**Working Principles of Prediction Markets through the Prism of a Bookmaker**

*What do ratios like 8: 1, 2: 1, 1.5: 1 mean in bookmakers? How to calculate the size of the fees using these ratios? How do the ratios change during the match and how are they related to the probabilities of winning athletes? What is more important for winning: luck, knowledge of sports or knowledge of mathematics?*

Sport is perhaps the first branch of human activity where organized prediction markets (PMs) appeared. The first sports PM is considered the Ancient Roman tote on horse racing. Its principles were simple: each bettor relies on the victory of a crew, and according to the results of the competition, all the money collected is divided between those who guessed the winner.

Winning on a simple tote is not difficult: just a little understanding of the sport and roughly imagine which of the crews has advantages. Regardless of the size of your bet, on average you will win more than lose: it is enough that among other viewers there will be a certain percentage of simpletons who make bets only by chance. You do not have to be an expert — just have a basic knowledge.

But modern bookmakers that accept bets on competitions have trickier rules. They make set conditions for the inequality of between athletes or teams in advance. Now it’s harder to win — it’s not enough just to have basic knowledge anymore.

Let’s suppose that you want to bet $100 on your favorite team in a match. You come to the bookmakers, and they show you the ratios for different teams. Your team has a ratio of 1.25. This means that if it wins, you will receive $125: $100 of the original bet and $25 modest win. But some other teams have higher ratios: 1.75, 3, 10. If you put $ 100 on a team with a ratio of 10, and it wins, then you will get $1000 back! But the probability of this is small — such factors are usually given to weak teams.

In which case is the risk justified? How do ratios correlate with winning probabilities? What percentage of bets are bookmakers taking as commissions and is it worth contacting them?

Let’s try to figure it out.

**How spherical bookmakers set ratios**

Before you start accepting bets, bookmakers (usual experts in sports) evaluate the probabilities of all possible outcomes of the competition. For example, in a football match, there are 3 such outcomes: the winning of the first team, the winning of the second team and a draw. Each team has a certain reputation — how often they won in the past, what are their strengths and weaknesses.

Let’s suppose that a football match is to be between teams A and B, where team A is stronger. Suppose that bookmakers estimate the probability of winning team A at 0.6 (i.e. 60%), team B at 0.2 (20%), and another 0.2 (20%) left for a draw. The sum of all the probabilities is 1 (100%).

In order to calculate the ratios from these probabilities, the bookmaker must also take into account the fee that it wants to get in his pocket. Let’s say this is 0.1 of all bets (10%). Then the ratios for the different outcomes will be as follows:

- For the team A: 1 / (0.6+ (0.6 * 0.1)) = 1 / 0.66 = 1.51
- For the team B and draw: 1 / (0.2+ (0.2 * 0.1)) = 1 / 0.22 = 4.55

Now, the one who wagers $100 on the team A will receive $151 ($ 51 net win) in case of winning, and the one who wagers on the team B or draw will receive $455 ($355 win). Losers simply lose bets.

**How real bookmakers change ratios**

Technically, the bookmaker is not obliged to strictly follow the above-mentioned scheme. Its purpose is not to accurately determine the probabilities of winning, but to maximize profits.

If there is public excitement around a match in favor of one of the teams, its ratios can be intentionally underestimated, as most fans will bet on it anyway.

Also, if the bookmaker accepts bets throughout the match, then the ratios will change over time. It is obvious, for example, that any goal greatly changes the balance of power. Even if at the beginning of the match the ratios and chances of the teams to win seemed equal, after the first goal the bookmaker will reduce the ratios for the leading team and increase for the lagging one.

**How to determine the probabilities of team wins and calculate the bookmaker fee**

Imagine you came to a football match between teams X and Y. You are told that for the X team the ratio is 1.33, for the Y team it is 5, for a draw, it is 4. We will designate them with the letters X, Y, and Z respectively. The probabilities of these outcomes, which we denote by P [X], P [Y], P [Z], will be approximately equal to the reciprocal of the ratios:

P [X] ~ 1 / X = 0.75 = 75%,

P [Y] ~ 1 / Y = 0.2 = 20%,

P [Z] ~ 1 / Z = 0.25 = 25%

The sum of these values exceeds 100% and is equal to 120%. This additional 20% are the fees of the bookmaker, which is quite a lot. Many bookmakers have much lower fees. For many bettors, this figure is enough not to take risks. But if you decide to continue, now you can get more accurate values of the probabilities, reducing the figures by 20%. In our case, we have

P [X] = 63%, P [Y] = 16%, P [Z] = 21%

It is important to understand the real meaning of these quantities. These are not at all some “objective,” “scientifically based” probabilities of outcomes. These are just the estimated probabilities from the bookmaker experts’ perspective. And even then, if you do not take into account the amendments for a possible stir in favor of one of the teams. Your opinion regarding the chances of teams may vary. And this is the key to your potential winnings.

**How to understand if the game is worth the candle**

Suppose you found the fee bearable. In the end, if you correctly guess the outcome of the match (out of three possible), you will still be in the black. But guessing is not so simple: unlike the Ancient Roman tote, you are betting not against amateurs, but experts.

Do not be afraid of such a big word as “expert.” Of course, if you bet at random, then this is an ordinary lottery. In it, you are more likely to lose than to win since there is a fee. It is worth playing if you have your own opinion about the chances of the teams, and it differs from the opinions of the bookmakers.

**You have a good chance of winning in two cases:**

- If you know the team better than the expert. And it is possible. Not a single person knows the features of all the teams in the world. If you have been following the X and Y teams for a long time, then, perhaps, you are stronger than an expert who is obliged to follow everyone immediately.
- If the ratios are distorted for the sake of public opinion and/or due to an inadequate influx of bettors of one of the teams in this match.

In either of these two cases, your opinion may be more accurate. If you think that the winning of one of the teams is more likely than it follows from the ratios — bet on it.

As the statistics of real bookmakers show, the vast majority of fans win only by chance. But there is 2–3% percent of connoisseurs who win more often than lose. As a rule, it is they who make maximum bets and sometimes make the game their main source of income.

Also, note that you can win at sports betting not only thanks to the knowledge of sports but also thanks to the knowledge of mathematical strategies that increase your total final win in a series of matches.