# The Wavy-Ruler Effect on NFL Bets

King Yao is the author of Weighing the Odds in Hold‘em Poker, and Weighing the Odds in Sports Betting. He uses his experience from making millions in financial derivative markets and translates it into gambling. Since he left his trading position in 2000, he has been playing poker and betting on sports. He travels to Las Vegas frequently, especially during football season.

Prospective sports bettors should know the basic principles of sports betting. You should not be betting online or anywhere else without this fundamental knowledge.

Projecting game lines for 16 NFL games is not easy. Who knows what the teams will look like by December?

Imagine holding a ruler horizontally. Wave it up and down. The part by your fingertips moves a little; the other end moves much more. This is a good analogy for projecting football lines for a team over an entire season. The points on the ruler close to your finger have some variability, but not much. The points on the other end have much greater variability. Although the farther points have greater variability, the center of the wave for them is the same as for the points closer to your fingers. You are expecting greater variability in the lines of the latter weeks of the season, but that variability will be around your original expectation. If your projections are good, then you may expect something like this in Week 1 vs. Week 16:

Week 1: you project DAL Even Money

Expected variability with 80% confidence interval: DAL +2.5 to DAL -2.5, with the midpoint around DAL pick ’em.

Week 16: you project DAL Even Money

Expected variability with 80% confidence interval: DAL +7 to DAL -7, with the midpoint around DAL pick ’em.

In both Week 1 and Week 16, you expected the midpoint to be DAL pick ’em. However, you expect greater variability in Week 16 than Week 1. In general, if you have a good midpoint, the variability can help you or it could hurt you, with equal chance; it does not increase or decrease the EV of your wager.

### Other Considerations

The time value of money, opportunity costs and sportsbook risk are concerns for any futures wagers, including RSW totals.

Time value of money

You are giving the sportsbook the money in August, and you will not get paid until late December or early January. Instead of making the wager, you could have placed the money in an interestbearing account or another investment.

Opportunity cost

During the season you may find an opportunity to make a positive-EV wager for a large amount of money, but be illiquid due to much of your funds being tied up in futures bets. If that happens, the futures bets have cost you money.

Sportsbook risk

Sportsbook risk is also important to consider. How sure are you that you will get paid in late December or early January when your wager is a winner? In Nevada, the risk is negligible. The risk can be greater outside of Nevada.

Value of a Half Win

Sometimes different sportsbooks have different totals for the same team. One sportsbook may have the total at 9, while another has the total at 9.5. The vigorish (“vig”) attached to the over and under compensates for the difference in the total. Compare the money lines for the different totals to see which is the better wager. As a rule of thumb, the probability that an NFL team wins the same number of games as its total (assuming it is a whole number) is about 20%. See chapter 11 in Stanford Wong’s book Sharp Sports Betting for more information on this topic.

As an example of the 20% rule of thumb, if the total on the Chicago Bears RSW is 9, then the Bears have about a 20% chance of winning exactly 9 games. Thus the Bears are expected to win more than 9 games 40% of the time and fewer than 9 games 40% of the time.

## Bears Wins Probability

More than 9 40%

Exactly 9 20%

Less than 9 40%

Those numbers can be used to value totals of 9.5, 9 and 8.5. With a total of 9.5, under 9.5 includes both percentages in “Exactly 9” and “Less than 9”; under 9.5 equals 60% (20% + 40%). With a total of 8.5, over 8.5 includes both percentages in “Exactly 9” and “More than 9”; over 8.5 equals 60% (20% + 40%).

With a total of 9, if the Bears win exactly 9 games, then all bets are refunded. Thus when considering over 9 or under 9, the important percentages are “More than 9” and “Less than 9.” Each has a 40% chance, so they are equally likely to happen; each wager has a 50% chance of winning or losing when the pushes are not considered. It is useful to think this way because all wagers are expressed in terms of the money line.

Here is the formula when ties are refunded. Then convert this percentage into a money line to see if the line put up by the sportsbook has value.

% Over X wins

= % More than X wins / (Total % without pushes)

In the case of the Bears:

% Over 9 wins

= % More than 9 wins / (Total % without pushes)

= 40% / (40% + 40%)

= 50%

= +100 in the money line

Below is a summary of the probabilities and the equivalent money line for the Bears.

Wager Prob Money Line

Over 9.5 40% 150

Under 9.5 60% -150

Over 9 50% 100

Under 9 50% 100

Over 8.5 60% -150

Under 8.5 40% 150

When the line is around even money (50%), a 10% change is worth about 50 cents in the money line. This can be seen in the table above by comparing over 9.5 to over 9 and over 9 to over 8.5. This relationship breaks down a bit when the initial probability is not 50%, but it is still a useful rule of thumb to use for RSW wagers.

Here is an example. You want to bet the under in the Bengals RSW. Which of these two wagers has better value?

Bengals under 9.5 -130

Bengals under 9 +100

Adding 50 cents to under 9.5 -130 shows that it is equal to under 9 +120. Therefore, under 9.5 -130 is superior to under 9 +100.