When to call on an advantageous restart option

1.1Literature review

A number of other sports also present situations in which a coin toss winner is given the option to either select an advantageous position earlier, or defer selection until a later point in the match. In the National Football League (NFL), a coin toss is performed at the beginning of the game. The coin toss winner can choose between (a) kicking off or receiving a kick at the start of the game (in which case the opponent will have the same choice at the beginning of the second half); or (b) kicking off or receiving a kick at the start of the second half (in which case the opponent has the same choice at the beginning of the game). A review of the literature was unable to find any scholarly publications examining the impact of this coin toss, although numerous articles in the popular press [Brustein (2011), Trotter (2012), Lisk (2012), Matuszewski (2014)] have discussed this topic, with Matuszewski noting that, in 2014, the team winning the coin toss chose to defer its choice 65 percent of the time. Trotter reported that, over the interval September 2010 - November 2012, the team winning the coin toss won 325 games and lost 324 games; moreover, coin-toss winning teams electing to defer won 55% of their games, while coin-toss winning teams that did not defer won 47% of their games.

In the event that an American football game ends regulation time with a tied score, a coin toss is again utilized. The structure of the overtime differs between the NFL and NCAA. Prior to the 2010 season, in the NFL, the coin toss winner could elect to either kickoff or receive a kick, with the first team to score winning the game. The rule was slightly changed beginning with the 2010 playoffs, such that if the first possession ends in a field goal, the other team is given possession and an opportunity to match (with a field goal, thus extending the game) or better (with a touchdown, thus ending the game) the field goal. These modified overtime rules were then applied to regular-season games beginning with the 2012 season. Jones (2004) found that, over the period 1974-2002, teams winning the overtime coin toss won 52% of games, with coin-toss losing teams winning 43% of the time, and 5% of games ending in a tie (note that this analysis considered only games played under the rules in which the first team to score in overtime won the game). In NCAA football, each team is given an opportunity to begin with possession of the ball on the opponent’s 25-yard line. The coin toss winner can elect to either begin the first overtime on offense or on defense. After each team has had one such possession opportunity, the overtime period ends, with the team scoring more points in this overtime period the winner. Should the score be tied at the end of this overtime period, a second overtime period is utilized, now the team that began the first overtime period on defense begins the second overtime period on offense; the process continues until a winner is determined. Rosen andWilson (2007) found that there is some advantage for the coin toss winner to elect to begin the first overtime period on defense, although there are situations in which this strategy is not preferred. Rotthoff (2012) contrasted the impact of these differing overtime structures on NFL and NCAA teams’ decision-making late in regulation time, and found that NFL teams are more likely to perform risky plays (ostensibly, in an effort to avoid overtime), whereas NCAA teams do not appear to make significantly different decisions late in a game, when contrasted with those made earlier in the game.

Soccer also uses a coin toss as part of the procedure for determining a winner in the event of a tie game, for matches in which it is necessary to determine a winner, as in tournament play. In such a situation, should the game end with a tie score, a coin toss is utilized to determine the order in which teams will kick during a penalty shoot-out. When penalty shoot-outs were first instituted in 1970, the winner of the coin toss was required to kick first; this rule was changed in June 2003 to give the winner of the coin toss the choice whether to take the first or second shoot-out kick. The teams then continue to alternate penalty kicks in the same order, until each team has had the opportunity for five kicks (or until one team gains an insurmountable advantage); the team with more successful penalty kicks is the match winner. Should the teams have an equal number of successful penalty kicks after five attempts each, they continue to alternate penalty kicks in the same order, until one team has more successful kicks in an equal number of opportunities. Apesteguia and Palacios-Huerta (2010) examined a set of 129 penalty shoot-outs from 1970 to June 2003, and found a statistically significant (p = 0.017) advantage for the team kicking first. This study also observed that, for a small sample of post-June 2003 matches, the team winning the coin toss elected to kick first in every case but one. A later study by Kocher, Lenz, and Sutter (2012) reexamined this issue, considering the same time period (1970-2003), but including a broader sample of 540 matches (including all of the matches considered by Apesteguia and Palacios-Huerta (2010)); this paper found a small, but not statisticallysignificant (p = 0.15) advantage for the team kicking first. Kocher et al. (2012) attributed this difference in results to differences in the sampling processes used by the two studies.

Despite wrestling being a very widely-practiced sport, with 180 national wrestling federations represented at United World Wrestling (the sport’s international governing body) and inclusion in the International Olympic Games program, the author is not aware of any published econometric analysis addressing decision-making in wrestling.

2.1Impact of winning coin toss

We begin by examining the impact of winning the coin toss on the match outcome. For this question, this study considers a randomized natural experiment. As discussed in Manski (1995), such a situation guarantees internal validity, because the conditions for causal inference are satisfied. Thus, we are confident that these results allow us to attribute differences in the match outcome to the treatment under consideration, namely the results of the coin toss.

The wrestler who won the coin toss ended up winning in 290 matches and losing in 308 matches. However, these simple counts do not account for a particularly important piece of information: the relative scoring differential in which the wrestler who wins the coin toss finds himself at the end of the first period.

Consider the group of wrestlers who were wearing the green ankle band during the match. To examine the impact of the coin toss on the match outcome, we performed a logistic regression using a binary response Did Green win match? (=1 if yes, =0 if no); and two predictor variables:

The results from this analysis appear in Table 1. This analysis suggests that the score differential at the end of the first period is extremely unlikely to have no effect on the ultimate match outcome (p = 0.000), but that winning the coin toss does not, in itself, have a significant effect on the ultimate match outcome (p = 0.196). Moreover, the odds ratio of 0.77 suggests that, holding the Green score differential at a fixed value, the odds of Green winning the match when Green wins the coin toss are 23% lower than the odds when Green loses the coin toss, although at the 95% confidence level this is statistically indistinguishable from the coin toss being associated with no change in the odds of Green winning. The odds ratio of 2.35 suggests that, holding Green’s coin toss result at a fixed value, a one-unit increase in Green’s score differential increases Green’s odds of winning the match by 135%. One may conclude that, having controlled for the match score at the end of the first period, winning the coin toss does not significantly impact a wrestler’s likelihood of winning the match.

2.2Impact of deferring choice of restart position

It is possible, however, that this lack of an effect for the coin toss outcome may be due to wrestlers making a sub-optimal choice after winning the coin toss. We observed in our data that the coin toss winner elected to defer in 283 matches (47.3% of occasions), and to begin the second period in the position of his choice in 315 matches. This suggests, at a minimum, that wrestlers do not have a uniform preference for deferring. Contrast this with the experience in soccer and the NFL, each of which demonstrate a strong preference (not deferring in the case of soccer, and deferring in the NFL). Wrestlers who did not defer ultimately won in 172 matches and lost in 143 matches, while wrestlers electing to defer ultimately won on 118 occasions and lost in 165 matches. A two-way contingency table analysis reveals an odds ratio of 0.59 for wrestlers who defer, suggesting that the odds of the coin toss winner winning the match when the coin toss winner defers are 41% lower than the odds when the coin toss winner does not defer. But again, these simple counts do not account for the relative scoring differential in which the wrestler who won the coin toss finds himself at the end of the first period.

We examined the impact of the defer/do not defer choice by performing another logistic regression, this time using a binary response Did coin toss winner win match? (=1 if yes, =0 if no); and two predictor variables:

The results from this analysis appear in Table 2. Again, we observe that the score differential at the end of the first period is extremely unlikely to have no effect on the ultimate match outcome (p = 0.000). However, we now observe that the decision to defer the choice of restart position has a considerable negative effect on the ultimate match outcome for the coin toss winner, with p = 0.073. The odds ratio of 0.69 suggests that, holding the coin toss winner’s score differential at a fixed value, the odds of the coin toss winner winning the match when the coin toss winner defers are 31% lower than the odds when the coin toss winner does not defer, although at the 95% confidence level this is statistically indistinguishable from the coin toss winner’s choice to defer being associated with no change in the odds of the coin toss winner winning the match. The odds ratio of 2.33 suggests that, holding the coin toss winner’s decision to defer or not at a fixed value, a one-unit increase in the coin toss winner’s score differential increases the coin toss winner’s odds of winning the match by 133%. Said differently, having controlled for the match score at the end of the first period, should the coin toss winner elect to defer his choice of restart position until the beginning of the third period, he has significantly (at the p = 0.1 level) reduced his likelihood of winning the match. What is most surprising about this finding is that, as discussed above, wrestlers did not demonstrate a clear preference for notdeferring.

A collinearity test was performed to determine if these two predictor variables are independent, this test found tolerance value of 0.9854 (or, equivalently, a Variance Inflation Factor of 1.0148), suggesting that these predictor variables are not significantly correlated with one another. However, to perform additional testing examining the relationship between these predictors, we performed two additional analyses: a difference of means t-test, contrasting the average score differential for those wrestlers who deferred with the average score differential for wrestlers who did not defer; along with a logistic regression, with the defer decision as the binary response and the match score difference as the sole predictor. The t-test found a significant difference, at the p = 0.01 level, between the mean score differentials at the end of the first period for the wrestlers who did defer (who, on average, were losing by 0.25 points) and those who did not defer (who, on average, were winning by 0.37 points). The logistic regression analysis found an odds ratio of 0.91, suggesting that a one-unit increase in the coin toss winner’s score differential decreases the coin toss winner’s odds of deferring by 9%. Because the 95% confidence interval for this odds ratio was [0.85, 0.97], which does not include the value 1, this effect is statistically significant at the p = 0.05 level. The evidence of dependence between the coin toss winner’s choice to defer and the score differential potentially confounds the interpretation of the logistic regression explaining the coin toss winner’s odds of winning the match and the coin toss winner’s choice to defer and the score differential.

An analysis of this question is further complicated by the fact that the coin toss winner’s decision is potentially confounded with other unmeasured factors that are not captured in the score differential. Said differently, it is possible that there are other, unobserved, factors that differentiate between two wrestlers that, each having won the coin toss with an identical score differential, would make one more likely to win the match when deferring and the other more likely to win the match by not deferring. Examples of such potential confounding factors include the relative stamina and fatigue of the opponents, or perhaps events that occurred in the first period that did not result in points being scored but demonstrated a relative advantage (such as physical strength) for one wrestler.

However, consider the practical implications of such potential confounding factors. We have observed that the coin toss winner’s odds of winning the match are 31% lower when he defers, and yet 47% of coin toss winners defer. Were all wrestlers homogeneous, it would be clear that any wrestler who defers is not making a rational choice. However, were there some heterogeneity in the population of wrestlers, as in the case of confounding factors, with some wrestlers truly increasing their likelihood of winning by electing to defer, then the odds of winning must be even worse for all other wrestlers who defer, making their decision even more ill-founded. This suggests that the possibility of confounding factors only strengthens the finding that many wresters are making sub-optimal decisions.

It is possible that a wrestler who defers his choice to the beginning of the third period may not even have the opportunity to make this choice because the match could end during the second period. In our data set, 36 of the 598 matches ended during the second period, 31 due to fall and 5 due to technical fall (match termination which occurs when one wrestler has earned a 15-point advantage over his opponent). A wrestler could potentially modify his deferral strategy in light of this fact by not deferring, if he thought that he might be able to achieve a victory by either fall or technical fall in the second period, and thus deny his opponent the opportunity to select the match restart position. And in fact, out of this set of 36 matches, there were 4 matches in which the coin toss winner was winning by 5 or more points at the end of the first period, in each case the coin toss winner did not defer (although, somewhat surprisingly, one of these was an instance in which the coin toss winner lost the match by fall in the second period).

In order to strengthen our confidence in the finding that a decision to defer reduces a wrestler’s likelihood of winning the match, a second analysis was performed. In this analysis, the outcome of a wrestling match was modeled as a 1-step stochastic process, with the single transition occurring between the end of the first period and the end of the match. To simplify the language for the remainder of this paper, we will refer to the wrestler who won the coin toss as the red wrestler, irrespective of which color ankle band this wrestler was actually wearing during the match. Define the following states for the match status at the end of the first period:

Observe that the vast majority of matches were in the Close state at the end of the first period. This is not surprising because the NCAA Championship tournament includes only those wrestlers who were successful at qualifying tournaments or awarded an at-large bid; the intent is that the entrants in this tournament are the best collegiate wrestlers in the United States, and thus we should expect that the contestants in each match are generally of comparable skill. Because a match is terminated when one wrestler gains a 15 point advantage over his opponent, our state definitions do not need to consider any cases in which the score differential is greater than 14 points at the end of the first period (further, there were no such matches in our data set).

We are interested in determining whether the decision to either select the match restart position at the beginning of the second period or to defer this choice until the start of the third period has a significant effect on the match result. Define p_ij as the probability that a wrestler in state i at the end of the first period transitions to state j at the end of the match, wherej can be one of two alternatives: win the match or lose the match. It is possible that the transition probability between states i and j depends upon the decision of the red wrestler to either select the restart position at start of second period or to defer selection of the restart position until start of third period. Let p_ij (k) denote the probability that a red wrestler in state i at the end of the first period transitions to state j at the end of the match, having made decision k, where k = 2 denotes selecting restart position at the start of the second period, k = 3 denotes deferring selection of restart position until the start of the third period.

We can use our data set to compute estimates of these transition probabilities. Define n_ij (k) as the number of transitions between states i and j for the red wrestler’s decision k. The estimates pˆij(k) of these transition probabilities p_ij (k) are:

Tables 3 and 4 present these estimates for the winner of the coin toss, as a function of his decision to select the match restart position at the start of the second period (does not defer) or at the start of the third period (defers), dependent on state i (score differential state at the end of the first period).

Table 3

Match result for winner of coin toss, when selecting restart position at start of second period (k = 2)

	nij (k = 2)		pij (k = 2)
	Red win	Red loss	Red win	Red loss
Lbig	0	6	0	1
Lmed	1	24	0.040	0.960
Close	135	112	0.547	0.453
Wmed	26	0	1	0
Wbig	10	1	0.909	0.091

Table 4

Match result for winner of coin toss, when deferring selection of restart position until start of third period (k = 3)

	nij (k = 3)		pij (k = 3)
	Red win	Red loss	Red win	Red loss
Lbig	0	10	0	1
Lmed	3	26	0.103	0.897
Close	87	126	0.408	0.592
Wmed	25	3	0.893	0.107
Wbig	3	0	1	0

The test for transition probability matrices provided in Anderson and Goodman (1957) can be used to determine whether the decision to either select the match restart position at the beginning of the second period or to defer this choice until the start of the third period has a significant effect on the match result. The null hypothesis is that, for each (i, j) pair, p_ij (k) = p_ij  ∀ k. The likelihood ratio criterion for each initial state i can be computed as:

When the null hypothesis is true, the distribution of -2ln (λ_i) is chi-squared with (J - 1) * (K - 1) =1 degree of freedom, where J represents the total number of possible states at the end of the match and K represents the total number of possible decisions.

Table 5 presents these test statistics and corresponding p-values for each state i at the end of the first period, comparing Tables 3 and 4. Observe that for the state Close, the probability associated with this test statistic is less than 0.01, allowing us to reject the null hypothesis at the 1% level and conclude that for this state, the likelihood of match victory is significantly impacted by the decision to defer or not. Given that our estimates for this set of states (i =Close and j =Red Win) are p_ij (2) =0.547 and p_ij (3) =0.408, we conclude that the red wrestler enjoys a statistically significant increase in his likelihood of victory in this state by not deferring and selecting the match restart position at the start of the second period. For state Close, an analysis of the two-way contingency table reveals an odds ratio of 0.57 for wrestlers who defer, suggesting that the odds of the red wrestler winning the match when he defers are 43% lower than the odds when the red wrestler does not defer. State Wmed shows a similar relationship, with statistical significance at the 5% level, of a benefit to the red wrestler by not deferring. Note that there are zero observations in which a wrestler in state Wmed elected to not defer and subsequently lost the match, preventing the calculation of an odds value for this state. For state Lmed, the data show a slight advantage to the red wrestler if he defers his choice until the start of the third period, but this difference is not statistically significant. For state Wbig, only one wrestler lost the match, this being the aforementioned wrestler who lost by fall in the second period. For state Lbig, the red wrestler lost every match, regardless of his choice to defer or not.

Taken together, the logistic regression analysis and transition probability matrix analysis suggest that, having accounted for the match score differential at the end of the first period, the wrestler who wins the coin toss can improve his likelihood of winning the match by not deferring and selecting the match restart position at the start of the second period, with this effect being strongest for Close matches (absolute score differential no greater than two points at the end of the first period).