# Rating System and Match Forecasting

As we know, sporting events are more complex than dice or roulette, and the true chance of one result or another occurring can never be determined mathematically from first principles. This is not to say, however, that mathematics cannot play a part in sports forecasting or prediction. Of course, many punters either dislike or distrust numbers, and prefer to work with qualitative information instead, like the latest injuries list, the expected weather, and other influencing factors on a forthcoming event. This is to be commended, because without an in-depth knowledge of each event in question, a punter is unlikely to succeed over the long term. However, what mathematics does offer the punter is a simple and effective means of analysing a sports event quantitatively, providing an estimation of the true chances of the possible outcomes and the identification of a betting edge.

Defining a probability function for the outcome of a sporting event is no easy task given the many influencing factors (or variables) that determine the end result, none of which can be determined exactly. That is not to say that a probability function does not exist, simply that the number of variables that would need to be described by it makes its calculation impossible from first principles. Like the weather, the evolution of sporting contests is just too complex to predict exactly. Instead, the punter must resort to estimation of the probability distribution of the possible results for each sporting contest. For dice, this would mean counting the number of times each number is thrown (although there’s hardly much point to that since the probability function can easily be calculated from first principles – it’s just 1/n where n is the number of sides). For sports, it means analysing historical data. Of course, one doesn’t need a degree in statistics to appreciate the importance of studying past performance in sports prediction. Every punter intuitively knows that to predict the outcome of the next event between two players or teams, it is necessary to study their past records and to somehow provide a measure, or rating, of the superiority of one competitor over another. The end product of such an approach is often called a rating system.

Of course, there is no guarantee that past performance of one form or another provides a reliable estimate of future probabilities. Nevertheless, there is ample evidence from the world of sports to suggest that a past performance record frequently provides an excellent indicator of future events, so much so that the use of such data forms the basis of most odds compilation by the bookmakers. The key to gaining an edge over the bookmaker then becomes an issue of finding better and more relevant data with which to build a more accurate rating system for sports prediction.

A rating system merely provides a quantitative measure of the superiority of one player or team over their opposition in a sporting contest. Such superiority is determined by analysing and comparing one or more aspects of past performance for each of the sides. For Liverpool versus Manchester United, this might include results in their most recent games (against other teams), goals scored in those games, relative performance home and away, league points and positions, and perhaps performance against each other for their most recent clashes. For tennis matches, it might also be relevant to study past performance on different surfaces since some players perform better than others depending on what they are playing on. Some players also seem to be better suited to playing other players than their world rankings might suggest, because of the way their playing styles match up.

More sophisticated rating systems may try to accommodate the quality of the opposition played. One well known rating system which does this is called the Rateform, the most famous of which is the Elo Rating System, named after Professor Arpad Elo, who first designed his system for chess competitions, but which has since been adapted for use in many different sports. After every game, the winning competitor takes points from the losing one. The total number of points gained or lost after a game is determined by the difference between the ratings of the winner and loser. In a series of games between high-rated and low-rated competitors, the high-rated one is expected to score more wins. If the high-rated competitor wins, only a few rating points will be taken from the low-rated competitor. However, if the lower rated competitor wins, many rating points will be transferred. For sports where draws are possible, the lower rated competitor will also gain a few points from the higher rated one, which conversely will lose a few in the event of a tie. This makes the rating system self-correcting. A competitor whose rating is too low should, in the long run, do better than the rating system predicts, and thus gain rating points until the rating reflects the true playing strength.

For any rating system, a judgement must be made about how far back in time past performance might be considered relevant. For head-to-head records, this may involve several years, although when analysing team sports one has to consider the change in personal. There seems little point in including a match played 15 years earlier when every player in both sides was different to the squads today. For a game like Liverpool v Manchester the only case for including such historically old data into the analysis would be on the grounds of motivation: whatever the team line up, these two sides have a history of wanting to beat each other more badly than for most other games.

Ratings calculated for each competitor in a sporting contest are then used to produce a match rating. The most obvious and typically used approach is simply to calculate the difference between the two ratings. For example if Andy Murray has a rating of 129 and Roger Federer a rating of 142, then the match rating will be 13 in favour of Roger Federer. Once we have a match rating the next step is to estimate from it the chances of each possible result occurring, from which betting predictions can then be made. To achieve this, a match rating must in some way be translated into a probability distribution for the possible results in a sporting contest. The most basic approach is to analyse the results of all previous matches in our data set where the match rating was 13 in favour of one competitor over another.

Suppose that having calculated match ratings for all historical matches we find 100 of them where the match rating was 13, and of these 58 resulted in a win for the higher-rated competitor and 42 for the lower rated competitor. Our probability distribution for match ratings of 13 is the 0.58/0.42 (or 58%/42%), and from this we can calculate what, on average, the fair betting odds should be for such matches. A probability of 0.58 translates into a betting price of approximately 1.72; for a probability of 0.42 it is 2.38. Armed with this information we can now check the bookmakers to see what odds they are offering for Murray and Federer. We find, for example, that Federer is available at a best price of 1.66 whilst that for Murray is 2.50. Since we’ve estimated from out rating system that Federer’s fair price should be 1.72, betting less that this would mean we have no edge, and since 1.66 is the best available price for him, there is consequently no value to be found for him. By contrast, Andy Murray’s price of 2.50 is better than our estimated fair odds of 2.38, giving us a theoretical edge of 1.064 (or 2.50/2.38). That is to say, if our probability estimations based on our rating system are accurate we should expect to make on average a little over 6 pence for every £1 stakes on a match like this. Of course we know that Murray, and our bet, will either win or lose, but this does not change the relevance of the ratings and value analysis we have performed. Remember, it’s all about the long term. If Murray should lose, this does not necessarily invalidate the forecasting methodology we have used; he may simply have been unlucky, perhaps losing a couple of key points in the final tie break which swung the result against him. We won’t know whether our forecasting method is accurate and potentially better than the one the bookmaker is using until many such matches have been played and we can the analyse our betting returns. Naturally, this is something that every punter using a rating system should do. There’s no point in continuing with something in betting if it doesn’t work.

The above example describes a match rating for a simple two-way betting proposition, typical in sports like tennis. Things become a little more complicated for sports where the draw is also an option (although this just means the calculation of another percentage figure), and particularly where there is a distinct advantage for a home competitor relative to the away competitor, as is prevalent in football. Arguably, a match rating of 13 in favour of a home football team is not going to show the same probability distribution of results as a match rating in favour of the away team. For most football competitions, the home sides generally win twice as many games as the away sides. Consequently, this bias needs to be taking into account. The most convenient way is to perhaps define the match rating as the superiority of the home team over the away team, rather than the stronger team over the weaker team. In this way we can have negative as well as positive match ratings, when the away team is rated more highly than the home team. Liverpool at home might be rated 117 whilst Manchester United – away – could be 137. The match rating would then be -20.

The possibilities for designing, testing, implementing and verifying match rating systems are endless and we could produce volumes and volumes of material on the subject. A summary of a typical approach to the development of a simple football match rating system is described by Football-Data at http://www.football-data.co.uk/ratings.pdf. But for now, so long as we understand that the relative abilities of sporting competitors can be translated into probabilities of different possible outcomes and their associated betting odds, we are on the road to finding an edge over the bookmaker.