**Basic concepts, bookmakers, vig and odds**

Betting is nearly as old as man. One could go as far as to say that betting is part of human nature. The game of Morra - with its origins in Ancient Egypt -, casinos, and predictions about sport are all part of this diverse world. Betting also appears in philosophy, including in the extremely well-known work of French mathematician and philosopher Blaise Pascal, who argued it was better to wager on the existence of God. In a way, his concept is useful. As Pascal himself wrote "...if you win, you win all; if you lose, you lose naught. Then do not hesitate, wager that He is..."

Here, we deal with far less weighty matters, but it is still worth spending a while on them, without getting into a debate about a concept that clearly has its fair share of detractors.

A bet involves a wager - that is, the thing you place or the potential loss - and the winnings, which changes depending on the rules governing the bet, received by the winner as compensation for the risk taken, that is, for the potential payoff.

Pascal argued that by betting that God did not exist, one lost everything (i.e. eternal beatitude and infinity) but won nothing. Hence, not having faith was, in essence, a less favourable bet.

There is a fairly well-known idiom that goes "the game isn't worth the candle". This is used when the required sacrifice (the wager) will not produce, even if one wins, a proportional benefit (winnings). Expressed in our context, one could state that the risk taken is not suitably compensated for by the potential payoff or gain.

The risk taken is simply the probability of losing, that is, the outcome is contrary to what the punter hoped. We now have our first, fundamental concept for betting: probability.

In some cases it is very easy to calculate the probability of a given event occurring and, consequently, what potential payoff would justify the bet or the risk taken.

For example, when tossing a coin, we know that the probability that the coin will land on one side or the other is the same. If this is expressed as percentages, then it is 50% for heads and 50% for tails. It is thus fairly easy to see why, for every Euro bet, the potential payoff must be equal to at least one Euro. "At least", the first problem. When the potential payoff covers the risk taken, one talks about a *fair bet*. Sticking to the example above, if the potential payoff is €0.90, this would make the bet *unfavourable*, while a potential payoff of €1.10 would make it *favourable*. The law of large numbers teaches us that, by tossing the coin an infinite number of times, it will fall 50% on one side and 50% on the other side.

Thus, bets are favourable or not even before one wins or loses them. The potential payoff of a bet is determined by the *odds*. The odds are expressed in various ways, but they indicate, in relation to the wager (or bet), the amount of the payoff.

In number-based games - the lottery or roulette - one can calculate exactly the probability that each event has of occurring (e.g. drawing a number or a series of numbers) and thus also what are fair odds for each scenario. However, it is well known that in such games, the casino (*bank*) holds the advantage. Why? Simply because the odds offered are not fair and thus ensure the person betting is always in an unfavourable position. The risk taken by the punter will never be fairly compensated and the difference between the probability that an event occurs and the odds paid is the bank's vig (or vigourish). The vig is thus nothing more than the margin of advantage that the house keeps for itself and ensures, with extremely large numbers, enormous profits. Since the bank's vig is determined mathematically, there is no sure way of winning such games consistently.

Let's look as some examples. The payout for hitting a number in roulette is 35 times the amount wagered. Using the most common forms for expressing odds, one could say that it is paid at 35/1 or 36.00. This means, for every €1 bet, one receives €35 back plus the €1 bet. However, there are 37 and not 36 numbers on the table. Fair odds would be 36/1 or 37.00. Thus, in this case, the bank's advantage over the gamer is 2.7%. If the ball is spun an infinite number of times, all of the numbers will come up the same number of times and so the bank will always have an advantage of 1/37. In other words, it will win one unit for every 37 spins of the roulette, which is exactly 2.7%. It might seem like a very small margin - when compared to the margin enjoyed by the bank in other games, like the lottery, it is - but it allows casinos across the world to make huge sums of money because of the enormous number of bets placed.

Wasn't this supposed to be an introduction to sports betting? Definitely. I thought, though, that it would be useful to cover some of the fundamental concepts before looking in detail at the specific nature of betting on sports.

Regulated sports betting can be divided, roughly, into two main categories:

**Pari-mutuel or tote betting** is normally used for horse and dog racing or for Jai Alai, but can potentially be used for any type of event. Football pools fall into this category. In this case, the amount wagered is set, but the payoff is not. The odds are thus not set prior to the event, but only afterwards. All the bets placed are pooled together to make the total prize money, from which the percentage owed to the bank (20-40%, sometimes even over 60%) is taken. The remainder is then divided among the winners, meaning the odds, while still existing, are only calculated afterwards and will be higher the more unlikely, according to the punters, the eventual outcome is.

**Fixed-odds betting**, which is the main focus of this article, is the type of bet that was referred to earlier, where the odds are set in advance and thus become the agreement between the bank (or bookmaker) and the punter. In this case, the punter or bettor knows what he can lose and win, if the outcome is right. As we noted before, the odds can be expressed in various ways, without the substance changing one bit. The main ones are the decimal (mainly used in Europe), fractional (mainly used in the United Kingdom and Ireland) and American formats (common in the US), although a number of other varieties - Malay, Indonesian, Hong Kong - are also found. For more information, please read **this article**.

The key difference between sports and number-based betting is that, for the former, it is not easy to calculate the probability of an event occurring. Indeed, it is impossible to determine in advance the probability that, for example, AC Milan will beat Inter-Milan in the Milan Derby or that Barcelona will win the Champions League. It is possible to make assessments or estimates.

The job of doing this falls to bookmakers, or more precisely, to the oddsmakers. Oddsmakers are normally genuine sports experts who work for bookmakers and often work under a risk manager who decides on the financial exposure for this or that event or for specific customers.

Yes, but the vig? The bookmaker's favourable position? This clearly exists and can change, theoretically, from 1/2% to 40/50% and even more on certain more complex events. The advantage is expressed through the bookmaker's margin. The odds, as was noted, express the probability that an event occurs. The odds are linked to a given probability that, in general, is expressed as a percentage.

For odds in decimal format, the basic formula for converting the odds into the related percentage is:

1/o where o is the odds in decimal format

For example, for the odds 3.00, the percentage is 1/3.00 = 33.33

Thus, taking a hypothetical Milan Derby with three possible outcomes and assigning these the same likelihood, then fair odds are worked out by dividing the total of the probabilities (100%) by 3.

Therefore, 100/3 = 33.33

As we have seen, 33.33% is expressed by the odds 3.00, meaning fair odds for the Inter-Milan match would be 3.00 3.00 3.00.

However, if a bookmaker were to accept bets on a fair odds basis, it would have no mathematical advantage. Yet, it also has such a margin, like a casino, although with a different implication.

Sticking to the Inter-Milan Derby, the odds, minus this margin (vig), could be:

2.90 2.90 2.90

The formula to calculate the bookmaker's vig is:

1-{100/[(100/Odds1)+(100/OddsX)+(100/Odds2)]}

thus we have:

1-{100/[(100/2.90)+(100/2.90)+(100/2.90)]}

or 1-0.9666 = 0.0333

In this case, the vig, expressed as a percentage, is 3.33%.

In practice - although this is very rare - if the bookmaker received the same amount bet on each of the three options, e.g.

Inter €1,000 at 2.90

Draw €1,000 at 2.90

Milan €1,000 at 2.90

it would take in a total of €3000 and it would payout, in each case, €2900 with a profit of €100, which is precisely 3.33% of 3000.

As I said, this is not common, but it is the basic mechanism underlying bookmaking.

How can one know if the bookmaker's assessment is correct? One cannot know in advance. Indeed, the major difference between a casino and a bookmaker is that, despite both beginning with an advantage, the first has the mathematical certainty of keeping its margin, while the latter faces the risk of loss. Particularly sharp and well-prepared punters can beat the bookmaker by exploiting any errors - small or big - which are feared by bookmakers the world over. Thus, in sports betting, the punter has the real possibility of winning. Clearly, this relates to a small group of particularly skilful, clever and informed people. Indeed, one calls them professional punters.

Just as a punter can come to ruin by losing everything - or building up debts - so too can a bookmaker go bankrupt. It's not common, but it can happen. By contrast, a casino cannot fail unless the amount of gambling done there is so low that is does not generate enough income to cover the substantial management costs.