Who's Line is It Anyways?
Part 1 of a series on the algebra of bookmaking, lines, and sharp betting.
Most of the time here, we talk about metrics. That is, we’re mostly interested in ways we can measure things that happen in sport. Measurement is essential because it allows us to better understand what is happening in the games. Measurement is tied up intimately with our knowledge about sport. And taking a clear eyed look at measurement is how we take a clear eyed look at the games themselves.
This post is something different. This post is about the fundamental theory of betting and line-making. In it, we don’t talk about how to measure anything. Rather, we talk about how lines work. How oddsmakers set lines and how professional betters—which oddsmakers are—determine whether or not to bet the lines that are offered.
This post is very algebra heavy. You won’t have to solve any equations. But you will be challenged to think in terms of abstract values. We’ll even use some calculus terms like limit.
But the reward for using this framing is large.
If you can stick with this post and work through the algebra. You’ll understand at a fundamental level how oddsmakers are thinking about setting lines, how sharps—professional bettors—think about betting, and where to look for opportunities to make money betting.
If that’s something you want to do.
What’s a line?
The line for a bet is part way between truth of how likely an event is to occur and the widespread opinion about how likely that event is to occur.
Market theory says that enough informed participants in a market, the market should converge to the truth. However, sports betting neither has informed participants nor liquid markets.
Participants are emotionally invested in their teams, misinformation about sports teams relative strength is rampant in the media, and engagement typically happens at the qualitative level—only a few people dig into quantitative value.
Further, betting markets are not liquid. Big dollar, profitable sports bettors are blocked from placing wagers—or highly throttled in the degree to which they can wager. This means that the market—to the extent its a market at all—does not fully account for all of the information.
Oddsmakers don’t need to care about this.
Oddsmakers job is to set a line that will attract good action.
Setting the stage, and initial observations.
To understand how lines operate, we need to think of a few different variables:
t - the true outcome
q - the amount of public money in play
r - the amount of sharp money in play
aq , ar - The ratio of public (and sharp) money on the A side
v - the Vegas line
From here we can start drawing relationships between the variables.
As v gets farther from t, r increases faster than q.
As v gets farther from t, ar increases faster than aq.
This is to stay, because the sharps are by definition sharp, they will notice a line that is very wrong and bet it heavily.
As corollaries
The limit of s, as v goes to t, is 0.
The limit of ar as v goes to t, is .5
Which is to say, that as the oddsmaker gets the line more and more correct, the sharps will both dial back their betting and bet more evenly.
In contrast, we have two corollaries about public money.
The limit of q, as v goes to t, is greater than 0.
The limit of aq, as v goes to t, is greater than .5.
This is to say, public money will be in play even if Vegas sets the line perfectly.
But if oddsmakers do something extreme…
As the distance between v and t grows, q will increase and aq will approach 0 or 1.
When the line is very wrong, people will notice and take advantage.
Given that v≅t, the oddsmaker maximizes its profit when q and r are the largest.
Because of the VIG, oddsmakers have an inherent edge, so they will make the most money when q and r are large.
But because we also know that as v approaches t, r goes to 0. And oddsmakers wants r to not be zero. So v will not—in most cases—equal t.
The Vegas line will not in most cases represent the true likelihood of a team winning a game.
In fact, oddsmakers are incentivized to set v maximally far from t, such that it maximizes q+r, and keeps the quantity ar x r + aq x q - (1-ar) x r - (1-aq) x q below, say, 10%.
That is not always possible.
For example, consider last Super Bowl, where the money and the public heavily favored Kansas City. The Chiefs received 75% of the bets and 70% of the money—even though the relative strength of talent was on the 49ers.
In this situation, Vegas could hardly move the line to be more Chiefs friendly—this would expose them too heavily to sharp money. Rather, they tried to keep the line as close to “true” as possible and remove the sharp money from the field—leaving only the public money which is massively positive return for oddsmakers.
Kelly Strategy
But how does oddsmakers know what the pros are going to do?
Professional betters follow the Kelly Strategy, or similar rules, with their bets. The Kelly Strategy supposes that we can find an optimal size for our bets by comparing two quantities: the probability of winning and the ratio of return.
The higher the probability of winning a bet, and the higher the ratio of return, the more money we should wager on that bet.
Mathematically, a “Kelly Bet” is formalized as follows:
Where the terms are defined as follows:
k is the correct “Kelly bet”, or the fraction of your bankroll you should wager
p is the probability of winning the bet
b is the return on the bet
How are these numbers influenced by our oddsmakers numbers above?
We know that as the distance between t and v increases, that professional bettors put more money in play. This occurs because oddsmakers are either increasing p while holding b constant, or increasing b while holding p constant. The former condition occurs for spread bets, while the latter occurs for money line bets.
Consider a hockey game between the Buffalo Sabers and the Detroit Red Wings. We know that the true likelihood of Buffalo winning the game is 60% and that they should be favored by 1.5 goals.
If oddsmakers move their line from Buffalo being favored by 1.5 goals to only being favored by 0.5 goals, the Kelly Strategy tells us to bet this game more than previously because p is now higher than it was before. In this example, Buffalo had a 50% chance of winning by 1.5 goals and a 60% chance of winning by 0.5 goals. So p has gone up 10%.
Assuming a standard -110 VIG, this changes our Kelly Bet size dramatically. We go from a no-bet (-0.06) to a massive pro-Buffalo bet: 15% of our bankroll, or about 7 units!
If instead, oddsmakers move the line from the standard -110 VIG to +115, they have now changed b. As b increases from 0.91 to 1.15, our bet size increases from no-bet (still -0.06) to a meaningful bet of 3 units (0.065).
In fact, oddsmakers moving the money line from -110 to +105 is enough to tip a 50/50 proposition from a no-bet into a 1 unit bet. This type of change goes unnoticed by even regular public bettors—but it changes the math for the professionals and the expected return from a line for oddsmakers.
So, the oddsmakers find themselves in a position where they can control what the professional bettors are going to do. And they can exploit what the public is going to do through the VIG or minor line tweaks. Therefore, oddsmakers can move individual lines to make sure that each game satisfies their own Kelly principles.
After all, oddsmakers cannot control the outcomes of games any more than sports fans. In every game, there is randomness. What oddsmakers must do, then, to avoid going broke, is make sure that each line is Kelly satisfying. That the amount of money they are exposed for on a given line is is appropriate to the likelihood that they win.
This is what creates the Pros versus Joes dichotomy in sports betting. Oddsmakers use the Pros to cover the risk of the Joes.
Again, let’s take a look at the last Super Bowl. San Francisco should have been about a 3.5 point favorite, for implied odds of about 65%. This means that at even money—assuming for simplicity that books lose about as much on a KC win as they win on a KC loss—our Kelly Bet size should be about 15 units, or 30% of our total bankroll. Being a bit more conservative and assuming KC wins 40% of the time, our Kelly Bet size is still 10 units—20% of our bankroll.
So if we make the assumption that $200 million of the $300 million in Super Bowl bets FanDuel took last year was on the point spread, FanDuel would need to have around $400 million in cash to support the size of their position comfortably.
So what is a line, and what does that mean for you?
To oddsmakers, lines are the vehicles by which they adjust their financial stake in the outcome of sporting events.
This means that at times it is not only possible, but strategically imperative for an oddsmaker to have a line that is wrong — because having a line that is correct could be catastrophic to one’s business.
As an individual, you can take advantage of that by looking for lines that are wrong.