Ratings

Each team are given ratings associated with their strength. This enables us to work out which teams are better and by how many goals. Ratings for each team were solved using a customised regression model.

Firstly, we calculated the current ELO ratings for each team. ELO is a rating system that applies weightings to different matches depending on its importance and adjusts based on the victory margin. These ELO rankings are a good reflection of current form. However, sometimes ELO puts too much emphasis on the recent results (more will be mentioned about ELO Adjustments later).

To limit ELOs weaknesses, we have created a different optimise regression model that is able to convert actual group matches 1x2, Asian Handicaps and Goal Lines odds into three variables. Home expected goals for the match, away expected goals and the draw correlation. The draw correlation shows us the relationship between expected home and away goals; thus, allowing us to adjust the probability of a draw accordingly. Using these three variables we apply a Bivariate Poisson Distribution to gain an accurate score grid. A score grid represents all of the probabilities of every potential score.

We are able to convert the ratings into match probabilities and compare to the score grid we just calculated from the match odds. The aim of this regression model is to reduce the predicted error between:

The actual score grids and predict based on the ratings.

Difference between the ELO ratings historic and new adjusted ratings that account for match odds.

Comparing the actual expected goals and predicted expected goals based on the new ratings.

The regression model then gives us the optimal solution for the new ratings. We can now go onto simulating the World Cup.

Simulating a match

We first need to calculate the score grid for each game. This is calculated one of two ways:

For early group games we use the actual odds since these are incredibly accurate. We use the optimise regression model to convert the 1x2, Asian Handicaps and Goal Lines odds into the three variables mentioned above. The reverse engineering of this score grid gives us an accurate representation of the probabilities of each score line occurring.

The second method used to calculate the score grid is found by translating the new ratings into expected goals home and away then applying the Bivariate Poisson distribution to predict the score grid (more about this in the next section).

Once we have obtained the score grids, we have essentially assigned each score a probability. By selecting a random number between 0 and 1, we can decide the score the game finished in our simulation. If the random number lies in the probability bucket, then this is the assigned match score. Trivially, the probability bucket sizes of each score are dependent on the likelihood of an event occurring.

Bivariate Poisson

Bivariate Poisson process allows us to calculate the probability of an event occurring considering the three inputs. Expected goals home and away, and the draw correlation. You can learn more about how you can apply Bivariate Poisson in sport here.

To predict the match result of the next game we need to first use a formula that we mathematically solved to convert ratings into supremacy (difference between expected home and away goals).
sup = ratingCoeff x (ratingHome - ratingAway)

Total Goals are predicted using historic data and the relationship between ratings and supremacy (sup). We use a quadratic which we solved earlier to predict total goals (TGs)
TGs = 0.1593(sup)² - 0.0005(sup) + 2.5492

We can easily convert supremacy and total goals into expected goals by the simple formula:
Home xGs = (TGs + sup)/2
Away xGs = (TGs - sup)/2

Draw correlation has been adjusted based on the phase of the tournament (excluding the solved draw correlation matches from the group). We are starting at correlation 0.1 for the group, 0.18 for knockout matches and 0.2 for extra time markets.

Applying these three variables to the Bivariate Poisson distribution we are able to calculate the score grid and thus simulate the match the same way as mentioned before.

ELO Adjustments Throughout the World Cup

ELO is a way of adjusting the ratings to reflect current form. After the first game has finished, we must update the rankings based on the simulated result. Trivially, if a team is winning by large margins, they are building confidence and momentum throughout the tournament and this needs to be captured. Therefore, since we calculate the match probabilities using the ratings, we must adjust them after each match and recalculate the score grids.

The new ratings are calculated based on the result of the previous game using the following formula:
Rₙ = R₀ + K x (W - Wₑ)

Rₙ is the new rating

R₀ is the old rating

K is the weight constant which equals 60 for the World Cup since this game is the most important the national teams will play

K is adjusted based on the goal difference N of the simulated match. K is increased by half if the game was won by two goals, or by 3/4+(N-3)/8 if the game was won by more than 3 goals.

W represents the results for that team (1 = Win, 0.5 = Draw and 0 = Loss)

Wₑ is the win expectance for that team, which is the draw no bet probability. This can be calculated by Wₑ = 1/(10^(-dr/400)+1) where dr equals the difference in the two teams ratings before the match.

Tie Breakers for Group Matches

Tie breakers for groups are applied so that if points are tied, we calculate 'goal difference', followed by 'goals for'. If we still have a dead heat situation, a random number will decide position. This is adequate because the probability on any further tie breakers is very small and no value will be added to the simulation.

Extra time

Should the random number select any of the draw possibilities for normal time we will simulate extra time.

In extra time we create another score grid similar to the normal time; however, with only 38% of the match goals and with the increased draw correlation.

We then simulate extra time and if this again ends in a draw then we will go to penalties. We have added a Penalty bias coefficient in favour of the home team of 0.15. To simulate penalties, we have used the formula:
0.5 + PB x (dnbHome - 0.5) to calculate the probability of Home Team winning the shootout. We select another random number to decide the outcome of penalties.

PB represents Penalty Bias which equals 0.15

dnbHome the draw no bet probability of a home win.

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