Are English Premier League Teams Paid Like Bureaucrats? An Incentive Analysis of Monetary Match Importance *

Abstract

Teams playing in the top English football league—English Premier League—face three types of financial incentives: rank-based prize money, qualification to European competitions, and relegation. For each type of incentive, we compute the monetary importance of each match for each team. A team's monetary match importance (MMI) is the difference between the team's expected award in the event it wins versus loses the match. We find that strong EPL teams have low average MMI relative to other teams. They are paid like bureaucrats as they face no risk of relegation, yet still receive a large and guaranteed award. We also show that total EPL MMI could be greatly increased by having a league with competitive balance.

Keywords

monetary match importance game importance monetary incentive EPL rank-based awards contest design competitive balance

Introduction

There is much variation across countries, sports, and over time in how sports organizations share between participants the revenues collected from broadcasting rights. Top North American leagues lean toward equal sharing rules, although the teams winning the most games receive additional revenue by participating in some variation of a playoff system to crown the league's champion. Conversely, European association football (or soccer, henceforth referred to as football) leagues use a mix of revenue sharing and merit-based awards in addition to the more punitive cost associated with relegating the worst performing teams to a league with a lower level of play (Speer, 2023).

Although many have debated how leagues should distribute their proceeds, few have looked at how this choice impacts teams’ incentive to win games. Past studies have estimated the impact of winning a game on a single outcome of interest such as winning a tournament, being relegated from the competition, or qualifying for a follow-up competition. When a league employs multiple types of financial incentives, as is common in football, it is natural to ask how much each component contributes to a team's incentives to win and whether the incentive to win games differs systematically for weak and strong teams. We study these questions in the context of the English Premier League (EPL).

The EPL collects more than £2B from broadcasting revenues each season, and distributes the majority of this bounty among the 20 participating teams through a set of financial awards.¹ EPL teams face three types of rank-based incentives: the team's final rank at the end of each season determines a team's pay-per-rank cash award, the opportunity to compete in ancillary tournaments organized by the Union of European Football Associations (UEFA), and the threat of relegation to the lower-tier English Football League Championship (EFL). The left panel on Figure 1 plots the award amount as a function of final rank for each of the three awards for Brighton & Hove Albion in the 2017–18 season. While the first two awards are the same for all teams, relegation incentives are team specific as it operates as claw-back system rather than as an end-of-season cash payment: a relegated team forfeits its right to compete in the EPL for at least one season and cannot return until it qualifies for a promotion from the EFL. An independent contribution of this work is to use a value function approach to compute the cost associated with relegation.

Figure 1.

EPL Award and Incremental Impact of a Win.

We detail how the EPL's financial awards incentivize teams to win games. We compute each team's monetary match importance (MMI), for each of the 38 matches it plays each season, as the difference between the team's expected awards in the event it wins versus loses a match. MMI is team- and game-specific: it measures the monetary return for a given team to win a given game. We argue that it can contribute to the understanding teams’ fine-tuned game-by-game decisions through various trade-offs (e.g., game-specific player selection, within-game player substitution, choice of high- or low-effort strategies, etc.).

We follow related literature to compute a teams’ final rank conditional on game outcomes (Geenens, 2014; Goller & Heiniger, 2023; Lahvicka, 2015; Lei & Humphreys, 2013; Scarf & Shi, 2008; Schilling, 1994). To illustrate our approach, the right panel on Figure 1 plots the cumulative probability that Brighton & Hove Albion achieves a given rank conditional on winning (solid gray curve) and conditional on losing (dashed gray curve) its match on 10 March 2018. Prior to this date, each team has played 29 matches and very little has been mathematically determined; only four teams are mathematically certain to avoid relegation; and several UEFA bids remain mathematically attainable, however unlikely, for all teams. Brighton & Hove Albion, our exemplar team, can improve its expected final rank if it wins the day's match. For example, it is about 35 percentage points more likely to end the 2017–18 season amongst the top 12 teams with a win compared to a loss (vertical black line for rank 12).

While the literature has looked at the impact of a win on final rank, we go a step further and attach a monetary value using the three types of EPL financial incentives: for example, we calculate that the monetary stakes for Brighton & Hove Albion for its 10 March 2018 match is equal to £4.1 M, computed as the sum of the incremental probabilities to achieve a given rank (again, the vertical black lines of the right panel of Figure 1) multiplied by the incremental award from reaching that rank (the steps of the left panel). Repeating this calculation for each team and match yields our measure of monetary match importance (MMI) for each of the 760 team-games in an EPL season.

The value of the MMI computed in Figure 1 is team-game specific and is also conditional on the outcomes of the matches that took place prior to 10 March. Taking the expectation of this value at the beginning of the season (unconditional on all previous match outcomes) provides a measure of the expected importance of the 10 March match for Brighton & Hove Albion: this second measure of team-game importance can be averaged across all games played by Brighton & Hove Albion which represents the team's average incentive throughout the season. Our study of team incentives focuses on this team-level measure of MMI.

We show that, although relegation incentives generate high absolute levels of MMI when compared to pay-per-rank and UEFA awards, this is not the case when one considers MMI per British pound sterling of award. In fact, UEFA awards generate the highest level of MMI per pound of award. The contribution of UEFA awards on total MMI, however, is small because the EPL purse is large compared to expected UEFA awards.

We distinguish two types of teams: strong teams (about one-third of the league, which is mostly comprised of the teams referred to as the “Big Six”) that are never relegated and are perennial contenders for qualification to UEFA tournaments; and weak teams that have little hope of qualifying for any UEFA tournament and are instead often relegated. We argue that strong EPL teams are paid like bureaucrats, receiving large, guaranteed awards while facing little risk of relegation.

We then investigate whether the EPL could increase MMI with competitive balance, maintaining the incentive framework the EPL currently uses (rank-based awards and a system of promotion and relegation). We evaluate the MMI that would result from a league of homogenous EPL teams (i.e., all teams have the same chance to win and lose each game). Under this scenario, total aggregate MMI increases by a factor of 4.6. The EPL incentives are weak under current team heterogeneity relative to a counterfactual league with competitive balance. This further demonstrates that EPL teams are paid like bureaucrats.

This paper is organized as follows: the rest of this introduction relates our work to the literature; Section 2 reviews the data and offers a detailed description of the financial incentives faced by teams; Section 3 presents a formal definition of match importance; Section 4 decomposes our measure of match importance by award category, season, team, and rank. The exposition requires an extensive use of notation to capture the three types of incentives used in the EPL and to explain the different decompositions. For the sake of clarity, Table A1 in the Appendix summarizes the notations used in the paper.

Contribution to the Literature: Understanding Incentives in EPL

This work makes three main contributions. Our first contribution is demonstrating that teams face very different incentives to win games. This analysis complements the literature that focuses on team's investment at the beginning of the season, focusing largely on wage expenditure (Garcia-del-Barrio & Szymanski, 2009).² In contrast, we take teams’ heterogeneity in player investments as given. These investments largely determine teams’ strengths and win probabilities. Our focus is on the incentive to modify gameplay strategies to win each game as the season progresses. For example, a strong team may may be able attract top players by offering high wages, but given this investment, our analysis shows that it faces little incentive (low MMI) to invest in each individual game. A weak team that is at risk of relegation throughout the season, however, faces stronger incentives (high MMI) to win games.³

A second contribution is to demonstrate that heterogeneity in team strength negatively impacts the average incentives across the whole league. This points to a previously undocumented benefit of competitive balance (defined here as having teams of equal strength) that has been overlooked in the literature; Késenne, 2000): competitive balance is not only connected to uncertainty of game outcomes and fan demand (Szymanski, 2022), but it also increases the overall incentive across all teams in the league (i.e., higher average MMI).⁴

Finally, this work contributes to the literature that investigates whether game importance influences real outcomes by developing a straight-forward methodology to measure the financial importance of each game in the EPL. Incentive theory says that teams should respond to changes in MMI over the season. In fact, there is much circumstantial evidence that teams do not select its best players in less important games.⁵ One could use our measure of game importance to study whether it influences other team decisions. For example, Courty and Cisyk (2024) shows that concussion injuries increase with match importance in American football and Goller and Heiniger (2023) studies the impact of match importance on public interest. Our new MMI measure could be used in future research to study a variety of game-specific outcomes in the EPL (e.g., fouls or yellow and red cards).

EPL: Data, Awards, and Stylized Facts

EPL Background and Data

Each EPL season, 20 teams compete in a double round-robin tournament, playing a total of 380 matches that take place over a 38-week season. Our main sample covers 2,660 matches from seven seasons (2015–16 to 2021–22), with 31 distinct teams competing in at least one EPL season. Three sets of data were collected from publicly available sources (see Appendix A2.1). First, we collect the 2,660 realized EPL match outcomes: a match may end with a home team win, an away team win, or a home and away team draw. Second, we collect the details of the financial award structures for the seven seasons. The EPL and UEFA allocate domestic and international broadcast revenues to teams based on team performance. We record the award formulas that describe team payment conditional on final EPL rank or stage of ancillary UEFA tournament. The third data source is the probability of each match outcome. For each season, we collect the Elo values of each team (Elo,1967). The Elo values are used to calculate the probability of a game ending in a home win, loss, or draw in a process described in Appendix A2.2.⁶

This work focuses on the distribution of the EPL and UEFA revenues which come largely from selling broadcasting rights.⁷ Teams also receive revenues from advertising, sponsorship, and match-day sales which represent 40 percent of total team revenues.⁸ The influence of these revenues on MMI is more difficult to evaluate because one does not know the sensitivity of these revenues to team performance as precisely as one does for EPL and UEFA ones. Instead, we focus on the league's influence on MMI through the choice of award structure.

Due to promotion and relegation, our panel is unbalanced, with only one third of the 31 teams having no missing observation (Table 1 shows that only 11 teams were part of the EPL in all seven seasons). This can be problematic, for example, when we compare average team awards as the EPL award purse has increased by 70 percent during our sample period and the UEFA award purse by 50 percent (see Table 2). The awards of the teams that compete at the end of the sample are inflated relative to those that compete at the beginning. To deal with this problem, we normalize the yearly award values by the season's award purse. These normalized values can be meaningfully compared across years.

Table 1.

League MMI by Seasons for PPR, UEFA, REL and Total (Values Relative to Pay-per-Rank Purse $σ$ ).

Season	$M (P P R)$	$M (U E F A)$	$M (R E L)$	M
2015–16	0.099%	0.053%	0.123%	0.275%
2016–17	0.103%	0.056%	0.114%	0.274%
2017–18	0.088%	0.052%	0.136%	0.276%
2018–19	0.091%	0.041%	0.164%	0.296%
2019–20	0.089%	0.040%	0.124%	0.254%
2020–21	0.094%	0.056%	0.118%	0.269%
2021–22	0.098%	0.053%	0.110%	0.261%
Average	0.095%	0.050%	0.127%	0.272%

Note. For $M (R E L)$ , $R E L (i)$ is computed using Equation A2. All MMI values are reported as a percentage of the season's pay-per-rank award purse $σ$ .

Table 2.

Total Purse By Season (Millions of British Pounds).

Season	Pay-per-Rank Purse $σ$	Pay-per-Rank Fixed Award $π$	Pay-per-Rank Variable Award $α$	UEFA(r) r = 1..4	UEFA(r) r = 5,6	UEFA(r) r = 7
2015–16	1,553.54	65.87	1.24	20.03	4.51	2.26
2016–17	2,238.33	93.47	1.94	20.11	4.72	2.36
2017–18	2,260.27	94.67	1.93	27.48	5.87	2.93
2018–19	2,297.10	96.63	1.92	28.12	6.01	3.01
2019–20	————————————————–N/A—————————————–
2020–21	2,362.36	97.56	2.16	28.86	6.94	4.99
2021–22	2,402.84	100.60	2.06	28.86	6.94	4.99

Note. Total pay-per-rank purse excludes facility fees awarded exogenously from final rank.

EPL Financial Awards

An EPL team's award depends on its ranking at the end of the season. Very stylistically, the expected benefit that team i receives from ending the season in rank r is composed of three sources:

E P L (i, r) = P P R (r) + U E F A (r) - d_{r = 18, 19, 20} R E L (i)

(1)

where

r \in {1, \dots, 20}

such that

r = 1

represents the

E P L

's champion, and

d_{r = 18, 19, 20}

is an indicator variable equal to one if team i concludes the season in any of the bottom three ranks. The team ranked r receives ‘pay-per-rank’ prize money,

P P R (r)

, composed of a guaranteed participation payment,

π

, and a merit payment that decreases as final rank is further from

r = 1

P P R (r) = π + α (20 - r)

The total pay-per-rank purse, $σ$ , is the sum of the pay-per-rank awards, $Σ_{r = 1}^{20} P P R (r) = 20 π + 190 α$ , and roughly corresponds to the EPL broadcasting revenues from the games played during the regular season. Although the absolute awards vary from season to season, the normalized values do not: each team receives about 4.2 percent of the season's purse ( $\frac{π}{σ} ≅ 0.042)$ and each additional rank is worth 0.08 percent of the seasons purse ( $\frac{α}{σ} ≅ 0.0008$ ).⁹

In addition, the teams ranked highest at the end of the EPL season participate in a sequence of ancillary competitions organized by UEFA. Performance in each competition determines the payment for that stage and the team's eligibility to progress to the next stage. The greatest expected awards are available to participants of the UEFA Champions League, followed by the UEFA Europa League, and, since season 2020–21, the UEFA Europa Conference League. Teams ending the EPL in higher ranks are seeded in more valuable tracks (Csató, 2020). We denote $U E F A (r)$ as the expected financial value from competing in a UEFA competition after ending the EPL season in rank r. A simplified expression for $U E F A (r)$ is a step function with three steps for $r = {4, 6, 7}$ as represented on Figure 1.¹⁰ Table 2 reports the value of the $P P R$ purse ( $σ$ ), the value of the fixed and variable awards, and the three steps in the $U E F A (r)$ function. The UEFA purse is much smaller than the $P P R$ one. The ratio of the two purses is highest in season 2015–16, when the value is only 0.058.¹¹

The last source of EPL incentives is created through the system of promotion and relegation between the EPL and EFL.¹² At the end of each season, the three lowest EPL teams are relegated to the EFL, while the top three EFL teams are promoted to the EPL.¹³ Relegated teams are not eligible to compete in the EPL the following season. Instead, they must compete in the EFL which yields much lower financial reward. In this sense, relegation operates as a claw-back system (Dehaan et al., 2013). In the season(s) that follow relegation, the relegated team bares the cost of not being eligible to compete in the EPL. We denote by $R E L (i)$ the financial loss suffered by team i when it is relegated, which is best thought of as forgone future awards.¹⁴

Result 1:

EPL teams face three types of incentives: pay-per-rank, UEFA qualification, and relegation.

The cost of relegation for Brighton & Hove Albion in the 2017–18 season, plotted in Figure 1, is about £70 M. This is estimated using a value function approach: because a relegated team is eligible to earn its promotion back to EPL after at least one season in the EFL, the loss from being relegated next season is equal to the difference in present value from competing in the EPL versus EFL. The cost of being relegated at the end of the current season is equal to this value discounted by one period:

R E L (i) = β (V_{i, P} - V_{F})

where

β

is the discount factor,

V_{i, P}

(resp.

V_{F}

)is the present value from competing in the EPL (resp. EFL).

V_{i, P}

is equal to the one-season value of competing the EPL, denoted

v_{i, P}

, plus the continuation value obtained the following season which is conditional on the league the team plays in:

V_{i, P} = v_{i, P} + β ((1 - ρ_{i, r}) V_{i, P} + ρ_{i, r} V_{F})

(2)

where

ρ_{i, r}

is team

i

's relegation probability. Team

i

's one-season value of competing in the EPL is equal to

v_{i, P} = \sum_{r = 1}^{20} (P P R (r) + U E F A (r)) f_{i} (r)

, where

f_{i} (r)

is the probability that team i ends the season in rank r. The same reasoning determines the present value of competing in the EFL as:

V_{F} = v_{F} + β ((1 - ρ_{i, p}) V_{F} + ρ_{p} V_{i, P})

(3)

where

ρ_{i, p}

is team

i

's promotion probability, and for the sake of simplicity, we assume that the one-season value of competing in the EFL,

v_{F}

, is the same for all teams.¹⁵ We obtain team

i

's cost of relegation:

R E L (i) = δ_{i} (v_{i, P} - v_{F})

(4)

where

δ_{i} = \frac{β}{1 - β (1 - ρ_{i, p} - ρ_{i, r})} \in [\frac{β}{1 + β}, \frac{β}{1 - β}]

can be interpreted as the (discounted) lost EPL time due to relegation, and

v_{i, P} - v_{F}

is the one period EPL premium. The cost of relegation varies across teams because teams have different expected benefit from competing in each league and because teams have different promotion and relegation probabilities.

Result 2:

Relegation incentives operate like a claw-back system: a relegated team loses the EPL premium for the time that it is not eligible to participate in the EPL.

Table 3 reports the relegation and subsequent promotion probabilities for the 2017–18 EPL teams, the teams’ expected pay-per-rank rewards, and estimated relegation costs (expressed as a percentage of the pay-per-rank award purse in the 2017–18 season). Implied relegation durations are short (around one season). This is because a strong team is likely to be promoted the season following relegation while a weak team is likely to be relegated next season if it is not this season. On the high end, we obtain relegation values in the range of 2.2 and 3.1 percent (equivalent to £50 M and £70 M in the 2017/18 season for Southampton and Huddersfield Town, respectively). Speer (2023) uses a regression discontinuity approach and report a relegation cost of about £159 m across the five European leagues. Speer (2023), however, applies only to the subset of teams that are relegated and thus cannot be used here because it does not deliver team-specific relegation costs.¹⁶ Our structural value-function approach imposes a much lower data requirement and can be applied to any team, including those that face little chance of relegation.

Table 3.

Relegation Costs in Season 2017–18.

Team	Relegation probability $ρ_{i, p}$	Promotion probability $ρ_{i, r}$	Implied duration $δ_{i}$	$v_{i, P}$	$v_{F}$	$R E L (i)$
Tottenham Hotspur	0.00%	100.00%	0.90	5.63%	0.42%	2.61%
Manchester City	0.00%	100.00%	0.90	5.58%	0.42%	2.57%
Chelsea	0.00%	99.99%	0.90	5.67%	0.42%	2.66%
Manchester United	0.00%	99.97%	0.90	5.55%	0.42%	2.55%
Arsenal	0.01%	99.94%	0.90	5.52%	0.42%	2.52%
Liverpool	0.01%	99.89%	0.90	5.49%	0.42%	2.50%
Everton	0.54%	97.63%	0.92	5.20%	0.42%	2.28%
Leicester City	1.87%	93.22%	0.94	5.07%	0.42%	2.24%
Southampton	4.24%	87.77%	0.97	4.96%	0.42%	2.22%
West Ham United	7.52%	80.77%	1.01	4.87%	0.42%	2.27%
Stoke City	9.91%	78.34%	1.01	4.82%	0.42%	2.24%
Bournemouth	12.21%	71.98%	1.05	4.78%	0.42%	2.36%
Swansea City	13.75%	69.59%	1.06	4.76%	0.42%	2.38%
West Bromwich Albion	14.98%	67.70%	1.07	4.74%	0.42%	2.40%
Crystal Palace	15.90%	66.35%	1.07	4.73%	0.42%	2.42%
Burnley	21.51%	58.49%	1.10	4.67%	0.42%	2.52%
Newcastle United	24.23%	55.90%	1.10	4.65%	0.42%	2.53%
Watford	34.39%	44.15%	1.12	4.57%	0.42%	2.71%
Brighton & Hove Albion	46.21%	34.00%	1.09	4.49%	0.42%	2.81%
Huddersfield Town	92.70%	2.17%	0.94	4.24%	0.42%	3.08%

Team Rankings and Awards

Table 3 reports summary statistics for the 31 teams in our sample. Team performance varies greatly across teams. Some teams regularly achieve high ranks and others perform systematically poorly. The same seven teams consistently participate in the UEFA Champions League during our sample. The second right-most column shows that the pay-per-rank award is distributed remarkably evenly across teams and most teams receive around four to six percent of the pay-per-rank award purse ( $σ$ ) each season.

Monetary Match Importance: A Theory of Rank-Based Incentives

This section derives team-game MMI and total league MMI for an archetypal rank-dependent award that distributes purse amount $σ_{0}$ to N teams, based on final ranking. Let $μ_{r}$ denote the share of the award purse distributed to the teams that rank r or better, and $μ = (μ_{1}, . ., μ_{N})$ is the corresponding $N$ -dimentional vector, with $\sum_{r = 1}^{N} μ_{r} = σ_{0}$ . Here $μ_{N} / N$ is the guaranteed payment distributed independently of team rank, and $μ_{r} / r$ is the incremental award given to a team that moves from rank $r + 1$ to r for $r < N$ . Ignoring ties, the award given to the team that ranks r is $A (r) = \sum_{j = r}^{N} μ_{j} / j$ . Under this notation, the $P P R (r)$ award used by the EPL, for example, fits in this setup with: $N = 20$ , $σ_{0} = σ$ , $μ_{20} = 20 π$ , and $μ_{r} = r α$ for $r < 20$ .

The N teams that are paired in games. Teams play $g = [1, G]$ rounds of games and for simplicity we assume that all games in a round are played simultaneously. Team $i$ 's final rank in the event it wins game g is denoted by ${\tilde{r}}_{i g w}$ . This is a random variable that is conditional on the history of all games that have happened up to game g, and whose realization depends on the yet unknown outcomes of all games that take place after game g. Similarly, ${\tilde{r}}_{i g l}$ is team $i$ 's final rank when it loses game $g$ . For each team-game, the two random variables describe the distribution of team $i$ 's final rank, conditional on all the information available up to game g and conditional on the outcome of game g. This approach is standard in the literature studying the importance of a game on a given outcome of interest (Lahvička, 2015; Lei & Humphreys, 2013; Scarf & Shi, 2008; Schilling, 1994). MMI for team i in game $g$ is the marginal benefit from winning instead of losing game g and is denoted as $E_{g} (A ({\tilde{r}}_{i g w}) - A ({\tilde{r}}_{i g l}))$ , where the sub-index g denotes the fact that the expectation is conditional on the history of all games that have happened up to game g. It measures the importance of game g for team i just before team i plays game g. The unconditional expectation measures the importance of game g for team i at the beginning of the season which is the focus of this paper:

m_{i g} = E (A ({\tilde{r}}_{i g w}) - A ({\tilde{r}}_{i g l}))

(5)

This measure of team-game importance can be averaged across all games for a given team, to obtain $\frac{1}{G} Σ_{g = 1}^{G} m_{i g}$ , which delivers the team's average incentive to invest in each specific game played over the entire season.¹⁷ The sports literature has used variations of these measures of match importance to assess, for example, audience interest or contest design (Goller & Heiniger, 2023; Scarf et al., 2009). The conditional measure of match importance is supported by the economics of incentive in tournament theory (Lazear & Rosen, 1981). According to this view, the league wants to maximize team effort, and in the canonical tournament model used in the literature, team i supplies an effort level equal to $E_{g} (A ({\tilde{r}}_{i g w}) - A ({\tilde{r}}_{i g l}))$ in game $g$ ¹⁸ A league that pursues the objective of maximizing total team effort is observationally equivalent to one whose objective is to maximize conditional MMI across all games.

Team-game MMI is measured in monetary units where $m_{i g} \in [0, σ_{0}]$ with $m_{i g} = 0$ when game g has, for example, no impact on team $i$ 's ranking and $m_{i g} = σ_{0}$ , for example, if $A (1) = σ_{0}$ and team $i$ becomes the (uncontestable) top team by winning game g (a very simple example of this would be a ‘winner-takes-all’ contest with two teams). Averaging Equation 5 across all teams and games, we obtain total monetary importance for the entire tournament:

M (μ) = \frac{1}{N G} Σ_{i = 1}^{N} Σ_{g = 1}^{G} m_{i g}

(6)

Consider again the example of a single-elimination, ‘winner-takes-all’ tournament with four teams this time, that are equally strong. In this scenario $M (μ) = \frac{2}{3} σ_{0}$ .¹⁹ At the opposite end, we have $M (μ) = 0$ in a tournament that pays teams a fixed participation award independent of rank. Such an award scheme generates strong incentives to qualify into the tournament, but no monetary incentive to win games once qualified, and, thus, no incentive to improve one's final ranking.

The value of $M (μ)$ depends on both the award function and on the team's change in ranking after a win instead of a loss. We re-write Equation 5 using the integration by parts formula. To do so, let $F_{i g l} (r)$ denote the cumulative probability function of random variable ${\tilde{r}}_{i g l}$ . We similarly define $F_{i g w} (r)$ for ${\tilde{r}}_{i g w}$ . $F_{i g w} (r)$ (respectively, $F_{i g l} (r)$ ) is the probability that team i ranks in the top r teams at the end of the season if it wins (respectively, loses) game g. The difference between these two distributions is denoted $Δ F_{i g} (r) = F_{i g w} (r) - F_{i g l} (r)$ . This is the impact of a win in game g on the chance to rank r or above at the end of the season. A visual representation of these three concepts is displayed on the right panel of Figure 1. We re-write Equation 5 through the following proposition:

Proposition 1:

$m_{i g} = E \sum_{r = 1}^{N - 1} \frac{μ_{r}}{r} Δ F_{i g} (r)$

In the simple case where the incremental awards are constant, $μ_{r} = r α$ for $r < N$ , we obtain $M (μ) = \frac{α}{N G} \sum_{i, g} r_{i g}$ where $r_{i g}$ is the rank match importance of game g for team i, defined as teams i 's expected rank gain if it wins instead of loses game $g$ :

r_{i g} = E ({\tilde{r}}_{i g w} - {\tilde{r}}_{i g l}) = E \sum_{r = 1}^{N - 1} Δ F_{i g} (r)

(7)

Rank match importance is small if teams vary a lot in quality. When this is the case, each team's expected final rank has a narrow range prior to the season's commencement. Final rank is determined by a team's ability and one would expect $r_{i g} \approx 0.$ The opposite occurs when many teams are tied in ability and a team's win has a large impact on its final ranking. In our sample, the average $r_{i g}$ is equal to approximately one across all teams and games. We call this the “one-win, one-rank” property.

Table 4 reports the values of MMI for different payoff functions. With constant incremental awards, an interesting case occurs when one-win-one-rank holds. This implies that $m = M = α$ which corresponds to the third row of Table 4. The last line illustrates an award that shares the total purse equally between the top r teams.

Table 4.

Payoff Function and MMI.

Payoff function	$m_{i g}$	$M (μ)$
$A (r)$	$E \sum_{r = 1}^{N - 1} \frac{μ_{r}}{r} Δ F_{i g} (r)$	$\frac{1}{N G} E \sum_{r = 1}^{N - 1} \frac{μ_{r}}{r} Σ_{i = 1}^{N} Σ_{g = 1}^{G} Δ F_{i g} (r)$
$μ_{r} = r α$ , $r < N$	$α r_{i g}$	$\frac{α}{N G} \sum_{i, g} r_{i g}$
$μ_{r} = r α$ , $r < N$ under one-win, one-rank	$α$	$α$
Top r teams receive $\frac{σ_{0}}{r}$ each ( $μ_{j} = 0$ for $j \neq r$ )	$\frac{σ_{0}}{r} E Δ F_{i g} (r)$	$\frac{σ_{0}}{r N G} E Σ_{i = 1}^{N} Σ_{g = 1}^{G} Δ F_{i g} (r)$

Table 1 illustrates the problem with fixed awards: $μ_{N}$ does not contribute to MMI. Applied to the $E P L$ pay-per-rank incentives ( $μ_{20} = 20 π$ , and $μ_{r} = r α$ for $r < 20$ ), and assuming one-win, one-rank, Table 4 implies that removing the fixed payment $π$ increases league pay-per-rank MMI by a factor 5.5.²⁰

Equation 6 and Proposition 1 offer one decomposition where total MMI is broken down into team-match components. We offer another decomposition that demonstrates the importance of each rank: Instead of decomposing MMI into match components (summing over g as done in Equation 6 we decompose MMI into rank components (summing over $r$ ) as follows:

Corollary 1:

$M (μ) = \frac{1}{N G} \sum_{r = 1}^{N - 1} μ_{r} \sum_{i = 1}^{N} m_{i r}$ where $m_{i r} = E \sum_{g = 1}^{G} \frac{Δ F_{i g} (r)}{r}$ is the expected importance of rank r for team i.

Note that the importance of match g for team i, denoted by $m_{i g}$ , is different from the importance of rank r for team i, denoted by $m_{i r}$ . The measure $m_{i r}$ is independent of the award vector. Summing over teams, we obtain that the overall importance of rank r is $\sum_{i = 1}^{N} m_{i r}$ , and $M (μ)$ is high when the award allocates a large share of the purse to important ranks.

Application to the EPL

This section applies the results of the previous section to the three EPL awards. The application is straightforward in the case of $P P R$ and $U E F A$ incentives. One replaces the vector $μ$ with the appropriate values: for $P P R$ awards, we have $μ_{r}^{p p r} = r α$ for $r < 20$ and $μ_{20}^{p p r} = 20 π$ ; and for $U E F A$ awards, the function $U E F A (r)$ has only three steps, at ranks four, six and seven. We have $μ_{4}^{u e f a} = 4 (U E F A (4) - U E F A (6))$ , $μ_{6}^{u e f a} = 6 (U E F A (6) - U E F A (7))$ , and $μ_{7}^{u e f a} = 7 U E F A (7)$ , where the values of $U E F A (r)$ are taken from Table 2, and $μ_{r}^{u e f a} = 0$ for $r \neq 4, 6$ or $7$ . The application of Section 3 to $R E L$ incentives, however, requires adapting the analysis to capture the claw back nature of relegation, and this will be done in Propositions 2 and 3.

We use a simulation approach to compute each MMI component. For each season, we simulate the 380 game outcomes 50,000 times using the match outcome probabilities obtained from the Elo values. Each iteration corresponds to a possible season history. For each team-game, we construct the team's average award across the 50,000 histories, conditional on the team winning or losing the game. We compute the team-game MMI using Equation 5 and replacing the expected awards with the average value from the simulation for each of the three award components. See Appendix 2.2 for details.

MMI Decomposition by Award Category

This section decomposes a season's MMI by award category. For the sake of keeping notations simple, we use $M (P P R)$ to denote $M (μ^{p p r})$ and do the same for the other two awards. Proposition 1 says that $M (P P R) = \frac{α}{760} Σ_{i = 1}^{20} Σ_{g = 1}^{38} r_{i g}$ and $M (U E F A) = \sum_{r = 4, 6, 7} \frac{μ_{r}^{u e f a}}{760 r} Σ_{i = 1}^{20} Σ_{g = 1}^{38} Δ F_{i g} (r) .$ The Appendix extends the analysis to relegation incentives with the following result:

Proposition 2:

$M (R E L) = \frac{1}{760} \sum_{i = 1}^{20} R E L (i) \sum_{g = 1}^{38} Δ F_{i g} (17)$ .

Table 1 reports the league's total MMI by seasons for the three award components. Again, here all MMI values in a given season are reported as a percentage of that season's pay-per-rank award purse $σ$ . The average MMI varies from season to season, and this holds for the three types of incentives. This is explained by the fact that relative team strengths vary from season to season.

In Table 1, $M (P P R)$ ranges from 0.088 to 0.103 percent with an average of 0.095 percent. This is close to the approximation obtained using the formula in Table 4 under the “one-win, one-rank” assumption (recall from Table 4 and discussion surrounding Table 2, $M (μ) = α$ and $\frac{α}{σ} = 0.085 %$ ). In contrast with pay-per-rank, which has a constant marginal reward for each rank, UEFA rewards are only available for the top ranks. Table 1 reports that $M (U E F A)$ ranges from 0.040 to 0.056 percent with an average of 0.050 percent, or about half of the value for $M (P P R)$ . Finally, we turn to relegation incentives: we expect $M (R E L)$ to be higher than $M (P P R)$ as a relegated team loses both the EPL fixed payment (net of any parachute payment) and the incentive-based pay-per-rank component until it is until promoted again. In fact, Table 1 says that $M (R E L)$ has an average of 0.127 percent, about one third larger than $M (P P R)$ . We conclude that relegation incentives are larger than the incentives from pay-per-rank prizes or bids to UEFA tournaments, and this is because the EPL distributes more than 80 percent of its total purse in fixed payments (see Table 2).

Result 3:

Relegation incentives generate high levels of MMI.

Next, we look at the average MMI per £1,000 of award. That is, we do not normalize the MMI by the same value (total pay-per-rank purse, $σ$ ) as done in Table 1. Instead, we normalize each value by the award amount dedicated to each award. It is straightforward to do so for pay-per-rank and UEFA incentives. In the former case, we use the variable component of the pay-per-rank purse as per Table 2.²¹ In the latter, we use the expected value of the UEFA awards. These amounts are respectively £391 M and £134 M for the 2021–22 season.²² We obtain that pay-per-rank delivers an average of £5.8 MMI per £1,000 of award and £9.6 for UEFA.²³

For relegation incentives, we use the guaranteed award component in pay-per-rank (£2,012 M for 2021–22) because this accounts for the majority of the loss experienced upon relegation. One should keep in mind, however, that this is a lower bound on the award funds contributing to relegation incentives and adding the pay-per-rank and UEFA awards would only reinforce the point we are about to make. With this lower bound, we obtain that relegation incentives deliver only £1.5 MMI per £1,000 of award.²⁴ Thus, the three types of awards can be ranked by level of efficiency with UEFA the highest, followed by pay-per-rank, then relegation.

Result 4:

Relegation generate low MMI per pound of award relative to pay-per-rank and UEFA awards; UEFA awards generate the higher level of MMI per pound of award.

MMI Decomposition by Team

Next, we turn at the analysis of MMI by team. We use the same formulas as in the previous section but omitting the summation over team. The $P P R$ incentive for team 1, for example, is $\frac{α}{G} \sum_{g = 1}^{G} r_{1 g}$ . Table 5 presents the decomposed MMI values by team, measured in percentage units of pay-per-rank purse ( $σ$ ), and averaged across all seasons the team is in sample.

Table 5.

MMI Values by Team (Values Relative to Total Pay-per-Rank Purse σ).

Team	$P P R$	$U E F A$	$R E L$	$S u m$
Cardiff City	0.092%	0.001%	0.479%	0.572%
Brighton & Hove Albion	0.107%	0.007%	0.288%	0.402%
Huddersfield Town	0.061%	0.000%	0.339%	0.400%
Fulham	0.100%	0.004%	0.296%	0.399%
Norwich City	0.105%	0.005%	0.284%	0.394%
Hull City	0.090%	0.001%	0.298%	0.389%
Watford	0.107%	0.005%	0.275%	0.387%
Middlesbrough	0.094%	0.002%	0.287%	0.383%
Aston Villa	0.104%	0.010%	0.263%	0.376%
Brentford	0.103%	0.004%	0.239%	0.347%
Sheffield United	0.107%	0.017%	0.219%	0.343%
West Bromwich Albion	0.108%	0.010%	0.223%	0.341%
Wolverhampton Wanderers	0.109%	0.043%	0.173%	0.326%
Sunderland	0.116%	0.011%	0.198%	0.325%
Leeds United	0.108%	0.034%	0.178%	0.321%
Newcastle United	0.114%	0.014%	0.190%	0.319%
Bournemouth	0.113%	0.010%	0.195%	0.318%
Burnley	0.114%	0.019%	0.172%	0.305%
Crystal Palace	0.113%	0.018%	0.158%	0.290%
Swansea City	0.117%	0.029%	0.127%	0.274%
Southampton	0.112%	0.046%	0.114%	0.272%
West Ham United	0.114%	0.036%	0.113%	0.263%
Stoke City	0.115%	0.040%	0.102%	0.256%
Leicester City	0.108%	0.062%	0.059%	0.229%
Everton	0.109%	0.056%	0.057%	0.223%
Arsenal*	0.069%	0.134%	0.002%	0.205%
Tottenham Hotspur*	0.076%	0.123%	0.005%	0.204%
Manchester United*	0.066%	0.135%	0.001%	0.203%
Chelsea*	0.062%	0.117%	0.001%	0.181%
Liverpool*	0.056%	0.091%	0.002%	0.148%
Manchester City*	0.040%	0.056%	0.000%	0.096%

denotes a “Big Six” team.

Table 6.

Awards by EPL Team, Seasons 2015–16 to 2021–22.

Team	EPL appearances	UEFA Champions League bids^a	UEFA Europa League bids^b^,^c	Total EPL prize share^d	Average EPL prize share^e
Manchester City*	7	7	0	5.74%	5.74%
Liverpool*	7	6	0	5.60%	5.60%
Tottenham Hotspur*	7	5	2	5.54%	5.54%
Manchester United*	7	4	3	5.53%	5.53%
Chelsea*	7	5	1	5.53%	5.53%
Arsenal*	7	1	5	5.42%	5.42%
Leicester City	7	1	2	5.30%	5.30%
West Ham United	7	0	3	5.04%	5.04%
Everton	7	0	1	5.02%	5.02%
Southampton	7	0	1	4.82%	4.82%
Crystal Palace	7	0	0	4.77%	4.77%
Newcastle United	6	0	0	4.11%	4.80%
Burnley	6	0	1	4.04%	4.72%
Watford	6	0	0	3.92%	4.57%
Bournemouth	5	0	0	3.37%	4.72%
Brighton & Hove Albion	5	0	0	3.33%	4.67%
Wolverhampton Wanderers	4	0	1	2.92%	5.11%
Aston Villa	4	0	0	2.61%	4.57%
West Bromwich Albion	4	0	0	2.60%	4.55%
Stoke City	3	0	0	2.03%	4.73%
Swansea City	3	0	0	1.98%	4.62%
Norwich City	3	0	0	1.81%	4.22%
Leeds United	2	0	0	1.37%	4.79%
Sheffield United	2	0	0	1.33%	4.66%
Huddersfield Town	2	0	0	1.25%	4.36%
Sunderland	2	0	0	1.23%	4.32%
Fulham	2	0	0	1.23%	4.32%
Brentford	1	0	0	0.68%	4.79%
Hull City	1	0	0	0.62%	4.36%
Cardiff City	1	0	0	0.62%	4.36%
Middlesbrough	1	0	0	0.61%	4.27%
Median	5	0	0	3.33%	4.67%

Notes. ^a In 2016–17, Manchester United is awarded a 2017–18 UEFA Champions League bid for winning the 2016–17 Europa League despite finishing with a rank of 6; in this season, a total of five teams are awarded 2017–18 UEFA Champions League bids and only two teams are awarded 2017–18 UEFA Europa League bids. ^b Beginning in 2020–21, two EPL teams (e.g., ranks 5 and 6) are awarded bids to UEFA Europa League and one EPL team (e.g., rank 7) is awarded bid to UEFA Europa Conference League. ^c In 2019–20, Arsenal is awarded a 2020–21 UEFA Europa League bid over Wolverhampton Wanderers due to winning the 2019–20 FA Cup despite the former finishing with a rank of 8 and the latter with a rank of 7. ^d Column 5 sums a team's EPL prizes over the seven seasons and divide it by the total EPL purse from Table 2. ^e Column 6 does this by season and takes average over the seasons the team is part of the EPL. * denotes a “Big Six” team.

The values of average MMI vary widely across teams for UEFA and relegation incentives, but not so for pay-per-rank (the coefficient of variation in the MMI measures is 1.1, 0.7, and 0.2 for UEFA, relegation, and pay-per-rank, respectively). This is because all teams (but the top-ranked team) can improve their rank and earn a constant marginal benefit. Conversely, UEFA tournaments are achievable only for strong teams, and relegation incentives matter most for weak teams.

Result 5:

The main source of MMI is from UEFA tournaments for strong teams and relegation risk for mediocre teams.

A team that is unlikely to fall in one of the bottom-three ranks faces low relegation incentives, and very few teams have this privilege: only eight teams have an average relegation incentive of less than one tenth of a percent. Other teams face a non-negligible chance to be relegated with huge financial consequences. Even teams that are never relegated during our sample period still have significant relegation incentives. For example, Crystal Palace and Southampton have relegation incentive values of 0.158 and 0.114 percent, respectively (equivalent to an average of £3.8 M and £2.7 M per game in the 2021/22 season, respectively). This is because these two teams continually risk concluding the season at a rank of 17 or worse.²⁵ The remaining teams have typically high values of relegation incentives and are occasionally relegated. To conclude, relegation is the most important incentive, but some teams seem completely immune to it.

This above finding delivers an answer to whether EPL teams are paid like bureaucrats. The last column of Table 5 orders teams with the teams generating most MMI at the top: stronger teams, including the teams that comprise the “Big Six,” are listed near the bottom. These teams have relatively small MMI under the current incentives because they receive a large, guaranteed payment with no risk of losing it, with almost no relegation incentives which is the most powerful source of incentives across all teams.

Result 6:

Strong EPL teams are paid like bureaucrat because the guaranteed award has no impact on their MMI.

Relegation and pay-per-rank incentives are positively correlated across teams ( $ρ = 0.36$ ) and both measures are negatively correlated with UEFA incentives ( $- 0.85$ and $- 0.57$ respectively). This helps capture the divide between strong and mediocre teams. Strong teams care only about UEFA qualification but not about pay-per-rank or relegation while the reverse holds for mediocre teams.

MMI Decomposition by Rank

Corollary 1 applies to $P P R$ and $U E F A$ incentives. It does not matter for the decomposition by rank whether $μ_{r}$ originates from the pay-per-rank or UEFA purse. We have $M (P P R) + M (U E F A) = \frac{1}{780} \sum_{r = 1}^{19} (μ_{r}^{p p r} + μ_{r}^{u e f a}) \sum_{i = 1}^{20} m_{i r}$ . Next, we extend the analysis to $R E L$ incentives.

Proposition 3:

$M (R E L) = \frac{1}{780} \sum_{r = 1}^{20} (μ_{r}^{p p r} + μ_{r}^{u e f a}) \sum_{i = 1}^{20} n_{i r} + k$ , where $n_{i r} = δ_{i} \frac{F_{i} (r)}{780 r} \sum_{g = 1}^{38}$ $Δ F_{i g} (17)$ and k is a constant that does not depend on $μ$ .

$n_{i r}$ is the importance of importance of rank r for team i due to relegation incentives. Relegation incentives influence MMI through the change in the probability to be relegated, $Δ F_{i g} (17)$ , imposing an impact on the future chance to receive the pay-per-rank and UEFA awards. Taken together, Corollary 1 and Proposition 3 deliver the full decomposition by rank:

M (E P L) = \frac{1}{780} \sum_{r = 1}^{20} (μ_{r}^{p p r} + μ_{r}^{u e f a}) (Σ_{i = 1}^{20} (m_{i r} + n_{i r})) + k

(8)

Award component $μ_{r}^{p p r} + μ_{r}^{u e f a}$ has two effects on MMI. First, it has a direct effect captured by $Σ_{i = 1}^{20} m_{i r}$ , which is the MMI value of allocating one British pound sterling toward rewarding the top r teams. It takes a high value when a win increases the probability (across all teams and games) to end the season at rank r. Second, $μ_{r}^{p p r} + μ_{r}^{u e f a}$ also increases the cost of relegation, imposing an indirect effect on MMI, captured by $Σ_{i = 1}^{20} n_{i r}$ .

Figures 2 plot the values of $Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ , as a function of r, and the sum of the two. The former is decreasing overall with a local trough for $r = 6$ . This says that top ranks are more contested and the local minimum at rank six shows a gap between the top tier teams and the rest of the league. As should be expected, $Σ_{i = 1}^{20} m_{i 20}$ is equal to zero because the guaranteed award does not have any impact on $M (P P R) + M (U E F A)$ . On the other hand, $Σ_{i = 1}^{r} n_{i r}$ is monotonically increasing in r because worse ranks have a greater impact on the risk of relegation. Looking at the entire range of ranks, relegation incentives take lower values than pay-per-rank and UEFA incentives (the two functions range from zero to two versus zero to 12, respectively). This confirms Result 4 that relegation incentives are less effective per pound of award.

Figure 2.

$Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ .

Figure 3 decomposes the function $m_{i r}$ and $n_{i r}$ by seasons and by teams. The season decomposition reveals that although the function $Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ vary from season to season, their general shapes do not. Some seasons, however, have significantly higher levels of $Σ_{i = 1}^{20} m_{i r}$ for the best ranks and this happens when the fight for at the top of table is highly contested. The team decomposition plots $m_{i r}$ and $n_{i r}$ for each team. Not surprisingly, these functions vary a lot across teams. Each team reaches a different maximum value and does so for different values of r. The functions are decreasing for strong teams, confirming the earlier conclusion that current pay-per-rank incentives do not incentivize strong teams because they fail to reward top ranks. The decomposition by team shows that the guaranteed award incentives only weak teams.

Figure 3.

$Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ by Season and by Team $m_{i r}$ and $n_{i r}$ .

Last, we consider a counter-factual hypothetical version of the EPL that has perfect competitive balance. That is, we compute a new set of $Δ F_{i g}$ assuming that all teams are equally likely to win each match they play. Figure 4 plots the new values of $Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ for this hypothetical league composed of equally strong teams. Note that these new values are much higher than the values in Figure 2. The function $Σ_{i = 1}^{20} m_{i r}$ is decreasing with $Σ_{i = 1}^{20} m_{i 1}$ having a value of close to 30, while $Σ_{i = 1}^{20} n_{i r}$ is equal to 10 for all ranks. We compute the league's MMI and find that having competitive balance increases MMI by a factor of 4.6 (a balanced league has an MMI of 1.244 compared with the average actual MMI of 0.272 from Table 1). This offers another reason why EPL teams are paid like bureaucrats:

Figure 4.

$Σ_{i = 1}^{20} m_{i r}$ and $Σ_{i = 1}^{20} n_{i r}$ with Competitive Balance.

Result 7:

Team heterogeneity within the EPL greatly reduces MMI.

Conclusion

We measure teams’ incentive to win games using a new measure, called monetary match importance (MMI), defined as the difference between a team's expected award in the event it wins versus loses a match. We compute the MMI for the three types of incentives used in the EPL: pay-per-rank, UEFA qualification, and relegation. Relegation incentives generate high absolute levels of MMI but low MMI when measured per British pounds sterling of monetary award. Strong and weak teams respond differently to the three types of EPL incentives: weak teams are incentivized largely by relegation incentives while strong teams are incentivized by UEFA incentives, although the magnitude of the latter is small due to the relatively small expectation-weighted UEFA financial stakes. Strong teams are paid like bureaucrats because they do not risk losing the large, guaranteed EPL award as most other teams do. We also show that introducing competitive balance (equal team strength) while keeping the current EPL award would increase MMI by a factor of 4.6.

This work proposes an incentive-based approach to analyze the rule that dictates how a leagues’ revenues are distributed. This approach has been pursued with much success in other economic fields, where the award designer pursues a well-defined objective (i.e., personnel economics). While we focus here on average team and league MMI over the season, future work could exploit the large within-season variation in a team's MMI to study whether team- and game-specific investments depend on MMI.

In our application to sports, we assumed that audiences are interested in watching teams that are eager to win games. But there is more to sports contest than average MMI as assumed in this work. To start, fans may not care only about how hard teams try to win games. Having some games with low MMI, for example, may be detrimental to the league if fans find these boring. Fans may also care about suspense and other dimensions of a sports competition (Chan et al., 2009). In addition, sports leagues are concerned about how the award allocation influences competitive balance (Szymanski, 2006) or may want to reward the teams that contribute most to broadcasting revenues (Bergantiños & Moreno-Ternero, 2020) and these teams may not necessarily be the teams that try hardest to win games.²⁶ Integrating these approaches opens interesting questions for future research.

Supplemental Material

sj-docx-1-jse-10.1177_15270025251348181 - Supplemental material for Are English Premier League Teams Paid Like Bureaucrats? An Incentive Analysis of Monetary Match Importance

Supplemental material, sj-docx-1-jse-10.1177_15270025251348181 for Are English Premier League Teams Paid Like Bureaucrats? An Incentive Analysis of Monetary Match Importance by Jeffrey Cisyk, Pascal Courty, and Amin Kouhbor in Journal of Sports Economics

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Jeffrey Cisyk

Pascal Courty

Supplemental Material

Supplemental material for this article is available online.

Notes

Author Biographies

Jeffrey Cisyk is an economic consultant at Charles River Associates in Chicago, Illinois, where he specialises in anti-trust economics. His research interests include the application of economic principles to the sports and entertainment industry.

Pascal Courty is a professor of Economics at the University of Victoria. He specialize in sports economics and cultural economics.

Amin Kouhbor is a PhD candidate at the University of Victoria.

Appendix 1. Proofs and generalization

Table A1.

Notations.

Section 2.2: EPL awards
$i \in {1, 2, \dots, N}$	Team
$r \in {1, 2, \dots, N}$	Rank, where $r = 1$ is the highest rank
$E P L (i, r) = P P R (r) + U E F A (r) - d_{r = 18, 19, 20} R E L (i)$	EPL award
$d_{r = 18, 19, 20}$	Indicator variable for final rank $r \in {18, 19, 20}$
$P P R (r) = π + α (20 - r)$	Pay-per-Rank award
$π$	Fixed pay-per-rank award
$α$	Variable pay-per-rank award (a.k.a. “merit money”)
$σ = Σ_{r = 1}^{N} P P R (r) = 20 π + 190 α$	Pay-per-rank purse
$U E F A (r)$	Expected UEFA award
$R E L (i) = δ_{i} (v_{i P} - v_{i F})$	Cost of relegation
$δ_{i} = \frac{β}{1 - β (1 - ρ_{i p} - ρ_{i r})}$	Lost EPL time (discounted) due to relegation
$β = 0.9$	Discount factor
$ρ_{i p}, ρ_{i r}$	Probability of promotion/relegation
$v_{i P} = \sum_{r = 1}^{20} (P P R (r) + U E F A (r)) f_{i} (r)$	One-season expected EPL award
$f_{i} (r), F_{i} (r)$	PDF,CDF of team $i$ 's final rank
$v_{F}$	One-season expected EFL award
Section 3: Model
$g = {1, 2, \dots, G}$	Games; $G = 2 * (N - 1)$ in double round robin
$μ = (μ_{1}, . ., μ_{N})$	$N$ -dimensional award vector
$A (r) = Σ_{j = r}^{N} μ_{j} / j$	Award function
$σ_{0} = Σ_{r = 1}^{N} μ_{r}$	Total award purse
$F_{i g w} (r)$ , $F_{i g l} (r)$	CDFs of team $i$ 's final rank conditional on winning/losing game $g$
$Δ F_{i g} (r) = F_{i g w} (r) - F_{i g l} (r)$	Impact on team $i$ 's final rank from wining versus losing game $g$
$M (μ) = \frac{1}{N G} Σ_{i = 1}^{N} Σ_{g = 1}^{G} m_{i g} = \frac{1}{N G} \sum_{r = 1}^{N - 1} μ_{r} \sum_{i = 1}^{N} m_{i r}$	Monetary match importance
$m_{i g} = \sum_{r = 1}^{N - 1} \frac{μ_{r}}{r} Δ F_{i g} (r)$	Monetary match importance of game g for team $i$
$m_{i r} = E \sum_{g = 1}^{G} \frac{Δ F_{i g} (r)}{r}$	Expected importance of rank r for team $i$
$r_{i g} = \sum_{r = 1}^{N - 1} Δ F_{i g} (r)$	Rank match importance of game g for team $i$
Section 4: MMI Decompositions ( $N = 20$ , $G = 38)$
$M (P P R) = \frac{α}{760} Σ_{i = 1}^{20} Σ_{g = 1}^{38} r_{i g} = \frac{1}{760} \sum_{r = 1}^{19} μ_{r}^{p p r} \sum_{i = 1}^{20} m_{i r}$ $M (U E F A) = \frac{1}{760} \sum_{r = 4, 6, 7} μ_{r}^{u e f a} Σ_{i = 1}^{20} Σ_{g = 1}^{38} Δ F_{i g} (r) = \frac{1}{760} \sum_{r = 1}^{19} μ_{r}^{u e f a} \sum_{i = 1}^{20} m_{i r}$ $M (R E L) = \frac{1}{760} \sum_{i = 1}^{20} R E L (i) \sum_{g = 1}^{38} Δ F_{i g} (17) = \frac{1}{760} \sum_{r = 1}^{20} (μ_{r}^{p p r} + μ_{r}^{u e f a}) \sum_{i = 1}^{20} n_{i r} + k$
$n_{i r} = δ_{i} \frac{F_{i} (r)}{r} \sum_{g = 1}^{38} Δ F_{i g} (17)$	Importance of rank r for team i through relegation

Appendix 2. Data and Simulations

References

Bergantiños

Moreno-Ternero

J. D.

(2020). Sharing the revenues from broadcasting sport events. Management Science, 66(6), 2417–2431. https://doi.org/10.1287/mnsc.2019.3313

Bergantiños

Moreno-Ternero

J. D.

(2024). The economics of sharing the revenues from sportscast. Mimeo. Universidade de Vigo.

Buraimo

Forrest

McHale

I. G.

Tena

J. D. D.

(2022). Armchair fans: Modelling audience size for televised football matches. European Journal of Operational Research, 298(2), 644–655. https://doi.org/10.1016/j.ejor.2021.06.046

Chan

Courty

Hao

(2009). Suspense: Dynamic incentives in sports contests. The Economic Journal, 119(534), 24–46. https://doi.org/10.1111/j.1468-0297.2008.02204.x

Courty

Cisyk

(2024). Sports injuries and game stakes: Concussions in the National Football League. Economic Inquiry, 62(1), 430–448. https://doi.org/10.1111/ecin.13173

Csató

(2020). When neither team wants to win: A flaw of recent UEFA qualification rules. International Journal of Sports Science & Coaching, 15(4), 526–532. https://doi.org/10.1177/1747954120921001

Dehaan

Hodge

Shevlin

(2013). Does voluntary adoption of a clawback provision improve financial reporting quality? Contemporary Accounting Research, 30(3), 1027–1062. https://doi.org/10.1111/j.1911-3846.2012.01183.x

Elo

(1967). The proposed USCF rating system, its development, theory, and applications. Chess Life, XXII(8), 242–247.

Garcia-del-Barrio

Szymanski

(2009). Goal! Profit maximization versus win maximization in soccer. Review of Industrial Organization, 34, 45–68. https://doi.org/10.1007/s11151-009-9203-6

10.

Geenens

(2014). On the decisiveness of a game in a tournament. European Journal of Operational Research, 232(1), 156–168. https://doi.org/10.1016/j.ejor.2013.06.025

11.

Goller

Heiniger

(2023). A general framework to quantify the event importance in multi-event contests. Annals of Operations Research, 341(1), 1–23. https://doi.org/10.1007/s10479-023-05540-x

12.

Hall

B. J.

Liebman

J. B.

(1998). Are CEOs really paid like bureaucrats? The Quarterly Journal of Economics, 113(3), 653–691. https://doi.org/10.1162/003355398555702

13.

Késenne

(2000). Revenue sharing and competitive balance in professional team sports. Journal of Sports Economics, 1(1), 56–65. https://doi.org/10.1177/152700250000100105

14.

Krumer

Megidish

Sela

(2020). The optimal design of round-robin tournaments with three players. Journal of Scheduling, 23, 379–396. https://doi.org/10.1007/s10951-019-00624-8

15.

Lahvička

(2015). Using Monte Carlo simulation to calculate match importance: The case of English Premier League. Journal of Sports Economics, 16(4), 390–409. https://doi.org/10.1177/1527002513490172

16.

Lazear

E. P.

Rosen

(1981). Rank-order tournaments as optimum labor contracts. Journal of Political Economy, 89(5), 841–864. https://doi.org/10.1086/261010

17.

Lei

Humphreys

B. R.

(2013). Game importance as a dimension of uncertainty of outcome. Journal of Quantitative Analysis in Sports, 9(1), 25–36. https://doi.org/10.1515/jqas-2012-0019

18.

Rohde

Breuer

(2016). Europe’s elite football: Financial growth, sporting success, transfer investment, and private majority investors. International Journal of Financial Studies, 4(2), 12. https://doi.org/10.3390/ijfs4020012

19.

Scarf

P. A.

Shi

(2008). The importance of a match in a tournament. Computers & Operations Research, 35(7), 2406–2418. https://doi.org/10.1016/j.cor.2006.11.005

20.

Scarf

Yusof

M. M.

Bilbao

(2009). A numerical study of designs for sporting contests. European Journal of Operational Research, 198(1), 190–198. https://doi.org/10.1016/j.ejor.2008.07.029

21.

Schilling

M. F.

(1994). The importance of a game. Mathematics Magazine, 67(4), 282–288. https://doi.org/10.1080/0025570X.1994.11996232

22.

Speer

J. D.

(2023). The consequences of promotion and relegation in European soccer leagues: A regression discontinuity approach. Sports Economics Review, 1, 100003. https://doi.org/10.1016/j.serev.2022.100003

23.

Szymanski

(2006). Uncertainty of outcome, competitive balance and the theory of team sports. In: Handbook on the economics of sport (Chapter 62, pp. 597–600). Edward Elgar Publishing. https://doi.org/10.4337/9781847204073

24.

Szymanski

(2022). Are footballers rewarded for luck? A surprise test. A Surprise Test (October 30, 2022).

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.04 MB