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Subject: Re: Math is your friend

An Observer
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Date Posted: 09:58:27 10/04/18 Thu
In reply to: voy vey 's message, "Math is your friend" on 07:56:23 10/04/18 Thu

Voy Vey,

Great post. Thanks for the reference to the Atomic Football website.

I agree with you that Atomic's numbers seem low. Here's a more precise way to look at probabilities for Dartmouth and Princeton.

P (Dartmouth finishes undefeated) + P (Princeton finishes undefeated) + P (Winner of D/P game loses at least one other game) = 1

So Atomic is saying that the probability of the winner of D/P loses at least one other game = 100.0 - 4.4 - 35.5 = 60.1%

I think that 60.1% is too high.

I would estimate the following:

P (Dartmouth finishes undefeated) = 0.25
P (Princeton finishes undefeated) = 0.5
P (winner of Dartmouth-Princeton loses at least one other game) = 0.25

I overestimated both Dartmouth and Princeton purposefully just to keep the math easier. I'm implicitly giving Princeton a slightly better than 2/3 chance of beating Dartmouth at home. I say *slightly* better because I think Princeton has a bit more risk in its other games because it still has to face Penn (which Dartmouth has already beaten) and the Tigers play Harvard in Cambridge, while the Green gets the Crimson in Hanover.

Ballpark, I'm also implicitly estimating that the winner of the D/P game is roughly a 95% favorite to win each of its other six games. That is, 0.95 * 0.95 * 0.95 * 0.95 * 0.95 * 0.95 = 0.735, which is approximately 1 - 0.25.

As I said earlier, I think the reality is that the D/P winner will be less than a 95% favorite in each of those games, especially against Harvard and, in the case of Princeton, Yale. But it keeps the numbers easier to calculate.

Lastly, I think you've coined a term which should catch on, The Multi-Trophy Game.

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[> [> [> Subject: Re: Math is your friend, Second Iteration

An Observer
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Date Posted: 10:18:21 10/04/18 Thu

As I fiddle with the numbers, I would offer this revision:

P (Dartmouth finishes undefeated) = 0.2
P (Princeton finishes undefeated) = 0.4
P (winner of Dartmouth-Princeton loses at least one other game) = 0.4

This second version keeps the numbers fairly simple, keeps Princeton at slightly better than a 2/3 probability to beat Dartmouth at home and then implies that the winner of the D/P game is a bit greater than a 90% favorite to win its other six games. 90% is probably still too high, but this is close for nice round numbers.

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[> [> [> [> Subject: Re: Math is your friend, Second Iteration

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Date Posted: 12:24:01 10/04/18 Thu

When I signed up for this Board, I was told there would be no math.

Therefore, I will withhold all analytically-based predictions until the season ends.

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[> [> [> [> [> Subject: I agree

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Date Posted: 13:49:41 10/04/18 Thu

this stuff is way over my head "that's why you play the game"

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[> [> [> [> Subject: Re: Math is your friend, Second Iteration

voy vey
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Date Posted: 17:01:39 10/04/18 Thu

An Observer,

I think your revisions to the original Atomic numbers are very reasonable.

To my eye, this site -- and also Massey Ratings, which similarly posts projected spreads AND win probability https://www.masseyratings.com/cf/fcs/ratings -- tend to understate win probability as the spread increases.

For example, Atomic says Dartmouth is an 8-point favorite over Cornell, and thus has about a 73% chance of winning the game. (Massey has D by 9 points, or 76%.)

I am inclined to think that in the real world, 8- or 9-point favorites win the game far more than 3 times out of 4.

So, the chances of winning out are probably higher than Atomic estimates, and I think your numbers are in the ballpark.

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[> [> [> [> [> Subject: Appropriate Relationship Between Betting Spread and Win Probability

An Observer
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Date Posted: 18:20:03 10/04/18 Thu

Voy Vey,

Thank you for your thoughtful, detailed response. I am very intrigued by your observation that win probabilities are underestimated as spreads increase. That is the kind of thing which, if substantiated, can be exploited.

I think that, in general, if you can find investing or betting markets where certain assets or bets have investors or bettors with different risk/reward instincts or goals, arbitrage opportunities are sometime available.

These opportunities are not that frequent, but when they exist, they tend to recurring.

I am going really think about this because I think you may be right, but I'm still unsure.

Slightly related sidebar: In my opinion, sometimes there is a disconnect between the betting spread and the over/under.

For example, last Friday night, Princeton and Columbia opened at Tigers -11, then expanded to Tigers -13 by kickoff. Meanwhile, the over/under opened at 60 and stayed there.

To me, the opportunity was betting Princeton -13 and taking the under. Specifically, I bet two units on Princeton and one unit on the under. So I was really expressing the strong view that if Princeton could not cover the 13, it would not be due to a scoring shootout in which Columbia starting its second string quarterback and without its best player Wainright would get into a scorefest and put up huge numbers. That worst case scenario would lead to my losing both legs of the bet.

So I was really betting one unit (Princeton does not cover, but the total is an Under) to win either one unit (Princeton covers, but the total is an Over) or three units (Princeton covers and the total is an Under).

At halftime, the score was 30-10 and it looked like Princeton would cover and the total would be an Over. Fortunately, Surace began substituting liberally midway through the third quarter as well as taking the air out of the Princeton offense.

I got very lucky when, by the fourth quarter, Princeton played to run out the clock. Both legs of the bet paid off when Princeton covered 45-10, coming in with the Under.

I think your observation about the win probabilities being understated as the spread increases is very, very intriguing. If the discrepancy is large enough, we should be able to arbitrage between the spread and the money line. Thank you for the thought-provoking observation.

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[> [> [> [> [> [> Subject: On second thought...

voy vey
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Date Posted: 20:27:09 10/04/18 Thu

I agree that IF my supposition is correct, the money line could be exploited. However...

These sites seem to confirm the relationship between spread and win probability expressed by Atomic and Massey. It doesn't "seem" right to me, but here it is:




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[> [> [> [> [> [> Subject: By the way, regarding your aside

voy vey
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Date Posted: 21:04:09 10/04/18 Thu

Your thought process on Princeton/Columbia is interesting, and totally the opposite of what my gut reaction would be.

If I saw a line move from P -11 to P -13 with no change in the O/U, my instinct would be to bet either Columbia +13, or the over. The logic there being: if the book thought that Princeton + Columbia + 11 = 60 was appropriate, then adding two to the line means the likelihood of the total remaining under 60 is diminished, thus creating opportunity on the "over" bet. And similarly, with a static O/U, there's less "room" for Princeton to cover the higher spread within the total of 60, thus signaling that taking the points is an opportunity.

Feel free to tell me why my logic is flawed. And obviously, it is, because you did just the opposite and won both bets. Lucky for me, I don't have access to Ivy lines at the site where I have an account.

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[> [> [> [> [> [> [> Subject: Re: By the way, regarding your aside

An Observer
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Date Posted: 21:32:52 10/04/18 Thu

Voy Vey,

Thank you very much for providing that interesting research on the relationship between the point spread and the win probability. I love reading that kind of analysis, just love it.

On the Princeton-Columbia game, I think that your gut reaction to the point spread moving but the O/U total staying static is, all else the same ("ceteris paribus" as I recall from college), the logical and appropriate reaction.

I was not arbitraging between the spread and the total as much as I was expressing a view of the game itself. I didn't know how much credit to give the Princeton offense for posting eye-popping numbers against its first two non-conference opponents.

I knew that Columbia had a very good run defense. I liked Princeton -13 a lot, but I was afraid of a 28-21 kind of score if the Columbia defense showed up as stout as in its non-conference games.

The one thing that I had a firm opinion about was that, if the game got into a high scoring shootout, Columbia could not cover because their starting quarterback would probably not play and their best offensive player Wainright definitely would not play.

So I put together the two-legged trade to try to isolate my belief that Columbia could not cover if the game was high-scoring.

As you may recall if you watched the telecast, the game started out with the Columbia returner taking the opening kick-off down the sideline to the Princeton seven-yard-line. I had suggested my two-legged bet to two friends to co-invest. As Columbia returner streaked past the coverage and the Princeton kicker made a lame, half-hearted attempt to shove him out of bounds, I thought to myself, "My friends think I'm an idiot."

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