Euro 2020: England or Italy? GANGAN PRATHAP The 51st and final match of Euro 2020 was played on 11 July 2021 at the Wembley Stadium, London. At extra time (a.e.t.), the teams were tied 1-1. Perhaps matters should have ended there, and the trophy shared, for two years each if allowed. But the rules needed it to be settled through a penalty shoot-out. It was 3-2 in favour of Italy. A cruel end to a cruel season, delayed by the pandemic. Euro 2020 saw 24 teams play a total of 51 matches starting from 11 June to 11 July 2021. Of these, 36 matches were played in a round-robin stage, with four teams each in six groups for a total of 36 matches. Sixteen teams qualified for the knockout stage: 8 matches in the round of 16, 4 matches in the quarter-finals, two more in the semifinals and a grand final to determine the top team. At the end of the day, a total of 141 points were shared, and another total of 142 goals were scored, discounting both the points and the goals in the penalty shoot-outs. That is, if two teams were tied a.e.t., the points for the match were shared at one each. A win of course earned 3 points. More than 50 years ago, an Indian mathematician named Ramanujacharyulu (C. Ramanujacharyulu, Analysis of preferential experiments, Psychometrika, 3 (1964), pp. 257-261) introduced a protocol which allows us to reconcile the results of a group of teams and their matches playing a tournament, and choose the winner, or rank them in an order. Each team meets the other team (the paired comparison) once or several times and registers a result each time – a win, a tie or draw, or a loss. That is each of the matches is considered to be an experiment, and the result, an expressed preference for one team over the other. With this knowledge, we can construct the tournament matrices based on goals, or on points. This tournament metaphor can be used to evaluate the Italy-England face-off. To illustrate the protocol, we choose Group B because there is a text-book Win-Loss loop. Table 1 shows the Points and Goals Tournament Matrices in this group B at the end of the round-robin stage. Belgium and Denmark proceeded to the knock-out stage. The Win-Loss loop is seen thus: Denmark>Russia>Finland>Denmark, where > signifies a clear win. All three teams are tied on points. Denmark goes through as the second team from this group on goal difference; here P-W. In the Ramanujacharyulu (henceforth Ram) protocol, the rowtotals or row-sums (let us call this P) give an idea of the “power” of each team and the column-totals or column-sums (we will call this W) give the “weakness” of each team. Ramanujacharyulu’s "most-balanced" point of view is that one should try to find out who can combine the greatest ability to win with the least susceptibility to lose. That is, "in tournaments one may be interested in locating the really talented man in the sense that he has won over the largest number of opponents but simultaneously he has been defeated by only a few opponents." Ram’s approach allows us to assemble the totality of the 51 matches. There will now be a 24 x 24 matrix of all the teams from the six groups. To the 36 matches in the round-robin stage we add the results from the 15 matches in the knock-out stage. There will be two 24 x 24 matrices, one based on the sharing of the 141 points scored and the other based on the 142 goals scored. Here, we reiterate what we do. At the knockout stage, if the teams are tied even after extra-time, the result is decided using penalty-shoot-outs. In applying Ram’s method, we shall assume that the match is actually tied (one point each is given to each team) and the goals position is that at this tied stage. This will of course mean that the teams that go to the final will have played 7 matches while the 8 that were kept behind will have played only 3 matches each. The P values (i.e. the row-totals) and the W values (the column-totals) now measure the “power” and “weakness” of each team. The P/W ratio (we call this the PowerWeakness Ratio) and the ratio (P-W)/(P+W) (we shall call this the Normalized PowerWeakness Difference) are dimensionless measures of the “quality” of the team. We can do this for the Points table or the Goals table. Note that PWD is a one-to-one monotonic transformation of PWR. PWD has the attractive feature that it is always bounded between -1 and 1 but this is not true of PWR which has no upper bound. At this stage, we should understand that the simple row-sums and column-sums assume that each team is given the same weight. Thus, in Group B, Belgium gets the same 3 points for beating Denmark, Finland or Russia. Ramanujacharyulu pointed out that the weightage can be changed iteratively, taking consideration of the “quality” of the team, leading to an eigenvalue problem. Effectively, this is done by multiplying the citation matrix by itself recursively until convergence is reached in both the power and weakness dimensions. This yields the weighted values of P and W, and the ratio of these values is PWR. This graphtheoretical procedure has considerable mathematical elegance: it handles the rows (power) and columns (weakness) symmetrically although the matrix, to start with, was necessarily asymmetrical. One more point is worth noting. At the group stage, within each group we have what is called a “complete” tournament in that all the teams are connected as they have all played each other. At the assembled level, where all 24 teams are taken together, we have an incomplete tournament: 8 teams have played 3 matches each, 8 teams have played 4 matches each, another 4 have played 5 matches each, 2 teams have played 6 matches each, and England and Italy have played 7 matches each. However, because of the symmetrical handling of the rows and columns both PWR and PWD are dimensionless ratios which measure the “quality” of performance of the teams. This graph theoretical approach is now used routinely in many situations (e.g., Google Page Rank, social choice theory, bibliometrics, etc.) and is called recursive iteration or repeated improvement. There is yet another philosophical quibble here. We now also have a choice of dealing with the points matrix or the goals matrix, and we shall see that each choice will lead to a different result. This is true of most complex problems in the social sciences – there is no such thing as a ground truth. Choosing goals over points leads to an alternative truth, both equally valid. The Euro 2020 rules lead to yet another result, etc. Our computations show that the recursive iteration converges reasonably well for both matrices. Fig. 1 shows the two-dimensional dispersion of Power-Weakness Ratio (PWR) and Normalized Power-Weakness Difference (PWD) for the iteratively weighted points tournament matrix. We find Italy, Belgium, and England to be the three teams that stand out on a points basis at Euro 2020. Belgium’s presence should not surprise us. It won handsomely in Group B. Of the five matches it played, it lost only to Italy. Fig. 2 shows the two-dimensional dispersion of PWR and PWD for the iteratively weighted goals tournament matrix. Now there are four countries that stand out: at the top are England, Belgium, Italy, and Denmark. From our experience with the points table, Belgium’s presence is not a surprise. Denmark’s presence is not so easily explained. It lost narrowly to England. ---------------------------------------------------------------------------------------------------------------Dr Gangan Prathap is associated with the A P J Abdul Kalam Technological University, Thiruvananthapuram. He was earlier at the National Institute for Science Communication and Information Resources (NISCAIR), New Delhi and more recently with the National Institute for Interdisciplinary Science and Technology (NIIST), Thiruvananthapuram. Address: 56 Pebble Gardens, Njandoorkonam, gangan_prathap@hotmail.com Chempazhanthy PO, Thiruvananthapuram-695587; Email: Table 1: The Points and Goals Tournament Matrices in Group B at the end of the roundrobin stage. Belgium and Denmark proceeded to the knock-out stage. We see what is called a Win-Loss loop: Denmark>Russia>Finland>Denmark, where > signifies a clear win, and so all three are tied on points. Denmark goes through on goal difference; here P-W. Group B POINTS Denmark Finland Belgium Russia P POWER Denmark Finland Belgium Russia GOALS P Denmark 0 0 0 3 3 Denmark 0 0 1 4 5 Finland 3 0 0 0 3 Finland 1 0 0 0 1 Belgium 3 3 0 3 9 Belgium 2 2 0 3 7 POWER Russia 0 3 0 0 3 W 6 6 0 6 18 Russia 1 1 0 0 2 W 4 3 1 7 15 Fig. 1. The two-dimensional dispersion of Power-Weakness Ratio (PWR) and Normalized Power-Weakness Difference (PWD) for the weighted points tournament matrix shows the top three countries at Euro 2020. Fig. 2. The two-dimensional dispersion of Power-Weakness Ratio (PWR) and Normalized Power-Weakness Difference (PWD) for the weighted goals tournament matrix shows the top four countries at Euro 2020.