What is Vig and Overround?

(NB: I wrote another post on this topic in 2017, though there I use the alternative definition of overround)

Bookmakers are business people and the prices they offer include a profit margin, which is sometimes referred to as the 'vig' or 'vigorish' in the prices they offer. The vig is the guaranteed profit that the bookmaker will make provided that the prices offered attract wagers on all outcomes in the desired proportions (more on which later).

Overround is a related - and often conflated - term and is defined as the sum of the reciprocals of all prices in a given market. Some definitions of overround subtract 1 from this sum, mainly because it makes the maths a little simpler in some areas.

Mathematically, overround (o) and vig (v) are related by the following equations:

o = (2v - 1)/(1 - v)

v = (o-1)/o

(Note that here, in both equations, I've used the definition of overround that does not subtract 1.)

From a punter's viewpoint, betting markets with lower overround and lower vig are to be preferred as they offer higher rewards for a given level of skill or luck.

Empirically, the overround in a betting market tends to vary depending on:

The level of competition that a bookmaker faces. Competition drives vig down.
The number of outcomes on which bets can be made within the market (eg a line market has only two outcomes on which bets can be made, but a margin market may have 30 or more). More outcomes tend to lead to higher vig, probably as recompense for higher risk on the part of the bookmaker, real or perceived.
The remaining time between when the market is being viewed and when the contest about which the market has been framed takes place. Vig is a form of insurance and the further away in time a contest is, the less certain is its outcome, so contests further away in time tend to carry higher vig. For any single contest, bookmakers tend to start with markets incorporating higher levels of vig which they progressively reduce as match time nears.

In the line and head-to-head markets typical overrounds are 105 to 108%. In margin markets they can be 140% or higher.

Head-to-Head Market Example

We'll start with a simple example for a head-to-head market, for which we'll ignore draws.

Assume that the market is as follows: Team A $1.30 vs Team B $3.55. The overround for this market is 1/1.30 + 1/3.55, or 105.1%, so the vig is (1.051-1)/1.051, or 4.85%.

In what way is that 4.85% a profit margin?

Well, if the prices of $1.30 and $3.55 induce wagers on each team equal to the reciprocal of the relevant price times the overround then the bookmaker will keep $4.85 of every $100 wagered, guaranteed.

The proportions required are 1/(1.30 x 1.051), or about 73%, on Team A, and 1/(3.55 x 1.051), or about 27%, on Team B.

So, imagine that $100 has been wagered in total in this market, $73.20 on Team A and $26.80 on Team B.

If Team A wins, the bookmaker pays out $73.20 x 1.30 and retains the $100 outlayed for a total profit of $4.85
If, instead, Team B wins, the bookmaker pays out $26.80 x 3.55 and retains the $100 outlayed for a total profit of $4.85.

(Incidentally, the two proportions used above to determine the desired amount to be wagered on each team from a bookmaker's viewpoint are also a measure of a team's implicit probability of victory.)

(Simplified) Margin Market Example

To convince ourselves that the relationship between overround and vig holds for markets where more than 2 outcomes are possible let's consider the following, more complex, margin market.

Team A wins by 30 points or more: $4.00
Team A wins by 1 to 29 points: $2.00
Drawn game: $51.00
Team B wins by 1 to 29 points: $3.50
Team B wins by 30 points or more: $8.00

The overround for this market is 1/4.00 + 1/2.00 + 1/51.00 + 1/3.50 + 1/8.00, or 118.0%, so the vig is (1.18-1)/1.18, or 15.3%.

From the bookmaker's viewpoint, the optimal wagering proportions are, again, the reciprocal of the price of the relevant bet multiplied by the overround, which gives:

Team A wins by 30 points or more: 1/(4 x 1.18) = 21%
Team A wins by 1 to 29 points: 1/(2 x 1.18) = 42%
Drawn game: 1 /(51 x 1.18) = 1.7%
Team B wins by 1 to 29 points: 1/(3.5 x 1.18) = 24%
Team B wins by 30 points or more: 1/(8 x 1.18) = 11%

Again let's assume a total wagering pool of $100.

Now, if Team A wins by, say, 40 points, then the bookmaker pays out $21.18 x 4 and retains the $100 wagering pool for a net profit of $15.28. The return to the bookmaker therefore is 15.3%, which is the vig.

If you perform this same calculation for the other possible outcomes you should come up with the same net profit.

To reiterate then, a market with an overround of o will produce a guaranteed profit of v for a bookmaker provided that the wagers on each outcome are ideally apportioned.

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What Is Vig and Overround?