The fact that over/under win totals come with moneylines provide no real complication for the betting public.   Convert the moneylines to a return, ask whether the probability is better than the return and bet accordingly.  But, what of rest of the public who want to read the over/under’s as Vegas’s expected wins?  How can you convert the combination of the moneylines and O/U’s to an expected win value?  To be concrete Michigan is at 7.5 (-140 over/+100 under) according to the odds just released.  Penn State is at 8.5 but (+100/-140).  How far apart are they?  My results suggest Michigan’s expected number of wins is just a little higher at 7.6, while Penn State’s is a ½ win lower than its O/U at 8.0.  In 2013, there were a few teams whose moneylines were so extreme that they implied expected wins a full win less than the O/U.

Let p denote the probability of winning more games than the O/U.  Then the implied expected number of wins is p E[wins|wins>O/U] + (1-p) E[wins|wins<O/U] where the expected values can be read as the average number of wins when a team comes above or below the over/under.  The probability p is calculated from the moneylines and Table 1 contains estimates of these averages, which depend on the level of the over/under.

The way to interpret the moneylines depends on whether they’re positive or negative.  If positive, the line gives you the extra amount you win if you bet $100.  If negative, it gives you the amount you have to bet to win an extra $100.  Let p(o) and p(u) denote the probabilities read directly from the moneylines, and denote by m as the absolute value of the number in the moneyline.  If the moneyline is positive, then the probability is 100/(m+100); if negative it’s m/(100+m).  In Michigan’s case, p(o) = 140/(100+140) = 0.583 and p(u) = 100/(100+100) = 0.500.  If you think the probability that Michigan wins more than 7-1/2 games is greater than 58.3%, bet the over; if you think the probability is less than 50%, bet the under. 

Note, however that p(o)+p(u) is greater than one.  That’s Vegas’s juice to make money.  We want a fair probability for our purposes, so we convert them by setting p = p(o)/(p(o)+p(u)).  In Michigan’s case, p = 0.583/(0.583+0.500) = 0.538.

Table 1 contains the applicable expected values for various over/under's calculated under two different methodologies.  The first method takes actual over/under’s from 2010-2013.  The second method takes Chris Stassen’s data on preseason magazine ratings and converts them as a decent ex-ante estimate of an O/U.  Then for each I calculate the average wins when the teams come in above and when they come in below the totals.  For my calculations below I then take an average of the two estimates.  In Michigan’s case, the applicable expected values are  E[wins|wins>O/U] = 9.20, which is the midpoint of the range and E[wins|wins<O/U] = 5.75, likewise.  Weighting by 0.538 gives 7.6 wins.

To be sure my estimated expected wins in table 1 could be improved upon, and the method as a whole could be augmented by arguing that the expected values should vary with the moneyline.

Table 2 gives expected wins for all Big Ten teams and Notre Dame for those over/under’s provided by Kegs ‘N Eggs.  Note that Notre Dame’s expected wins is 0.7 below the O/U.

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