content below posted 00:05SUN04AUG02 cst
Copyright Crescent Connections, 2002

Know the Vig:
A Mathematical Analysis of Sportsbook Odds in a Parimutual Setting

For the purposes of this discussion we define the take as the total dollar amount collected by the
house for a particular sporting event. We define the return as the dollars paid to the winner(s) of
such an event. The cut is the take minus the return and the vig is the cut as a percentage of the take.

If you are like us, you want to know what you are paying for the privilege of gambling. Most bettors do. We all know the cost of betting against the point spread. We lay down 11 to win 10. The cost of playing, the vig, is the percentage of the take that the house keeps before it returns the cash to the winners. For a spread bet, the vig is 4.55% because for every 22 units the house collects, it returns 21 to the winners and keeps 1 for itself. 1/22 equals .0455.

But what about those futures bets that we all play before the season starts? You know the ones, where all the teams are listed with particular odds of reaching, say, the Super Bowl or the AFC championship or the NBA finals. These are, in fact, parimutual events, where the public lays down a certain amount on each entry in a race toward a pre-established goal. The house sets the odds for each entry based on how much the public bets. The idea is equivalent to horse racing, where by law the track pockets between 11 and 14 percent, and the odds for each horse are determined by the remainder of the pool. The big difference in other sports, particularly in offshore betting, is that there are no such guidelines. We don't know what the vig is.

Let's take a look at a set of odds that were kindly published by our friends at a well-established offshore sportsbook. These are the odds for the AFC championship race.

Odds of Winning the 2002-2003 AFC Championship
Pittsburgh Steelers4/1
Miami Dolphins11/2
Indianapolis Colts6/1
Oakland Raiders6/1
Denver Broncos8/1
New England Patriots8/1
Tennessee Titans10/1
Cleveland Browns11/1
Buffalo Bills15/1
New York Jets18/1
Kansas City Chiefs20/1
San Diego Chargers25/1
Baltimore Ravens30/1
Jacksonville Jaguars30/1
Cincinnati Bengals35/1
Houston Texans75/1

As a gambler, the first thing you notice when you look at a list like this is that some teams are way undervalued. We're not here to talk about that. Let's leave that for another day. Today, we want to know what the vig is. What's the cost of doing business? At first glance it's impossible to tell because we only know the odds. We can't tell how much the house is holding back.

If you look a little closer you can see that the information is there, its just buried in the odds. The key in determining the vig is to realize that the odds are skewed for each entry in the list so that for any entry that wins, the house will still keep the same cut of the total take. Of course, we'll never know what the actual take is, or the return for that matter, but we can come up with theoretical numbers, representations of the actual take and return in units. First, we look at the return.

To determine a theoretical return we simply look at the longest odds on the board, and then assume that one bettor bet one unit on the entry. If the odds of the longest shot are 120/1, for instance, then we can quantify the return as 121 units - the one unit our bettor played plus the units returned to him by the house if he wins.

The theoretical return is a factor of the actual return but we have no idea what the multiplier is. In other words, we don't know what units we are using because we don't know how much cash is bet on the longshot. We simply used a value of one unit as our bet amount. In actuality, our unit is the total dollar amount wagered on the longshot. The good news is that we don't need to know what our unit is because we aren't interested in finding out how much was bet, we just want to know the vig.

Having a return puts us halfway there but we are still left with determining the theoretical take. Since we know the return, we can determine how much, in units, each group of players put up for each entry in the field. To do this, we look at the odds of each entry. If one entry has odds of 10/1, we know from the fact that since 121 units are being returned, that the total number of units bet on that entry is 11, the number needed to assure that the bettors get their money back plus the remainder of the return at the specified odds. From the example cited it can be derived that the units bet on any entry in the parimutual field equal the return divided by 1 plus the odds, or in this case, 121 divided by 11 (1 plus 10).

We now have the information needed to determine the theoretical take. Remember, as we stated before, the odds for each entry are skewed equally by the house with respect to the vig, so when we add up all of the units for each of the entries we have an accurate representation of the total house take, in units. For posterity we offer the following mathematical formula for determining the take based on any set of odds in a parimutual setting.

Once we have the take, it's easy to subtract the return to get a representation of the house cut, again in units, and its a simple matter to divide the cut by the take to determine the vig.

Let's review the steps needed to determine the vig when given the odds in a parimutual setting:

  1. Assume one unit is bet on the longshot, then use those odds to determine the return, in unknown units.
  2. Use the return from step 1 to determine the units bet for each entry.
  3. Add up the units bet for all of the entries to determine the take.
  4. Subtract the return from the take to get the cut
  5. Divide the cut by the take and multiply by 100% to determine the vig.

Now lets walk through a simple example to serve as both a guide and as a proof. Consider a parimutual event involving 3 entries, A, B, and C, where the dollar distribution and the take is as follows:

A $90
B $60
C $30

Take = $180

This is the information the house sees. Now lets suppose that the house determines that their vig should be 1/6, or 16.7 %, and they cut $30 from the take and return $150 to the players. Their adjusted odds read as follows:

A 60/90 = 2/3
B 90/60 = 3/2
C 120/30 = 4/1

Now we'll apply our method. First, we determine the return, in units, by adding 1 to the longest odds. For this example we add 1 to 4 to set the theoretical return at 5. Then we calculate the units wagered on each entry as follows:

A @ 2/3 => 5 / (1 + 2/3) = 3 units
B @ 3/2 => 5 / (1 + 3/2) = 2 units
C @ 4/1 => 5 / (1 + 4/1) = 1 units

Take = 6 Units

We see that our theoretical take is 6 units, and given the return of 5 units we calculate the cut as 1 unit. So we end up with an actual vig of 1/6, or 16.7%. We have determined the vig just by looking at the odds, without knowing the actual take or return. In fact, though we know from the example that our unit is $30, we'll never know what it is in actual practice.

A neat side effect of this analysis is that we can also determine the true odds of each entry in the race. Since we have a representation of both the take and the units bet we can reset the odds back to the public's actual preference. In the case of our example, we can see the true odds, at a vig of Zero:

A a 3 unit bet nets 3 = 1/1
B a 2 unit bet nets 4 = 2/1
C a 1 unit bet nets 5 = 5/1

We'll leave it as an exercise to compare these odds to the true odds dictated by the original dollar distribution from our example.

Now for some real fun. Lets go back to the race for the AFC Championship and apply our method to determine the actual vig percentage charged by the sportsbook, and the true odds, as determined by the public. We'll spare you the calculations and present the info in tabular form.

Odds of Winning the 2002-2003 AFC Championship
Pittsburgh Steelers4/115.211/2
Miami Dolphins11/211.715/2
Indianapolis Colts6/110.98/1
Oakland Raiders6/110.98/1
Denver Broncos8/18.411/1
New England Patriots8/18.411/1
Tennessee Titans10/16.914/1
Cleveland Browns11/16.315/1
Buffalo Bills15/14.721/1
New York Jets18/1425/1
Kansas City Chiefs20/13.627/1
San Diego Chargers25/12.934/1
Baltimore Ravens30/12.540/1
Jacksonville Jaguars30/12.540/1
Cincinnati Bengals35/12.148/1
Houston Texans75/11101/1
Return => 76
Take => 102
Vig => 25.5%

There you have it. At this sportsbook the vig you pay for futures in the AFC Championship race is 25.5%. Is it fair? We'll leave it up to you to decide. We will reveal that we have performed the analysis on several sets of odds from several sportsbooks and the vig seems to converge to 35%, so this list is pretty impressive. We saw one set that topped 50%, and another that was as low as 12%. Can you imagine the ramifications if you applied this method and ended up with a negative vig?

Anyway, as we said before, the first thing that jumps into your mind when you see a futures list is that some teams appear undervalued, so go ahead and bet to your heart's content. Just remember, you now have the knowledge to determine the vig.