k | eloratings.net | elofootball.com |
---|---|---|

60 | World Cup finals | Champions League, UEFA Supercup |

50 | continental championship finals and major intercontinental tournaments | Europa League, Champions League qualification |

40 | World Cup and continental qualifiers and major tournaments | Europa League qualification, domestic leagues, domestic cups and super cups |

30 | all other tournaments | domestic league cups |

20 | friendly matches | not covered yet |

I then stumbled upon this interesting blog entry on

*pena.lt/y/*, a site covering sport metrics, and found a very interesting graph about the relationship between $k$ and the error of the prediction. The minimum is around $k=15$. Upon realizing that the weighting factor I use for most of the games, 40, is not even on the graph, I decided to run my own calculations to see if I get similar results.

The two main formulas in the calculation are:

- Two-way win expectancy of the home team; $dr$ is the difference in ratings before the match:

$$WE_{h}=1/{10^{-{dr}/400}+1}$$

- Points exchanged, with $gdm$ being the goal difference multiplier (another metric on which
*pena.lt/y*has an interesting different take); outcome takes values 1 (win), 0.5 (draw) and 0 (loss):

$$RΔ_h=k ⋅ gdm ⋅ (outcome-WE_h)$$

The dynamic between these formulas is clear: Given a certain $WE_h$ before the game, $RΔ_h$ assumes a value which changes the post-game ratings of the teams and subsequently influences the $WE_h$ of the next game. That is, $k$ does not only alter the magnitude of the rating delta $RΔ_h$, but also changes the subsequent prediction. A smaller $k$ reduces the convergence speed and avoids very high probabilities for favorites and very low probabilities for underdogs, which means that extreme events become more likely. For example: Two way probabilities of 90:10 could become 87:13 for a lower $k$.

**Results**

The question is then:

*Does it improve the prediction to assume slightly lower (higher) probabilities for the favorite (underdogs) of a given match?*

To this end, I apply least squares as indicated by the

*pena.lt/y*blog entry. The residuals $R$ are plainly the difference between prediction and outcome.

$$R=WE_{h}-outcome$$

Eventually, I take the average of the squared residuals RSS as my metric for the prediction error:

$$error(k,n)={RSS}/{n}$$

Since my database currently contains over 470,000 games since 1955, it would be too time consuming to run the whole history for different specifications of $k$. Hence, I limit the analysis to $n$ in (1000; 5000; 10,000; 20,000; 30,000; 40,000; 50,000) and run it for $k$ in (5,10,15,20,25,30,40,50).

The simulations confirm the main result of

*pena.lt/y*, namely that the error minimizing

**$k$ is around 15**. The size of the dataset is, however, not specified, which is unfortunate as there seems to be a clear relationship between the size of the dataset and the prediction error: With increasing $n$ the prediction error is steadily decreasing. Apart from that, with increasing $n$ the error minimizing $k$ is falling as well. While for a smaller samples, $n=1000$ or $n=5000$, the optimal $k=25$ and $k=20$, respectively, it is always $k=15$ for samples between $n=10,000$ and $n=50,000$.

What I can take from this: My $k$ is evidently too high and for the next historical re-run I will adjust it accordingly. I will, however, maintain the differentiation between different types of competition, as teams clearly put a higher value on Champions League than e.g. on domestic competitions. I might reduce the relative importance of super cups, though, as they often don't carry a higher importance than, say, league cups.

**Tanking**

Quite often I see Champions League participants lose a lot of Elo points in a league game prior to a Champions League contest, supposedly due to fielding a B team. Since the expected Elo rating exchange in any given contest equals zero (meaning that teams - on average - play according to their strength), I will test whether this is true for league or cup games directly prior to a Champions League game.