One of the core businesses of Bayes Esports is creating betting odds for our shareholder Sportradar. At least, that's what we usually say to keep things simple. The reality is, we have never created odds in our entire existence! Instead, we deal with probabilities, true and perceived. So what's the difference between probabilities and odds?

Let's start with the most straightforward one. Most people have learned about probabilities in school: your probability of getting a 6 when rolling the dice is 1/6, your probability of heads is 1/2, and so on. Probabilities are a way of saying how likely something is to happen, and the sum of the probabilities of all possible outcomes is always 1.

In betting this is called the true probability. In esports it is the actual probability that a team will win the match given the current game state. Calculating this probability is what our data science team does for a living: They take game data from old games, train a machine learning model with it, and have it predict the outcomes to the best of its ability. Our algorithms are very accurate and their predictions are the closest you can get to the true probability of the outcome at any given point in the match.

How does a probability become odds? Let's say that we are watching a game of Dota 2 and our models predict that Radiant will win the map with probability 0.75. This means that the win probability for Dire is 0.25 - remember, the probabilities of all outcomes must sum up to 1! It also means that Radiant is three times as likely to win as Dire. So if you bet on Dire and they win, you should win the triple of your stake.

This is exactly what you see in decimal betting odds. In the above example, you'd be offered 4.0 for Dire and 1.33 for Radiant to win. The numbers here signify how much you will get back for each 1€ you bet, including your stake. Bet 1€ on Radiant and get 1€ + 0.33€ winnings back. Bet 1€ on Dire and get 1€ + 3€ winnings. Of course, if you lose the bet, you lose your stake and get nothing. That’s how the bookmaker makes money.

Since these odds were calculated using the true win probability for both teams, let’s call them “true odds”. In the real world, these odds don’t exist. You will never be offered 1.33/4.0. Instead, you will probably see something like 1.3/3.7. Let's convert these numbers back to probabilities and discuss what they mean. We can get the probabilities implied by these odds by calculating

Applying this formula we get to 0.77 win chance for Radiant and 0.27 win chance for Dire. These are higher than our original true probabilities! We call them the implied probabilities, since they are implied by the betting odds. These probabilities no longer sum up to one: 0.77 + 0.26 = 1.04. These additional 0.04 are the so-called bookmaker's take, also called overround. This is the amount of money the bookmaker is guaranteed to make, provided bets are made in proportion to the implied win probabilities. In our our example, if the bookie receives 0.27€ on Dire and 0.77€ on Radiant, they will always have to payout 1€ to whoever wins the bet, and pocket the remaining 4 cents.

Of course this will not always work out for a particular bet. Let's say we put 1€ on Dire against only a 0.77€ bet on Radiant. If we win, the bookie will make a loss despite the overround. But remember that a bookmaker offers a multitude of bets on a variety of games. So if they lose on one, they can make up for it on the others. They will also make sure to get enough bets on Radiant or to limit how much we can bet on Dire to minimize their exposure. This is called balancing the book and is crucial if a bookmaker wants to stay in business long term. In fact, that is why they are called book-makers.

It's helpful to be able to calculate the overround and the underlying true probabilities from decimal odds. Here is what you need to do:

1. first, calculate the implied probabilities using the formula above
   
2. next, sum up the implied probabilities, substracting 1 from that gives you the overround
   

1. finally, your true probabilities are given by this formula:
   

You can see the overround very clearly if you look at draws. A draw means that the true probability of either team winning is 0.5. So the betting odds for both teams are also identical. The "true odds" would be 2.0 on either team, but we will see anything from 1.95 to 1.7 depending on match and bookmaker. All of these correspond to equal win probabilities for both teams - but odds of 1.7 mean that both teams have 0.59 chance of winning. The overround is 0.18.

An overview of different bookmakers' odds with the true probabilities, given in percent, and overround (called “key” here) for a recent esports match. Image provided by Sportradar.

There is yet another probability that we must take into account: The perceived probability of an outcome. This is the probability that you, the punter, assign to a team winning. It is always subjective and sometimes diverges from the true and the implied probability quite a lot. A common example of this is when fans overestimate their preferred team. To stay with our example, let’s say the odds on Dire are 3.7, which implies a win probability of 0.27. If your perceived probability of Dire winning is 0.4, you will absolutely take these odds! But what if you thought Dire has no chance at all to win this game? Then no payout would entice you to place money on them and you’d bet on Radiant instead.

As we said before, the bookmaker needs to balance their books and receive a proportional amount of bets on both sides of the match. So it is their job to offer odds that are not only accurate, but also interesting enough for punters to engage. Ultimately, it’s the combination of true and perceived probabilities, as well as risk management, that makes betting odds. And this, in turn, makes betting a difficult and very interesting field.