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Two-Leg Games

Most of the inter-league competition in European cups is played in two-leg matches. If the standard Elo formula is applied to these games, a serious problem can arise: A team winning its first leg by a high margin can afford to lose the second leg by a narrow margin and go through to the next round. This will be preceived as an overall victory, the Elo system however will count this as a victory and a loss, which could result in an overall loss in Elo points.
This ranking will apply a different calculation for 2-leg matches. The first match will be considered as a standard game. The second leg won't. After the second leg, the aggregate result of the two games will instead be decisive for the Elo ranking.
Obviously, an aggregate result of 2 games is more significant than a single match, just how much more significant?
Let us consider the result of a football game as a combination of actual team strength difference and randomness. If two teams play against each other once, both factors will play a role that determines the result. If these two teams play each other a second time, the strength difference will be as big as in the first game. The randomness though can play out in any direction. In my eyes, this phenomenon is similar to Signal averaging. We can interpret the strength difference as singal, the randomness as noise and the Signal-to-Noise-ratio as significance. Signal averaging implies that the SNR increases with the square root of the number of measurements. Analogically, the significance of a result increases with the square root of the number of games. The Elo points exchanged in a classic 2-leg-match will thus be multiplied by the square root of 2.
Let's have a look at the adjustments for goal difference and single matches or 2-leg games:


Goal DifferenceFactor (1 game)Factor (2 games)
011.414
111.414
21.4142
31.7322.449
422.828
52.2363.162
nn½(2n)½


Likewise, the probabilities for winning a 2-leg-match change. A stronger team is more likely to beat a weaker team over 2 legs than in a single match. Calculations show, that the Elo difference has to be multiplied by the square root of 2 to obtain the result expectancy for a 2-leg-match.

Result Expectancy Single MatchResult Expectancy Two Legs90%80%70%60%50%40%30%20%10%Elo points difference-400-2000+200+400

To avoid that the first leg is counted twice, the points exchanged from the first leg match will be subtracted from the exchanged points from the aggregated result. Hence the Elo points exchanged from the second leg will represent the difference of the aggregate outcome of the 2 legs and the result of the first match.
However, if a team wins both legs, these 2 games will count independently. Otherwise, a team could lose points despite winning a match.