Page 1 MAREK M. KAMINSKI Host Paradox How Strong are Soccer Teams? ‘Host Paradox’ in FIFA’s ranking.* Marek M. Kaminski University of California, Irvine Abstract: FIFA's ranking of national soccer teams is littered with paradoxes (or non- intuitive properties). The most surprising phenomenon is that it underrating the teams of the main championship tournament hosts. This “Host Effect” follows from the absence of hosts in preliminaries and the resulting restriction of hosts’ pre-tournament matches to friendlies (friendly matches) that enter the FIFA system with low weights. Various models built with the FIFA data allow estimating the magnitude of the effect at 16.5-35.4 positions. Keywords: football, soccer, FIFA ranking, social choice paradoxes, Euro 2012. Department of Political Science i Institute for Mathematical Behavioral Science, University of California, 3151 Social Science Plaza, Irvine, CA 92697-5100, U.S.A.; email: [email protected]; tel. 9498242744. * Barbara Kataneksza, Grzegorz Lissowski and Marcin Malawski provided helpful comments. Center for the Study of Democracy provided financial support. Page 2 MAREK M. KAMINSKI Host Paradox 1. Introduction: FIFA ranking FIFA (Fédération Internationale de Football Association) is the international governing body of national soccer associations. It organizes the World Cup and coordinates the activities of six regional federations that supervise local championships and friendlies (friendly matches) of their members. Based in Zurich, FIFA is managed by 25-member strong Executive Committee headed by the President. One of FIFA’s high profile activities is the monthly announcement of the ranking of the best national soccer teams. The positions are noted by the media and affect sponsors’ generosity. In addition, the position in FIFA’s ranking affects the team’s chances in drawing opponents in preliminaries and main tournaments of various Cups. Higher positions in the ranking are associated with lower expected quality of opponents. FIFA’s ranking evolved over time. The most recent 2006 version takes into account the results of the official matches the national team played in the past four years, the opponent’s position in the ranking, the strength of opponent’s federation and the match’s importance.1 For every game the team receives corresponding points. Then, the average score is calculated for the past 12 months, previous 12 months, etc; the average is lowered for teams that played fewer than five matches. The position in ranking is the function of weighted average score for the past 48 months. A detailed description of the procedure and its components is as follows: 1) Score for a match: For every match the team receives the number of points equal to the product: P = M × I × T × C 1 Procedures, scores and ranking positions are quoted from FIFA’s website (2012a). Page 3 MAREK M. KAMINSKI Host Paradox where the factors are calculated according to the following rules: • M (points for the match’s result): 3 for victory, 1 for tie and 0 for defeat; when penalty shots where used, the winner receives 2 points and the loser 1 point. If preliminaries include a two-match game, and the results are symmetric, the result of the second match is disregarded and the points are assigned as if penalty shots were applied; • I (importance) depends on the match’s category and is equal to: 1 – a friendly or a minor tournament 2.5 – preliminaries to World Cup or a federation’s cup 3 – main tournament of a federation’s cup 4 – main tournament in World Cup; • T (opponent’s strength) depends on the most recent position of the opponent in FIFA ranking and is equal to (200–opponent’s position in the ranking). Exceptions: the ranking’s leader has the strength of 200 and the teams ranked from position 150 downwards have the strength of 50; • C (correction for federation’s strength) is equal to the average strength of the team and its opponents’ federations. The strength of every federation is calculated from its members results in the three most recent World Cups. At the end of 2011, the value of C was equal to: 1 for UEFA (Europe) and CONMEBOL (South America) 0,88 for CONCACAF (North America, Central America and the Caribbean) 0,86 for AFC (Asia and Australia) and CAF (Africa) 0,85 for OFC (Oceania) Page 4 MAREK M. KAMINSKI Host Paradox 2) Average score for the past year (beginning at a certain point and ending exactly 12 months later) is calculated as an arithmetic mean from all matches if the team played at least five times. With a smaller number of matches, the average is multiplied by 0,2×number of matches played. 3) Position in ranking r at a given moment represents the total weighted sum of points over the past four years according to the formula: R = P-1 × 0.5 P-2 × 0.3 P-3 × 0.2 P-4 where every component P-i is a weighted score for matches played over the period: P-1 – last 12 months; P-2 – from 12 to 24 months back; P-3 – from 24 to 36 months back; P-4 – from 36 to 48 months back; 2. Paradoxes The formula of FIFA’s ranking is vulnerable to „paradoxes.” Here, the term „paradox,” made popular in voting theory and social choice theory by books by Brams (1975) and Ordeshook (1986), denotes a situation when the ranking behaves contrary to our basic intuition. In other words, the function that assigns to the relevant soccer statistics a ranking doesn’t satisfy certain properties that are interpreted as “obvious,” “desired” or “fair.” Social choice theory socialized us to a situation when certain desired properties Page 5 MAREK M. KAMINSKI Host Paradox cannot be satisfied at the same time (Arrow 1951). Whether one can find an equivalent of Arrow’s Theorem for ranking teams remains an open question. Below, I start by providing several examples that illustrate a few striking paradoxes. The analysis of the main paradox follows. It is easy to point out certain components of the FIFA formula that would generate surprise or protest with most fans. For instance, the number of points doesn’t depend on whether a team plays at home or not. Thus, in a match of a similar rank, a team receives more points for defeating Qatar (the controversial organizer of 2022 World Cup and #93 in the ranking at the end of 2011) than for a tie with Brazil (#6) played on the famously intimidating Estádio do Maracanã in Rio. The ranking’s quirks make certain teams reach very high position against common sentiment placing them lower. For instance, in September 1993 and July and August 1995, Norway was second while in 2006 USA was fourth. Sometimes paradoxical results happen systematically. 2.1. Violation of Goal Monotonicity: losing a goal increases the score An apparently obvious property is that the score used by FIFA should increase or stay constant with every goal won by a team. Under certain conditions, this property of “Goal Monotonicity” is violated. Team A plays with Team B in two-match competition for advancing to the next round. A defeats B at home 2:0, and then loses 0:2.2 Let’s analyze the consequences for A of losing the second goal: By losing 0:1 A would advance to the next round receiving zero points for the lost match. When A loses the second goal, the score becomes symmetric (2:0 and 0:2), and the result of the second match is decided by penalty shots. But this increases A’s score for this match (versus 2 Example from football-rankings (2012). Page 6 MAREK M. KAMINSKI Host Paradox receiving zero points for defeat 0:1) since under two possible outcomes A wins some points. If A loses, it receives the relevant number of points with a multiplier 1; if A wins, the multiplier is 2. In both cases the number is positive. As an effect, losing the second goal by A automatically increases A’s FIFA score for the match and possibly its position in the ranking! A mirror problem appears for Team B that receives more points for a match won 1:0 than for winning a second goal, regardless of the result of penalty shots. The “Goal Monotonicity” paradox took place in Jordan-Kyrgyzstan preliminaries. On October 19, 2007, Jordan lost 0:2 and ten days later beat Kyrgyzstan at home 2:0. For winning 2:0 and then winning penalty shots Jordan received 284.75 points while for winning only 1:0 it would receive substantially more, i.e., 427.125 points.3 A similar problem was noted when Australia beat Uruguay when competing for advancement to the 2006 World Cup. In general, similar problems appear always when the result of a two- match competition is settled with penalty shots. It is not easy to eliminate the above paradox. The core problem is that penalty shots are an additional “mini-match” that takes place after two games that ended with symmetric results. If penalty shots affect the score, then we get problems similar to those described above. If penalty shots do not affect the score, then the fact that one team overall beat the second one is disregarded. The number of problems is greater. In all examples offered below, we assume that all teams played at least five matches in every twelve-month period used for calculating the means and exactly five matches in the last period; that previously played matches will not be re-classified to a different period after the ranking is modified and that no other matches were played in the period between the announcement of old and new rankings.