Sharp Consensus Line: How to Build a Fair Line Yourself

By · author of the MIT-licensed teachersbettextbook library ·

One bookie's number is one opinion. Nine of them, pooled properly, is the closest thing to a true price anybody gets.

Meet your classmates

Same five, plus every other bookie in the school.

Nine boards, one race

Bookie Brian is not the only bookie in this school. Eight other people are writing a price on the same race. Not one of them agrees with him.

That is fine. Nobody knows what Chalk’s real chance is. Every board is a guess with a fee stapled on.

You want one number out of the nine. Not a pick. A price: what the race is worth before anybody’s cut.

Take the cut out, then average right

Each board posts two prices and they add up to more than 100%. The extra is the vig. Brian’s pair adds to 104.3%.

So de-vig every board on its own, first. Pool the posted prices and you are pooling nine fees along with nine opinions.

Then average in log-odds, not in percent. A plain average of percentages drags everything toward 50%, and it drags hardest near the edges.

The votes are not equal either. Tight, deep and recent counts for more than wide, thin and stale. And two boards copying each other are one opinion, not two.

One of them has a typo

Look along the row and you will find it. Eight boards land within a few points of each other. One sits 14 points off.

That is not information. That is a thumb on a keyboard. Leave it in and it moves the answer 1.6 points, so it goes before anything gets pooled.

Sometimes nothing is left worth pooling. Two boards is not a consensus. Nine that violently disagree is not one either. When the numbers will not carry an answer, we say so instead of printing a confident one.

To grade a board, leave it out

This one took me three goes to see. You cannot ask whether Brian’s price is any good by comparing it to a pool Brian helped set.

He voted on his own grade. His number pulled the benchmark toward his number, so of course he comes out looking fair.

So you pool everybody except him. Same rule for every board. One benchmark each.

Here are all nine, on one race.

nine boards pricing one race Brian −108/−110 49.8% Dev −120/+102 52.4% Maya −118/+100 52.0% Otis −122/+103 52.7% Gil +150/−190 37.9% Ruth −115/−102 51.4% Kai −119/+101 52.2% Noor −124/+105 53.2% Tess −116/−101 51.7% Average Joe Bookie Brian +EV Evan takes the first price writes one of the nine does the pooling FAIR LINE · EIGHT BOARDS 51.9% fair price on Chalk: −108 POOLING ALL NINE 50.3% the typo is still in there GRADING BRIAN · SEVEN BOARDS 52.2% his own board left out Nine boards. One race. Nine different numbers. One board has a typo. Averaged in, it drags the answer 1.6 points. Drop it. Pool the eight that qualify. That is the fair line. To grade Brian’s board, pool the other seven. Not his own vote.
Real arithmetic, not a drawing. Each board shows the pair it posted and, under it, that board on its own with the vig taken out. Chalk and Pip never find out. Gil’s board is 14 points off the middle of the pack, which is a thumb on a keyboard rather than an opinion, so it is dropped before anything is pooled. The pool here is a plain unweighted mean in log-odds so you can redo it yourself; the live board weights each venue, and that part we keep. The last panel is the whole point: Brian posted −108 on Chalk, which needs 51.9% to break even. Graded against the eight-board pool he is in, that price comes out at 0.00 points. Graded against the other seven, the same price comes out at +0.31 points.

Same price, two answers. The one that counted Brian’s own vote is the wrong one.

Brian posted −108 on Chalk. That price needs the race to land 51.9% of the time to break even. Pool the eight boards that qualify and the fair number is 51.9%, so his price grades out at nothing. Leave his board out and it is 52.2%, so the same price sits 0.3 points better than fair.

That gap is small. It is also not random. Skip the step and every edge on the row comes out a little too small, in the same direction, every time.

The method is all here. What we keep is which live lines have earned the right to lead, and how much each one counts. That is learned from closing line value and from what actually happens. The method is open. The weights are the edge.

Want more math?the Nerd Corner

Nerd Corner

Advanced material. Nothing above depends on it. The full specification, every constant included, is on the methodology page.

Step 1: take the vig out of every book

A posted line carries the book’s margin. That’s why its two implied probabilities q1,q2q_1, q_2 sum past 100%:

q1+q2=1+vig>1  ⇒  pi=qiq1+q2  (proportional de-vig)q_1 + q_2 = 1 + \text{vig} > 1 \;\Rightarrow\; p_i = \dfrac{q_i}{q_1 + q_2}\ \ (\text{proportional de-vig})

That’s the crude version. The power method corrects favorite–longshot bias, and Shin’s method models insider-shaded prices.[1][2] Full walkthrough: how to de-vig odds. De-vig each book to its fair pip_i, and track how much the three methods disagree: an ambiguous de-vig earns less trust.

Step 2: one book is not the truth, and neither is a plain average

The classic move: crown one sharp book (Pinnacle) as the benchmark. But that feed is legally fraught to source, and a single book is one point of failure: stale, shaded, or gone tomorrow. Averaging every book is worse: soft books systematically shade the popular side, so a plain average gets dragged toward the recreational crowd. You want a pool that leans on the honest lines and discounts the rest.

Step 3: anchor on a line you trust

The move that decides everything. Pick the sharpest price in the market (real money on both sides, hold thin for the right reasons) and let it lead. The trap: low vig alone does not mean sharp. Soft books compress their hold to 2%–4% on exactly the high-volume public games they shade hardest, so weighting by thin margins hands the shaded crowd the loudest voice. Here, take the anchor as given and build around it.

Step 4: don’t let copycats vote twice

Plenty of “different” sportsbooks run the same odds engine and post identical prices. Ten copies of one line masquerade as ten independent opinions and manufacture false confidence. Collapse near-identical lines into a single vote before you pool.

Step 5: average in log-odds, around the anchor

Combine in log-odds (logit) space, not plain probability. Turn each fair probability into a log-odds value, weight-average, convert back:

zˉ=∑iwi zi∑iwi,zi=ln⁡ ⁣pi1−pi\bar{z} = \dfrac{\sum_i w_i\, z_i}{\sum_i w_i}, \qquad z_i = \ln\!\dfrac{p_i}{1 - p_i}
pA=ezˉ1+ezˉp_A = \dfrac{e^{\bar{z}}}{1 + e^{\bar{z}}}

The logarithmic pool is the statistically correct way to merge probability estimates. A plain average is provably underconfident.[3] Averaging one number and mapping it back guarantees the two sides sum to 100% with no vig.

Then clip outliers around the anchor, not the crowd’s middle. Center the clip on the crowd and it “corrects” the sharp line as the outlier. Backwards. Center it on the anchor and the shaded pack gets discounted instead. That’s why Step 3 does the heavy lifting.

Step 6: a confidence number that can’t lie about certainty

Every fair line ships with a trust signal: how tightly the independent lines agree, and how many independent opinions survive the copycat collapse. When no line is worth anchoring on, the honest move is to say so and cap the confidence, not dress up a recreational average as truth.

Step 7: to grade a book, take that book out of the pool

The six steps above build one fair line. The moment you turn around and ask is this book’s price any good?, that line stops being the right yardstick. The book you are grading helped set it. Score a book against a pool it belongs to and you are measuring it partly against itself. The heavier its weight, the worse it gets: a venue carrying half the pool drags the fair number toward its own price, so its line always looks about right and whatever edge it really offers is understated.

The fix is to rebuild the pool once per book, leaving that book out, and grade each price against its own version. That is leave-one-out grading, and it is standard practice anywhere a model is scored on data it was fitted to.

This has a visible consequence: the fair line printed on a game and the number behind that game’s +EV mark are not the same number, on purpose. The printed one pools every venue. The one behind a mark (and the one the bet sizer starts from) pools every venue except the one being graded. On our board they usually sit within a point of each other, and the panel names which venue it left out.

Leaving one venue out is a small correction on a deep pool and a large one on a thin pool. Drop from four venues to three and a single outlier in what remains can move the fair line several points, enough to make a mediocre price look like an edge. The honest guard is to require both readings to agree: our board only marks a price +EV when it beats the leave-one-out fair and the all-venue fair. If the two disagree, the disagreement is the finding, and nothing gets a badge.

How a venue earns its weight

Every step above takes the weights as given. This is where they come from, and it is the one part of the method we do not publish the numbers for.

A venue does not get a big weight for being famous, or for being cheap. It gets one by being right, measured two ways over a whole season of settled markets:

  • Closing line value. Did this venue’s price beat the closing benchmark, on average, across everything it quoted? That is the same CLV the report card grades a person on, pointed at a venue instead.
  • Log-loss. CLV says a price was better than the close. Log-loss asks the harder question: when this venue said 70%, did the thing happen about 70% of the time? A venue can beat the close and still be badly calibrated, and log-loss is what catches it.
log-loss=−1N∑j=1N[ yjln⁡pj+(1−yj)ln⁡(1−pj) ]\text{log-loss} = -\frac{1}{N}\sum_{j=1}^{N} \Big[\, y_j \ln p_j + (1-y_j)\ln(1-p_j) \,\Big]

where pjp_j is what the venue said before market jj settled and yjy_j is 1 if it happened and 0 if it did not. Lower is better. Confident and wrong is punished hardest, which is the property you want: a venue that says 95% and is wrong should hurt more than one that shrugged and said 55%.

One discipline holds the whole thing up. No lookahead. A weight used to grade a Sunday game may only be fitted on markets that had already settled before that Sunday. Score a venue with weights fitted on the outcome you are grading and every number that comes out is beautiful and worthless. It is the same failure as step 7, one level up.

The method is on this page. The fitted weights and the correlation structure between venues are not, and that is deliberate: the technique is standard and the scoreboard is the part that took a season to build.

Building one fair line, end to end

Worked example: build a fair line around a line you trust

Trusted anchor: −148 (2.4% vig)  ⇒  fair 58.6%\text{Trusted anchor: } -148\ (2.4\%\text{ vig}) \;\Rightarrow\; \text{fair } 58.6\%
Book B: −150 (2.6%)  ⇒  fair 58.8%\text{Book B: } -150\ (2.6\%) \;\Rightarrow\; \text{fair } 58.8\%
Book C: −162 (3.5%, soft)  ⇒  fair 60.3%  (shades the fav up)\text{Book C: } -162\ (3.5\%,\ \text{soft}) \;\Rightarrow\; \text{fair } 60.3\%\ \ (\text{shades the fav up})
Two more soft books  ⇒  ≈60%\text{Two more soft books} \;\Rightarrow\; \approx 60\%
Naive average of all five  ⇒  59.7%\text{Naive average of all five} \;\Rightarrow\; 59.7\%
Log-odds pool, anchored on your line  ⇒  59.4%  (fair −146)\text{Log-odds pool, anchored on your line} \;\Rightarrow\; 59.4\%\ \ (\text{fair } -146)
The soft books lean the favorite up. Anchoring pulls the fair number back to 59.4% against the crowd’s 59.7%. Swap the anchor and the number moves. That’s why it’s the part worth earning.

The pool, on one page

De-vig every book, pool in log-odds, convert back. The two sides sum to 100% with no vig. The weights are where the judgment lives.

Where the technique comes from

None of this is ours to invent. It’s standard statistics. Taking the vig out with a model of insider-shaded prices is Shin’s method.[1][2] Merging the de-vigged lines in log-odds is a logarithmic opinion pool, the geometric-mean-of-probabilities rule.[3]

  1. Shin, H. S. (1993). “Measuring the Incidence of Insider Trading in a Market for State-Contingent Claims.” The Economic Journal 103(420), 1141–1153. The de-vig (Shin’s method).
  2. Shin, H. S. (1992). “Prices of State Contingent Claims with Insider Traders, and the Favourite-Longshot Bias.” The Economic Journal 102(411), 426–435.
  3. Genest, C. & Zidek, J. V. (1986). “Combining Probability Distributions: A Critique and an Annotated Bibliography.” Statistical Science 1(1), 114–135. The logarithmic opinion pool.

The papers give you the technique. Which live lines anchor the pool on our board, learned from closing-line value over time, is the piece we keep.

See the sharp consensus live on the board →Free. The board publishes the fair line, its confidence, and book agreement for every game.

The pool as a product, and a number for agreement

Two extras. Neither changes the recipe above.

First: each book’s de-vigged probability becomes a log-odds value zi=ln⁡ ⁣(pi1−pi)z_i = \ln\!\left(\dfrac{p_i}{1 - p_i}\right), and the pool averages those. Exponentiate the average and addition turns into multiplication. The pooled fair line is a weighted geometric mean of odds ratios, one factor per independent book.

Odds-ratio product form

pA1−pA=∏i(pA,i1−pA,i) ⁣wi/∑jwj\dfrac{p_A}{1 - p_A} = \prod_{i} \left( \dfrac{p_{A,i}}{1 - p_{A,i}} \right)^{\! w_i / \sum_j w_j}
Each book’s fair odds ratio, raised to its share of the weight. Multiply the factors and you get the pool’s own odds ratio, the same number Step 5 produces.

Odds ratios are multiplicative (a coin flip’s is 11, a heavy favorite’s is 99), so splitting differences on the probability scale is the wrong arithmetic. The product form also makes Step 4 visible: ten copycat feeds are the same factor repeated ten times, one opinion with an inflated exponent.

Why the product disagrees with the average

Book 1: fair 90%  ⇒  0.900.10=9\text{Book 1: fair } 90\% \;\Rightarrow\; \dfrac{0.90}{0.10} = 9
Book 2: fair 50%  ⇒  0.500.50=1\text{Book 2: fair } 50\% \;\Rightarrow\; \dfrac{0.50}{0.50} = 1
Equal weights  ⇒  9×1=3  ⇒  p=33+1=75%\text{Equal weights} \;\Rightarrow\; \sqrt{9 \times 1} = 3 \;\Rightarrow\; p = \dfrac{3}{3 + 1} = 75\%
Plain probability average  ⇒  70%\text{Plain probability average} \;\Rightarrow\; 70\%
Equal weights, for illustration only. The geometric mean respects the odds scale: 75% where the plain average drifts to 70%. Live weights are not equal.

To hand-check a small pool of two equally weighted lines: the square root of the product of their odds ratios.

Second, agreement. Step 6’s raw ingredient is the weighted spread of the books’ log-odds around the pooled value zˉ\bar{z}:

Dispersion (agreement)

σz=∑iwi (zi−zˉ)2∑iwi\sigma_z = \sqrt{ \dfrac{\sum_i w_i\,(z_i - \bar{z})^2}{\sum_i w_i} }
ziz_i is each book’s de-vigged log-odds, zˉ\bar{z} the pooled value. Small σz\sigma_z: the independent lines agree. Large: the market is genuinely split and the fair line carries real uncertainty.

Log-odds, not probability points, on purpose: a one-point disagreement at 95% counts for far more than one at 55%. The worked example’s five books run from about 0.350.35 to 0.420.42, so σz≈0.03\sigma_z \approx 0.03, a tight market. The 90%/50% pair gives σz≈1.10\sigma_z \approx 1.10. That’s wide. Run it before trusting two fair lines the same amount.

Every formula here lives on the Formula Sheet for quick reference.

Check your understanding

Frequently asked questions

How is a sharp-consensus fair line calculated?

De-vig every book, collapse copycat feeds, then average in log-odds around a line you trust, clipping outliers around that anchor, not the crowd. The two sides sum to 100% with no vig. The one judgment call is the anchor.

Why not just use Pinnacle as the benchmark?

A single book is one point of failure. It can be stale or shaded, and third-party Pinnacle data is legally fraught to source. A multi-book pool anchored on the sharpest lines removes that dependency and degrades gracefully as books come and go.

Why not just average all the books?

Soft books systematically shade the popular side, so a plain average gets dragged toward the recreational crowd. Anchoring the pool (and the outlier clip) on the lines that price honestly is what resists that pull.

Why is the fair line on a game different from the number behind its +EV mark?

Because a book cannot be graded against a pool it helped set. The printed fair line pools every venue; the number behind a +EV mark pools every venue except the one being marked, so that price is not measured against itself. The two normally sit within a point of each other, and the board names the venue it left out. A price is only marked +EV when it beats both readings.

Which books do you treat as sharp?

That’s the part we keep. The technique works with any anchor you trust. Which lines have earned the anchor on our board, and how much each counts, is learned from closing-line value and outcomes. It’s the edge we don’t hand over.