Expected Value and Scenarios under different Markets

[uva3021]

It is absurd to explore the nature behind any sort of long-term expected value or growth models as it relates to the WNBA (obviously this relates to every sport but to fit the overall nature of my degeneracy I feel it necessary to purely focus on the WNBA). Notwithstanding this blatant absurdity and at its core a depressing and wasteful commander of the scientific mind, I decided it would be practical to at least mention some of the basics of Expected Value, Expected Growth, and its applications. I am not qualified to be the source of knowledge on this topic, I’m just trying to layout the foundations that are understood in typical matters of statistics and integrate the formulas into the world in which I operate.


Since the WNBA market isn’t as tightly purveyed as other markets, primarily the NBA and NFL, there are more opportunities to reduce your necessary break even point that is standard issue for the -110 spectrum of spread betting, which I will explain later. Now admittedly for me, having access to a variety of books that offer better margins and lower overrounds, I already have an advantage over most US bettors. Not to sound arrogant or betraying some ill-conceived notion of my position in the proverbial market pecking order, but the US is not the most ideal place for a gambler to reside. The value of investment is immediately reduced by mere locations that are happen to be within the scope of arbitrary policies.

However, as it has been pointed out, if you shop around, you can find positions that are better than the market, and rival the rest of the world, despite the limits, ironically, of an American residence. But I don’t want to expound on that notion here.


Let’s do a simple armchair analysis, by way of rational sentences followed by a brief single scenario analysis.


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Case 1:


The less potential losses can cost me, the lower my break even point is, and the higher my potential return.

Stating the obvious, yes? Pretend you make three wagers, but instead of the -$110 reduction per loss (and concurrently the $100 for every win), that number is -$107. In those three wagers, you happened to go 2-1 (in the observable history), for a profit of $93. Had you wagered at the typical 10% vigorish (-110 odds), your profit would only by $90. Just on three wagers you’ve made $3. Basic linear logical thought. A conecpt that could have been realized by sitting in chair in thought. Hence armchair analysis.


Case 2:


Sportsbooks 1 through 50 are all offering -6, Sportsbook 51 is offering -4.5.


The market price is for the most part, the market consensus. In some instances, Pinnacle Sportsbook is generally seen as the market setter. And from my perspective, a brief survey of line movements and most markets triangulate to Pinnacle.


So with the market price set at typical odds (-110), this implies each team has a 52.38% chance of winning, regardless of other ancillary factors. The principles behind implied winning percentage was discussed here, but both teams having the same implied winning percentage is equivalent to both teams having equal probability. This 50% probability (implied less the overround) is at the market price of -6. Since all betting options, conditions, and outcomes are stochastic in nature (devoid of all absolute certainty), results are eminently a random occurrence. Qualitatively sports betting events may be deterministic to a degree (in regards to overall behavior and the invariable nature of regression to the mean), but from a quantitative standpoint, even odds implies even probability.

Sportsbook 51, however, has -4.5. Standard operating procedure suggests each half point is worth 10 cents. There is another way of looking at this however, a way which is much more significant for the purposes of realizing all potential outcomes.


If -6 is -110, then by the typical rules of line movement, -4.5 is -110 minus 30 cents, which equates to -140, and the underdog is proportionally +120. Under the conditions of implied probability appropriated by the vigorish spectrum, starting with -110 at 50% chance of winning, this leads one to believe a 30 cent probability gain results in roughly a 6% increase in odds.


Implied Probability Favorite = -140 / (-140-100) = 58.33%
Implied Probability Underdog = 100 / (100+120) = 45.45%
Favorite Chance of Success = 58.33% / (58.33% + 45.45%) = 56.20%
Underdog Chance of Success = 1 – 56.20% = 43.80%


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Now that we have isolated the fair value odds of winning, we have uncovered a 6.2% market advantage in our favor merely as a factor of finding a better number.


With our two conditions now as a reference point for utilizing some form of calculated money management strategy, the framework is in place to find the optimal frequency for success if the same bet(s) were made in an infinite amount of situations over an infinite amount of time. Optimal here indicates the break-even point.


At length I’ll provide the necessary data in order to evaluate, specifically, the points of relevance that are favorable to create a +EV based on the information below. For now, this post will just serve as a quick introduction to expected value and expected value as a function of success when operating in the sports betting market.


Expected growth models are a much more sophisticated process and requires another post exclusively devoted to the dynamics, in order to explain the concept. It would serve the purpose to just find a more credible resource and pull quotes directly from that an apply it to this post as I see fit. But the whole spectrum of expectational mathematics in the sports betting market is for the most part an abstract formulation lest epistolatory efforts be applied for personal measures, so a concrete understanding and awareness of what it takes to practice sound money management, and fully utilize the resources available, can now be attached alongside the appropriate behavior needed to survive the madness.


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When you are given two possible outcomes in regards to one discrete random variable ($110 placed on a game) you can evaluate the measure of expected return with the 50/50 proposition that is presented. The wager here is marked as a random variable due to its nature of having two possible outcomes that are a function of the probabilities of each outcome, which formulates into an expected percentage return after the event has reached its conclusion. To express the aforementioned, known as the expected value of a wager, you multiply each outcome by its probability of success, and sum the products. This result can also be seen as the mean of the sample of data.


The random variable is typically notated by X accompanied by a subscript based on the number of outcomes. The probability of each outcome, p, then takes on the weight placed on the value of each outcome (X1, X2, X3, …Xk ).


Therefore the expected value with two possible outcomes, with subscript one of variable X as your current hold, can be expressed as :


EV(X) = p1X1 + p2X2


Where the sum of all values of ‘p’ must equal one.


With the the variable X and its concomitant qualifiers:


X1 = $100
X2 = -$110
p1 = 50%
p2 = 50%
EV(X) = $100 (50%) – 110 ( 50%)
EV(X) = – $5 = 5% expected loss


(This 5% expected loss for every wager is what induces the construction of very tall buildings that line the streets of Vegas.)


Certainly, one wouldn’t expect to lose $5 in any possible way under any one single condition involving the logistics behind this particular game. But imagine the game being played out one million times across one million alternative universes. There are only two potential outcomes here, a win or a loss, which result in a change in immediate income. Assuming each universe operates on the same equi-probable dimension, one team winning 500,000 times and one team winning the other 500,000 times would be the expectation, and result in a loss of $5,000,000. This 5% is on average to infinity.


Now if there are more than one outcomes, than the formula is only slightly adjusted but remains in accordance with the concepts of the process in establishing expected value.


Using the same method for a variable, k, amount of possible outcomes, the expression is notated thus:

Remember all values of ‘p’ should sum to a total of one.


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We can interact these expressions with the WNBA (or again with any sport but we are concerned with the WNBA) by implementing the numbers that are available from across the WNBA market. Here are the figures when applying the EV methods to a -107 WNBA market (available for US bettors at ABC Islands casino).


EV(X) = $100 (50%) – 107 ( 50%)
EV(X) = – $3.50 = 3.5% expected loss

And here is the expected value when finding an off-market number by 1.5 pts:


EV(X) = $100 (56.20%) – $110 (43.80%)
EV(X) = $8.03 = 8.03% expected return

Keep in mind, had you bought 1.5 pts, the expected value of the market number would now be:


EV(X) = $100 (56.20%) – $140 (43.80%)
EV(X) = -$5.11 = 5.11% expected loss

All the previous expected value numbers merit little commentary besides the obvious. But I will say this: Gamblers are irrational, and after seeing these figures materialize, it takes every fiber of my moral indifference to not disallow my expenses with such remunerations and attempt to find reason in perpetuation of the hobby.


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Some WNBA market generates do not administer their lines as tightly as other sports. Digibet, based on about a week’s worth of observation, operates with a modest amount of market awareness. Essentially, Digibet has a habit of setting their WNBA lines behind the current state of the line movement. However, their fixed juice of -118 is perhaps a way to counter any sort of resources that would otherwise be exerted towards deliberately following what the other books are doing in regards to the WNBA.


Fundamentally, the theory behind doing such is immaculately flawed. In order to truly validate this, one would have to research a considerable amount of WNBA games using the point differential as it relates to the line and line movement over a substantial amount of games. But conventional wisdom suggests not moving lines with the market is reductive.



Again let’s run a quick expected value model comparing their all too common 1.5 off line at -118 odds with the two conditions presented above along the same dimension.


1.5 pt off-line, -110 odds:
EV(X) = $8.03 = 8.03% expected return

1.5 pt buy, -140 odds:
EV(X) = -$5.11 = 5.11% expected loss

1.5 pt off-line, -118 odds (using the -140 implied winning percentage):
EV(X) = $4.53 = 4.53% expected return


Again an approximation of at least which numbers would result in a positive or negative expectation requires little exertion other than sitting in chair in thought. But seeing the precise numbers on screen, and having the ability to now calculate the exact value of each bet, may engender a more methodical approach to your overall positions.


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When thinking rationally, expected value is incomplete without evaluating the actual level of success. Success not in terms of team by team probability, but the success vs failure ratio of winning the specified wager.
Expected number of wins and losses is a simple and direct formulation, multiply the number of events by the probability of success. Given n events and a measure of probability, Success(S) equals:


Sn = np

The expected value calculations above did not include S as a factor in the sense that S is a variable determined by p, and p will always, with the simple formula above, equal S/n. But S has a dynamic quality for its value is observational. Values of S can be plugged in a manner in which one attempts to disclose certain conditions of n and p. Though, in theory, S here is virtually implied, and therefore not very meaningful.
There are times, though, when S, as a designation of desired output, or frequency of success, is a rigorous amplifier.


When the expected value is set to zero, and the sums of all the possibilities equal zero, this is what is referred to as the break-even point. So now the calculation is a function of success. Which can also be formed into a mathematical expression.


Keep in mind, only one of two possible outcomes dictates what is the optimal level of success in order to reach the break even point. Therefore an assumption can be made that only the one observed outcome of any of the possible outcomes in any alternative universe is statistically significant. Which in turn implies the value of S is imminently measurable as a percent, a number between 0 and 1.


Given:


p = probability, S = success, Xi as a factor of a one unit wager


EV(X) = 0
0 = p1X1S + p2X2(1-S)
0 = p1X1S + p2X2 – p2X2S
p2X2S – p1X1S = p2X2
S(p2X2 – p1X1) = p2X2





Using the original example from above, basically the calculated number of success where the resultant is the break-even point, with both sides having the same probability of winning and the odds are at the even -110:


0 = p1X1S + p2X2(1-S)
0 = .50S – .55 + .55S
.55 = 1.05S
S = 52.38%


Complications arise when more than two possibilities are introduced. The dynamics of the equation becomes much more complex, yet nonetheless can be expressed. Having k number of possible outcomes, where k > 2, across a variety of different bets that all hold the same inherent value is quite uncommon and not conducive to juxtapose onto a multi-universe dimension. Perhaps in soccer where k is three on a three way line this is yet applicable, of course with horse racing the odds that the same allocation or probability is distributed exactly the same as a prior field of horses is relatively slim. Regardless, the expression below assumes, k, total number of potential winners , is statistically significant only at the point where k is a success, therefore all other values the correlate with k used to measure the optimal success rate serve is mere ancillary sum holders to ensure efficacy of the primary variable S:





(Feel free to double check this particular equation, because it took me a while via pencil and paper to come up with the proper expression. I could not find it anywhere online, or a derivative of some sort. Perhaps I overlooked it.)


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All of the above numbers are an expectation, as mentioned before. The next step would be to aggregate the necessary data needed to find an observed expected value model using as big of a sample as possible. For the WNBA that spans back to 1997. And while extracting the data should be somewhat time consuming, once the data is serried into a manageable fashion than the data isolation process will ideally represent figures that are more or less an indication of market inconsistencies. Hopefully these inconsistencies are conveniently in line with any of a number of potential methods that can take advantage of market inconsistencies, and not purely be a random fluctuation of off numbers and linesmaker errors.
As mentioned before, expected growth and expected value are two different animals, and eventually I’ll add some insight into the various concepts that embody an expected growth model and what the model actually yields in terms guiding your ROI.


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Reference and tools:


http://www.chass.utoronto.ca/~cook/ecmb02/expectedvalue.pdf
http://en.wikipedia.org/wiki/Expected_value
http://www.sitmo.com/latex/

[IrishTim]

Did you write that whole thing? Doesn't seem like your usual style, guess you decided to get Socratic on us. :)

At any rate, well done. A few minor points:

- each WNBA half point is not worth 10 cents; 8c is a better general number, though your point about beating the market price remains valid
- in my opinion, the WNBA is a way more efficient market than it gets credit for. The reason in my estimation, is the ratio of square to sharp money.
- what is Digibet? Never heard of it.
- be careful using WNBA data back to '97. The changing of the game's structure (from 2 halves to 4 quarters) and the addition of the shot clock in its lifespan alter the distributions.

[uva3021]

yeah i wrote it, and used the latex app to make the formulas, comes in handy and is better than the mircrosoft office feature

Digibet is just a new book that has propped up that offers flat -118 WNBA lines that for the most part remain static, and they are open to US

yeah i decided to use data back to 2002

does efficient market you are referring to indicate a market that is unpredictable and adapts to the nature of market volatility as to remain unpredictable?

thanks for the resposne

[justdonating]

Very nice read UVA.......

[IrishTim]

Originally Posted by uva3021

yeah i wrote it, and used the latex app to make the formulas, comes in handy and is better than the mircrosoft office feature

Digibet is just a new book that has propped up that offers flat -118 WNBA lines that for the most part remain static, and they are open to US

yeah i decided to use data back to 2002

does efficient market you are referring to indicate a market that is unpredictable and adapts to the nature of market volatility as to remain unpredictable?

thanks for the resposne

Well -118 is pretty tough to beat, especially when there are some -05 WNBA outs around. But if they're not adjusting their lines I'm sure you could get some +EV plays vs. the market price.

re: markets, it's tough to predict any market although I believe the more efficient a market is, the more predictable it will be if you have a +EV model. What I mean by that is, by definition, efficient markets creep towards the fair value and generally reach that point by the time it closes. If you're model finds intrinsic value in these lines, generally the efficient market will agree and you'll beat the closing line.

Additionally, in my opinion, an efficient market is one that is not volatile. For instance in the AFL (something like $250 limits), you'll see lines move 3-4 points instantly. You don't see this as much in the NFL which is by most accounts the most efficient market. When RAS releases a play on a NCAAB total, it moves about 3 points. There is no equivalent of this in the NBA (Spiro moves NBA totals maybe 1-1.5 points). Why? Because the NBA market is more efficient than college basketball due to fewer teams/games, less volume, etc.

Sorry for getting off track, but I'm not entirely sure what you mean by a market that "adapts to the nature of market volatility as to remain unpredictable".