#Moneypuck Wednesday. How bookmakers make money using probabilities.

21 Nov

The blog post will be a little slow over the next few weeks because of final exams.  My plans to pursue some research to determine winning or losing teams will have to hold off until after 13 December.

However, I was thinking that I could still do some reading and sharing in the meantime.  Thinking of probabilities, I wondered how casinos determined their odds (probabilities of a team winning) and how they made money off of it.  So, I went to everyone’s favorite research site…Wikipedia.

My thought was that if they enticed people more with giving the team most expected to lose really good odds, then people would feel like taking a risk would be more worthwhile because of the payoff to loss ratio (risk taking is a whole subfield on its own in economics).  But, the bookie/casino would lose a ton of money if the underdog won and the payoff odds were too disproportionate.  I suppose there is an equilibrium between risk taking and odds making.

Anyway, here are the basics on odds….it is the probability a team will win based on certain situations:

In considering a soccer match (the event) that can be either a ‘home win’, ‘draw’ or ‘away win’ (the outcomes) then the following odds might be encountered to represent the true chance of each of the three outcomes:

Home: EvensDraw: 2-1

Away: 5-1

These odds can be represented as relative probabilities (or percentages by multiplying by 100) as follows:

Evens (or 1-1) corresponds to a relative probability of 12 (50%)
2-1 corresponds to a relative probability of 13 (3313%)
5-1 corresponds to a relative probability of 16 (1623%)

By adding the percentages together a total ‘book’ of 100% is achieved (representing a fair book). The bookmaker, in his wish to avail himself of a profit, will invariably reduce these odds. Consider the simplest model of reducing, which uses a proportional decreasing of odds.

The not-so-odd fact is that most oddsmakers do not work with a fair book, but they work with the concept of an ‘overround’.  Check out this example:

Home: 4-5
Draw: 9-5
Away: 4-1
4-5 corresponds to a relative probability of 59 (5559%)
9-5 corresponds to a relative probability of 514 (3557%)
4-1 corresponds to a relative probability of 15 (20%)

By adding these percentages together a ‘book’ of 1111763%, or approximately 111.27%, is achieved.

The amount by which the actual ‘book’ exceeds 100% is known as the ‘overround’:  it represents the bookmaker’s potential profit if he is fortunate enough to accept bets in the exact proportions required. Thus, in an “ideal” situation, if the bookmaker accepts £111.27 in bets at his own quoted odds in the correct proportion, he will pay out only £100 (including returned stakes) no matter what the actual outcome of the football match. Examining how he potentially achieves this:

A stake of £55.56 @ 4-5 returns £100.00 (rounded down to nearest penny) for a home win.
A stake of £35.71 @ 9-5 returns £ 99.98 (rounded down to nearest penny) for a drawn match
A stake of £20.00 @ 4-1 returns £100.00 (exactly) for an away win

Total stakes received — £111.27 and a maximum payout of £100 irrespective of the result. This £11.27 profit represents a 10.1% profit on turnover (11.27 × 100/111.27).

In reality, people use models of reducing more complicated than the model of “ideal” situation.

Sneaky!  The books are cooked!

Remember that when you are betting on games that you are not looking alone on the best probabilities, risk taking behavior and good payoffs.  Your bet is part of a larger over round scheme put together by smart math folks that are going to make money for the casino…and most likely make you lose yours.  Not to mention, the profits are made off by the combination of games and odds in a certain period of time.  That means there is the crazy combination of cooking the books for a whole set of games in all sports where profitability is maximized overall for the entire set of games.

On that note, anyone want to put a friendly wager on an over/under on the NHL agreeing to terms to end the lock out by the end of the year?

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