A Simple Protocol for FIFA World Cup 2022 GANGAN PRATHAP “The greatest show on earth” has just completed the group stage on 02 December 2022. In eight groups of four teams each, we saw 32 teams play 48 matches, starting from 20 November and ending with the final group stage match on 02 December. Two teams qualified from each group for the knockout stage. A very complex 8-step protocol is used to determine the teams to advance to the knockout stage (2022 FIFA World Cup - Wikipedia): 1. Points obtained in all group matches: Win: 3 points; Draw: 1 point; Loss: 0 points; 2. Goal difference in all group matches; 3. Number of goals scored in all group matches; 4. Points obtained in the matches played between the teams in question; 5. Goal difference in the matches played between the teams in question; 6. Number of goals scored in the matches played between the teams in question; 7. Fair play points in all group matches (only one deduction can be applied to a player in a single match): Yellow card: −1 point; Indirect red card (second yellow card): −3 points; Direct red card: −4 points; Yellow card and direct red card: −5 points; 8. Drawing of lots. All this could have been avoided if a simple head-to-head tournament matrix procedure had been adopted (https://link.springer.com/article/10.1007/BF02289722) based on the work of an Indian mathematician C Ramanujacharyulu (henceforth Ram’s tournament metaphor). In the 48 matches a total of 120 goals were scored. Note that in the group stage there is no recourse to penalty shoot-outs. With this knowledge, we can construct the tournament matrices based on goals, for each group. 1 Table 1 is an illustration of how the Tournament Matrices based on goals scored are calculated. We choose Group H for this purpose. Of all the groups this was the most closely contested. In 6 matches a total of 17 goals were scored. There was an interesting win-loss loop: Portugal beat Ghana (3-2), Ghana beat South Korea (3-2) and in turn South Korea beat Portugal (2-1). South Korea and Uruguay were tied in a goalless draw and had identical profiles: 1 win, 1 loss and 1 draw each for a total of 4 points each, and were identically tied on goal difference (0). The FIFA protocol chose South Korea instead of Uruguay. However, we see from Table 1 that Ram’s protocol chooses Uruguay instead of South Korea. Let us now spend some time to see how Ram’s metaphor works. In Table 1, there are three stages. In the “Power” stage, the row-totals or row-sums (let us call this P) give an idea of the “power” of each team. Thus, the vector product of each row of the 4x4 tournament matrix with a vector {x} of unit terms {1,1,1,1) gives the Goals For (GF) for each team in the row. In the second stage we use a transpose of the 4x4 matrix to get the “Weakness” matrix. The rows here give the column-totals or column-sums of the original matrix (we will call this W). Again, the vector product of each row with the unit vector gives the Goals Against (GA) or “weakness” of each team in Group H. Ramanujacharyulu’s "mostbalanced" point of view is that one should try to find out who can combine the greatest ability to win (most goals scored) with the least susceptibility to lose (least goals conceded). That is, "in tournaments one may be interested in locating the really talented man (sic) in the sense that he has won over the largest number of opponents but simultaneously he has been defeated by only a few opponents." The P values (i.e., the row-totals) and the W values (the column-totals) now measure the “power” and “weakness” of each team. We call the ratio (PW)/(P+W) the Normalized Power-Weakness Difference (PWD) and this is a dimensionless measure of the “quality” of the team. PWD has the attractive feature that it is always bounded between -1 and 1. At this stage, we should understand that the simple row-sums and column-sums assume that a goal scored by each team is given the same weight irrespective of how strong how weak the opponent is (that is the idea behind using a vector of unit terms). This is what we see in the column marked 1. We call this the raw or unweighted score. At this stage the ranking based on PWD scores is: Portugal (0.20), Uruguay and South Korea tied on (0.00), and Ghana (2 0.17). Ramanujacharyulu’s deep insight was that the weightage can be changed iteratively, taking consideration of the “quality” of the team, leading to an eigen-value problem. Effectively, this is done by multiplying the citation matrix by itself recursively until convergence is reached in both the power and weakness dimensions. Table 1 shows the operationalization of this step. This yields the weighted values of P and W and the corresponding PWD ratio after each iteration. This graph-theoretical 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. 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. We find from Table 1 that for Group H there is reasonable convergence after 6 iterations to produce a clearcut ranking: Portugal (0.07), Uruguay (0.06), South Korea (-0.03) and Ghana (-0.06). We can repeat this for the other seven groups. Some groups need more iterations for an acceptable convergence. Table 2 shows the rankings Group-wise based on Ram’s PWD score after 9 iterations. The teams that made the grade using the FIFA protocol is highlighted in olive green. In Groups A, D and H, Ram would have picked a different team. In Groups E and F the team orders would have been reversed. This very simple algorithm, which can be easily set up in an Excel sheet, is more rational and more intuitive than the 3 points for a win, 1 point for a draw, etc. procedure that FIFA now uses. ---------------------------------------------------------------------------------------------------------------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, Chempazhanthy PO, Thiruvananthapuram-695587; Email: gangan_prathap@hotmail.com 3 Table 1: The Goals Tournament Matrices in Group H at the end of the round-robin stage. Portugal and Uruguay should have proceeded to the knock-out stage. Instead, it was South Korea that joined Portugal following the FIFA logic! 5 6 South Korea 4 Uruguay 3 Ghana 2 Portugal 1 x Portugal 0 3 2 1 1 6 0.35 1.35 0.29 1.44 0.31 1.43 0.31 1.42 0.31 1.43 0.31 Ghana 2 0 0 3 1 5 0.29 1.41 0.30 1.42 0.30 1.39 0.30 1.41 0.30 1.40 0.30 Uruguay 0 2 0 0 1 2 0.12 0.59 0.13 0.61 0.13 0.61 0.13 0.60 0.13 0.61 0.13 South Korea 2 2 0 0 1 4 0.24 1.29 0.28 1.19 0.26 1.23 0.26 1.21 0.26 1.22 0.26 4 17 1.00 4.65 1.00 4.66 1.00 4.65 1.00 4.63 1.00 4.65 1.00 x X x X x X x X x South Korea X Uruguay x Ghana X Portugal POWER x X Portugal 0 2 0 2 1 4 0.24 1.29 0.28 1.24 0.27 1.23 0.26 1.24 0.27 1.24 0.27 Ghana 3 0 2 2 1 7 0.41 1.41 0.30 1.67 0.36 1.55 0.33 1.60 0.35 1.58 0.34 Uruguay 2 0 0 0 1 2 0.12 0.47 0.10 0.56 0.12 0.53 0.11 0.53 0.11 0.54 0.12 South Korea 1 3 0 0 x X x X x X x X x X x Uruguay South Korea 4 0.24 1.47 0.32 1.19 0.26 1.34 0.29 1.26 0.27 1.30 0.28 17 1.00 4.65 1.00 4.66 1.00 4.65 1.00 4.63 1.00 4.65 1.00 Ghana 1 4 Portugal WEAKNESS Portugal 0 2 0 2 1 0.20 0.20 0.02 0.02 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 Ghana 3 0 2 2 1 -0.17 -0.17 0.00 0.00 -0.08 -0.08 -0.06 -0.06 -0.06 -0.06 -0.06 -0.06 Uruguay 2 0 0 0 1 0.00 0.00 0.11 0.11 0.04 0.04 0.07 0.07 0.06 0.06 0.06 0.06 South Korea 1 3 0 0 1 0.00 0.00 -0.06 -0.06 0.00 0.00 -0.04 -0.04 -0.02 -0.02 -0.03 -0.03 normalized POWERWEAKNESS DIFFERENCE x X x X 4 x X x X x X x X x Table 2: The rankings Group-wise based on Ram’s PWD score after 9 iterations. The teams that made the grade using the FIFA protocol is highlighted in olive green. In Groups A, D and H, Ram would have picked a different team. In Groups E and F the team orders would have been reversed. Power Country Group Netherlands P-Raw Weakness W-Raw P-Wt PWD W-Wt Raw Wt 0.04 0.01 0.00 0.00 0.67 0.67 0.03 0.03 0.00 0.00 0.14 0.15 Senegal 0.04 0.03 0.00 0.00 0.11 0.02 Qatar (H) 0.01 0.06 0.00 0.00 -0.75 -0.66 England 0.08 0.02 0.02 0.01 0.64 0.62 0.02 0.01 0.00 0.00 0.33 0.00 Iran 0.03 0.06 0.01 0.02 -0.27 -0.17 Wales 0.01 0.05 0.00 0.01 -0.71 -0.88 Ecuador United States A B Argentina 0.04 0.02 0.00 0.00 0.43 0.22 Poland 0.02 0.02 0.00 0.00 0.00 0.09 Saudi Arabia 0.03 0.04 0.00 0.00 -0.25 -0.12 Mexico 0.02 0.03 0.00 0.00 -0.20 -0.17 France 0.05 0.03 0.00 0.00 0.33 0.28 Tunisia C 0.01 0.01 0.00 0.00 0.00 0.13 Australia 0.03 0.03 0.00 0.00 -0.14 -0.15 Denmark 0.01 0.03 0.00 0.00 -0.50 -0.31 Spain 0.08 0.03 0.16 0.06 0.50 0.44 0.03 0.03 0.12 0.08 0.14 0.18 Germany 0.05 0.04 0.12 0.13 0.09 -0.04 Costa Rica 0.03 0.09 0.08 0.20 -0.57 -0.45 Croatia 0.03 0.01 0.00 0.00 0.60 0.46 0.03 0.01 0.00 0.00 0.60 0.28 Belgium 0.01 0.02 0.00 0.00 -0.33 -0.23 Canada 0.02 0.06 0.00 0.00 -0.56 -0.41 Brazil 0.03 0.01 0.03 0.01 0.50 0.43 0.03 0.03 0.04 0.03 0.14 0.15 Cameroon 0.03 0.03 0.04 0.05 0.00 -0.12 Serbia 0.04 0.07 0.05 0.07 -0.23 -0.17 Portugal 0.05 0.03 0.09 0.08 0.20 0.07 Uruguay 0.02 0.02 0.04 0.03 0.00 0.06 South Korea 0.03 0.03 0.08 0.08 0.00 -0.03 Ghana 0.04 0.06 0.09 0.10 -0.17 -0.06 Japan Morocco Switzerland D E F G H 5