I fully apologize for having come off as condescending. Insofar as I have I've clearly acted unfairly and inappropriately and I do wholeheartedly apologize.
Nevertheless, just because I may have acted the asshole (and sorry again) should in no way serve to negate my argument.
I'm continuing here because in this one particular case there really is an obvious flaw in your logic, and even if you don't care to respond (and that's perfectly OK), I just want to make sure that the issue is abundantly clear for anyone else who might come across this thread in the future. It's an important point and is one that is often overlooked.
I do understand where you're coming from -- the OP's initial bet was almost certainly quite poorly conceived, and you quite reasonably want to drive that point home. I'm with you on that 100%. Nevertheless, the argument you're is advancing overstating the negative aspects by all reasonable mathematical conceptiosn.
While I truly do respect your statistical acumen, the issue at hand is just a very simple application of expected value theory. Vig is a book's expectation (or the negative of a player's expectation). If the player bets $100 on this wager, he
expects to lose about $4. The fact that part of the time the wager is resolved he breaks even, doesn't impact the mathematical concept of vig.
Your argument is that because roughly 57% of the time time his initial wager is returned to him, the vig he's paying,
conditioned on money being won or lost is higher. Strictly speaking you are indeed correct, but the point that future readers of this thread need to understand is that this is an inappropriate manner in which to consider expectation.
Consider the following probability vector call it
p (I'm just blindly using your numbers):
- Atlanta wins by any margin: 57%
- Chicago wins by exactly 1 run: 10.75%
- Chicago wins by 1 run: 32.25%
And consider the following payout wager vector,
y, corresponding to a 0.4167 unit bet on the Chicago run line and a 0.5833 unit bet on the Atlanta money line:
- +0 units
- Chicago wins by exactly 1 run: -1 units
- Chicago wins by more than 1 run: +0.1875 units
So what expectation does this represent? Clearly the expectation as calculated canonically definition would be
pTy = 57%*0 + 10.75%*-1 + 32.25%*0.1875 ≈ -4.70%.
Now your argument is that because no money changes hands in the case of the first outcome, that outcome should be ignored when calculating expectation. So based upon that definition what do we have? Well, expectation conditioned on no net push would be:
pTy / (1-p
1) = -4.70% / (1-57%) = 10.94%.
So firstly what we see is that even using the liberal definition of vig you've proffered (and in a moment I'll demonstrate why it's usage is inappropriate), the vig is "only" 10.94%, considerably less than the figure of 50% that you've posited.
Nevertheless, we still do need to consider the issue of which methodology makes more sense -- the one implying 4.70 vig or the one implying 10.94 vig. I'll argue that the first method is more sensible. I'm going to argue that in two ways: firstly I'll consider it from the perspective of an alternative investment by the book; and secondly from the perspective of the same reductio ad absurdum I used above:
Alternative investment argument
If the book were risk neutral and they had an opportunity to either book this set of bets or to lend out money at 5% (over the life of the bet) which one would they do? Obviously, if their expectation on the bet really were 10.94%, then of course they'd prefer to book the bet. But would this make sense?
No. If the structured bet pushes (as it does 57% of the time), it's not as if the book could retroactively lend out money at that 5% rate -- they would still have an opportunity cost of 5%. The fact that no money changed hands would be of no importance to the book. All they would care about is their gross expectation without regarding to the microstructure of that expectation.
Hence, the 10.94% figure does not meaningfully reflect the book's expectation.
Reductio ad absurdum
What if instead of wagering 0.4167 units on the Chicago run line and 0.5833 units on the Atlanta money line, the OP had instead wagered 0.41670000001 units on the Chicago run line and 0.58329999999 units on the Atlanta money line. How would that change his expectation? Well in this case the payout vector
y would look like:
- -0.0000571429 units
- Chicago wins by exactly 1 run: -1 units
- Chicago wins by more than 1 run: +0.187595 units
resulting in an expectation by the canonical method of
pTy = 57%*-0.0000571429 + 10.75%*-1 + 32.25%*0.187595 ≈ -4.70% which is two 3 decimal places is identical to the expectation calculated above.
But what about using your method? The important point to note is that because there are no zero payout outcomes your method is
identical to the canonical method. Why is this important? Well it's important because it suggests that if one were to use your method for calculating expectation a 0.0000000024% change in bet size would correspond to a 57% reduction in vig. Clearly this is logically inconsistent (if the reduction were meaningful, the a bettor could palpably reduce his vig simply by wagering a fraction of a penny more or less) and hence we need to reject the conception of vig you've advanced.
Anyway, I hope this makes sense.
paid for baseball picks
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Originally Posted by jover
