , while the Incumbent’s pay-out falls to A 2 . If the Challenger plays the Passive strategy while the Incumbent also plays the Passive
strategy, the drivers maintain their track position through the checkered flag. Provided that the Incumbent driver chooses the Aggressive strategy, the Challenger does better by not contesting the Incumbent and instead selecting the Passive strategy. This again produces the outcome in which the drivers maintain their track position through the race finish. If the Challenger plays the Aggressive strategy and the Incumbent is playing the Aggressive strategy, we observe the catastrophic result, D. Game theorists would recognize this pay-off matrix as being related to the classic game of chicken (Dixit and Skeath, 2004). A deviation from classic games of chicken is that the Incumbent driver possesses a weakly dominated strategy of driving Aggressively. That is, the
Incumbent driver always does at least as well by choosing the Passive strategy. The implication of this pay-off matrix design is that we produce a dominance-solvable Nash Equilibrium in pure strategies of the Incumbent choosing the Passive strategy while the Challenger chooses the Aggressive strategy. The pay-offs at this Nash Equilibrium are that the Challenger wins the race and receives A1 while the Incumbent takes second place and receives A 2 . The dominance-solvable Nash Equilibrium is not the only Nash Equilibrium in the game. In fact, the cell we arrive at when the Incumbent driver chooses to drive Aggressively while the Challenger chooses to drive Passively is also a Nash Equilibrium. In repeated games of chicken, players have an incentive to establish a reputation of selecting the equivalent of our “Aggressive” strategy in this game. For our pay-off matrix, we see the potential gain of establishing this reputation. If a Challenger is convinced the Incumbent will choose to drive Aggressively, the Challenger will choose the Passive strategy. Hence, the result of effectively establishing the reputation of an “Aggressive” last-lap driver for the Incumbent is that he will be able to secure the victory. For the moment, we will suspend discussion on the implications of playing this game in a repeated framework. This will be re-examined below in an evolutionary model. It is important to point out that these conclusions do not depend on the magnitude of the difference between A1 and A2 . All that is necessary is that A1 > A2 . As long as there is any difference between A1 and A2 there will be an incentive for drivers to choose the aggressive strategy. Von Allmen (2001) argues that the payment structure in NASCAR events appears to be designed to reduce risky driving behavior. The peculiar feature of the NASCAR payment structure is the relative equality of award amounts across finishers. That is, the first place finisher receives a far smaller portion of the total award payouts than is typical in other sport tournaments. The Sports Economics literature refers to this anomalous feature as the “NASCAR
exception to tournament awards.” Von Allmen suggests that near equal payments for all contestants lessens the return to engaging in risky driving behavior. Examining the pay-off matrix presented in table 1 above, we note that multiplying, dividing, adding or subtracting the returns by any positive constant factor does not change the incentives or Nash Equilibriums in the game. The Nash equilibria we find in the game are consistent with one driver risking the catastrophic outcome and this result is not affected by the magnitude of the pay-outs between the top two finishers. The only way to eliminate the incentive to choose the Aggressive strategy is to make A1 exactly equal to A2 . This implies that it is not possible for NASCAR planners to eliminate aggressive driving by making the payout “nearly equal.” The payouts would need to be exactly equal to achieve this result.1 An Evolutionary NASCAR Field In this section, we explore a slightly different model by assuming that rather than choosing to be aggressive or passive, drivers are simply born to race either aggressively or passively (i.e. drivers are born safe or wild). Taking the approach of evolutionary biology, we will examine the scenarios that make the safe drivers more successful and the wild drivers more successful. In particular, we will be considering the impact of the finish reward structure and how it impacts the sustainability of ingrained driver styles. Finally, we will evaluate the impact of NASCAR incentives such as bonus points for leading a lap and the number of laps lead. Let PI represent the probability that a driver assumes the incumbent status on the final lap. When matched against another safe driver, a driver that is born safe has the expected return of E S [R O = S ] = PI ⋅ A1 + (1 − PI ) ⋅ A2 = A2 + ( A1 − A2 ) ⋅ PI . If matched against a wild driver, the safe driver’s expected return is: E S [R | O = W ] = A2 . That is, regardless of whether the safe
driver is the incumbent or not, the wild driver will always win the race. Hence, a second place finish is always the outcome when a safe driver is pitted against a wild driver. A born wild driver’s expected return versus a safe opponent is: EW (R | O = S ) = A1 . Matched against a wild opponent, the wild driver’s expected return is: EW (R | O = W ) = D. That is, we observe the catastrophic outcome of a wreck. To determine the success rates of these inherent driving styles, we may analyze the driving style versus the NASCAR field. We let x represent the proportion of drivers in the NASCAR field that are inherently safe. This suggests a safe driver’s unconditional expected return is: E S (R ) = x ⋅ E S [R | O = S ] + (1 − x ) ⋅ E S [R | O = W ] = x ⋅ [A2 + PI ⋅ ( A1 − A2 )] + (1 − x ) ⋅ A2 . Similarly, a wild driver’s unconditional expected return is: EW (R ) = x ⋅ EW [R | O = S ] + (1 − x ) ⋅ EW [R | O = W ] = x ⋅ A1 + (1 − x ) ⋅ D. Our focus now becomes what NASCAR field composition will result in a wild driver experiencing greater success than safe drivers. Mathematically, this would be true when: EW (R ) > E S (R ) or x ⋅ A1 + (1 − x ) ⋅ D > x ⋅ [ A2 + PI ⋅ ( A1 − A2 )] + (1 − x ) ⋅ A2 . Re-arranging, we find a wild driver will be more successful when the population of safe drivers in the NASCAR field satisfies: x>
A2 − D . A1 − D − PI ⋅ ( A1 − A2 )
(1)
If the proportion of safe drivers in the NASCAR field satisfies the inequality in (1) above, we see that wild drivers will be more successful and would imagine wild drivers would be attracted to develop rides in the NASCAR field. This dynamic, of course, would mean safe drivers would be relatively unsuccessful and begin falling out of the NASCAR field. Hence, the proportion of safe drivers (x) would be declining. However, if x falls below the threshold identified in equation
(1), safe drivers would then become more successful and the NASCAR field would begin moving back towards a safer proportion of drivers. This suggests satisfying equation (1) with equality would result in a stable population of safe and wild drivers. Examining equation (1), we notice that if A1 = A2 the wild driver would never do better than a safe driver in any initial composition of NASCAR fields. Hence, we would expect that eventually the entire NASCAR field is comprised of safe drivers. In other words, an equal payment produces an entirely safe field of drivers. Next, we wish to examine the success of wild drivers when the reward system has regular steps. Let n represent the number of drivers, and define k ≡ A1 − A2 and n ⋅ k ≡ A1 − D. This implies the inequality in (1) becomes x >
(n − 1) ⋅ k n ⋅ k − PI ⋅ k
. The magnitude of the step cancels out of
the numerator and denominator leaving the result that aggressive drivers will be more successful when x >
n −1 . As PI → 1 , we see the stable population of safe drivers goes to 1 as well. On n − PI
the other hand, as PI → 0 the stable population of safe drivers falls. To keep our analysis very
1 simple, we may take n=2 and PI = . In this setting, we see the proportion of safe drivers that 2 2 establishes a sustainable equilibrium is: x = . Though simplistic, we do find a compelling 3 model to suggest that a stepped reward system does influence the driving field to become more risky. Notice that with this formulation of the model, the size of the step is immaterial provided the step size is uniform throughout the rewards. It does appear that we have found a very simple model to explain how a flat reward system impacts the evolution of a field of drivers. However, the NASCAR points and reward
system is more complicated than simply paying rewards solely based on a driver’s finish. Next, we consider the implications of bonus points for leading a lap and most laps lead. The Impact on the Sustainable Equilibrium When Bonus Points Are Available
We have demonstrated that an award system with steps between award amounts will result in less safe NASCAR events. No variation in award amounts by finishing order would result in an entirely safe driving field. While NASCAR award amounts are fairly uniform across finishing order, the point system does reward drivers for leading a lap, number of laps run, and leading the most laps. We now consider the implication of these point awards on our drivers’ strategic behavior. Inherently safe and wild drivers make the introduction of bonus points a perplexing initiative on the part of race administrators. This is because the bonus points simply amount to increasing the award payment steps by finishing order. The bonus points system essentially creates the same incentives as a system that increases the monetary award payout steps according to finishing order. To see this, note that the expected payoff to a safe driver facing a safe opponent take the form
(
)
E S [R O = S ] = PI ⋅ ( A1 + AM + T ⋅ AR + AL ) + (1 − PI ) ⋅ ( A2 + T ⋅ AR ) = A2* + A1* − A2* ⋅ PI , where AM is the monetary value of leading the most laps, AR is the monetary value of running a lap, T
is the total number of laps run, AL is the monetary value of leading a lap, A1* = A1 + AM + T ⋅ AR + AL and A2* = A2 + T ⋅ AR . Given the safe driver’s opponent is a wild driver, the expected return is A2* . The return to a wild driver competing against a safe opponent is A1* while the return to a wild driver competing against another wild driver is D. A wild driver will be more successful than a safe driver if:
(
[
] )
x ⋅ A1* + (1 − x ) ⋅ D > x ⋅ A2* + A1* − A2* ⋅ PI + (1 − x ) ⋅ A2* . Re-arranging, we find a wild driver will A2* − D . be more successfully when x > * A1 − D − PI ⋅ A1* − A2*
(
)
This expression is similar to the
inequality presented in equation (1). In fact, if we expand A1* and A2* , we have: x>
A2 − D + T ⋅ AR . A1 − D − PI ⋅ ( A1 − A2 ) + T ⋅ AR + (1 − PI ) ⋅ ( AM + AL )
(2)
Satisfying (2) with equality provides an equilibrium between safe and wild drivers in the NASCAR field. Notice that even if the monetary awards are equal (ie. A1 = A2 ), wild drivers will be in the NASCAR field provided AM or AL are positive and PI < 1. This suggests that even if the race administrators desired to create a safe driving field by enacting a completely uniform monetary awards structure, the existence of the bonus points system would prevent achievement of that goal. Our analysis to this point suggests that offering a perfectly flat monetary award structure with bonus points and other contingency awards is analogous to a disproportionate monetary award structure. The main implication of our model is that we would expect no difference in driver behavior confronted with either award structure. In the next section, we examine two particular NASCAR races held each season at the Daytona International Speedway. In one race, there is a highly disproportionate monetary award structure while in the second, the award structure is relatively flat. Our analysis above leads to the conclusion that the existence of bonus points and contingency awards would result in there being no difference in driving behavior between the two races, whereas the conventional wisdom is that the race with the highly disproportionate monetary award structure would be a less safe NASCAR event. The NASCAR Exception and Empirical Evidence
Recent theoretical work has suggested a possible explanation for the uniform payment structure of most NASCAR races is to keep NASCAR races safe. That is, if highly disproportionate awards are paid out, as is the case in most other sporting contests, drivers would have enhanced incentives to behave in such a way that puts the driver and others in serious danger. A problem with this suggestion, though it may be intuitively compelling, is that there is no empirical evidence of this phenomenon. It also seems to imply that NASCAR shares a common feature with professional wrestling as drivers are not engaging others as competitively as humanly possible in their sport. This finding would be quite damaging to the sport as it implies that the preeminent masters of the sport lack the competitive desire to win. To explore the empirical evidence, we analyze results from two races run annually at the Daytona International Speedway over the past 26 years: the Daytona 500, which is the biggest NASCAR race in any season, and the Pepsi 400. We have selected these races for an important reason. The Daytona 500 offers a highly disproportionate monetary award structure while the Pepsi 500 monetary award structure is relatively flat, which is consistent with the “NASCAR exception” described in the literature.2 The award pay-out by finish order for the 2005 Daytona 500 and the 2005 Pepsi 400 are illustrated in figure 1 below. The pay-outs for the 2005 Daytona 500 are indicated by the solid line, while the pay-outs for the 2005 Pepsi 400 are indicated by the short-dashed curve. In addition to the NASCAR race awards, figure 1 presents pay-outs for two Professional Golfers’ Association (PGA) tournaments. The pay-outs represented by the dotted curve are the 2005 PGA Championship, while the pay-outs represented by the long-dashed curve are the 2005 John Deere Classic awards. As the curves indicate, the Pepsi 400 has a fairly flat award structure (i.e. the NASCAR exception). However, note that the Daytona 500 is fairly typical of the
disproportionate pay-outs associated with other tournament award structures. According to the conventional wisdom, this disproportionate award structure should induce risky behavior as drivers have a larger incentive to take the top spots in the race. As noted above, it is intuitively appealing to believe that this disproportionate award structure should lead to “win at all costs” behavior. Risky driver behavior will be evidenced by the number of cars that leave a particular race due to accidents. To explore the risk-taking behavior of drivers, we examine the results of all NASCAR races at Daytona International Speedway over the past 26 years, including the number of cars that left each race due to accidents. Given the disproportionate award structure of the Daytona 500, we would expect more accidents at the Daytona 500 if the conventional wisdom is true. Separating the data into the past 26 runnings of the Daytona 500 and the past 26 runnings of the Pepsi 400 we can test for statistical differences in accidents as represented by the number of cars leaving the race due to accidents. We find that the Daytona 500 averages 5.846 accidents per running (or 0.0300 accidents per lap) with a standard deviation of 3.717 accidents. The Pepsi 400 witnessed an average of 4.692 accidents per running (or 0.0295 accidents per lap) with a standard deviation of 4.84 accidents. If we formulate a null hypothesis that there is no difference between cars leaving the races due to accidents, we fail to reject the null hypothesis at all reasonable levels of significance. In fact, the t-statistic is 0.093. This suggests there is no statistical evidence to suggest the Daytona 500, despite its disproportionate pay-out structure, has any more accidents than the Pepsi 400. We can also test variations of this hypothesis. For example, we may wish to test the hypothesis that one race has a higher occurrence of accidents than the other. If we wished to test H 0 : μ 500 ≤ (1 + θ ) ⋅ μ 400 versus H A : μ 500 > (1 + θ ) ⋅ μ 400 , figure 2 illustrates the value of the test
statistic given our observed data over a continuum of values for θ . The horizontal dotted line in figure 2 takes on the value of 1.96 at the 2.5% level of significance. This suggests that we may only reject the null hypothesis for values of θ less than -0.35. Statistically, we may only claim that the Daytona 500 has more than 0.65 times the accidents of the Pepsi 400. Clearly, the evidence indicates that there is not statistical evidence supporting the claim that the Daytona 500 has more accidents than the Pepsi 400. Alternatively, we could test H 0 : μ 500 ≥ (1 + θ ) ⋅ μ 400 versus H A : μ 500 < (1 + θ ) ⋅ μ 400 . The short dashed line in figure 2 indicates the value of -1.96. Values beneath this threshold are inconsistent with the null hypothesis at the 2.5% level of significance. As figure 2 suggests, we may safely reject this null hypothesis for values of θ in excess of 0.38. Thus, we have statistical evidence that there is no more than 38% more accident cars at the Daytona 500 compared to the Pepsi 400. Figure 2 demonstrates that regardless of the framing of the null hypothesis, any test with θ ∈ (− 0.35,0.38) would be inconclusive. Even though the Daytona 500 award structure is considerably more disproportionate than the Pepsi 400 award structure, we find no evidence of riskier behavior being engaged in by the drivers in the Daytona 500 compared to the Pepsi 400. We find the implication of this result is that the typically flat NASCAR award structure is not consistent with the notion of creating a safer race—or at least it has no empirical effect on race safety. Why the Flat NASCAR Award Structure?
Von Allmen (2001) suggests that another motivation for a flat award structure is to ensure parity across NASCAR competitors. If a steep award structure was used, drivers who enjoy greater early success will have expanded opportunities to make car investments that other drivers cannot. This implies that the drivers who enjoy early success are more likely to sustain
this success while drivers who have early setbacks will struggle. Perhaps more concerning for NASCAR planners is that drivers who fail to finish well in the early season may be forced to leave the field of drivers because of financial reasons. A flatter monetary award schedule may alleviate many of these concerns provided the award schedule can provide enough of a benefit for poor performing drivers to remain in the NASCAR field. However, as we demonstrated above, a perfectly flat monetary award structure will reduce the number of wild drivers in the field. This may make the race lead less contested throughout the race. To encourage top finishes, the bonus points provide an incentive mechanism to finish well as we demonstrated above. This implies a flat award schedule with bonus points will create a hotly contested race while ensuring NASCAR drivers will be able to remain in the field throughout the race schedule. We find this to be a much more compelling motivation for the typically flat NASCAR award structure. With small deviations in pay-outs across finishing order, even teams that do not finish well will be able to continue to compete in future NASCAR events. This ensures NASCAR will be able to maintain a competitive racing field. We also note that the disproportionate pay-out at the Daytona 500 has been increasing over time. This suggests that the NASCAR organization may desire some minimal pay-out to all teams. Once this has been achieved, the organization appears to be willing to offer a more standard disproportionate tournament payout structure to drivers. Again, this appears inconsistent with the simple desire to discourage drivers from valuing a win. Conclusions
Researchers have suggested a possible motivation for a flat award structure in NASCAR events is to enhance race safety. The theoretical models that we develop indicate that achieving
this objective would require a perfectly flat monetary award structure with no bonus points. Any deviation from this will create an incentive to engage in risky behavior. As NASCAR events do offer contingency awards and bonus points even when monetary awards are flat, we expect to observe no statistical difference between the numbers of cars leaving races with relatively steep payouts versus races with relatively flat payouts. While most NASCAR events do have a fairly uniform award structure, we have found that the Daytona 500 offers a disproportionate award structure. In our analysis, we found no evidence of drivers behaving differently when competing for the disproportionate Daytona 500 awards versus the award structure at the Pepsi 400 (which also takes place at the Daytona International Speedway). This result brings us to the conclusion that it is likely that other factors have motivated NASCAR to adopt the flat award structure at most events. A likely possibility suggested by von Allmen is the interest of maintaining a consistently large and competitive field of NASCAR drivers. As the maintenance of stock cars is expensive, disproportionate pay-outs at all racing venues would likely result in some drivers falling out of the field. The revenue generated by the Daytona 500 is sufficiently large that the award structure can be highly disproportionate while even the bottom finishers still do better than top finishers at other NASCAR race events.
Figure 1
Figure 2
Becker, Brian E. and Mark A. Huselid “The Incentive Effects of Tournament Compensation Systems” Administrative Science Quarterly 37 (1992): 336-350. Dixit and Skeath 2004 Lazear, Edward P. “Pay Equality and Industrial Politics” Journal of Political Economy (1989) vol 97(3): 561-580. Leeds and Von Allmen 2005 von Allmen, P. “Is The Reward System in NASCAR Efficient?” Journal of Sports Economics (February 2001): 62-79.
Endnotes 1
It is true that in a probabilistic formulization, reducing the difference between payouts would impact marginal decisions throughout the race. However if drivers think so cogently about probabilistic returns, it seems to reject the notion that they are acting irrationally as would be implied by engaging in a “win at all costs” strategy. 2 Results from all races run at the Daytona International Speedway are available at www.nascar.com.
