Fig 1.
How accurately do sportsbooks predict the median outcome?
(a) The distribution of margin of victory for National Football League matches with a consensus sportsbook point spread of s = 6. The median outcome of 4.26 (dashed orange line, computed with kernel density estimation) fell below the sportsbook point spread (dashed blue line). However, the 95% confidence interval of the sample median (2.27-6.38) contained the sportsbook proposition of 6. (b) Same as (a), but now showing the distribution of point total for all matches with a sportsbook point total of 46. Although the sportsbook total exceeded the median outcome by approximately 1.5 points, the confidence interval of the sample median (42.25-46.81) contained the sportsbook’s proposition. (c) Combining all stratified samples, the sportsbook’s point spread explained 86% of the variability in the median margin of victory. The confidence intervals of the regression line’s slope and intercept included their respective null hypothesis values of 1 and 0, respectively. (d) The sportsbook point total explained 79% of the variability in the median total. Although the data hints at an overestimation of high totals and underestimation of low totals, the confidence intervals of the slope and intercept contained the null hypothesis values.
Table 1.
The relationship between sportsbook point spread and true margin of victory.
Regular season matches from the National Football League occurring between 2002-2022 were stratified according to their sportsbook point spread. Each set of 3 grouped rows corresponds to a subsample of matches with a common sportsbook point spread. The “level” column indicates whether the row pertains to the 95% confidence interval (0.025 and 0.975 quantiles) or the mean value across bootstrap resamples. The dependent variables include the 0.476, 0.5, and 0.524 quantiles, as well as the expected profit of wagering on the side with higher likelihood of winning the bet for hypothetical point spreads that deviate from the median outcome by 1, 2, and 3 points, respectively.
Table 2.
The relationship between the sportsbook’s estimate of the point total and the actual total.
Matches were stratified into 24 subsamples defined by the value of the sportsbook total. The dependent variables are the 0.476, 0.5, and 0.524 quantiles of the true point total, as well as the expected profit of wagering conditioned on the amount of bias in the sportsbook’s total.
Fig 2.
Do sportsbook point spreads deviate from the 0.476-0.524 quantiles?
With a standard payout of ϕ = 0.91, achieving a positive expected profit is only feasible if the sportsbook point spread falls outside of the 0.476-0.524 quantiles of the margin of victory. The 0.476 and 0.524 quantiles were thus estimated for each stratified sample of NFL matches. Light (dark) black bars indicate the 95% confidence intervals of the 0.476 (0.524) quantiles. Orange markers indicate the sportsbook point spread, which fell within the quantile confidence intervals for the large majority of stratifications. An exception was s = 5, where the sportsbook appeared to overestimate the margin of victory. For two other stratifications (s = 3 and s = 10), the 0.524 quantile tended to underestimate the sportsbook spread, with the 95% confidence intervals extending to just above the spread.
Fig 3.
Do sportsbook point totals deviate from the 0.476-0.524 interval?
The 0.476 and 0.524 quantiles of the true point total were estimated for each stratified sample of NFL matches. For all but one stratification (t = 47, 95% confidence interval [41.59-45.42], sportsbook overestimates the total), the confidence intervals of the sample quantiles contained the sportsbook proposition. Visual inspection of the data suggests that, in the NFL betting market at least, sportsbooks are very adept at proposing totals that fall within the critical 0.476-0.524 quantiles.
Fig 4.
How large of a bias in the point spread is required for positive expected profit?
In order to estimate the magnitude of the deviation between sportsbook point spread and median margin of victory that is required to permit a positive profit to the bettor, the hypothetical expected profit was computed for point spreads that differ from the true median by 1, 2, and 3 points in each direction. The analysis was performed separately within each stratified sample, and the figure shows the results of the four largest samples. For 3 of the 4 stratifications, a sportsbook bias of only a single point is required to permit a positive expected return (height of the bar indicates the expected profit of a unit bet assuming that the bettor wagers on the side with the higher probability of winning; error bars indicate the 95% confidence intervals as computed with the bootstrap). For a sportsbook spread of s = 3 (dark black bars), the expected profit on a unit bet is 0.021 [0.008-0.035], 0.094 [0.067-0.119], and 0.166 [0.13-0.2] when the sportsbook’s bias is +1, +2, and +3 points, respectively (mean and confidence interval over 500 bootstrap resamples).
Fig 5.
How large of a bias in the point total is required for positive expected profit?
Vertical axis depicts the expected profit of an over-under wager, conditioned on the sportsbook’s posted total deviating from the true margin by a value of 1, 2, or 3 points (horizontal axis). The analysis was performed separately for each unique sportsbook total, and the figure displays the results for the four largest samples. A deviation from the true median of a single point permits a positive expected profit in all four of the depicted groups. For a sportsbook total of t = 44 (green bars), the expected profit on a unit bet is 0.015 [0.004-0.028], 0.075 [0.053-0.10], and 0.13 [0.10-0.17] when the sportsbook’s bias is +1, +2, and +3 points, respectively (mean and confidence interval over 500 bootstrap resamples).