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    My training is as an actuary – a skill set that finds its natural application in insurance. You might ask yourself what do insurance and betting have to do with each other? A type of insurance cover you might buy pays out a sum to your loved ones (or to your cat/dog/mum if you are alone) in case of death by end of year. What are the essential characteristics there? Well the probability of death and possibly the timing. Now consider a bet on whether a manager will be fired by end of season – the mathematical underpinnings are fairly similar. Indeed the concept of insurance developed as a subsidiary of betting – Lloyds was a coffee shop where punters used to bet on which ships will make the trip. This quickly was adopted as a form of insurance.

    What the hell is a bet?

    If you are good in high school / sixth form math, you can just go through this article I published (Shameless self promotion) but I will evade the math in my examples.

    To be able to understand what a bet is, I trust the best thing is to understand how a bookmaker makes real money. Let us use the example of a lottery instead as a simpler representation.

    Consider a lottery with ten tickets, each costing €1 (or £1). Let us assume there is only a prize for the winner and that all tickets are mutually exclusive, that is only one will win. Hence the total collected is €10. A lottery organizer would not give out €10 in winnings but less, let’s say €8. So the lottery organizer would receive €10, pay out €8 and profit €2 which is 20%.

    Now let us assume we have a horse race with ten horses. Let’s also assume all are equally strong and all are equally popular – so €1 was placed on each. The outcome is also mutually exclusive (only one horse will win) and the bet organizer would give out €8. The 20% is the profit made by the bet organizer, well the bookmaker.

    From the view of a bettor, they would have bet €1 with a potential return of €8 – earning €7 on top of the original €1 bet. This translates to European odds of 8 or British Odds of 7/1. European odds, also known as decimal odds, show the return on a one euro investment. British odds show the “profit”/”stake” ratio and are also called fractional odds.

    For example 1.5 means that for every €1 bet, you will get €1.50 back if the outcome occurs – essentially winning 50c. That means that for ever €2, your potential win is €1. The British odds would then be 1/2.

    There are also American odds as well as a multitude of Asian style-odds. However as the online betting market in the US is not as well developed and most Asian institutions tend to use decimal odds; a discussion of these would be superfluous.

    I will focus on decimal odds as a chief example but write fractional odds in brackets for those who are more familiar with that style. I find decimal odds easier to deal with as their inverse shows the implied probability (or maybe I am biased because not pro-Brexit!).

    Let me go back to the horse race example. If the bet was fair, the winner would get €10. The reason is that every horse has a one tenth chance of winning. It is basically given that ten horses multiplied by a tenth each is one whole. The sum of all probabilities should be one whole.

    Yet the odds being offered to the bettor are 8 per horse. The sum of all implied probabilities is ten horses multiplied by one eighth, which is 1.25. The sum of all probabilities here is more than one. That discrepancy (the 0.25) is called the betting margin or the vig or the spread. Whatever you want to call it, it shows the “unfairness” of the market itself. The larger it is, the higher the profit for the bookmaker.

    And it can never be negative, otherwise the bookmaker is expected to make a loss.

    Before you go all out and open your own betting shop, it is not as simple as creating a high betting margin. If it is too high, the odds are not competitive enough and people would simply bet with a competitor. Moreover, the bookmaker has no control on where the bets are placed, they could have just had one person bet €1000 on the Horse A and no bets elsewhere. In this case, the bookmaker is still expected to make a profit on average but can either end up losing €7,000 (if that horse wins, with a probability of 1/10) or profit €1,000. Hence the bookmaker is more interested in getting a balanced book, that is bets that are placed throughout the market, so as to get certain profits.

    The bookmaker needs to be a bit clever in calculating the true probability for each horse, their popularity and what the competition is offering. The good news is that you can somewhat use this to your advantage. Patience you must have my young padowan and read the rest of this.

    Value Bets

    A bookmaker’s aim is to balance the book – that is keep a number of bets on different outcomes such that they make a certain profit. Hence, one must think that there may be bets on which one can make a profit. The answer is yes and no, but mostly no.

    A bet which is worthwhile to place a bet on is called a value bet. The problem is that usually we use the benefit of hindsight to confirm our belief on whether it was a value bet. In simple terms a value bet is when the odds on that outcome offered on the betting market is higher than the actual perceived odds. Say for example the odds on a “Horse-with-no-Name” are 8 (7/1) but the actual probability of the Horse-with-no-Name is a fourth. Then the true unbiased odds should have been 4 (3/1). As the odds offered by the bookmaker are higher than the actual odds, one should bet on this horse. Indeed if this horse races in four similar matches, you are expected to win one. That is you would have bet €1 for four times, a total of €4, and received winnings of €8 – quite a hefty profit (100% on your bets). Of course the horse could have not won any of the four races, or it could have won all four. Well, the true probability of the horse’s chance is hard to evaluate even in hindsight.

    So there we have it, if the odds are higher than expected – you should bet on it! No Sh*t Sherlock.

    The difficulty and skill lie in deriving true odds for an outcome an event. If you have those skills, you might be better off working as a quantitative analyst with a bookmaker rather than betting yourself so do treat this approach with caution. However in essence, if you believe that some odds are too high, then it is worthwhile to bet on.

    Moreover the higher the discrepancy, the higher you could bet (always subject to what you can afford). So for example if the odds offered on a horse to win a race are 8 (7/1) but their true probability is one fourth (odds of 4 or 3/1) when you should bet an amount. Similarly you should bet on this horse if the probability of a win is two thirds (odds of 1.67 or 1/2) but if this was the case, you should bet even more as the discrepancy is higher. How much to bet is a subject of debate, but many tend to use some factor of the Kelly Criterion.

    Arbitrage and Cumulative Staking Methods

    We have established that value bets exist – you can find bets which are worthwhile enough to bet on despite the margin that bookmaker adds to odds to make them unfair. It is difficult to calculate and really depends on how efficient the market is.

    Market efficiency has been an academic interest for economists and financial analysts since its inception. This theory (or rather definition) eventually won a Nobel prize for categorising economic markets in three categories of efficiency : strong, semi-strong and weak. In strong markets, the value of any financial product (be it anything from an equity to a credit default swap) is fair – that is it reflects all public and private information. Practically this means you cannot beat the market. On the other side of the scale, a weak market does reflect the public information. Whether financial markets are efficient is subject to debate. Yet some of you might have heard of arbitrage in financial markets. Let me explain it to you using a simple example and see how this could translate to betting markets.

    Imagine someone wants to buy burgers for €3 each and someone else is willing to sell them for €2. Let’s assume there are no transaction charges and no other costs. It seems an easy profit here, get the buyer to pay you for 100 burgers and the seller to sell you the same amount (get them delivered to the buyer directly while at it). You’d pay out €200, receive €300 and profit €100. If you do it instantaneously – that’s arbitrage! It’s important that you are not holding the burgers, it’s not that you bought them, kept them and then sold them. Essentially arbitrage is free money, subject to some credit risk.

    How could this translate to betting markets? If you recall earlier, I mentioned that the sum of mutually exclusive outcomes should be greater than one for the bookmaker to make a profit. However different bookmakers may have different odds for the same outcome. So assuming we can pick the best odds from different bookmakers – if their addition is less than 1, then there is arbitrage.

    This concept is easier to follow with an example. Assume an event with three outcomes – say a soccer match that can end in a draw, win for the host (we call this a home win) or loss for the host (we call this an away win). Let’s say you can find odds (from different bookmakers) of 3.5 (5/2), 2.5 (3/2), 4 (3/1) for home win, draw, away win. The sum of probabilities is 1/3.5+1/2.5+1/4=0.9357. As this is lower than one, you can place a set of bets that guarantee you a win. Indeed if you place €28.60, €40, €25 on home win, draw and away win; you would have bet €93.60 but would win about €100 whatever the outcome. You can read the underlying mathematics here: www.sciencedirect.com/science/article.

    If this seems too easy, it’s because it is. It can exist, especially for some small markets where bookmakers may have had a big bet on an outcome and want to balance the book but do not expect it to be for a long time. Firstly there are many automated bots that scour the market for such cases (some sell their service but you’d need to be online at that time and have an account with the betting companies mentioned). Secondly most bookmakers receive a feed of odds for events rather than calculate it themselves and then adjust the odds according to their risk profile and market. That means that odds tend to be fairly similar (but not exactly the same as discussed later.

    Finally any bookmaker’s odds that are out of line in the market would be subject to arbitrageurs so they would be taking too much risk and not balancing their book. One also needs to consider that the bookmaker with very high odds might have an error in their odds and they just might cancel the bet prior to the event.

    Practically I am saying it’s not worthwhile. Do note that arbitrage methods for casino games means a completely different thing. It refers to staking methods to use in a casino (in no way am I recommending these). Similar staking methods are sometimes suggested for sports betting too. One technique is a Fibonacci, in which you use a series of stakes to bet until you win. Let me simplify it, it could be double your stake until you win on odds greater than 2. So assuming you lose five in a row but win the sixth one. You would have bet €1, €2, €4, €8, €16 and €32 on the first, second, …, Sixth outcome respectively. Let’s say the bet was on odds of 2.1 (11/10). Your profit is €32×2.1- (1+2+..+33) =€4.2. These methods eventually make you lose all your money once you get a run of bad events and you’d need a very large bank roll to sustain. The bookmaker probably has more money than you, so I wouldn’t bother.

    Now that I have commented that arbitrage isn’t worth pursuing and that cumulative staking methods are a gimmick, let’s discuss where there are potential avenues.

    Where can I find an edge?

    The efficiency of betting markets has been investigated academically as betting markets could be considered to represent financial markets in microcosm.

    The main bias that one notices is the favourite-longshot bias. In this academic paper in which we investigated about over one 160 thousand odds, we noticed that high odds tend to lead to bigger losses. That doesn’t mean that low odds lead to profit but a lower loss. This bias has been proved in many other papers, that essentially betting on long-shots is likely to lead to bigger losses – that is the odds on unlikely events are less fair.

    There are a multitude of explanations for this, which are beyond the aims of this article. However one rationale is that we are less able to understand events with very small probabilities and we tend to prefer to have a story to tell. That is we are keen to place a bet on high odds, even if it should be higher.

    The inability to understand unlikely events is taken a big advantage of in multiples or accumulators (see more here). An accumulator is one bet that pays of if a series of outcomes occur. Since there could be a number of outcomes, say 10 matches – the final payout would be very high. If odds are 2 (1/1) per outcome, the final payout would be 2×2×2…=1024 (1023/1). The expected profit for bookmakers if everyone bets on multiples/accumulators is much higher, i.e. it’s not worthwhile.

    Recent offerings pay out on a series of outcomes within the same event, for example Cristiano Ronaldo scoring twice while Juventus winning. The outcome tends to appear more likely but the odds typically offered are fairly lower than the true fair odds.

    We have established that bookmakers tend to have similar odds. However they tend to adjust according to their markets. For example a German-based bookmaker may have better odds for local matches as it needs to be competitive but lower odds on Germany winning a world cup as it is bound to get more bets on that than say in England. This discrepancy is unlikely to be high enough to lead to arbitrage.

    Odds also tends to change over time as new information is processed. It is therefore expected that odds closer to the start of an event tend to be better predictors of the outcome. If you see odds on an outcome going down, it may mean that it’s likely and worth a punt.

    Conclusion

    In conclusion it seems that the best items to bet on are boring outcomes with low odds, ideally just before the event starts. For example draws in soccer are considered boring and the hardest to predict. Similarly betting on a low amount of goals scored tends to be less lossmaking than betting on many goals.

    Yet, where is the fun in that? As a Leicester city FC (lcfc) supporter I must mention our story – we won the league despite odds of 5000:1. Now it was close to impossible to predict that at the start of the league. However mid-way through the English Premier League of that year, LCFC were top of the league and with great stats. Despite that, they were still not considered as favourites to win the league. No one could imagine it and the market was badly priced. I noticed similar effects during the world cup in Brazil, where the majority were expecting Brazil to win despite they were not impressing (https://www.um.edu.mt/article).

    So you don’t have to be just place boring bets, don’t be afraid to go against the grain. The most important part is, enjoy it and always bet responsibly.

    Simple Betting Guide Video

    MrCasinova.com created a video where sports betting is simply explained. Watch it below!

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    About the author
    author
    Dominic CortislinkedinAcademic Contributor

    Dominic Cortis PhD is an associate actuary and his research focuses on sports analytics as well as financial and betting derivatives. He is a lecturer in Actuarial Science and Insurance Risk at The University of Malta. If you would like to sleep, feel free to stalk his google scholar account and read his academic work (https://scholar.google.com/citations?user=Sl8_NpwAAAAJ&hl=en). He worked as a live sports trader early in his career back when odds were changed manually. He is mainly interested in solving problems using numbers, in whichever setting. These days most of his work is related to iGaming and insurance projects. Dominic sides with Leicester City FC and lived in Leicester during their mythical Premiership win. He is terrible at five-a-side football (despite owning a Maradona shirt) but cooks mean pancakes.

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