Sportsbook Infallibility and Line Movement

As tonight's Duke/St. John's game shows, the books do occasionally set bad lines. I've been thinking about this topic framed in terms of line movement. What causes the line move and, more importantly, how does it affect the expected value of both sides?

As I wrote yesterday, I've yet to be convinced that value does not exist on sides with reverse line movement. That seems like an odd null hypothesis at first glance. As I've mentioned previously, my H0 is mainly anecdotal with some small sample empirical evidence thrown in. Most other gamblers I think take the opposite null hypothesis, which makes sense without taking the book's logic into account. If you are betting on a dog, would you rather have +7 or +6.5, knowing that your EV falls by a certain amount by taking the 6.5?

Let's take yesterday's WVU-Notre Dame line as an example. The line opened Tuesday night at 8.5 and closed at 9.5. Without thinking too much about it, you obviously would have rather had WVU -8.5. The more important question, particularly if you are stuck at work all day while the line moves, is whether there is still value at 9.5, not whether you lost some EV in the move (which you obviously did). Using the half point calculator, a true line of -8.5 +100 works out to -9.5 +118.2. From that, you can infer that you are losing 4.18% in probability by taking the line at 9.5. That calculation also implies that you needed to have a >55% chance of that line hitting at 8.5.

So, is there a >55% chance of WVU covering at 8.5? I acknowldege that contrarians are working within a tight margin, but I think, because of the line movement, the margin is a bit looser in this case. The "public" was backing Notre Dame yesterday at a 63.13% clip (from close at Wagerline). Why would the books move the line, exposing themselves to 4.18% chance of a middle? I think the answer has to be they realized an error in the line and WVU was overvalued at 8.5. That is, there was a significantly greater than 50% chance that WVU covers 8.5, and sharps knew it and hammered the line, facilitating the move to 9.5. Even if big money was coming in, if the books were attracting equal action from sharps (thereby validating their line), they never would have moved it, no matter what the "public" was on.

Here is some math to back me up (all at -110 juice for the book). I'm going to go through the first example spelling things out, from there you should pick it up. As you look through the numbers, ask yourself if you were the bookmaker, would you move the line to change your EV or variance?

Example 1
Probabilities held constant at 60% chance WVU covers at -8.5 and a 40% chance that Notre Dame covers at +8.5

80% of the money is coming in on WVU
EV = .6 (-800+220) + .4 (880-200)
= .6 (-580) + .4 (680)
= -76

70% of the money is coming in on WVU
EV = .6 (-370) + .4 (470) = -34

60% of the money is coming in on WVU
EV = .6 (-160) + .4 (260) = +8

I could solve an equation for percentage of money that needs to be wagered on WVU for the books to break even, but I'm lazy and it's fairly obvious that the number is slightly above 60%.

Example 2
WVU is attracting a constant 70% of the money and the probabilities of a WVU cover vary

WVU covers 70% of the time
EV = .7 (-370) + .3 (470) = -118

WVU covers 65% of the time
EV = .65 (-370) + .35 (470) = -76

WVU covers 60% of the time
EV = .6 (-370) + .4 (470) = -34

WVU covers 55% of the time
EV = .55 (-370) + .45 (470) = +8

Clearly, if the books are taking a ton of money on WVU, they are at risk of taking loss. WVU needs to cover a little under 60% of the time for the books to take wash here.

Example 3
WVU is attracting a constant 60% of the money and the probabilities of a WVU cover vary

WVU covers 70% of the time
EV = .7 (-160) + .3 (260) = -34

WVU covers 65% of the time
EV = .65 (-160) + .35 (260) = -13

WVU covers 60% of the time
EV = .6 (-160) + .4 (260) = +8

WVU covers 55% of the time
EV = .55 (-160) + .45 (260) = +29

I have no idea how much risk the books tolerate. But I think it is pretty clear looking through these examples that action doesn't have to be all that one sided if the probabilities aren't even. So, why would a book move a line by a point if they weren't uncomfortable with the risk? And inherently, will there still be value in the new line? I'll leave the math to you, but it obviously depends on how one-sided the action was. Remember, for there to be value (without considering juice), the probability for WVU to cover at 8.5 has to be greater than 54.18%. Overall, I think when the line moves more than a point, it is a signal that value remains on the contrarian side.

Finally, how bad did the books screw up with their Duke/SJU line given this analysis? The books opened themselves up to a huge middle opportunity. There must have been a ton of action on Duke and they realized the Duke was probably going to cover.