The Push Is A Third Outcome And It Has To Be Priced
Published August 22, 2026 | Margin Distribution Model | MLB Prediction Data Lab
A win probability model returns one number. That number is sufficient to price a moneyline and nothing else. Tonight's largest position is a Houston Astros alternate run line of -1 at -180, and that instrument has three outcomes rather than two: Houston by two or more wins the ticket, Houston by exactly one refunds it, and any Athletics win loses it. Pricing it requires the distribution of the margin, not the sign of the margin. This model builds that distribution, backs the market's implied push mass out of the posted prices, and checks it against 1,935 completed games.
The Algebra Of A Three Outcome Ticket
Write the three probabilities as p2 for a Houston win by two or more, p1 for a Houston win by exactly one, and pL for any Athletics win. A push returns the stake, so it drops out of both sides of the expectation. The fair price of the -1 ticket is therefore p2 divided by the quantity p2 plus pL, and the push mass never appears in the numerator at all. It only shrinks the denominator's competitor.
That structure has a consequence people miss. Two teams with identical win probabilities can carry very different fair prices at -1, purely because one of them wins by a single run more often than the other.
Backing The Push Mass Out Of The Market
Houston is posted at -240 on the moneyline and the Athletics at +195. Those convert to 70.59 percent and 33.90 percent, summing to 104.49 percent. Normalizing removes the 4.49 points of margin and leaves Houston at 67.56 percent and the Athletics at 32.44 percent.
Now solve the alternate line. A price of -180 breaks even at 64.29 percent. Setting p2 divided by the quantity p2 plus 0.3244 equal to 0.6429 gives p2 of 58.40 percent. Subtract that from the 67.56 percent win probability and the market is pricing a push mass of 9.16 percent.
| Quantity | Value | Source |
|---|---|---|
| Houston moneyline | -240 (70.59 percent raw) | Posted |
| Athletics moneyline | +195 (33.90 percent raw) | Posted |
| Houston win probability, vig removed | 67.56 percent | Normalized |
| Alternate -1 price | -180 (64.29 percent break even) | Posted, this is the card price |
| Implied Houston by 2 or more | 58.40 percent | Solved |
| Implied push mass | 9.16 percent | Solved |
| Push mass as a share of Houston wins | 13.56 percent | Solved |
So the posted price says that if Houston wins tonight, there is a 13.56 percent chance the winning margin is exactly one run. That is the claim to test.
The Observed Margin Distribution
Across 1,935 completed regular season games in 2026 through August 21, the distribution of absolute winning margin looks like this.
| Margin | Games | Share |
|---|---|---|
| 1 run | 527 | 27.24 percent |
| 2 runs | 370 | 19.12 percent |
| 3 runs | 278 | 14.37 percent |
| 4 runs | 191 | 9.87 percent |
| 5 runs | 170 | 8.79 percent |
| 6 or more | 399 | 20.62 percent |
The single most likely non zero margin in baseball is exactly one run, and it is 8.1 percentage points more likely than the next bucket. This is the structural reason an alternate -1 is a different product from a -1.5 run line rather than a cosmetic variation on it. The refund is being purchased over the fattest slice of the distribution.
Houston Is Not A League Average Margin Team
The league rate is the wrong prior for this specific club. Houston has played 129 games and 24 of them were decided by exactly one run, a rate of 18.6 percent that is the lowest of the eight teams involved in tonight's four carded games. Of Houston's 65 wins, 13 came by exactly one and 52 came by two or more.
Take that 20.0 percent one run share of wins as the conditional estimate and reprice. Holding the normalized win probability at 67.56 percent, p2 becomes 54.05 percent and the fair break even for the -1 ticket becomes 62.49 percent, which corresponds to roughly -167.
The card price is -180. On Houston's own unconditional season history, the alternate line is about thirteen cents rich.
Why The Conditional Estimate Should Beat The Unconditional One
That verdict deserves a challenge, because the unconditional base rate mixes in every opponent Houston has faced, and tonight's opponent is the worst run prevention club in the sport by a wide margin. The Athletics are 49-80 with 746 runs allowed in 129 games, the most of any club on tonight's card and 207 more than the Tampa Bay Rays.
Margin distributions widen against weak opposition. The direct evidence here is unusually clean: Houston and the Athletics have met seven times in 2026 and the winning margin was 7, 11, 2, 4, 11, 5 and 4. Zero one run games in seven tries, against a league baseline that would have predicted about two.
The Eight Starters, Expressed As Rates
Every figure below is derived from the same four raw counts: innings, strikeouts, walks and home runs allowed. Rate form is the only way to compare a 26 start season against a seven start one.
| Starter | IP | K/9 | BB/9 | HR/9 | WHIP | ERA |
|---|---|---|---|---|---|---|
| Shane McClanahan, Rays | 102.1 | 8.27 | 3.08 | 0.70 | 1.12 | 3.25 |
| Brandon Young, Orioles | 117.2 | 7.34 | 3.21 | 1.15 | 1.35 | 3.52 |
| Jacob Lopez, Athletics | 88.0 | 8.28 | 4.70 | 1.53 | 1.53 | 5.22 |
| Hunter Brown, Astros | 68.2 | 9.44 | 5.24 | 1.18 | 1.30 | 3.67 |
| Drew Anderson, Tigers | 76.1 | 10.26 | 3.42 | 1.30 | 1.30 | 4.01 |
| Michael Wacha, Royals | 155.2 | 6.94 | 2.37 | 1.21 | 1.15 | 3.58 |
| Tanner Bibee, Guardians | 150.1 | 7.00 | 2.57 | 1.50 | 1.15 | 4.01 |
| Gabriel Hughes, Rockies | 37.2 | 7.88 | 3.11 | 0.72 | 1.38 | 6.69 |
Two entries in that table deserve a flag. Drew Anderson's 10.26 strikeouts per nine is the best rate on the board, but it comes from a swingman usage pattern rather than a rotation slot: five starts inside 76.1 total innings. And Gabriel Hughes's 0.72 home runs per nine sits on 37.2 innings, which is small enough that the interval around it is wider than the difference between him and anyone else in the column.
Applying The Same Logic To A Team Total
A team total is a count distribution rather than a margin distribution, but the discipline is identical. A projection of the mean is not an answer. What matters is the probability mass at or below four runs.
The ATLAS v1 team total model projects Detroit at 4.40 runs tonight. Detroit's season mean is 4.52 runs per game across 128 games. Fit a Poisson at that season mean and the model says Detroit should score four or fewer 52.83 percent of the time. Detroit's observed rate is 72 games out of 128, or 56.25 percent.
That 3.42 point gap is the signature of overdispersion. Run scoring in baseball is right skewed, because innings are not independent trials and a single crooked frame moves a game from the four bucket to the nine bucket. Poisson understates the low counts.
Carry the same adjustment forward. Poisson at the projected 4.40 gives 55.12 percent under 4.5. Add the 3.42 point empirical skew premium and the estimate lands near 58.5 percent. The card price of -139 breaks even at 58.16 percent.
An edge of roughly a third of a percentage point is not an edge. It is a coin flip with a rounding error attached, and it carries the second largest stake on the card at 1.5 units.
Where This Model Is Weakest
Three problems, stated plainly.
The first is that normalizing a two way market proportionally is a known approximation, and it systematically overstates favorites by a small amount. If Houston's true win probability is 66 percent rather than 67.56, the implied push mass falls and the -180 looks worse still.
The second is sample size on the conditional estimate. Seven head to head games is not a distribution. Zero one run finishes in seven tries is consistent with a true one run rate anywhere from about 0 to 35 percent at any reasonable confidence level, so it cannot be used to overturn the season base rate. It can only argue that the base rate is not the ceiling.
The third is that the model treats Houston's margin history as a property of Houston. Some of it is a property of the schedule. The Astros have a run differential of minus 42 across 129 games, having scored 587 and allowed 629, which is not the profile of a club that routinely blows opponents out. A team that has been outscored on the season is a strange candidate to lay a two run margin with, and the 65-64 record says so directly.
The Two Legs The Model Likes Better Than The Headline One
Tampa Bay at -126 breaks even at 55.75 percent. The Rays are 76-52 with 539 runs allowed, second only to Detroit's 500 among the eight teams playing tonight, and McClanahan's 0.70 home runs per nine is the best mark in the starter table. Baltimore is 62-67 with 613 allowed, and Brandon Young has surrendered 15 home runs in 117.2 innings. The counterargument is recent and real: Baltimore has taken five of the last six meetings and leads the season series 6-5.
Cleveland at -137 breaks even at 57.81 percent against a Colorado club that is 50-78, has lost four straight and has allowed 742 runs. The reservation is distributional rather than directional. Bibee's 1.50 home runs per nine is the second highest rate in the table behind Jacob Lopez, Coors Field carries a posted game total of 11, and a high variance environment compresses the advantage a better team holds over a worse one.
The Positions
Tampa Bay Rays moneyline -126, one unit. Houston Astros run line -1 at -180, two units, alternate line, a one run Houston win pushes. Detroit Tigers team total under 4.5 at -139, one and a half units. Cleveland Guardians moneyline -137, one unit. Five and a half units across four games.
The model's verdict on the headline position is split. The structure is right and the price is not generous. Choosing -1 over -1.5 buys back the fattest bucket in the entire margin distribution, which is correct instrument selection. Paying -180 for it asks for 64.29 percent when Houston's own unconditional history supports about 62.5 percent, and the seven game head to head file is the only thing arguing the gap closed.
What is the margin distribution model measuring?
It replaces a single win probability with the full distribution of winning margin, because an alternate -1 run line has three outcomes rather than two. The fair price is the probability of winning by two or more divided by that same probability plus the probability of losing, with the push mass removed from both sides.
What push probability is the market pricing tonight?
9.16 percent in absolute terms, or 13.56 percent of Houston win outcomes. That is solved from a normalized 67.56 percent Houston win probability and the -180 price on the alternate line. Houston's own season history puts one run wins at 13 of 65, which is 20.0 percent of wins.
Why does a Poisson fit understate a team total under?
Because run scoring is overdispersed and right skewed. Poisson at Detroit's 4.52 season mean predicts four or fewer runs 52.83 percent of the time, while Detroit has actually done it in 72 of 128 games, or 56.25 percent. The 3.42 point difference is the correction that has to be carried into any projection built on a mean.