Mathematics of Bookmaking - Overround On Multiple Bets

Overround On Multiple Bets

When a punter (bettor) combines more than one selection in, for example, a double, treble or accumulator then the effect of the overround in the book of each selection is compounded to the detriment of the punter in terms of the financial return compared to the true odds of all of the selections winning and thus resulting in a successful bet.

To explain the concept in the most basic of situations an example consisting of a double made up of selecting the winner from each of two tennis matches will be looked at:

In Match 1 between players A and B both players are assessed to have an equal chance of winning. The situation is the same in Match 2 between players C and D. In a fair book in each of their matches, i.e. each has a book of 100%, all players would be offered at odds of Evens. However, a bookmaker would probably offer odds of 5-6 (for example) on each of the two possible outcomes in each event (each tennis match). This results in a book for each of the tennis matches of 109.09...%, calculated by 100 × (6⁄₁₁ + 6⁄₁₁) i.e. 9.09% overround.

There are four possible outcomes from combining the results from both matches: the winning pair of players could be AC, AD, BC or BD. As each of the outcomes for this example has been deliberately chosen to ensure that they are equally likely it can be deduced that the probability of each outcome occurring is 1⁄₄ or 0.25 and that the odds against each one occurring is 3-1 (3/1 or 'three to one'). A bet of 100 units (for simplicity) on any of the winning combinations would produce a return of 100 × (3/1 + 1) = 400 units.

As detailed below, the actual return on any of these winning doubles is obtained by multiplying stake × ('odds plus one' from each single bet) together. Thus for a stake of 100 units we get a return of 100 × (5/6 + 1) × (5/6 + 1) = 336.11... units, representing odds of 2.3611-1 which is far less than the true 3-1. Odds of 2.3611-1 represent a percentage of 29.752% (100/3.3611) and multiplying by 4 for the total number of equally likely outcomes gives a total book of 119.01%. Thus the overround has slightly more than doubled by combining two single bets into a double.

In general, the combined overround on a double (O_D), expressed as a percentage, is calculated from the individual books B₁ and B₂, expressed as decimals, by O_D = B₁ × B₂ × 100 − 100. In the example we have O_D = 1.0909 × 1.0909 × 100 − 100 = 19.01%.

This massive increase in potential profit for the bookmaker (19% instead of 9% on an event; in this case the double) is the main reason why bookmakers pay bonuses for the successful selection of winners in multiple bets: compare offering a 25% bonus on the correct choice of four winners from four selections in a Yankee, for example, when the potential overround on a simple fourfold of races with individual books of 120% is over 107% (a book of 207%). This is why bookmakers offer bets such as Lucky 15, Lucky 31 and Lucky 63; offering double the odds for one winner and increasing percentage bonuses for two, three and more winners.

In general, for any accumulator bet from two to i selections, the combined percentage overround of books of B₁, B₂, ..., B_i given in terms of decimals, is calculated by B₁ × B₂ × ... × B_i × 100 − 100. E.g. the previously mentioned fourfold consisting of individual books of 120% (1.20) gives an overround of 1.20 × 1.20 × 1.20 × 1.20 × 100 − 100 = 107.36%.