Every player (and this applies not only to betting) wants to be a winner. However, if the wish of all bettors came true, the world of betting would collapse overnight. Players compete, and sometimes very fiercely, with each other, using their experience and skill. The winners get everything, the losers are left with their noses. But what are the consequences of this competition? Let’s try to figure it out together.

One study on the evolution of a typical betting market and determining the number of winners in it proposed a model for such an evolution. A professional, by definition, is a bettor with a positive expected long-term return on his bets with the bookmaker. At the same time, just successful players are not always professionals.

In most cases, professional bookmakers, such as Pinnacle, ignore most of the actions of ordinary players, whom they consider to be relatively unqualified. Information about whether this conclusion is correct will, of course, remain a trade secret. Here is what one of the well-known players in the sports betting market thinks about this (he posted his opinion on Twitter):

“More often than not, most bookmakers ignore the money of ordinary players. I was familiar with an extremely shrewd old school bookmaker who once told me: “I put ordinary people’s bets in my pocket as if they never existed.”

The aforementioned modeling, in addition, showed that an approximate reproduction of the observed changes in the odds becomes possible only if the proportion of professional players whose actions deserve the attention of bookmakers is relatively small. But why is this happening?

## From experience to professionalism

Most likely, most pros who make a living betting will agree that the following fact speaks about the skill of the player: he can accurately determine the lines or odds before the betting companies do it.

Suppose you were asked to determine the probability that CSKA will win against Spartak, or predict the total points in a basketball match between the Los Angeles Lakers and the Golden State Warriors. Will your predictions coincide with the bookmakers’ predictions, even with a small margin of error of 1-2%? It’s one thing to see a published rate and decide if it’s accurate, and quite another to predict the correct rate in the absence of points of view to take into account.

There is a widespread point of view according to which the majority of players are strongly attached to the expressed points of view about the likelihood of outcomes in sports competitions. In fact, this cognitive bias may explain the inefficiencies in the betting markets we mentioned earlier.

## Skill Competition in the Betting Market

The betting market is considered to be a competition with a bookmaker. If you manage to outperform the bookmaker, you win. This statement is true and false at the same time. Yes, sometimes the bookmaker takes a position on the outcome of a particular competition. If you beat the bookmaker, he will lose money and you will make a profit. However, in most cases the outcome of the competition does not matter to the bookmaker: whatever the result, the bookmaker will remain in positive territory.

We can say that the bookmaker is a fairly large player with an unfair advantage, that is, a margin. The betting market should be seen as a financial market (in which buyers and sellers operate) reflecting the opinions of many players (people who place bets and people who have opposing opinions). All of these players (including people like you) express their opinions through bets.

The bookmaker simply reflects the characteristics of the flow of this money by publishing the odds with the most profitable ratio of profit and loss. At the same time, the bookmaker pays attention to certain sources of information and ignores others that he considers insignificant.

If the published odds in one way or another reflect the opinions of betting market players about the likelihood that CSKA will beat Spartak, or that the Los Angeles Lakers and Golden State Warriors will score more than 210 points, it becomes clear that the players who were able to accurately assess the correct values of the probability have the highest chance of winning in this competition.

Bettors compete with each other using their relative skills. Accurately evaluating the correct values is not enough: it is important that the accuracy of your estimates exceeds that of your competitors. Roger Federer wins the French Open not because he plays tennis well, but because he plays tennis better than others. The player makes a profit not because he was right when placing a bet, but because he was more right than other players.

## Winner Takes All

Alas, in the long term, the truth of this statement is hidden behind the binary results of single bets, that is, winning or losing. A player who has made a profit as a result of winning one bet can easily get the illusion that in order to achieve success in the game, it is necessary to correctly determine the winners or beat the bookmaker.

But in the world of small samples, which is the world of bets, chance dominates: almost all events occur as a result of a coincidence. Signs of competition in relative skills only appear in the long run. The competition is repeated many times. People come into play again and again. Relative differences in forecasting skills accumulate. The likelihood that Federer will win one point is marginally greater than the chance that his opponent will, but this fact alone significantly increases the likelihood of Federer winning a five-set match.

In the long run, only the best players win. The betting game, like the poker game, is not just a competition of relative skills: it is a winner-take-all competition. Thus, a professional player is not necessarily completely professional. Such a player is relatively more professional than other people; relative professionalism helps him overcome the hardships of the disadvantage that the bookmaker puts him in. The bookmaker’s margin reflects well its danger as a competitor, strong and unpredictable.

In addition, the achievement of absolute professionalism by a player may not lead to an increase in his chances of winning if all other players by that time have also achieved absolute professionalism. Paradoxically, this could mean that the player’s chances of winning by any means other than luck diminish over time. Let’s consider a model that will allow us to predict the outcome of events in general.

## How to predict total points in an NBA match?

More often than not, people cannot intuitively determine the likelihood of an event. Just answer the question “Who will win – CSKA or Spartak?” much easier than determining the probability of winning each of these teams. Popular sports in which leagues such as the NBA are active give us the ability to look at numbers rather than percentages, since match totals are large enough.

From 2007 to 2019 NBA teams averaged 200 points per game. Let’s build a model in which we ask competing players to predict the number of points scored in an NBA match. In doing so, we will use the following conditional parameters:

- The “true” total points are 200.
- On average, the match ends after 200 points have been scored, but due to chance this average has some variance. The standard deviation is 17.5 and is approximately equal to the standard deviation of the observed total points in matches for which the bookmaker has set a betting line of 200 points.
- Eleven competing players predict the following total points: 190, 192, 194, 196, 198, 200, 202, 204, 206, 208 and 210.
- A random number generator determines the actual total of points. After that, players receive points, which are distributed according to the following principle.
- Each player gives to all other players a number of points equal to the difference between his predicted total and the actual total of points.
- Each player receives from all other players a number of points equal to the difference between their predicted totals and their actual total points. As you can see, the more accurate a player’s forecast is, the more points he gets in comparison with other players.
- The players’ competition continues for 10,000 matches. The total points for the eleven players are calculated after each match. The player with the most points is declared the winner.
- Monte Carlo simulation is used to determine the likelihood that a player will become the leader in points (i.e. the winner) in 100 matches, 1000 matches and 10,000 matches.

Graph 1 below shows the likelihood of a player winning over 100 matches based on Monte Carlo simulations. It is not surprising that most often the winner is the player who made the most accurate prediction – 200 points. This value coincides with the “true” (average) total points in matches. The probability that this player will become the leader in the number of points scored in 100 matches is more than 1/3. It is worth paying attention to the fact that even after 100 matches, the players making the least accurate predictions very rarely win: the six weakest players account for less than 2.5% of the winnings.

The next Graph 2 shows the probability of winning in 1000 matches. Only the top three players managed to take the lead, with 86% of victories being accounted for by the most effective player. As the number of rounds played increases, the relative advantage of the most professional players accumulates, which increases the likelihood of their victory. Albert Einstein is rumored to have called such a phenomenon as compound interest “the most powerful force in the universe.”

We do not show here the schedule for 10,000 matches. You can probably guess what this graph would look like. The player predicting the total of 200 points wins every time.

## The Paradox of Mastery

But what happens if the players’ inherent predictive skills are improved? We rebuilt the model, assuming that the players predict the following total points for each match: 195, 196, 197, 198, 199, 200, 201, 202, 203, 204 and 205. Ten players out of eleven have become more professional in absolute terms. The next Chart 3 shows the distribution of the winning shares:

Even though the best player, whose prediction matches the “true” total, leads in the number of wins in 100 matches, the likelihood that this player will become the leader has dropped from 34% to about 19%. Likewise, the likelihood of less effective players becoming leaders has decreased, despite their increased professionalism in absolute terms compared to the first model we built.

According to the first model, the player with the second highest number of wins predicted a total of 198 points; the probability of his winning was 22%. In the second model, this player predicts a total of 199 points. The probability of his victory dropped to 15%. At the same time, the likelihood of the weakest players becoming the leaders has increased. Now the six least efficient players account for almost a quarter of the wins.

Obviously, by the end of 1000 matches, the probability of winning the best players increases, but still not as strongly as in the first model ( Graph 4 ):

But what led to the transition from the first model to the second? First, in the second model, the winner takes all again, but events take longer. According to the results of 10,000 matches, 1.6% of victories were won by not the best bettor. Secondly, despite the fact that in the framework of the second model the players have become more professional in absolute terms, the variance of the totals predicted by them has halved.

Since the variance of the predicted values is usually a combination of the variance of skills and variance of luck, as the variance of skills decreases, the effect of variance in luck increases. Paradoxically, as the absolute professionalism of the players increases and, more importantly, the gaps between different levels of professionalism narrow, the impact of professionalism on the players’ results decreases. This is the paradox of mastery.

If bookmakers consider the vast majority of bettors to be non-professional gamblers, is it easier for relatively professional gamblers to win? Let’s put it this way: yes, such players will prevail over ordinary people. If the number of players with absolute skill increases by refining the model we are using, the variance in player skills is likely to decrease. The absolute mastery increases, and the relative differences decrease, albeit not significantly.

Even if the bookmaker ignores most of the actions of unqualified players, the lines and odds offered to them still reflect the actions of the professionals. The more markets you play in, the more likely you are to compete with professional bettors.

The following Graph 5 shows the results of a competition between three relatively professional players and eight relatively unqualified players. Unqualified players failed to take the lead in 1000 matches in any of the 1000 runs of the model used.

Let’s compare this distribution with the following: the number of professional players has increased, on average the absolute skill of the players has increased, but the relative differences between players have decreased ( Graph 6). Regular players keep losing, but the results of professional players are more determined by luck. The betting game is a zero-sum competition with the winner taking all. The smaller the skill gap between players (and even subgroups of players), the more luck determines what happens. Now you know why some professional gamblers say they find it harder to win.

## Can you beat the bookmaker?

We have already mentioned that the most powerful competitor of any player is the bookmaker. Perhaps it is for this reason that many believe that in order to make a profit, the player needs to beat the bookmaker. Bookmakers not only use the most efficient models on average, but can also reduce the effectiveness of player models, taking part of their profit in the form of margin.

For the most accurate information on the performance of bookmaker models, see Chart 7 . It compares the bookmaker’s lines with the actual totals of the matches. The data used, sourced from Sportsbook Reviews, consists of NBA close lines and total points accumulated in 15,508 NBA games from October 30, 2007 to May 5, 2019.

If bookmakers offer 190, 200 or 210 lines to players, on average the match ends after 190, 200 or 210 points, respectively. Yes, the results have random variance, which, as mentioned, deserves attention. Almost a third of the matches where bookmakers offered a 200-point line ended after less than 182 points or more than 218 points.

But in the long run, the bookmaker’s estimates are extremely accurate. To beat the bookmaker, you need to overcome his margin. If a player places bets at odds of −105 or 1.95 (estimated margin of 2.5%), he only needs to win 51.3% of the time to cover the costs. If the odds are -110 or 1.91 (with a 5% margin), the proportion of required winnings rises to 52.4%.

Taking into account the standard deviation of the observed results of the NBA matches, we get values of 0.5 points and 1 point, respectively. It may seem to some that these values are not so great. However, can you find several hundred or thousands of matches in which the bookmaker is more mistaken than you, and the gap between your estimates is equal to any of these values?

## Things to remember for a player who wants to become a professional

Betting can be a lot of fun, both from the process itself and from the funds won. At the same time, most people should still perceive the betting game as entertainment. But if you strive to make a living by betting, or at least want to pay with winnings for your annual vacation, it is important to remember the following:

- Betting is a relative skill competition between players. The player makes a profit not because he was able to accurately predict the future, but because his forecast turned out to be more accurate than the competitors’ forecasts.
- Bettor who has trained his skills may not be the winner if other players develop their skills in parallel with him. The less skill set a player possesses, the more often his success is determined by elementary luck.
- In the long run, the winners with the relatively best skills get it all. The player competition is iterative, that is, it is constantly repeated. As a result of a large number of repetitions, the existence of even small differences in the skills of the players leads to a very uneven distribution of the “prize pool”.
- Betting is a zero-sum game. The prize pool is not endless; it is limited by the amount of players’ bets. The losers give their money to the winners. Moreover, the former are much more numerous than the latter.

Have fun with the betting process and do not bet funds that you may not have enough for vital goals.