Alternate Regular Season Wins

I was trying to clean up my desk yesterday and came across a prop sheet from the Circa, obviously from before NFL week 1. It's not unusual to find props on total season wins, but they have alternate totals too. Let's look at Miami, for example.

The standard bet (closest to 50/50) is over under 9.5. They have the lines at:

Over 9.5 +110
Under 9.5 -130

Here are the alternate totals:

Over 8.5 -170
Under 8.5 +150

Over 10.5 +200
Under 10.5 -235

Let's assume the bets against 9.5 are fair, after squeezing out the juice. How shall we then analyze the value of buying or laying an extra win?

First, let's squeeze out the juice from the bets against 9.5.

If the over were fair at +110, it would imply a 100/210 = 47.62% chance of winning. If the under were fair, it would imply a 130/230 = 56.52% chance of winning. Take the total and you get 104.14%. Let's split the difference in those probabilities and use these as the fair odds:

Over: 0.457256
Under: 0.542744

Let's assume each game has the same probability of winning. Yes, I know, ridiculous assumption, but let's go with it for now. Using Excel's "goal seek" feature, we arrive at a probability of winning each game of 0.578750. The following table shows the probability of winning any number of games from 0 to 16, based on the binomial function. You will see the probabilities of 0 to 9 and 10 to 16 match those desired above.

Wins	Probability
0	0.000001
1	0.000022
2	0.000223
3	0.001428
4	0.006376
5	0.021022
6	0.052951
7	0.103926
8	0.160630
9	0.196166
10	0.188657
11	0.141378
12	0.080932
13	0.034213
14	0.010072
15	0.001845
16	0.000158
Total	1.000000

Next, let's look at the bets against over/under 8.5 wins.

The probability of over 8.5. is 0.653423064. At -170 odds, there is a 3.78% player advantage.
The probability of under 8.5. is 0.346576936. At +150 odds, there is a 13.36% house advantage.

Next, let's look at the bets against over/under 10.5 wins.

The probability of over 10.5. is 0.268600. At +200 odds, there is a 19.42% house advantage.
The probability of under 10.5. is 0.731400. At -235 odds, there is a 4.26% house advantage.

This makes doing a Polish middle look attractive. There is where you bet against two different sets of over/under lines, winning both ways in the middle. Here is what the odds look like there, betting to win one unit both ways on over 8.5 and under 10.5.

Wins	Net win	Probability	Return
0 to 8	-0.7	0.346577	-0.242604
9 or 10	2	0.384824	0.769647
11 to 16	-1.35	0.268600	-0.362609
Total	0	1.000000	0.164434

That is an expected win of 0.1644 units. Divide that by the 4.05 units bet and we get a player advantage of 4.06%.

It seems too good to be true.

You might come back and argue the assumption about each game having the same probability of winning. The alternative is a small number of critical games that could go either way and the rest likely wins or losses. That would only increase my argument about betting the numbers close to 9.5.

Where I think the flaw in all this is that there is a correlation in the probability of winning from game to game. That would cause the total wins to drift further away that what the binomial distribution suggests.

Does anyone have an old prop sheet of season wins from a prior season? It would be interesting to see how the actual totals differed from the estimates.

I'll shut up now and throw it open to comments.

Quote: Wizard

Does anyone have an old prop sheet of season wins from a prior season? It would be interesting to see how the actual totals differed from the estimates.

Here's a link to a few from the 2019 season; note these were released in mid-June, so the numbers could have changed by the time the season started.

Here is how they fared:

Overs:
Baltimore (14)
Buffalo (10)
Green Bay (13)
Houston (10)
Kansas City (12)
Minnesota (10)
New England (12)
New Orleans (13)
Oakland (7)
San Francisco (13)
Seattle (11)
Tampa Bay (7)
Tennessee (9)

Unders:
Atlanta (7)
Carolina (5)
Chicago (8)
Cincinnati (2)
Cleveland (6)
Dallas (8)
Detroit (3½)
Indianapolis (7)
Jacksonville (6)
LA Chargers (5)
LA Rams (9)
NY Giants (4)
NY Jets (7)
Philadelphia (9)
Pittsburgh (8)
Washington (3)

On the Nose:
Denver (5)

Depends on Where You Bet:
Arizona (5½)
Miami (5)

I looked at an ESPN contest where fans are picking which teams they think will win each game of each week. Some fans input their predictions for all the weeks of the season in advance, and ESPN allows you to see what fans have picked for every game of the season.


So, this is not perfect, but here are the contestant predictions, as of week 3, for every game of the the Miami Dolphins for 2021, a 17 game season. As you can see, if you total up the projected probability for every game of the season, the Miami team is expected to 8.72 wins against their schedule. But look at how much variation in projected win probability exists for each week!

Projections for 2021 Miami Dolphins (as of week 3) by pickers in the ESPN Pick'em contest.

Week	Win Prob.
1	0.46
2	0.14
3	0.06
4	0.65
5	0.51
6	0.87
7	0.8
8	0.08
9	0.87
10	0.11
11	0.91
12	0.7
13	0.86
14
15	0.88
16	0.19
17	0.17
18	0.46
Total	8.72

Now this is far from perfect in several ways, but you can see that even though Miami has about a projected average prob of 0.51 for winning any particular game in 2021, that the projections for any specific game range from 0.91 to 0.06 -and are rarely close to 0.51.

To be fair, this may be affected by the fact that Miami plays in a bipolar division: they are huge underdogs agalnst the Bills and huge favorites against the Jets. Even so, a binomial distribution based on an average win probability of 0.51 would probably look a lot different than a simulation over many trials in which each game was weighted according to the above probabilities.

Quote: ThatDonGuy

Denver (5)link to original post

You mean 7. Took me a while to figure out why my total wins weren't adding up.

Quote: ThatDonGuy

Here's a link to a few from the 2019 season; note these were released in mid-June, so the numbers could have changed by the time the season started.

Here is how they fared:link to original post

This is just what I needed, thanks.

Over the whole season, the total deviation in wins between actual and the over/under was 65 wins. That comes to 2.03 per team.

This table shows the difference between the over/under line on wins and actual wins in 2019.

Deviation in wins	Count	Probability
0	3	9.38%
0.5	3	9.38%
1	7	21.88%
1.5	4	12.50%
2	2	6.25%
2.5	3	9.38%
3	3	9.38%
3.5	2	6.25%
4	2	6.25%
4.5	1	3.13%
5	1	3.13%
5.5	1	3.13%
Total	32	100.00%

Wiz, any idea if they use similar analysis as yours when setting these lines?

Quote: Johnzimbo

Wiz, any idea if they use similar analysis as yours when setting these lines?

link to original post

Sport books are notoriously lazy in setting lines. Most of them copy the lines of somebody else who already set them. That leaves the job of somebody to do it first, which is often Pinnacle Sports. They do have very good people setting lines. How they do it, I don't know exactly, but I think they like to hire people with both a very solid knowledge of sports and math. I think I would blend in their offices in Curacao well.

I took a new look at doing a Polish middle of Miami this season, based on the 2019 data. Here is my revised return table.

Wins	Probability	Un 10.5	Ov 8.5	Net Units Won	EV
0	0.000000	1	-1.7	-0.7	0.000000
1	0.000000	1	-1.7	-0.7	0.000000
2	0.000000	1	-1.7	-0.7	0.000000
3	0.000000	1	-1.7	-0.7	0.000000
4	0.023438	1	-1.7	-0.7	-0.016406
5	0.039063	1	-1.7	-0.7	-0.027344
6	0.070313	1	-1.7	-0.7	-0.049219
7	0.085938	1	-1.7	-0.7	-0.060156
8	0.132813	1	-1.7	-0.7	-0.092969
9	0.148438	1	1	2	0.296875
10	0.148438	1	1	2	0.296875
11	0.132813	-2.35	1	-1.35	-0.179297
12	0.085938	-2.35	1	-1.35	-0.116016
13	0.070313	-2.35	1	-1.35	-0.094922
14	0.039063	-2.35	1	-1.35	-0.052734
15	0.023438	-2.35	1	-1.35	-0.031641
16	0.000000	-2.35	1	-1.35	0.000000
Total	1.000000				-0.126953

The -0.1270 is the expected units won, based on 4.05 units bet. So the expected value, based on the total amount bet is -0.127/4.05 = -3.13%.

Quote: Wizard

Let's assume the bets against 9.5 are fair, after squeezing out the juice. How shall we then analyze the value of buying or laying an extra win?

And if this assumption is wrong, then what?

I'm fairly confident there was an over 9 at -110 the same time as the u9.5 -130. Much of the summer there were even better numbers on alternate wins for Miami.

Quote: TomG

And if this assumption is wrong, then what?

I'm fairly confident there was an over 9 at -110 the same time as the u9.5 -130. Much of the summer there were even better n

I am not saying that the bet specifically on Miami is good or bad. I'm trying to address whether the idea of Polish or English middling the alternate lines is a good idea. I just picked Miami as an example. It was the first bet on the page to have an over/under line that didn't have a half. The math gets messier if we have to deal with pushes.

Quote: Wizard

I'm trying to address whether the idea of Polish or English middling the alternate lines is a good idea.

The answer is most undoubtedly yes. Maybe not with these numbers, but there were definitely numbers available this summer that were very good for both.

As a generality: I think most people are averse to going for those Polish middles (and not just because of the vaguely racist name). But if the numbers are good, then the bets are good, even if we can lose big.