First, it is notable that r(0 km) = r0 + r∞ >0.5 (p < 0.01), which means that the HA cannot be solely explained by the geographical/travelling distance d alone, as reported before (e.g. Carmichael & Thomas, 2005; Pollard, 2006, 2008; Legaz-Arrese et al., 2012). The HA is predominantly present, even if teams with minimal distance d (e.g. if they belong to the same city) play each other (Clarke & Norman, 1995; Ponzo & Scoppa, 2014). Familiarity with home conditions (Pollard, 2002), territoriality (Wolfson & Neave, 2004) or crowd effects (Johnston, 2008; Nevill et al., 1996) among other factors can cause this finding. However, this distance-independent contribution to HA has significantly (p < 0.01) decreased over time (see r* of datasets A and B in Table 1).
Since r∞ is negative throughout and non-zero (p < 0.01 for dataset A), an influence of the distance d on the HA is present (see Eq. 3). However, it also has significantly (p < 0.01) decreased (r∞,B < r∞,A). On the one hand, shorter distances d reduce the HA of the home team and on the other hand, the HA saturates for larger distances d. The total influence of distance is smaller compared to the distance-independent influences on HA, since |r∞| < |r*| is significant (p < 0.01) in the past (A) and the present (B). The results suggest that up to half (p < 0.01) of the total HA has been explicable by distance-related effects in the past (see βA in Table 1), while these effects have been roughly halved to an insignificant amount nowadays (see βB).
These findings will be discussed in the context of selected other countries. Pollard (1986) analysed the two highest divisions in England (1970–1981), a country with distances comparable to Germany, where he found lower but existing HA in London local derbies (394 matches) and no further influence of travel distance above 200 km (3496 London-related matches and 6274 others). These results are similar to the saturation and drop in HA for shorter distances reported here. Using Eq. 3 one might derive r∞ = −0.082 (r0 = 0.643) from the dataset he used, coinciding with the distance-dependent influence r∞= −0.076 ± 0.026 found here of dataset A (1964–1989) in Table 1. Appropriately, Clarke and Norman (1995) found an increase of HA with distance in the subsequent years (1981–1990) for the first soccer division in England. The HA reduction in local derbies has also been found in the Turkey Super League (Seckin & Pollard, 2007) from 1994 to 2005 (r∞ = −0.040 and r0 = 0.617 derived from the dataset they used) comparable with the value of r∞ = −0.018 ± 0.025 found here for dataset B (1990–2020). The same trend has been described for the Italian Seria A (Ponzo & Scoppa, 2014) from 1991 to 2012 (7398 matches) even between teams that share the same stadium (128 matches), probably cancelling out familiarity with home conditions as a factor in this case (Pollard, 2002). One limitation of the present study is that the number of same stadium derbies is unknown, which might reduce HA for the shortest distances (d ≈ 0 km). However, the number of same city derbies is only about 10% (40 of 389 matches with d < 20 km), thus playing a minor role here.
As a mathematical robustness test, we repeated our analysis for the percentage of points gained at home g with 3 (win), 1 (draw) and 0 (loss) points per match (3-point counting system), which is a common measure for many soccer divisions worldwide (e.g. in England since 1981, Turkey since 1987, Italy since 1993), including the German Bundesliga since 1995. As noted before, for the 2‑point counting system (1994 and earlier) the identity r = g holds. For the years from 1995 to 2020 (defined as dataset C as a subset of dataset B), we find very high correlations of c = 0.9997 between the two types of analysis for rC(t) with gC(t) as well as for rC(d) with gC(d). However, g is usually (except for the year 2019 with AA) slightly larger than the result value r by an offset of Δ3P = + 0.0083 ± 0.0026 ≈ (0.8 ± 0.3) % ≈ 1% on average (<gC> = 0.594 ± 0.003 = 59.4% and <rC> = 0.586 ± 0.003 = 58.6%), which has to be taken into account. All according fit parameters for g(d) = g0 + g∞ ⋅ exp(−d / d0g) equivalent to Eq. 3 (g0 = 0.597 ± 0.011 = 59.7%, g∞ = −0.021 ± 0.026 = −2.1%, d0g = (122 ± 313) km, g* = 0.076 ± 0.037 = 7.6%, gHA = 0.097 ± 0.011 = 9.7%, αg = (3.6 ± 4.4) %, βg = (22 ± 29) %) coincide well with fit parameters of r(d) (r0 = 0.589 ± 0.009, r∞ = −0.020 ± 0.025, d0 = (111 ± 276) km, r* = 0.069 ± 0.034, rHA = 0.089 ± 0.009, α = (3.4 ± 4.3) %, β = (22 ± 30) %) for dataset C and even with parameters of dataset B (1990–2020) within the margin of error (see Table 1 for comparison). As a mathematical consequence of this rule change, the percentage of points gained at home slightly increased statistically for all teams (up to Δ3P ≈ 1%). Thus, teams with comparatively higher HA were marginally advantaged in gaining points at home (Jacklin, 2005; Clarke & Norman, 1995) but not regarding match results (outcome). Since the difference Δ3P is small for this dataset, the main conclusions drawn from the match results r regarding HA are not affected and also apply for the points gained at home g. To note, Δ3P usually increases with more points per victory as well as for higher HA or AA (for example, one finds Δ3P = 1.6% for dataset A with higher HA, see Fig. 1) and thus might play a more relevant role for other studies concerning different divisions or sports.
Oberhofer et al. (2009) also analysed the German first division, but restricted themselves to years from 1986 to 2006 (6389 matches), which is partly comparable to dataset B here. The authors state that travelling is especially detrimental to away teams at short distances, which can be seen as another formulation for the drop in HA reported here (see Fig. 2). Further, they stated that the influence of distance over time (1986–2006) did not change significantly. Here, in contrast, the distance-dependent contribution r∞ significantly decreased (p < 0.01) from past (A) to present (B), indicating a reduction of the distance-dependent influence on HA. It is argued here that Oberhofer et al. (2009) did not detect this decrease, since they considered fewer seasons (only 21 compared to 57 here) and more recent seasons (which already exhibit less influence of distance). For the Superleague Greece (1994–2010), the influence of distance on HA has been reported to be insignificant, which has been explained by the specific and very short travelling distances in Greece (Armatas & Pollard, 2014). Correspondingly to the findings of our study, another reason might be their restriction to more recent years, as noted above. The distance-dependent contribution r∞ has decreased and lost significance over time (from p < 0.01 for A to insignificant for B).
For Brazil (Pollard et al., 2008), Australia (Goumas, 2014), Turkey (Seckin & Pollard, 2007) or the Balkan region (Pollard & Seckin, 2007) increases for the home team HA have been reported for the largest travelling distances of over 1000 and up to 5432 km (Goumas, 2014). These results have been attributed to climate conditions in different regions, time zones and jet lag, remoteness and ethnic or cultural differences. However, the largest distances for Germany are around 740 km (between the cities of Freiburg and Rostock), with low to no relevance of these mentioned factors. Accordingly, no such increases in HA for the largest distances have been found (see Fig. 2). However, to check for remoteness explicitly, the teams’ individual HA (calculated as the average result <rteam,home> of home matches divided by the average result of all matches <rteam> of the team to account for the different playing strengths of teams) has been compared with the average distance <dteam> travelled (not shown). Indeed, there are no outliers for the largest distances <dteam> and also slope (−0.006 per 100 km with s = ±260%) as well as correlation (c = −0.05) are even slightly negative and insignificant according to margin of error (the same is true for dataset A or B alone).
To summarise, the literature findings regarding the influence of distance on HA for the cited countries England (Pollard, 1986; Clarke & Norman, 1995), Germany (Oberhofer et al., 2009), Italy (Ponzo & Scoppa, 2014) and Turkey (Seckin & Pollard, 2007) are qualitatively comparable and understandable in context with the saturation behaviour proposed here (see Fig. 2). The abovementioned particular influences for large distances as suggested for Brazil (Pollard et al., 2008), Australia (Goumas, 2014), Turkey (Seckin & Pollard, 2007) or the Balkan region (Pollard & Seckin, 2007) do not play a role for Germany due to its limited extent. In possible contrast to Greece (Armatas & Pollard, 2014), significant influence of distance on HA has also existed in the past for the lowest distances; however, this has declined to an insignificant amount nowadays.
Looking at the underlying causes for the reduction of the distance-dependent influence detected here, altered balances in the distance-dependent factors travel fatigue (Pollard & Pollard, 2005a; Goumas, 2014) and away team fan support (Ponzo & Scoppa, 2014; Seckin & Pollard, 2007) have been suggested. For example, travel fatigue could have been reduced nowadays by less stressful travelling, more travel comfort (Pollard & Pollard, 2005a) or extended overnight stays (Brown et al., 2002) for relaxation, which might explain the reduction in distance-dependent HA over the decades found here. A travelling distance of around 100 km might have been a critical distance for notable travel fatigue in the past (see dataset A in Fig. 2) due to inferior travel facilities. Teams in Germany usually travel by bus to away matches, nowadays with notable comfort (Autobild, 2021), since various travel factors (bus size, seat width and comfort, travel speed and duration, vibration attenuation, paving quality, etc.) have improved over the decades (e.g. HOV, 2021; MAN, 2021). These long-term developments went along with increasing club budgets and Gross Domestic Product (GDP) of Germany (DeStatis, 2021). These financial and travelling possibilities are reduced in lower divisions, which consequently leads to higher HA (Leite & Pollard, 2018). In addition, general improvement of travel possibilities may also have increased accessibility for the away team’s fans to accompany their team in larger numbers. Thus, more fans of the away team would be present in the stadium, decreasing the home team’s HA via noise and crowd effects and thus altering the amount of important referee decisions (Carron et al., 2005; Nevill et al., 1996; Ponzo & Scoppa, 2014). However, the influence of the crowd on HA might not always be significant or relevant, as suggested by recent studies in the context of COVID-19 for Austria and England (Sánchez & Lavin, 2020) or other European countries (Wunderlich et al., 2021).
Furthermore, it is a well-known fact that intraspecific territorial aggression in vertebrates declines with distance up to a maximum distance from their territory centre (Myrberg & Thresher, 1974; Lorenz, 1966). Accordingly, the saturation of HA with distance found here may be linked to a reduction of the away team’s intraspecific aggression up to their (perceived) territory border, which could be around 100 km away from their home stadium. However, players as well as referees may act more professionally today, and are also better trained physiologically (Stolen et al., 2005) and psychologically. They may thus be less influenced by travelling, territorial influences, the surrounding of the playing field and crowd effects (e.g. noise), especially in higher divisions (Leite & Pollard, 2018). To train such behaviour has been proposed by Wolfson and Neave (2004) as a strategy for away team coaches (Pollard & Gómez, 2014). Thus, parts of the reductions of the distance-dependent as well as the distance-independent contributions to the HA may be due to developments that might be summarised as increased professionalism and internationalisation nowadays or, alternatively, altered idiosyncrasies of players (Thomas et al., 2004) and stadiums. Players might be less (emotionally) affiliated with a home stadium, location or city as well as less unnerved by foreign places and stadiums, since they more often change teams, clubs or places, or come from other countries, which might reduce familiarity with home conditions (Pollard, 1986) and territoriality (Wolfson & Neave, 2004; Neave & Wolfson, 2003). Accordingly, Pollard (2002) found that the HA is reduced when a team changes stadium. Thus, a change of territoriality over the decades may be an indirect cause (Thomas et al., 2004) for the reduction of the influence of distance here. To measure territoriality, salivary testosterone levels of players have been successfully used as an indirect marker, and it has been shown that testosterone levels are significantly higher before a home game than an away game (Neave & Wolfson, 2003). Indeed, there have been reports from European (Andersson et al., 2007; Carruthers, 2009) and North American countries (Travison, Araujo, O’Donnell, Kupelian, & McKinlay, 2007) that testosterone levels in men have been gradually decreasing for the last century in the general population overall. These findings could indicate a decline in territoriality, which could also be connected to a diminution of the distance-independent as well as the distance-dependent HA (e.g. if testosterone level differences between teams are correlated with distance according to intraspecific aggression).
Another factor is the increase of points per victory from 2 to 3, which had been identified as a main cause for the observed drop in HA in 1981 in soccer in England (regarding the ratio of numbers of home wins to away wins) by Jacklin (2005) due to lessened incentives of away teams to settle for a draw (Thomas et al., 2004). As noted above, this rule change alone imposes a mathematical increase of about 1% on the percentage of points gained at home g(d) (Clarke & Norman, 1995). In the men’s German first soccer division, this change happened between 1994 and 1995. However, this issue led only to a small drop of HA for a single year regarding normalised match result (Δr = −0.043 = −4.3%), while a decreasing trend of HA had already set in during the years before (see Fig. 1). This may indicate only a minor influence of this issue here. It is interesting to note that another larger drop of HA (Δr = −0.078 = −7.8%) is clearly visible in the year 1990, which is the year of German reunification. Never again after this year did the HA reach result values r(t) of 0.65 or above (see Fig. 1). The reunification also led to other socioeconomic alterations within Germany (e.g. Hesse et al., 2003) and might have reduced the sense of territoriality of the players (Wolfson & Neave, 2004) or accelerated the abovementioned internationalisation and professionalism within the soccer divisions. However, the largest drop of HA in a single year (Δr = −0.140 = −14%) in the whole history of the Bundesliga happened in 2019, reversing the HA into an AA for the first and only time (r = 0.430 < 0.5). In this year, the global pandemic of COVID-19 spread, also imposing social disruptions. During this time, many soccer matches worldwide (and also in Germany) were held without spectators. From the analysis of tens of thousands of those matches in European major and minor leagues, Wunderlich et al. (2021) found a significant reduction of referee bias and shots, which they attributed to omitted crowd effects such as crowd noise. However, they only found an insignificant lowering of HA, from which they concluded that there must be more important influences to HA than crowd effects alone. Similarly, Sors et al. (2020) claimed a disappearance of referee bias and HA in the absence of spectators for the two highest leagues in Germany, Spain, England and Italy in 2019. In contrast, Sánchez and Lavin (2020) found a change of HA (and also of AA) between playing with or without a crowd only for the former two countries (and no change of HA for the latter). Here, for comparison, spectators increased from 5.9 million in 1964 (about 25,000 per match) to about 13.3 million in 2018 (DFB, 2021). Due to COVID-related political decisions, spectators were reduced in 2019 down to about 9.1 million in 2019, when the aforementioned AA set in (in line with Sánchez & Lavin, 2020). In contrast, however, the HA recovered in 2020 (r = 0.551 > 0.5), even when spectators were reduced to less than 0.2 million. For the total dataset (1964–2020), we even find a high anti-correlation (c = −0.572 < 0 with p < 0.001) between r(t) and the number of spectators (not shown). Surprisingly, this is a hint that more spectators could even reduce HA, however superimposed by others covariates (Van Damme & Baert, 2019; Pollard & Gómez, 2013). Hence, the onset of the emerging crisis of COVID-19 with its accompanying socio-economic changes (rather than spectators or distance), possibly also provoking psychological effects, is correlated with the reduction (and reverse of) HA in 2019, as it has been shown that socioeconomic changes (such as crises and civil wars) may also influence the HA (Pollard & Gómez, 2013).