How Strong Are Soccer Teams? The “Host Paradox” and Other Counterintuitive Properties of FIFA’s Former Ranking System

Abstract

I investigate the paradoxes associated with the Fédération Internationale de Football Association (FIFA) point-based ranking of national soccer teams. The ranking has been plagued with paradoxes that incentivize teams to avoid playing friendly matches, i.e., matches that are not part of any official FIFA tournament or preliminaries, and applying other counterintuitive strategies. The most spectacular paradox was the dramatic underrating of the hosts of major tournaments. For a long time, host teams, which were absent from preliminary matches, would play only friendly matches that awarded few points. Here, I present three models that estimate the magnitude of the resulting “host effect” at 14.2–16 positions. Such an estimate counteracts the intuition that a large investment in hosting a tournament should result in an improvement in the host team’s standing. However, as discussed here, a given host’s low ranking could decrease interest in the tournament, and likely result in a major loss of advertisement revenue.

1. Introduction

I investigate the paradoxes associated with the former FIFA ranking. Among a variety of its counterintuitive properties was a very poor treatment of tournament hosts. The source of problems was the low weight assigned to friendly matches, or friendlies, i.e., matches that are not part of any official FIFA tournament or preliminaries, versus the preliminaries of both the World Cup and the regional Federation Cups (the multiplier of 1 versus 2.5). Hosts advance to main tournaments automatically; therefore, they do not play in the preliminaries, which typically started about two years before the main event. Thus, for approximately two years before the tournament, the host plays only low-scoring friendlies. Here, it is important to notice that a team’s ranking is the weighted sum of its scores over the course of four years (with the weights equal to 0.2, 0.3, 0.5, and 1, from the most distant year to the most recent, respectively). If a team only played friendlies over two years, the friendlies had a total weight approximately equal to 0.75 of all the team’s results of the past four years. Thus, paradoxically, even if a host scores very well in the friendlies, its position in the rankings is usually doomed to decline. FIFA has vaguely acknowledged that host teams have “less opportunity for getting more points” [1]. There are also various publications that describe the problem with respect to specific hosts; for example, Wang and Vandebroek [2] described how their ranking system and its variants would have avoided the host effect problem for the organizers of Euro 2012—Poland and Ukraine.).

Consider the 2018 soccer World Cup that took place in Russia. At the time, Russia was criticized by various countries for the annexation of Ukrainian Crimea and the “hybrid war” in Donbass. Ultimately, the presidents of some participating countries decided to skip the World Cup. Another problem was the poor standing of the Russian team. On 7 June 2018, just before the first match, the Fédération Internationale de Football Association (FIFA) rated Russian Sbornaya at position 70, the lowest ranked team of all 32 World Cup competitors. Critics pronounced that the Russian team was “arguably the poorest in the history of Russian football” [3], saying that Russia was lucky to be drawn in a relatively easy first-round group, but also that this was “even better news for Uruguay, Saudi Arabia and Egypt, because they get to face Russia—the worst Pot 1 team by a wide margin” [4]. In fact, Russia turned out to be a tough opponent and achieved considerable success in the tournament. For major competitions, teams are grouped into “pots”: Pot 1 includes the top-ranked teams worldwide (for the World Cup or preliminaries) or within each confederation (for Confederation Cups or preliminaries); Pot 2 includes the same number of teams ranked immediately lower, and so on. Each grouping includes one team from each pot.

Russia humiliated Saudi Arabia 5:0 and convincingly beat Egypt 3:1 in the first-group, round-robin phase of the 2018 World Cup. Ultimately, Russia became the first team to advance to the last-16 knockout phase. Among the eight Pot 1 teams, Poland and Germany failed to qualify, and Argentina did so only thanks to a last-minute goal. Russia’s “safety margin” was comfortable: they would have advanced due to their superior goal difference. Then, in two dramatic matches, Russia first beat the soccer superpower Spain in a penalty shoot-out, and then narrowly lost a penalty shoot-out to Croatia—an ultimate finalist that lost only to France. Russia was thus very close to advancing to the semifinals.

Better-than-expected performance of the host of a tournament is likely to be partly due to home-field advantage, an effect widely observed in major soccer national leagues (see Table 2 in [5]. However, the potential home bias effect comes during the tournament, whereas any substantial host’s slide in the rankings precedes the tournament. Thus, possible home bias present in tournament results does not explain the pre-tournament fall in the rankings (see Figure 1). Reviewing Russia’s record, we must ask: was it really an overperformance of the Russian team? Systems other than the FIFA’s ranking rated the Russian Sbornaya much higher: from 42 [6] to 45 [7], and 49 [8].

Similar cases of “overperformance” have happened in previous tournaments as well, and they also involved tournament hosts outperforming expectations. In almost all cases, the tournament was preceded by a long slide of the host in the ranking. As mentioned earlier, the problem was that FIFA’s ranking system seriously undervalued the friendly matches played by the tournaments’ hosts, whereas other teams played in highly valued preliminaries. The preliminaries for the 2018 World Cup started as early as 12 March 2015 (in Asia), and as late as 4 September 2016 (in Europe), and they lasted until 15 November 2017. Thus, in the time leading up to the 2018 World Cup, Russia had predictably been sliding in the world rankings (see Figure 1).

In this paper, I investigate this perverse “host effect,” which can lead to what I term a “host paradox”. The “host effect” was originally introduced by Kaminski [9]. The preliminary analysis dealt mostly with the performance of the Polish team and the data were limited to pre-2012 tournaments. In Section 2, I begin by reconstructing the details of the former FIFA ranking system and briefly discuss a few paradoxical traits of the ranking. In Section 3, using the data from past tournaments, I examine the magnitude of the host effect. Next, I discuss possible solutions to the paradox. Finally, I conclude with an assessment of the paradox’s consequences.

2. The Paradoxes and Strategic Manipulability of FIFA’s Ranking System

FIFA is the international governing body of 211 national soccer associations. Founded in 1904 and headquartered in Zürich, Switzerland, FIFA is managed by a 25-member executive committee, which is headed by a president. FIFA’s main activity is the organization of the FIFA World Cup, the preliminaries, and other tournaments. It coordinates the activities of six regional federations that supervise their members’ local championships and friendlies. The six federations include the Asian Football Confederation (AFC, with Australia included); Confederation of African Football (CAF); Confederation of North, Central American and Caribbean Association Football (CONCACAF); South American Football Confederation (CONMEBOL); Oceania Football Confederation (OFC); and Union of European Football Associations (UEFA).

Among FIFA’s highest profile activities is announcing its monthly ranking of national soccer teams. From this ranking, the media duly notes the positions of the teams, and their exact location—whether high or low—can affect the generosity of team sponsors. Moreover, a team’s ranking determines its chances of drawing opponents in the preliminaries and main tournaments of the various cups, including the World Cup, because of its allocation among different pots. Having a higher ranking implies a lower expected ranking of opponent. For example, consider team A from Pot 1 versus team B from Pot 2. A and B have the same chance of meeting any specific opponent from Pot 3 and higher pots. However, in addition, A will face one opponent from Pot 2 while B will face one opponent from Pot 1. The position in the ranking is a proxy for the teams’ level of strength; therefore, A will theoretically face a weaker opponent than B.

FIFA’s ranking system has evolved over time. Before a major rule change that occurred in 2018, the previous system (established in 2006) factored in the results of the official matches that the national team played in during the previous four years, their opponents’ ranking positions, the strength level of the opponents’ federations, and the importance of each match. Each team receives points for every game it plays. Then, about once a month, the teams’ average scores are calculated for the past 12 months, the 12 months before that, etc. A team’s ranking position reflects its weighted average score for the prior 48 months, a sort of a moving weighted average. The details of the procedure are presented in

Table 1. Factors affecting the scores for individual matches.

Factor	Symbol	Definition
Match score	ms	Victory: 3; tie: 1; defeat: 0; penalty shoot-out: 2 for winner, 1 for loser
Match weight	mw	Friendly: 1; preliminaries to World or Federation Cup: 2.5; Federation or Confederation Cup: 3; World Cup: 4
Opponent strength	st	r = 1, st = 200 For opponent ranked r: 1 < r ≤ 150, st = 200 − r r > 150, st = 50
Federation strength	fs	In June 2018: CONMEBOL—1; UEFA—0.99; other—0.85

Note: If preliminaries included a two-match game, and if the results were symmetric, the result of the second match was disregarded and the points were assigned as if penalties were applied. Supposedly, a change took place in 2012 regarding this aspect, but the exact new rules were never perfectly clear and could not be reconstructed.

. The procedures, scores, and ranking positions are drawn from FIFA’s website [10]. It should be noted that the procedure for ranking women’s soccer teams is different; see [11] for a comparison of the FIFA men and women ranking systems.

A score for every match was equal to the product of four factors characterizing the strength of the opponent (as a sort of inversed ranking) and its confederation, the match’s importance, and the match’s result:

$$sm=ms\times mw\times st\times fs$$

(1)

The average score for the year (beginning at a certain date and ending exactly 12 months later) was equal to the arithmetic mean of all matches if the team played at least five times. With a smaller number of matches (i.e., between 0 and 4), the average was multiplied by 0.2 × m. The position in the ranking, r, at a given moment represented the total weighted sum of points over the previous four years according to the formula:

$$s=s{m}_{-1}+0.5 s{m}_{-2}+0.3 s{m}_{-3}+0.2 s{m}_{-4}$$

(2)

where sm_−i was the average score of the matches played over the period between 12(i − 1) and 12i months previously.

2.1. The Model

The problem of ranking certain objects based on individual information was first formalized by Arrow [12] and was implicitly present in various earlier works of welfare economists. Arrow’s simple setup involved assigning linear social orderings to all profiles of individual linear orderings. In the Arrowian approach, there was no “true” underlying social ranking, but only the aggregation of preferences; on the other side of the continuum, there are problems such as Condorcet jury, where the task is to find the underlying true outcome. Soccer and other sport rankings occupy the position somewhere between the two extremes: there is no true deterministic ranking of team’s strength. The ranking should be probably treated as some probabilistic device that provides estimates of teams’ strengths or a semi order, where only big differences in the scores are meaningful. It is intuitively clear that a strong team paired with a weak team would have a better chance of winning, but it is also clear that this relation is not deterministic. Empirically, the situation resembles selecting a competition winner based on experts’ judgments. There is some arbitrariness in the judgments, but they are also not completely detached from candidates’ performance (see, e.g., [13]). Empirical analysis of factors affecting match results such as home bias is performed using success functions [5].

FIFA’s soccer rankings consider much more complex information than preference profiles. Moreover, the constraints on certain relevant parameters are not well defined. The informal model formulated below is “excessive” in that it covers an excessive domain that encompasses all possible combinations of information components, but also some combinations that would not happen in reality. The ranking is defined for a specific date and aggregates the information over a period of time. The parameter “time” assumes all dates from the prior 48 months as values. Our goal is to define all components that are necessary for the application of Equations (1) and (2). The model’s contribution is in providing a formal delineation of all possible FIFA-like rankings.

We start with a non-empty and finite set of countries C, which is partitioned into non-empty subsets called federations. Each federation C_i is assigned a weight 0 < w_i^t ≤ 1 that may assume different values in time. The exact formula for determining federation weights is unknown.

A match is a vector that includes two different countries from C, their federation strengths, their individual strengths, the match weight (see “Match weight” in Table 1), the score, and the time.

The domain D includes all FIFA profiles d that are vectors with their subsequent positions defined as follows:

• Date of ranking.
• Set of countries C.
• The initial ranking of all countries from C, 48 months before the ranking’s date.
• The initial scores of all countries from C, 48 months before the ranking’s date.
• Exact dates of recalculating the ranking (once a month).

All matches over the 48-month period are considered (the assumed minimal constraint is that no country can play two or more matches on the same day).

A FIFA score function $S:\mathit{D}\to {ℝ}^C$ assigns a numerical score to all countries from C according to Equations (1) and (2). A FIFA ranking $F:\mathit{D}\to {ℝ}^C$ is a function that assigns a weak ordering to all countries from C that for each specific profile d represents S(d).

In a generic championship tournament (such as the World Cup or Confederations Cups), the rankings are fixed for the entire tournament’s period. Then, all matches played by a team enter the calculation of the next ranking. Typically, half of the teams end by playing only group-stage matches, and the others advance for play-offs. The weights for such matches are substantial (4 for the World Cup and 3 for Confederations Cups); therefore, the impact of tournaments on the final rankings is often substantial.

We assume that C does not change over time, which is a slight simplification because new countries are added or countries that cease to exist, such as Czechoslovakia, the Soviet Union, or Yugoslavia, may be excluded. The model formulated above includes many FIFA profiles that could not happen in real calculations of the ranking. For instance, no team could play matches on every day over 48 months. Another excess is created by the fact that the algorithm for changing the confederation strengths is unknown. Probably, it is an endogenous function of the results of matches of the members of a given federation. However, the purpose of the model was to encompass all actual FIFA profiles; the excess is impossible to eliminate due to the lack of information on the constraints on profiles, such as the number of matches a team can play.

Due to the complexity of information involved in FIFA rankings, the informational connection with Arrowian social choice is not very strong. The feature connecting the two models is the vulnerability to paradoxes of the FIFA function and Arrow’s social welfare functions. The excessive character of the model does not allow the formulation of any global paradoxes, because such a paradox could be due to the excess of the model that could not happen in actual calculations. However, it is possible to identify and discuss local paradoxes. Such paradoxes involve some subsets of FIFA profiles that are characterized with parameter combinations, which are known to be possible (in some cases, which actually happened). The paradox appears when the ranking functional F is applied to some subset or subsets of profiles and returns results from some other subset or subsets of profiles that have some counterintuitive properties. A sample of local paradoxes is discussed below.

2.2. Paradoxes and Strategic Manipulability

FIFA’s web page declares that “The basic logic of these calculations [the rankings] is simple: any team that does well in world football wins points which enable it to climb the world ranking” [10]. Unfortunately, FIFA’s ranking system has sometimes violated this and other simple properties; in other words, it was vulnerable to paradoxes.

Voting paradoxes were first investigated by Condorcet and Dodgson (see [14,15]). The term “paradox” was made popular in social choice theory, and especially in its political science applications, in books by Brams [16] and Ordeshook [17]. The idea denotes a situation where a ranking behaves contrary to basic intuition, i.e., it does not satisfy certain properties that would be informally considered “obvious”, “desired”, or “fair”. FIFA’s ranking has generated multiple paradoxes and multiple incentives for strategically sensible, but also apparently counterintuitive behavior.

Vulnerability to paradoxes is often closely related to vulnerability to manipulation. Farquharson [18] suspected that Arrow’s theorem implies manipulability of all reasonable voting methods; this suspicion was later proved by Gibbard [19] and Satterthwaite [20]. This is also the case with FIFA’s ranking. Lasek et al. [21] listed several methods for the “optimization” of a team’s position in the FIFA rankings that include choosing the number of matches, selecting the ideal opponents, avoiding friendlies, and creating score-improving coalitions. Wang and Vandebroek [2] further analyzed such strategic opportunities; as did Kaiser [22,23] in another sport.

Certain features of the FIFA formula were bound to generate criticism from soccer fans. For instance, the number of points did not depend on whether a team played at home. Thus, in a match of similar importance, a team would receive more points for defeating Qatar at home (the controversial organizer of the 2022 World Cup, and, as of 7 June 2018, ranked 98th) than for a tie with Brazil (ranked 2nd) played in the famously intimidating Estádio do Maracanã in Rio de Janeiro. Using the ranking positions and other data as of 7 June 2018, for any team, victory over Qatar would generate 102 × 3 × 0.85 = 260.1; however, a tie with Brazil would generate 198 × 1 × 1 = 198. The problem arises both with disregarding who was the host of the match and with the relatively high score of the weaker opponent.

Another problem is with the scores attached to friendlies. It is well understood that friendlies provide fewer points; therefore, a team that did well in the preliminaries could strategically avoid playing friendlies. For instance, Romania was criticized before the 2018 World Cup preliminaries for strategically playing only one friendly (and being seeded). Similarly, Poland was criticized before the 2018 World Cup for not playing friendlies until the ranking seeded them in the top pot. Thanks to such quirks, a team could climb the rankings despite common wisdom that would have placed that team much lower. Notably, in September 1993, as well as in July and August 1995, a relatively weak team of Norway was ranked second, whereas in April 2006, the United States was ranked fourth. When we reverse the ranking dates, in September 1993, the United States was ranked 26th, whereas in April 2006, Norway was ranked 40th.

Various paradoxical results happen systematically that are less obvious than “strategic match avoidance” and can be analyzed formally.

2.3. The Violation of Weak Goal Monotonicity

Intuitively, one might assume that a team’s score would increase, or at least stay unchanged, with each additional goal won by that team. However, until 2012, FIFA’s rules sometimes violated this property, which may be called “weak goal monotonicity”. A violation happened when preliminaries included a two-legged, home-and-away fixture. When the results were symmetrical, the result of the second match was disregarded, and the points were assigned as if penalty shots were applied.

For all teams $A, B \in C$, let us denote the score of team A from a match between A and B when A scores a goals and B scores b goals as $s{m}^{a,b}$. A more formal definition of this property is as follows:

Weak goal monotonicity: For all teams$A, B \in C$and for all matches between A and B,$s{m}^{a+1,b}\ge s{m}^{a,b}$.

As an example, I provide the following scenario. Team A plays against Team B in a two-match competition to advance to the next round. A first defeats B at home 3:0, and then loses 0:3. If the match had ended at 0:2, A would have advanced to the next round, receiving zero points for the lost match. When A concedes the third goal, the score becomes symmetrical (3:0 and 0:3), and the result of the second match is decided by penalties. However, crucially, we find that A’s score for this match would increase with the conceding of a third goal, regardless of the result of the penalty shoot-out: If A loses the penalty shoot-out, they receive some points with a multiplier of 1; on the other hand, if A wins, the multiplier will be 2. In both cases, we see that the number for A is positive, instead of receiving zero for losing 0:2. As an effect, conceding the third goal by A automatically increases A’s FIFA score for the match and, paradoxically, possibly its position in the rankings. Conversely, the mirror problem occurs for Team B, in that it would receive more points for a match won 2:0 than for winning 3:0, regardless of the result of the penalty shoot-out.

This “weak goal monotonicity” paradox has appeared in several matches. In the Jordan–Kyrgyz Republic preliminaries on 19 October 2007, Jordan lost 0:2, and 10 days later, beat Kyrgyz Republic at home 2:0. Let us calculate the actual and hypothetical score for Jordan from the second match. The weights are as follows:

• 0.85—Opponent confederation’s strength.
• 67—Opponent’s strength (Kyrgyz Republic on 24 October 2007 was ranked 133).
• 2.5—Match’s weight (World Cup qualifier).
• 2—Match’s score (Jordan won 2:0, and then it won penalty shots 6:5).

Jordan received a score of 284.75 for this match, i.e., the product of the four numbers listed above. For winning only 1:0, the last number on our list (the Match’s score) would have been equal to 3, and Jordan would have received substantially more—0.85 × 67 × 2.5 × 3 = 427.125 points [24]. A similar problem was noted when, on 12 July 2011, Saint Lucia defeated Aruba 4:2 after having lost previously, also by a score of 4:2. Furthermore, in November 2005, Australia beat Uruguay in a penalty shoot-out after first losing 0:1, and then winning 1:0. Losing the goal guaranteed Uruguay the same score as losing on penalties, or a better score in the case of winning the penalty shoot-out. Apparently, FIFA fixed this problem, although no proclaimed correction of the rules can be found [25]. For a comprehensive discussion of the problem, see [24].

In general, similar problems always appear when the result of a two-match competition is settled by penalties. Penalties constitute an additional “mini-match” played after two symmetrically completed games. If a penalty shoot-out affects the score, then we encounter difficulties such as those described above. If the penalties do not affect the score, then the fact that one team beat the second one overall is disregarded.

2.4. Examples of Paradoxes and Their Strategic Consequences

Certain FIFA’s ranking local paradoxes are associated with non-intuitive shifts in the ranking positions and violations of different versions of monotonicity (see Kaminski 2012). The following examples involve subsets of FIFA profiles with certain parameters fixed; under suitable ceteris paribus clauses, they illustrate the ranking’s potential for generating paradoxes. Each of the examples discussed below can be formulated as an appropriate property or axiom that is violated.

In the examples below, we assume for simplicity that: (a) all teams (Reds, Blues, and possibly Greens) played exactly five matches in the prior 12-month period; (b) all matches played in the previous 48 months would not be reclassified into a different period after the ranking is modified; (c) no other matches were played between the time of the old and new rankings by the teams in the examples or any other teams; and (d) the federations’ strengths are equal to 1. The scores are rounded in the usual way.

Example 1.

Winning a match moves the winner behind the loser.

The paradox is that, initially, Reds are ranked higher than Blues. In a friendly match, Reds defeat Blues, and as a result, Reds fall behind Blues in the ranking. For all teams $R, B \in C$, let us denote R winning a match against B as RwB. Let us denote the ranking of R and B before their match as r(R) and r(B), and the rankings after their match as r*(R) and r*(B).

Strong ranking reversal: For all teams$R, B \in C$and for all matches, $\left[r\left(R\right)<r\left(B\right)\&RwB\right]\to r^{*}\left(R\right)<r^{*}\left(B\right)$.

The numerical example illustrating the violation of this property is as follows:

• Reds: #20 in the ranking; 12 m points: 1100; total points in all previous periods: 200.
• Blues: #30 in the ranking; 12 m points: 200; total points in all previous periods: 1050.

According to our assumptions, Reds are ranked higher than Blues with the total of 1100 + 200 = 1300 points versus 200 + 1050 = 1250 points. Now, in a friendly match, Reds defeat Blues. Let us calculate the scores of both teams in the next ranking using our assumptions:

• Reds receive 200 + 5/6 × 1100 + 1/6 × 170 = 1145.
• Blues receive 1050 + 5/6 × 200 + 1/6 × 0 = 1216 2/3.

The next examples present the numbers that generate the described paradoxes in more concise formats. The rubrics in the examples are defined as follows:

• Team—the name of team.
• Ranking—team’s ranking.
• Score (last)—the average score from last year (five matches).
• Score (3)—the average score from three previous years.
• Final—the final score after the match or matches.

Example 2.

Ranking leaders doomed to fall after the match regardless of the result.

Reds and Blues occupy the top two positions in the ranking. After their friendly match, a different team, Greens, is ranked first, regardless of the result. The property that is violated here may be formulated in a slightly more general way without assuming any specific positions for the three teams. Again, r(.) denotes the ranking before a match between R and G and r*(.) denotes the ranking after the match. We assume that the only change in the parameters that happens is a single match between R and B.

Automatic exchange of leaders: For all teams$R,B,G\in C$, if $r\left(R\right)<r\left(G\right)\&r\left(B\right)<r\left(G\right)then{r}^{*}\left(R\right)<r^{*}\left(G\right)or{r}^{*}\left(B\right)<r^{*}\left(G\right)$.

A numerical example in

Table 2. Automatic exchange of leaders.

Team	Ranking	Score (Last Year)	Score (3)	Final
Reds	1	1000	1000	no more than 1932
Blues	2	990	1000	no more than 1933
Greens	3	950	1000	1950

shows how this property may be violated.

Greens did not play a match; therefore, their score remained the same and was equal to 1950. The ranking’s leader, Reds, received a post-match score of, at most, 1932 points (in the best-case scenario of winning against Blues), whereas Blues received, at most, 1933 (when winning against Reds). Let us provide the explicit calculations for Reds’ best-case scenario:

The final score is equal to 1000 + [5/6 × 1000 + 1/6 (3 × 198 × 1 × 1) = 1932

In both cases, Greens have the highest score and become the new ranking leader.

Example 3.

Coalitional manipulations increase the scores of both teams.

This property demands that when two teams play two consecutive matches, then it is impossible for them to jointly manipulate the scores. For two teams $R,B\in C$, let us call the results of a pair of matches between R and B a scenario, i.e., a scenario includes two coordinates where the first one represents the result of match 1 for R (win, tie, loss) and the second one represents the result of match 2 for R. For any two scenarios Sⁱ, let us denote the points score by team R under Sⁱ as Sⁱ(A).

Coalition-proofness: For all teams$R,B\in C$and any two scenarios S¹, S², $[S^1\left(R\right)>S^2\left(R\right)\left]\to [S^1\left(B\right)\le S^2\left(B\right)\right]$.

The property says that R and B can never manipulate the results of the two matches to simultaneously increase their scores as compared with some alternative scenario.

This problem appears always when, assuming the unchanged parameters between the two matches, in S¹ there are two ties and in S² R wins once and B wins once. In such a case, the higher multiplier for winning is equal to 3, whereas the combined multiplier for the two ties is equal to 2; this means that agreeing to one loss and one win versus two ties increases both teams’ total scores by 50%.

Example 4.

Tournament punishes the winners.

Teams Reds, Blues, and Greens play in an informal round-robin tournament in one group. Ranking before the tournament: Reds–Blues–Greens; results of the round-robin: Reds win, Blues are second, and Greens are third (RtBtG); ranking after the tournament: Greens–Blues–Reds.

Without unnecessary formalizing the rules for a round-robin tournament, we use the common rules that the teams are ranked according to their total points and the tie-breaking rule is the surplus of goals (which is sufficient for our example). In this example, r(.) denotes the rankings before the tournament and r*(.) denotes the ranking after the tournament. We assume that the only change in the parameters that happens are three pairwise matches among R, B, and G.

Tournament reversal: For all teams$R,B,G\in C$and any tournament T,$\left[RtBtG \& r\left(R\right)<r\left(B\right)<r\left(G\right)\right] \to \neg (r\left(R\right)>r\left(B\right)>r\left(G\right)$.

In our example of how the Tournament Reversal property may be violated, we assume that the results of the matches were as follows: R:G 2:0; R:B 0:1; B:G 0:1. Thus, the tournament results are RtBtG. Additionally, the initial rankings are $r\left(R\right)<r\left(B\right)<r\left(G\right)$. If we calculate the score of Reds after the tournament, we will derive $\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\times 1000+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$7$}\right.\times 3\times 197+500\approx 1299$, with the usual rounding. The results of calculations for the remaining teams are presented in

Table 3. Tournament reversal.

Team	Ranking	Score (Last)	Score (3)	Final
Reds	1	1000	500	1299
Blues	2	780	700	1343
Greens	3	560	900	1385

After the tournament, the ranking of Reds, Blues, and Greens was reversed.

.

Example 5.

Winning consecutive matches causes a slide in the ranking behind the losers.

Reds play matches in an informal play-off tournament; these are classified as friendlies. With each consecutive win, the position of Reds in the ranking declines behind the position of the opponent who lost. Now, $r^0\left(R\right), r^1\left(R\right), r^2\left(R\right)$ represent the initial ranking of Reds, their ranking after the first match, and their ranking after the second match, respectively; w¹, w² denote winning in the consecutive matches. We assume that the only change in the parameters results from two consecutive matches of R against B, and R against G.

Play-off cannot reverse rankings inversely to results: For all teams$R,B,G\in C$,$\left[r^0\left(R\right)<r^0\left(B\right) \& R{w}^{1}B \& r^1\left(R\right)<r^1\left(G\right) \& R{w}^{2}G\right]\to \neg \left[\left(r^1\left(R\right)>r^1\left(B\right)\right] \& \neg \right[\left(r^2\left(R\right)>r^2\left(G\right)\right]$.

Initially, Reds are ranked higher than Blues. In a play-off, they beat Blues, and after the match, they slide below Blues in the ranking. After this match, Reds are still ranked higher than Greens, but after winning the play-off against Greens, they slide below the Greens. The last column “Scores” represents the teams’ total scores before the first match, after the first match, and after the second match, respectively (see

Table 4. Play-off reverses rankings inversely to results.

Team	Ranking	Score (Last)	Score (3)	Scores
Reds	1	1000	200	1200/1132/1084
Blues	2	290	900	1190/1142/1142
Greens	3	250	900	1130/1130/1108

).

The common core problem for many examples discussed above is the low score assigned to friendlies, and the fact that the means for the four years are computed in the ranking independently. If a team’s high ranking depends mostly on an outstanding previous year, then even a very high victory in a low-value friendly with the ranking leader may ruin that team’s position. On the other hand, for a team with a weak previous year, even a defeat may be negligible.

The paradox described in Example 3 (Coalition-proofness) is based on another property of the index. Namely, the total number of points acquired by both teams in a match is not constant. In the case of a tie, the total is 1 + 1 = 2, whereas in the case of one team’s victory, it is 3. This creates some room for coalitional manipulation described in this example.

An obvious strategic incentive facing a team trying to maximize its position in the ranking is “match avoidance” or “match delay” to avoid playing low-scoring friendlies or to move them to a different period for calculating ranking positions. Under some conditions, both strategies could be also optimal for a host team to maximize its position in the rankings. For instance, in Example 5, an obvious incentive for a rank-maximizing team is to lose their first match and become eliminated from further matches. In Example 2, rank-maximizing teams should not be playing a match at all, or possibly register such a match as being played between their reserves rather than the top national teams.

3. Estimation of the Host Effect

An example of how the host paradox has affected teams is provided by the matches played in 2011 by Poland, a co-host of Euro 2012. Out of 13 matches, Poland won 7, tied 3, and lost 3. In these matches, they beat strong opponents, such as Argentina (ranked 10th at the end of 2011) and Bosnia and Herzegovina (ranked 20th); lost in close games with Italy (ranked 9th) and France (ranked 15th); and tied in their matches with Greece (ranked 14th), Germany (ranked 2nd), and Mexico (ranked 21st). Overall, 2011 was a good year for Poland, and much better than the previous two years (in 2010, Poland’s victories, ties, and defeats were 2, 6, and 3, respectively). Despite 2011 being a very good year for Poland, the team ended 2011 ranked 66th with 492 points—only slightly better than their finish in 2010, which was a terrible year for them, where they ranked 73rd, and lower than their final ranking in 2009, 58th, before the start of the preliminaries. Meanwhile, Ukraine, the other co-host of Euro 2012, was punished by FIFA’s ranking system as well. At the end of 2009, Ukraine was ranked 22nd; in 2010, the team fell to 34th; and ended 2011 in 55th place despite having quite a good year.

The difference in the weights for the friendlies versus the preliminaries is responsible for this disconcerting fall in rankings if everything else is held constant. In Section 4, I recalculate hypothetical scores for three recent tournament hosts (Russia, Poland, and Brazil) using three different corrections to the ranking to illustrate how they would have been ranked much higher if they had not been punished with the low weight attached to friendlies.

The estimation of the host effect presents substantial methodological challenges. First, the data did not come from a “representative sample,” but rather, were generated by three processes: the nonrandom selection of the host, the nonrandom selection of matches, and the random process of the results of matches with an unknown distribution. I believe that the first process could have affected the analysis in a noticeable way—mostly for the World Cup—whereas the second process did not introduce any systematic bias (see [26] on sample selection bias in the FIFA rankings). Second, in my calculations, I used ranking positions, rather than FIFA scores, which created a problem due to the ordinal measurement. However, the paradox in the host effect that I am interested in is caused by a host’s low position in the rankings. Any result obtained with FIFA scores would have to be translated into a mean ranking loss, which would recreate the ordinality problem.

Below, I discuss three alternative methods. The data used in this analysis included 26 hosts of eight World Cups, nine Asian Cups, and nine Euro Cups. After the change in the ranking methodology in 2006, FIFA recalculated the teams’ rankings back to the year 1993, which meant that the tournaments which took place from 1994 onward were included. Certain regional tournaments were omitted: specifically, the CAF and CONCACAF take place biannually, and their preliminaries are held in the same year as the main tournaments, which creates a short life span for a potential host effect. Additionally, the CONMEBOL has a small number of members and no preliminary rounds, and the OFC has lower-ranked teams, with 11 official members; the best team is New Zealand, ranked 119th.

3.1. Average Dip in the Rankings

Considering the above, the most obvious question to ask is: what is the average dip (i.e., drop in positions) in the rankings caused by the host paradox? To begin to answer this question, we assume the following notation for the key variables:

T—year of the tournament.

r_T—last ranking before the tournament (in the case of unclear timing, we use the ranking from the month immediately preceding the month of the tournament).

r_P—the ranking immediately preceding the start of host’s confederation preliminaries.

r_T+4—the first ranking in January or February of year T + 4.

$\Delta =r_T-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(r_P+r_{T+4}\right)$—the estimated individual host effect.

At the time of r_P, there were no negative consequences of future preliminaries, whereas at the time of r_T+4, the preliminaries were too old to be included in the calculations. The mean of both rankings was used to smooth random variations in the teams’ levels of performance and the potential effect of the growing FIFA membership. There is no good reason to assign speculatively unequal weights to both rankings. Finally, r_T is the last ranking before the tournament, where the preliminaries still carry heavy weight; this is publicized in the media and is often used as a measure of the host’s strength (see Appendix A). A bar over a variable denotes its mean value.

Table 5. The average positions and changes in the FIFA rankings for the hosts of major tournaments since 1994.

Tournament	Number of Hosts	${\overline{\mathit{r}}}_{\mathit{P} }$	${\overline{\mathit{r}}}_{\mathit{T} }$	${\overline{\mathit{r}}}_{\mathit{T}+4 }$	$\overline{\mathsf{\Delta }}$
All major	26	50.4	64.7	50.6	14.2
Euros	9	25.4	42.2	23.9	17.5
World Cup	8	32.0	36.1	17.0	11.6
Asian Cup	9	89.6	112.4	100.4	17.4

Note. Rounding resulted in slight discrepancies in the means. Nonparametric binomial tests were performed, with r_T > r_P and r_T > r_T+4 counted as “successes” and r_T < r_P and r_T < r_T+4 counted as “failures” (any ties were disregarded; q signifies the probability of success). The p-values for the one-sided test “H₀: q = 0.5” versus “H_A: q > 0.5”: Euro + World + Asian p < 0.001; Euro p < 0.001; World Cup p ≈ 0.008; Asian Cup p ≈ 0.008. Appendix A provides data for all 26 hosts. See the introduction to Section 3 for the explanation of the choice of tournaments.

displays the calculations of ${\overline{r}}_{P }$, ${\overline{r}}_{T+4 }$, and ${\overline{r}}_{T }$ for the Euros, the Asian Cup, the World Cup, and all three competitions combined. Hypothesis 1 operationalizes the conjecture of the host effect and states that a positive fall in the ranking equal to $\overline{\mathsf{\Delta }}$ can be identified:

Hypothesis 1 (H1).

$\overline{\mathsf{\Delta }}$> 0.

The overall average dip in rankings, as represented by $\overline{\mathsf{\Delta }}$, is equal to 14.2. For all cases, the estimated dips are positive and are in double digits; additionally, the p-values are small despite low counts for the regional tournaments. After conducting calculations, we find that Hypothesis 1 is corroborated in all cases.

The exact estimates for specific tournament types must be treated with caution due to the small counts. We see that the smallest host effect appears in the World Cup—a change of +11.6. One mitigating factor here might be the fact that, except for Russia, all recent World Cup host teams were very strong in their respective federations. One or two years before the World Cup, all eight teams participated in main regional tournaments, where it was relatively easy to qualify (see the analysis of Brazil in Section 4.3). Additionally, six out of the eight host teams also took part in the previous World Cup. Thus, the host effect was partially offset by the nonrandom process of selecting the hosts, i.e., the higher probability of the World Cup being hosted by regionally strong teams, which have more opportunities to play in heavily weighted matches than weak teams, and which also lose less often due to not playing in preliminaries. Moreover, one year before the World Cup, FIFA also organizes a small Confederations Cup. In this competition, both the host and the world champion play the champions of the regional confederations. This gives the strong host teams a relatively easy opportunity to earn heavily weighted points against the champions of smaller confederations.

Many more teams play in the regional Confederation Cups than in the World Cup. Thus, nonparticipation in the World Cup preliminaries is offset to a greater degree by participation in the Confederation Cups than vice versa. One would expect a stronger effect for the hosts of regional preliminaries than for the hosts of the World Cup. The data corroborate this conjecture.

We find that a strong host effect appears for the Asian Cup (+17.4). Here, one can speculate that the effect is relatively less diluted by participation in the other main tournaments, because the teams in the Asian Cup are substantially weaker than the teams of the organizers of either the World Cup or the Euros (see Appendix A). Every World Cup host except for Russia took part in the previous Confederation Cups; however, no Asian Cup host, except for China, participated in the previous World Cup. China gained nothing from their participation because they earned no points and scored no goals. Thus, the hosts of the Asian Cup earned their ranking points only in federation preliminaries, federation main tournaments, the World Cup preliminaries, and in friendlies. The only non-systematic effect influencing the results was nonparticipation in the regional championships. Additionally, the teams participating in the Asian Cup occupy lower positions in the ranking, where fewer points are needed on average for climbing the ranking. Thus, participation in the preliminaries leads to greater average gains than for competitors in the World Cup.

In the case of the Euros, organized by the UEFA, the high host effect of 17.5 could have been somewhat reduced by the fact that five out of the nine hosts participated in the previous World Cup. Similarly to the World Cup, the hosts of the Euros have much stronger teams than the hosts of the Asian Cup, and they also have more chances of playing highly valued matches. Nevertheless, playing friendlies decreases the average score quickly for high-ranked UEFA teams. Given the typically high-ranking positions of European teams, the Euro effect seems to be most consequential.

3.2. Estimated Dip as a Function of a Team’s Position

Another question, implicitly present in the earlier discussion, is whether the dip depends on a team’s strength. A strong team has more chances of playing in highly valued tournaments than a weak team due to the higher probability of advancing to the regional competitions and the World Cup. The point differences among the strongest teams are greater on average than such differences among weaker ones. Moreover, a strong host may additionally benefit from the FIFA’s Confederations Cup, which is held one year before the World Cup. All these aspects suggest that the ranking of a strong team is less vulnerable to the host effect.

An assessment of how the host effect depends on the host’s initial ranking would allow for a more precise estimation of the losses. Thus, an examination of the scatter plot should help us quickly see whether the relationship is linear (see Figure 2). For the alternative explanatory variable r_T+4, all results are nearly identical. Due to strong multicollinearity (Spearman’s rho correlation 0.965), r_T+4 and r_P cannot be used jointly. A polynomial regression returned similar results as well.

From Figure 2, we see that the scatter plot shows a clear linear relationship; the slope is slightly greater than 1 (see

Table 6. Rankings before the tournament, predicted by the ranking of the preceding preliminaries (regression r_T = B × r_P + C).

Variable	Coef.	SE(B)	t	p	No of Obs.
rP	1.078	0.098	10.94	0.000	25
constant C	12.1	6.18	1.95	0.063

Note. R² = 0.84; Adj R² = 0.83; F(1, 23) = 119.78. See Appendix A for data for all 26 hosts.

), which means that the expected loss of positions in the ranking is slightly increasing with the increase in the team’s initial position. The slides of stronger teams are surprisingly consistent, whereas the slides of weaker teams display more variation in the graph. This phenomenon may be caused by two factors. First, relatively larger differences in scores among stronger teams and some “crowding” in scores among weaker teams (smaller typical differences between consecutively ranked teams) may result in greater sensitivity to changes in the score among lower-ranked teams. Second, high-ranked teams are likely to play both in the World Cup and the Confederations Cup, which may alleviate the losses associated with being the host of one of them. As mentioned above, the Confederations Cup offers extra opportunities for stronger teams to play highly valued matches, and a greater number of such matches somewhat counterbalances the negative effects of friendlies.

Longer vertical distances of the data points located in the middle from the regression line suggest a heteroscedasticity problem, which is confirmed by tests. Thus, we know that the estimated standard errors are biased, and that the OLS results should be treated as representing the parameters of the population of all hosts thus far, but that one should be cautious when interpreting them as estimates of the general relationship. Heteroscedasticity arises, at least partially, from the one-sided constraints of the ranking. Specifically, the fact that high-ranked teams can fall in the rankings, but that their rise is limited (and, of course, for the first-positioned team, it cannot increase at all). Heteroskedastic two-step GLS estimation returns a significant constant of 4.42 and a borderline significant exponential coefficient of 0.023. However, the estimates used this way are mostly unreasonable; for example, for r_P = 25, $\widehat{r}$_T ≈ 10.7 (negative host effect); for r_P = 50.4 (mean), $\widehat{r}$_T ≈ 96 (very large mean effect); and for r_P = 60, $\widehat{r}$_T ≈ 289 (beyond the range of r_T). The ordinary least squares (OLS) intercept has substantial variance; therefore, a linear regression suppressing the intercept was also performed. Suppressing the intercept to zero would be equivalent to the assumption that, for high-ranked teams, the host effect is almost nonexistent, which has no justification in either theory or data. The estimate of a mean host effect in this case was similar. The problem with heteroscedasticity may be also related to the fact that both variables are essentially ordinal; the scatter plot of the scores could have looked differently.

Additionally, we find that the change in the host effect is surprisingly weakly dependent on the initial team’s position. The intercept of 12.1 and the slope of 1.078 mean that the host team which has a certain position in the rankings at time P could be expected to slide down in the last pretournament rankings by 12.1 plus 0.078 times its pre-preliminaries ranking. Thus, if one knows the host’s ranking r_P, one can estimate that the position at time T is increased by 12.1 + 0.078 r_P. The average estimated dip in the rankings according to this method is 16.0. The results corroborate that stronger teams do, in fact, slide less than weaker ones, but also that the difference is quite small (Table 6).

3.3. Comparative Analysis of FIFA Rankings and Alternative Rankings

Although FIFA’s ranking system is the most popular one, there are alternative rankings that use different methodologies for grading national soccer teams. If the host effect is caused by factors specific to the FIFA ranking system, we should not be able to see it in other ranking approaches. Conversely, if some external intervening factors are responsible for the host effect, we should be able to identify them in the alternative rankings as well.

Table 7. The average positions and changes in the FIFA and two alternative rankings Elo and rankfootball.

Ranking	${\overline{\mathit{r}}}_{\mathit{P} }$	${\overline{\mathit{r}}}_{\mathit{T} }$	${\overline{\mathit{r}}}_{\mathit{T}+4 }$	$\overline{\Delta }$
FIFA (repeated from Table 5)	50.4	64.7	50.6	14.2
Elo	52.1	50.9	52.8	−1.5
Rankfootball	53.6	52.5	51	0.1

Note. Nonparametric binomial tests run for r_T > r_P and r_T > r_T+4 counted as “successes” and r_T < r_P and r_T < r_T+4 counted as “failures” (any ties were disregarded; q signifies the probability of success). The p-values for the one-sided test “H₀: q = 0.5” versus “H_A: q > 0.5”: ELO—0.93; rankfootball—0.17. Note that four years after the tournament, hosts’ FIFA rankings returned exactly to the pre-preliminary levels. For the methodology behind the alternative rankings, see [7,8].

repeats the values of the FIFA aggregate rankings for the tournament hosts in the first row and shows the respective indicators compiled from the rankings produced by two alternative systems: Elo and rankfootball. The Elo ranking system, named after the Hungarian physicist and chess player Árpád Élö, is most notably used for ranking chess players. Alternative rankings are cited using the websites Elo [7] and rankfootball [8]. For Elo, the rankings were available for the beginning of years T − 2 and T + 4, and for the last month preceding the tournament in year T for all cases except at T + 4 for the recent hosts—France 2016 and Russia 2018. For rankfootball, the rankings were available for the beginning of years T − 2, T, and T + 4 from the end of 1996, and except at T + 4, for the recent hosts—again, France 2016 and Russia 2018.

The calculations of the two alternative ranking systems shown in Table 7 indicate that they are immune to systematic host-like effects. We find that the average rankings of all three systems are quite similar for the times P and T + 4, when the host effect was not present. However, Elo and rankfootball provide similar averages for time T before the beginning of the tournament, but FIFA shows a big dip. Both alternative estimates of the size of the host effect are insignificant.

4. Solutions

The host paradox could be eliminated, or substantially reduced, with various simple solutions, as described below. It is notable that FIFA had the means to introduce a solution to the paradox as a set of rules independent of the ranking’s properties. We describe three such solutions and perform simulations how they would affect the positions of three recent hosts, i.e., Russia in the 2018 World Cup, Poland in the 2012 Euros, and Brazil in the 2014 World Cup. For Poland and Russia, the elimination of the host effect would have placed them much higher in the rankings, but still well-below the top. For Brazil, the effect lowered their position in the ranking by a small number of positions, but without it, Brazil would have been a clear 1st in the world for long time instead of being ranked in the middle of the top 10.

4.1. Russia: Freezing the Host’s Score

A simple method to deal with the host paradox would be to freeze the host’s score at about the time the preliminaries start. In fact, the European confederation UEFA applies an equivalent solution. The UEFA uses its proprietary scoring system to assign teams to pots and to allocate club tournament spots to club teams. The rules for the UEFA ranking make an explicit provision for the tournament hosts:

“In the case of an association that has hosted UEFA’s Euro or FIFA’s World Cup final tournament during one of the reference periods as mentioned under Annex D.1.2 and therefore has no points from the respective qualifying competition, the points earned in the most recent qualifying competition in which the association has taken part are used.”

[27]

The preliminaries to the 2018 World Cup started almost three-and-a-half years before the main tournament, and the most important European preliminaries started approximately two years before the tournament. The exact moment of the freeze could be subject to discussion, but for the main regional confederations, this could occur with the start of the preliminaries and end with the last match of the preliminaries. About six months before the tournament, after the preliminaries are over, the calculation could be unfrozen and the score computed as if the period of freeze did not exist. About three-and-a-half years after the tournament, the host’s score would be back to the usual calculations, i.e., with the use of matches from the previous four years. Freezing could also be annulled if a host declares this desire in advance.

Freezing would certainly eliminate the effects for hosts of not playing in the preliminaries; however, a negative aspect of freezing would be that it would prevent any legitimate changes in the host team’s strength from being represented in the ranking. Moreover, an advantage is that it is a relatively simple and computationally uncomplicated.

Calculating the impact of freezing on hosts’ ranking positions is straightforward. If Russia had held its position from before the preliminaries, it would have been ranked 38 instead of 70 in June 2018. If instead of using the position in the ranking we used Russia’s score of 728 points from before the preliminaries, it would have given the Russian team position a slightly higher position: 33 in June 2018. It would not have affected Russia’s place in the World Cup group competition, but it would have given it a higher position at Euro 2020, both in the preliminaries and in the final competition.

4.2. Poland: Substituting Friendlies with Preliminaries

The second possible solution is more complicated, but it also has the advantage of using actual recent scores. This solution is motivated by the following question: What would a team’s position have been if some of the friendly matches had been assigned a higher multiplier to compensate for the higher multiplier of 2.5 used in the preliminaries?

For Poland, had it played all its 2010 and 2011 matches in the Euro 2012 preliminaries with identical results, then, at the end of 2011, it would have received approximately 1039.5 points instead of just the 492 it did receive since the scores from all 2010 and 2011 matches would be multiplied by 2.5 (the weight attached to preliminaries) instead of 1 (the weight attached to friendlies). The change in weights would have contributed 547.5 points to the total score. With 1039.5 points, Poland would have been in 11th place instead of 66th, which would have situated it just behind Argentina (which had 1067 points) and just ahead of Denmark (which had 1035 points).

Using the multiplier of 2.5 for all matches would be too generous to the host team, because the number of all friendly matches is usually greater than the average number of matches in preliminaries. Let us now estimate the modified score for Poland for the years 2010 and 2011—the time when the preliminaries took place—under the assumption that the multipliers for all friendlies are increased by the same number in such a way that the total surplus of weights over 1 is equal to the average total surplus of weights over 1 for teams that participated in the preliminaries. In more detail:

(1)

The points and positions in the rankings of all other teams remain unchanged.

(2)

The points of the Polish team are recalculated as follows: each of the 26 friendly matches Poland played in 2010–11 receives the increased multiplier equal to 1 + (9.725 × 1.5)/26 = 1.56, where 1 is the original weight assigned to friendlies; the number 9.725 represents an average number of preliminary matches for the European teams (see explanation below); 1.5 is the extra weight that is assigned to preliminary versus friendly matches; 26 is the total number of friendly matches that Poland played in 2010–2011).

Some actual opponents in friendlies, such as Mexico or Argentina, were non-European and could not be in the same preliminary group; in the case of friendly opponents such as Germany, France, and Italy, only one team could be in the same group with Poland in the preliminaries. This aspect is disregarded because an implicit assumption is that specific teams are less important and because the results in friendlies are only proxies for actual results. Moreover, the effects of recalculations on other relevant host countries are also disregarded. Out of five such hosts, only recalculation of the scores for Ukraine could potentially send Poland one position lower.

The possibly reduced incentive to play in a friendly is also disregarded. This may make it easier for weaker teams to score well against teams that are stronger but less motivated, and it may also allow teams to experiment with reserve players.

For point (2), the weights were calculated using the average number of matches played by the European teams that played in groups of six or five, as well as with some additional rounds. Since 51 teams played 248 matches, the average is equal to 9.725 (given that there are two teams per match). The weight of 1.56 uniformly distributes the extra weight of 14.59 from 9.725 hypothetical preliminary matches to 26 actual friendly matches.

Under such assumptions, Poland’s scores would have been as follows:

2008: 288.74 (actual score for 2008, unchanged).

2009: 171.4 (actual score for 2009, unchanged).

2010: 347.52 (estimated), instead of the actual score for 2010 of 222.8.

2011: 389.1 (estimated), instead of the actual score for 2011 of 249.4.

The total number of points at the end of 2011 would have been equal to (288.74 × 0.2) + (171.4 × 0.3) + (347.52 × 0.5) + 389.1 ≈ 672. Such a score would have placed Poland in 39th position in the December 2011 ranking—a full 27 positions higher than the actual position. This ranking for Poland would have been close to the rankings of Elo (38th, [7]), RoonBa (23rd, [28]), rankfootball (31st, [8]), CTR (33rd, [6]); and AQB (28th, [29]), all of which are FIFA’s ranking competitors.

4.3. Brazil: Disregarding Some or All Friendlies

Probably the simplest solution would be to entirely disregard the friendlies played by the host over a certain period. We will calculate the effect of disregarding the friendlies on the performance of the Brazilian team, the host of 2014 World Cup, for one year preceding the tournament. This is quite conservative given that the preliminaries in all federations started much earlier. In CONMEBOL, the preliminaries started as early as October 2011.

Brazil seems like an outlier in our analysis. Before the preliminaries, it was ranked #3; before the 2014 Cup, it was ranked #4; and four years after the tournament, it was ranked #2. Thus, on average, the host effect subtracted a relatively small 1.5 from the average positions before and after. However, a detailed analysis reveals that the negative effects on Brazil’s ranking were more subtle and somewhat hidden in the other rankings before the World Cup. Over the year preceding the 2014 World Cup, Brazil occupied much lower positions in the ranking, between 6 and 11. The host effect was masked by a spectacular performance of the Brazilian team in the 2013 Confederations Cups, which consolidated the high position just before the 2014 World Cup.

Let us consider the CONMEBOL preliminaries to the 2014 World Cup. They started early in 2011, and South American teams played a staggering number of 16 highly valued matches. The top-performing teams rose in the rankings, whereas Brazil slid from #5 on 19 October 2011, to a shockingly low #22 on 6 June 2013, just before the start of the 2013 Confederations Cup. Argentina, Colombia, Uruguay, and Ecuador were ranked higher. Then, Brazil won all five matches in the 2013 Confederations Cup played with the federations’ winners, including famously beating #1 ranked team Spain 3:0 in the final on 30 June 2013. This victory elevated Brazil, arguably the best team in the world at that time, only to #9 in the world. The low position in the ranking was dragged down by 15 friendlies that Brazil played earlier.

Let us calculate what would have been the effect of disregarding one year of friendlies from July 2013 to June 2014 on the last ranking before the 2014 World Cup. The average for the previous year would be equal to the average from five victories in the Confederations Cup with very highly ranked rivals. This would have given Brazil approximately 1673 points instead of the approximately 827 that it received when the friendlies were counted. The ranks of the five defeated teams were as follows: Italy (19), Spain (1), Japan (33), Mexico (18), Uruguay (20). After disregarding tiny corrections for Confederations’ strength, we obtain Brazil’s estimated 2013 score as 3 × 3 × (181 + 200 + 167 + 182 + 180)/5 ≅ 1673. The weighted score for the previous three years would have stayed unchanged at 415; 1673 points would have elevated Brazil to #1 position with 2088 points, with a huge margin over Spain, the leader on 5 June 2014, which had accumulated 1485 points. Brazil faced very strong incentives to strategically abandon playing any friendlies before the 2014 World Cup. Even if Brazil had reduced the number of friendlies from ten to two and lost both, it would have been ranked #1 with approximately 1610 points.

After the 2014 World Cup, Brazil would have been the ranking leader as well. Brazil’s performance in this tournament was not spectacular, although finishing in fourth place in the world would have been considered by most teams a great success. Brazil won four matches, tied one, and lost two. The performance was worth of 9114 points total, which equaled 1302 points per match. If the ten friendlies that Brazil played before the World Cup were disregarded, in the first post-tournament ranking on July 17, 2014, Brazil would have received 528 more points in addition to its 1241 for the total of 1769, ahead of #1 Germany’s 1724 and well ahead of #2 Argentina’s 1606. Given that Brazil played very well in tournaments and preliminaries after the 2014 World Cup, it would have stayed on top for a long period of time.

5. Conclusions

As discussed in this paper, the host paradox has produced results that were deeply counterintuitive. The FIFA ranking was based on counting weighted scores assigned to each of the previous one-year periods (see Equation (2)). The yearly scores were the averages of a team’s performance in its matches, with weights given for the match’s result, the opponent’s ranking position and federation strength, and the match’s importance. The highest weights of 4 were attached to the World Cup matches; other tournament matches had a weight of 3 and all preliminaries had a weight of 2.5; a very low weight of 1 was given to friendly matches (see Equation (1)). The fact that hosts of major tournaments were admitted without preliminaries made them play low-weight friendlies instead of preliminaries for long periods of time. This led to paradoxical effects. Instead of climbing in the ranking—as one would expect of the host—hosts slid substantially because they had no opportunity to acquire high scores from matches even if they were winning with highly ranked opponents.

The size of the host effect was estimated using two methods. The average dip in the host’s position between the start of preliminaries and the beginning of the tournament was 14.2. Regression estimates show an average dip of 16. At the same time, the competing rankings of ELO and Rankfootball showed minor and statistically insignificant changes in the hosts’ positions (see Table 7).

Countries hosting soccer tournaments invest massive amounts of money to present themselves in the best possible light. The cost of the 2018 World Cup in Russia was estimated to be between USD 14.2 billion and USD 20 billion [30], whereas the economic impact was estimated to be USD 30.8 billion over the 10 years from 2013 to 2023. Major soccer tournaments provide an opportunity for a thriving democracy to promote its achievements and for an autocracy to soften its image. This results in a broader stream of money going to the host team. One would expect that a host team would benefit financially and, on average, improve its quality of play and its position in the rankings. Leeds and Leeds [31] found that having hosted a World Cup in the past strengthens a country’s FIFA score by adding 218 points on average; however, the estimate was not significant (see also [26,32,33]. In reality, however, instead of the tournament host teams climbing slightly (as predicted by the Elo ranking system), they began their steep slide down: according to our estimates, to between 14.1 and 16 positions lower (see Figure 3).

As shown in Figure 3, we see that over the first six months, little happens. Then, there is an approximate one-and-a-half-year steep decrease. Finally, the last six months—after the end of the preliminaries—is quiet once again. The period with the steepest drop coincides with the typical timing of the preliminaries; some kinks in the graph are likely attributed to other tournaments taking place at about the same time.

The host effect is especially important for host teams that may slide several positions down. For non-hosts, the effect is usually much weaker because it involves a small number of relevant hosts that would have been assigned higher positions in the ranking. For some non-hosts, it may result in jumping up in the rankings over one or two hosts, but for most, their ranking position would not be changed because they were above or below all hosts under all scenarios. Deterioration in the host’s position negatively affects the organizers, and this has a real potential for the team to lose sponsors and their financial backing. Moreover, perhaps the most salient effect is the loss of interest in the tournament, not only by sponsors, but also by the fans. Here, I provide some anecdotal evidence. In June 2012, I traveled to Poland while Euro 2012 was under way. While there, I engaged in a casual conversation with a cabbie about the chances of the Polish team. His response started with a resigned statement: “Mister, they are so low in FIFA’s rankings that nothing will help them.” Later that same day, I was talking with my father, who repeated this gloomy prognosis, using the same FIFA rankings to make his point. From these conversations and my own reflections, it seems clear that a low position in the rankings may lead both fans and sponsors to underestimate their team’s chances and decrease their interest in the tournament.

The host effect has led to substantial fluctuations in the host team’s position, both before and after a tournament. The effect also contradicts FIFA’s intention to provide a universal and objective tool for evaluating the strength of the teams [34]. A team’s low ranking translates into its lower chances in the next preliminaries, because the lower-ranked teams are placed in lower pots for the draw, where they can expect to face stronger opponents. In the preliminaries for the 2010 World Cup, CONCACAF, CAF, and UEFA used FIFA rankings from various months preceding the draw for separating teams into different pots: for the 2010 World Cup, the October 2009 rankings were used; for the preliminaries for the 2012 Olympics, CAF used rankings from March 2011 [35]. Before the 2018 World Cup, one could observe the strategic behavior of some of the teams to maximize their positions in the rankings.

In the main tournament, hosts are usually seeded in the top pot, and this mechanism provides some crude compensation for their lower position. However, the erroneous placing of the hosts reduces the rankings’ power for predicting the results of matches. The FIFA rankings’ predictive power was estimated as worse than almost all other alternative rankings, including the Elo ranking [36]. Luckner et al. [37] found that “prediction markets” outperformed the FIFA ranking system in terms of forecast accuracy. The FIFA rankings were found to be somewhat accurate in predicting the success of the subsets of the top teams in the World Cup finals [38], but the authors offered no comparison with other methods.

Prior to the 2018 Russian World Cup, some high-profile commentators made public comments disregarding the Russian team. This type of disparagement of teams is not unusual. Before Euro 2012, for example, a typical opinion of the time was offered by Peter Schmeichel, a former Danish goalkeeper of Polish descent, and a coach of Manchester United, who belittled the Polish team: “[In Euro 2012, the] 15 best European teams will play and also Poland—the 28th team in the rankings” (Poland was designated as 28th in Europe, and 75th in the world around the time of this interview.) [39]. Although Poland did not perform significantly well in that tournament—failing to qualify from the group only by one goal—it did soon start climbing in the rankings from its 65th position in mid-2012 to 35th four years later, until it reached its all-time high of 5th in 2017, soon after the low scores associated with playing preliminaries disappeared. Russia advanced from its pre-2018 World Cup 70th position to position 37 on 16 September 2021. Similarly, France climbed in four years after being the host from 20th to 2nd, and Ukraine climbed from 50th to 29th.

Although appearing to be a “neutral” tool that promotes “objective” standards in evaluating national teams, FIFA’s rankings dishearten host fans and discourage sponsors. They certainly react negatively to the low position of their home team in the ranking, and the negative publicity associated with pessimistic comments in the media. However, the flaws in FIFA’s ranking system are not impossible to eliminate or reduce. Three obvious solutions would be to freeze the host’s position for approximately one-and-a-half to two-and-a-half years, to disregard some friendly matches, or to introduce higher weights for friendlies played by the hosts. Although FIFA did introduce a new, modified ranking system in 2018, it did not explicitly deal with the host paradox. However, according to FIFA’s official document, “The ranking of host nations who do not play competitive qualification matches in the period before championship competitions will not be as severely or negatively impacted with the SUM [new] formula as with the previous one”. It remains to be seen whether it offers a sensible solution and can limit the negative effects associated with their point-based rankings. The new ranking is based on the idea behind the Elo ranking and adds or subtracts points to the team’s ranking based on the importance of match, its result, and its estimated expected result. In the last paragraph discussing the ranking’s properties, we read that, “5. The ranking of host nations who do not play competitive qualification matches in the period before championship competitions will not be as severely or negatively impacted with the SUM formula as with the previous one. Thanks to the sum-of-points calculation method, successful results in friendly matches will result in point gains more substantive than the existing formula currently allows.” [40]. Another salient feature of the new ranking is that friendly matches have very low weight, which is between 10 and 12 times lower than the weight given to World Cup matches. The solution may generate some other problems, but for the tournament hosts, it is close to freezing their scores.