# A general framework to quantify the event importance in multi-event
contests111 We are grateful for helpful comments and feedback from Nora
Bearth, Enzo Brox, Hannah Busshoff, Christian Deutscher, Riccardo Di
Francesco, Michael Knaus, Alex Krumer, Michael Lechner, Fabian Muny, Gabriel
Okasa, and Carina Steckenleiter. The usual disclaimer applies.

Daniel Goller1,2, Sandro Heiniger2
1Centre for Research in Economics of Education, University of Bern
2Swiss Institute for Empirical Economic Research, University of St.Gallen

###### Abstract

We propose a statistical framework for quantifying the importance of single
events that do not provide intermediate rewards but offer implicit incentives
through the reward structure at the end of a multi-event contest. Applying the
framework to primary elections in the US, where earlier elections have greater
importance and influence, we show that schedule variations can mitigate the
problem of front-loading elections. When applied to European football, we
demonstrate the utility and meaningfulness of quantified event importance in
relation to the in-match performance of contestants, to improve outcome
prediction and to provide an early indication of public interest.

Keywords: Incentives; Event importance; Multi-event Contest; Front-loading;
European Football;

##  1 Introduction

Incentives are an important tool for motivating people to exert effort. In
many environments, from the workplace to sporting contests, incentives are put
in place to ensure that invested efforts are optimised to achieve predefined
goals Lazear (2000); Laffont Martimort (2002). Changing incentives directly
translates into altered performance or success probability Rosen (1985);
Prendergast (1999). In contests where the reward depends solely on the outcome
of a single event, incentives are provided directly through the potential
rewards. In multi-event contests, the translation of contest rewards into
incentives for single events is not directly observable. Moreover, single
events may be unequally important for obtaining the final rewards, e.g.
performance in an interview is rated higher than the previous assessment
centre test, or the results of previous events lead to momentum for subsequent
events.

As this transmission of multi-event contest rewards into incentives reflects
the (personal) expected rewards, incentives vary not only between events but
also between participants. Understanding these disparate and possibly
asymmetric incentives in multi-event contests is essential as it could lead
participants to strategically allocate their efforts Preston Szymanski (2003).
Asymmetric individual incentives may have spillover effects on the outcome
probabilities of all other participants, which in turn could lead to
potentially unbalanced or unfair contests.

In this work, we propose a general statistical framework to quantify the
importance, or implicit incentive, of single events in complex multi-event
contests for each participant individually – the event importance (EI). The EI
measures the capability of an event to change a participant’s (expected)
reward for a contest. Our approach is based on two steps. First, we calculate
the probability distribution for each contestant to reach certain end-of-
contest rewards based on an outcome model determining the outcome
probabilities for every single event. Second, the importance of a single event
is determined through the changes in the end-of-contest reward probabilities
with respect to the possible outcomes of this particular event. If the
probabilities to reach the final rewards are changed substantially, the
importance of this event for the participant is high and our methodology
returns a high event importance measure.

The proposed framework generalises previous approaches Schilling (1994); Scarf
Shi (2008); Buraimo . (2022) in that it is suitable for any contest design and
any number of participants and is not specific to any particular contest
environment. Moreover, it allows for participant-specific reward structures
and both the reward structure and the schedule can change dynamically during
the contest. Crucial for practical usage is that the proposed statistical
procedure can also be used in situations in which the importance of the single
event potentially plays a role in the determination of the event’s outcome –
this can be accounted for by calculating the specific event importance in an
iterative procedure.

Our statistical framework can be applied in a variety of practical use cases,
like competing pharmaceutical companies developing a drug for the same medical
indication, presidential elections which are held in a series of local
elections, a job or promotion contest among applicants or workers, or sports
tournaments.

To showcase our methodology, we use our framework in two applications. In the
first, we apply the framework to the US presidential primaries to examine the
problem of front-loading: Earlier elections are known to have a greater impact
on the outcome of the nomination process, which is why several states are
pushing for earlier election dates. We analyse the Democrats’ electoral
schedule for the 2020 primaries and compare it with two alternative
hypothetical schedules, one sorted by the number of delegates and one
randomised. In this analysis, we find that the positioning of a state’s
election in the schedule substantially affects its impact on the outcome of
the nomination – indicated by higher event importance measures. A comparison
of the different schedules shows that the problem of front-loading can be
mitigated by arranging the schedule according to the number of delegates in
the states and completely eliminated by a random scheduling.

In a second application of our framework to the double round-robin tournament
structure in football leagues we provide explicit measures of the EI that
express implicit incentives for teams. In this setting, the relevance of a
particular match with respect to the team’s expected rewards varies
substantially, even though every match is actually awarded the same number of
points. Hence, the importance of a match varies over the season and between
teams. This leads to pairings between teams with potentially very different
incentives that change the presumed probabilities of winning.

We demonstrate the meaningfulness of the derived values by analysing their
relationship to various observable characteristics of the matches. The
integration of the EI information into a prediction model improves the
accuracy of match outcome forecasts. We show that bookmakers do not fully take
into account the team-specific importance of events in their prediction model.
Furthermore, a positive interrelation can be drawn between the estimated
importance of the match and the public’s interest in certain matches in the
form of larger stadium attendance and social media engagement for more
important matches. For the in-match activity of the players and the outcomes
of the match, we observe a comprehensive pattern suggesting that teams
approach more important matches with a more aggressive, direct, and successful
playing style.

Both the event importance values and replication code for the applications are
publicly available on Harvard Dataverse Goller Heiniger (2022).222For the US
presidential primaries, the EI estimates are published for all states and
territories, for the three different scheduling scenarios. The published EI
estimates for the European football leagues cover the seasons 2009/10 through
2018/19 and all seven leagues, i.e., the German ’Bundesliga 1’, the Dutch
’Eredivisie’, the Spanish ’La Liga’, the French ’Ligue 1’, the English
’Premier League’, the Portuguese ’Primeira Liga’, and the Italian ’Serie A’.
The rest of the paper is structured as follows. Section 2 discusses related
literature. Section 3 explains the proposed statistical method. Section 4
applies the framework to the front-loading in US primaries and the application
to double round-robin tournaments appears in Section 5. Section 6 concludes.

##  2 Related literature

This work mainly refers to two types of literature, (a) the importance of
specific events and attempts to quantify them and (b) the literature on
incentives in contests. The literature investigating the role of (explicit)
incentives on performance generally finds that higher incentives increase
performance Ehrenberg Bognanno (1990); Prendergast (1999); Lazear (2000).
However, it is important to distinguish between effort- and skill-based tasks;
in the latter, strong incentives can lead to performance decrements, a
phenomenon known as choking-under-pressure Ariely . (2009); Cohen-Zada .
(2017); Goller (2022). While most studies, and all of the studies mentioned
above, consider individual incentives, the role of team-based incentives on
performance may be different Alchian Demsetz (1972).

The importance of specific events in multi-event contests can be found in
several fields of literature. The order of action literature finds that the
outcome of the contest is influenced by the order of the events which has been
shown in musical contests Ginsburgh van Ours (2003), song contests de Bruin
(2005), or judicial decisions Danziger . (2011). More specifically, several
works focus on potential advantages in the first event in (usually sequential)
contests, like in R&D Harris Vickers (1987), sports Apesteguia Palacios-Huerta
(2010), or elections Klumpp Polborn (2006).

Research documents the differential importance of sequential elections in US
presidential primaries. Surprising wins in early states led to momentum
effects in the 2004 primaries Knight Schiff (2010). In their work they find an
unbalanced influence on the final result for voters in early and late
elections. klumpp2006primaries model this first-winner advantage for primaries
– known as the New Hampshire effect – giving an explanation for the more
intense campaigning in early elections. With more influence in the nomination
process in the early events front-loading, i. e. states moving their elections
to earlier dates, is well documented Mayer Busch (2003). ridout2008importance
find more attention of the candidates to the states the earlier their
elections are. Moreover, they find that scheduling is more important than the
delegates’ count.

The first approaches to determining an event’s importance use rather
simplistic measures Jennett (1984) and basic contest structures Schilling
(1994). Most influential was Schilling1994’s general idea of defining and
calculating event importance in terms of how the probability to reach a final
goal varies for different event outcomes. Recently, more sophisticated
approaches have emerged, for instance, Scarf2008 simulating probabilities of
final contest rewards conditional on two different event outcomes.
Lahvicka2015 and buraimo2022armchair build on the ideas of Jennett1984 and
Schilling1994 but simulate final standings in the ranking in a Monte Carlo
simulation to estimate the importance of single events. A different objective
is followed in Geenens2014: The importance of a match with regard to its
influence on the final contest outcome is investigated. This has an
interesting use case to investigate contests from the neutral spectator’s
perspective but is conceptually very different from the importance of a match
for the specific contestant.

The drawback of all the discussed approaches is that they are specific to a
certain type of contest that is prevalent in sports, i. e. a fixed number of
event outcomes and one specific reward (e.g., winner-takes-it-all contests).
This does not encompass more complex or dynamic contest designs and reward
structures, which are common in society. The approach we propose in the
following section provides the flexibility to handle a variety of practical
applications with a variety of contest and reward structures.

##  3 The event importance

###  3.1 Introduction to the general framework

The event importance measures the difference between the contest reward
probability distributions induced by the possible outcomes of a single event.
If the probabilities for the final rewards vary substantially with the
differential outcomes of the examined event, its impact on the tournament
reward is large and a high event importance measure is attributed.

To quantify the importance of a particular event we hence require the
probability distribution of the contest rewards conditional on each possible
outcome of the investigated event. To determine the probability distributions,
our framework sets the outcome of the examined event accordingly and solves
the remainder of the contest by successive evaluation of the outcome model.
Subsequent to the examined event, whose outcome is set by the framework, all
entities (outcomes, covariates, schedule) are subject to the probabilistic
outcome model. The successive application of the outcome model until the end
of the contest generates the probability distribution for the contest reward
conditional on the initial outcome.

There are six valuable attributes of our approach: First, by evaluating the
reward probability from the perspective of every contestant individually, the
event importance measure is specific to every participant and not the event
itself. Second, we do not impose narrow restrictions on the contest setup.
Since the contest reward probability distribution is conditional on each
possible event outcome, we only need to assume a finite number of contestants,
a finite number of possible outcomes for an event, and a finite schedule for
the contest. Moreover, the tournament rewards have to be measurable based on
the outcomes of all single events.

Third, the reward structure can be any function of all single event outcomes
or a final contest ranking if such exists, e. g. close-by ranks can be grouped
together if valued equally. Fourth, the framework is not restricted to a
specific schedule. As the probability distributions are calculated through a
successive evaluation of all events in the contest, the reward probabilities
encompass all the essential features of the schedule as well, e. g. early
elimination of participants in the contest or differences in information sets
induced by events held in parallel or sequentially. Fifth, the framework is
not tied to a specific distance metric to calculate the difference in the
reward probability distributions. Sixth, if the outcome model is not known, it
can be estimated on training data using any well-suited statistical method.

In the following, the details of how the described framework can be
implemented to determine the event importance values in a general case are
outlined.

###  3.2 Technical implementation

####  3.2.1 Notation

This section defines the notation used to describe the general framework.
Table 1 summarises the notation as a quick reference. Upper case letters
denote random variables, lower case letters denote their realisations or other
exogenous variables, and calligraphic letters are sets. Multi-character names,
such as EI or function names, are evident choices.

Table 1: Notation $t\in\mathcal{T}$ | Time $t$ in contest schedule $\mathcal{T}$
---|---
$\mathcal{T}_{t^{-}}=\bigcup_{t^{\prime}\leq t}t^{\prime};\ \mathcal{T}_{t^{+}}=\bigcup_{t^{\prime}>t}t^{\prime}$ | Sub-schedule up to (-) or after (+) time $t$
$e_{t,i}\in e_{t}$ | An event $e_{t,i}$ held at time $t$
$k\in\mathcal{K}_{e}\subseteq\mathcal{K}$ | Contestant $k$ participates in event $e$
$x_{e}=\bigcup_{k\in\mathcal{K}_{e}}x_{e,k}$ | Information on all contestants in event $e$
$\mathbcal{Y}_{e}$ | Set of all possible outcomes for event $e$
$y_{e}\in\mathbcal{Y}_{e}$ | Realised outcome of event $e$
$\text{out}(x_{e})=\bigcup_{\mathbcal{Y}_{e}}P[Y_{e}=y_{e}|X_{e}=x_{e}]$ | Probabilistic outcome model
$\left\\{\mathcal{T_{t^{+}}},x_{t^{+}}\right\\}=\text{gen}\left(\mathcal{T}_{t^{-}},x_{t^{-}},y_{t^{-}}\right)$ | Outcome-dependent schedule and covariates
$r_{k}=\text{rew}_{k}\left(\bigcup_{\mathcal{T}}y_{e}\right)$ | Reward for contestant $k$ after contest $\mathcal{T}$
$\text{EI}_{e,k}=\text{dist}\left(\bigcup_{\mathbcal{Y}_{e}}r_{k,y_{e}},\text{out}(x_{e})\right)$ | Event importance for contestant $k$ in event $e$
Note: For better readability, we omit the subscripts of $e_{t,i}$ if a
notation is not tied to a particular event but holds for any arbitrary event
$e$.

The contest is held along a finite schedule $\mathcal{T}$. Because of the
implicit chronological ordering of $\mathcal{T}$ we can define the notation
$\mathcal{T}_{t^{-}}=\bigcup_{t^{\prime}\leq t}\,t^{\prime}$ and
$\mathcal{T}_{t^{+}}=\bigcup_{t^{\prime}>t}\,t^{\prime}$, denoting the sub-
schedule up to and after time $t$. Multiple events $e_{t,i}$ can be held
simultaneously at time $t$, in this case $e_{t}=\bigcup_{i}e_{t,i}$. A finite
set of contestants $\mathcal{K}$ participate in the contest of which a subset
$\mathcal{K}_{e}\subseteq\mathcal{K}$ participate in event $e$. For each
event, a set of covariates $x_{e}=\bigcup_{k\in\mathcal{K}_{e}}x_{e,k}$ and
its outcome $y_{e}=\bigcup_{k\in\mathcal{K}_{e}}y_{e,k}$ is observed.333In
this context, observed refers to recording an outcome within the framework and
not to the actual outcome of an event that may have been observed. The outcome
$Y_{e}$ of event $e$ is a random variable that follows a conditional
probabilistic outcome model
$\text{out}(x_{e})=\bigcup_{\mathbcal{Y}_{e}}P[Y_{e}=y_{e}|X_{e}=x_{e}]$
which, in case out() is not known, is approximated by
$\widehat{\text{out}}()$. In the description of the general framework we
assume w.l.o.g. that $\text{out}_{e}()$ is known and uniform. The cases of an
approximated outcome model $\widehat{\text{out}_{e}}()$ or event specific
outcome models $\text{out}_{e}()$ can both be handled in the general
framework.

The chronological feature of the events further allows to define the sets of
covariates $x_{t^{-}}=\bigcup_{e,t^{\prime}\leq t}x_{e_{t^{\prime}}}$ and
$x_{t^{+}}=\bigcup_{e,t^{\prime}>t}x_{e_{t^{\prime}}}$ which combine all
information on events and participants taking place either up to or after time
$t$. The analogous operation on the outcomes defines $y_{t^{-}}$ and
$y_{t^{+}}$. In settings where parts of the covariates $x$ and/or the schedule
$\mathcal{T}$ depend on past outcomes, they are generated at run time based on
the previous outcomes by
$\left\\{\mathcal{T}_{t^{+}},x_{t^{+}}\right\\}=\text{gen}\left(\mathcal{T}_{t^{-}},x_{t^{-}},y_{t^{-}}\right)$.
After the full contest $\mathcal{T}$, the probability distribution of the
final rewards is determined according to the valuation function
$r_{k,y_{e}}=\text{rew}_{k}\left(\bigcup_{\mathcal{T}}y_{e}\right)$ which can
be individually specific for every contestant $k$. The event importance
$\text{EI}_{e,k}=\text{dist}\left(\bigcup_{\mathbcal{Y}_{e}}r_{k,y_{e}},\text{out}(x_{e})\right)$
for contestant $k$ in event $e$ is the difference between the multiple
probability distributions of the final rewards measured by any distance
measure dist(). The distance function can incorporate the outcome
probabilities $\text{out}_{e}(x_{e})$ of the starting event as weights.

####  3.2.2 Algorithm

Algorithm 1 describes the computation of the event importance value for a
competitor $k$ in event $e_{t,j}$. Readers that are less familiar with the
pseudo-code notation can consult the literal translation of the algorithm in
Appendix A.

Data: $e_{t,i},k,t,\mathcal{T},x,\mathbcal{Y}_{e_{t,i}}$

Result: Event importance for competitor $k$ in event $e_{t,j}$

1 begin

2 forall  _$\mathbcal{y}_{e_{t,i}}$ in $\mathbcal{Y}_{e_{t,i}}$_ do
$\;\triangleright$ We denote all conditional dependence on
$\mathbcal{y}_{e_{t,i}}$ by *

3 $y_{e_{t,i}}^{*}\leftarrow\mathbcal{y}_{e_{t,i}}$

4 forall  _$e_{t,j}$ in $e_{t}$ with $j\neq i$_ do

5 $y_{e_{t,j}}^{*}\leftarrow\text{out}\left(x_{e_{t,j}}\right)$

6

7
$\left\\{\mathcal{T}_{t^{+}}^{*},x_{t^{+}}^{*}\right\\}\leftarrow\text{gen}\left(\mathcal{T}_{t^{-}},x_{t^{-}},y_{t^{-}}^{*}\right)$

8 forall  _$t^{\prime}$ in $\mathcal{T}_{t^{+}}^{*}$_ do

9 $y_{e_{t^{\prime}}}^{*}\leftarrow\text{out}(x_{e_{t^{\prime}}}^{*})$

10
$\left\\{\mathcal{T}_{t^{\prime+}}^{*},x_{t^{\prime+}}^{*}\right\\}\leftarrow\text{gen}\left(\mathcal{T}_{t^{\prime-}}^{*},x_{t^{\prime-}}^{*},y_{t^{\prime-}}^{*}\right)$

11

12
$r_{k,\mathbcal{y}_{e_{t,i}}}\leftarrow\text{rew}_{k}\left(\bigcup_{\mathcal{T}}y_{e}^{*}\right)$

13

14
$\text{EI}_{e_{t,i},k}\leftarrow\text{dist}\left(\bigcup_{\mathbcal{Y}_{e_{t,i}}}r_{k,\mathbcal{y}_{e_{t,i}}},\text{out}(x_{e_{t,i}})\right)$

15 return _$\text{EI}_{e_{t,i},k}$_

16

Algorithm 1 Event Importance

####  3.2.3 Approximation of the probability distributions

By the subsequent evaluation of the outcome model, the probability
distribution of the rewards at the end of the season can be determined,
independent of the contest design. However, a large amount of sequential
events opens an extremely large number of possible outcome paths which causes
numerical problems if their probability would be evaluated exactly. For an
outcome model that depends on past outcomes, the outcome paths can
additionally become very complex. For this reason, it is often appropriate to
perform a Monte Carlo simulation to approximate the probability distribution
of the final rewards. Each run simulates one path for the remaining contest
according to the outcome model and the generated covariates/schedule
information at run time. With an adequate number of $N_{\text{MC}}$ Monte
Carlo runs, the estimated probability distribution and hence the event
importance values become sufficiently close to the exact values.

In a simulation that estimates the EI values for all events consecutively, a
chronologically backward iteration over the events allows for the reuse of
already evolved paths as they can be merged with respective previous outcome
to longer paths and thus reduce the computational complexity. In this case,
$N_{\text{MC}}$ is an upper bound for the number of actually performed runs in
each step and at the same time, a lower bound for the number of runs the event
importance estimate is premised on.

####  3.2.4 Iterative approximation of event importance

In many applications, the event importance is an integral part of the outcome
model, e. g. when the importance can be interpreted as an incentive for the
contestants to provide effort that in turn influences the outcome of the
event. Independent of whether the outcome model is known or not, it
encompasses the event importance values which are not available beforehand.

To determine the unknown event importance values, Algorithm 1 is at first
executed with an approximate outcome model that does not feature the event
importance in the variable set. This returns an initial approximation of the
desired EI values. A subsequent iterative application of Algorithm 1 with the
full covariate set including the preliminary EI variables updates the event
importance estimates accounting for their own impact through the outcome
model. This iterative procedure can be continued until a predefined stopping
criterion is reached. The application in Section 5 is an example of an outcome
model which includes the event importance in the covariates. Algorithm 2 in
Appendix C.3 illustrates how the iterative procedure is implemented in the
context of the application.

####  3.2.5 Distance functions

To measure the difference between the probability distributions on the contest
rewards, an appropriate distance function needs to be chosen. In simple
applications with only binary event outcomes and a binary reward scheme, the
difference between the contest-winning probabilities conditional on the event
outcome is a straightforward choice as the distance function.

More complex cases which feature either multiple possible event outcomes or
multiple rewards require a statistical distance function. For most settings,
the Jensen-Shannon divergence (JSD) is an appropriate distance function to
cope with multiple discrete probability distributions Lin (1991). It is a
common distance measure Nielsen (2020) with desirable properties, such as
allowing for a weighting of the probability distributions as well as being
bounded and symmetrical. The JSD measures the difference in the Shannon
entropy between the probability distributions which implies that it does not
have an intuitive linear interpretation. If such an interpretation is of
relevance, other candidates such as the total variation distance can be
applied.

##  4 Application: Front-loading in US primaries

###  4.1 Introduction

Presidential primary elections in the United States are held by the Democratic
and the Republican party to determine the presidential election nominees. Both
parties follow a similar procedure where each state, every permanently
inhabited US territory, and party members living abroad444For the ease of
readability, all entities are henceforth labelled as states. are attributed a
certain number of votes (pledged delegates). In addition to the pledged
delegates, selected party officials’ have additional votes (unpledged
delegates) that are not tied to states’ election results. In order to be
nominated for the presidential elections, the candidates in the primaries must
receive the majority of the delegate votes.

Each state holds its election or caucus on an individually chosen date, and
the election results determine how its delegates vote. Several states can vote
on the same day, e.g. on "Super Tuesday" about one third of all delegate votes
are determined. Due to the partially sequential nature of the primaries, it
may happen that later elections become irrelevant to the outcome of the
nomination if a candidate has already received more than half of the total
delegate votes. Moreover, the first elections are of greater importance as
they reveal voters’ preferences and influence later elections through their
results. These two features of the electoral process lead to a long-known and
unresolved problem of front-loading Mayer Busch (2003); Ridout Rottinghaus
(2008).

From a state’s perspective, an early election date can increase its influence
in the primaries. If all states are considering moving their elections to
earlier dates, a solution must be found to regulate the timing of state
elections that takes into account the different importance of the dates.
Currently, additional delegates are granted for late election dates, but these
do not provide sufficient incentive for states to resolve front-loading. In
the following analysis, we compare the US Democrats’ 2020 election schedule
with two proposed election schedules, namely randomising the election dates
and arranging the states according to their delegate count. The aim of the
analysis is to find out whether this leads to a more balanced distribution of
the importance of elections for the individual states that is less driven by
the timing of the elections.

###  4.2 Setup

We utilise the actual schedule and reward structure of the 2020 democratic
party presidential primary elections. The ordering of the elections and the
number of delegates rewarded by the election are displayed in Table 2 in
Appendix B. For ease of exposition, we simplify the model of the election
process by discarding unpledged delegates, implementing only winner-takes-it-
all elections, and engaging only two candidates $i\in\\{0,1\\}$.

The reward function of the contest is given by winning the primaries, i. e.
obtaining the majority of the delegates’ votes. We model the state’s election
as a representative voter facing a binary choice model with random utility.
The utility function (1) is composed of four components: a) the fixed
reputation $\\{\eta_{0},\eta_{1}\\}=\\{0.5,0\\}$ of the candidates, b) the
match between state preferences $\rho_{s}\sim\mathcal{N}(0,1)$ and the
candidates’ positions $\\{\rho_{1},\rho_{2}\\}=\\{-1,1\\}$, c) spillover
effects $\zeta_{i,s}$ of previous elections. d) a standard Type I extreme
value error term $\epsilon_{i,s}$. Under the assumption, that the
representative voter always chooses the candidate with maximum utility, the
setting describes a conditional logit model McFadden (1974) with outcome
probability $\pi_{i,s}$ that candidate $i$ wins the election in state $s$ as
described by Equation (2).

Spillover effects from the results of early states on future elections occur
if the share of obtained delegates’ votes in prior states differs from the
expected share solely based on the candidates’ reputation. The spillover
effect as defined in (3) is an example of dynamic covariates in the outcome
model that have to be re-evaluated when new election results are determined.

$\displaystyle\psi_{i,s}$
$\displaystyle=\eta_{i}+\zeta_{i,s}-\frac{\lvert\rho_{i}-\rho_{s}\rvert^{2}}{2}+\epsilon_{i,s}$
(1) $\displaystyle\pi_{i,s}$
$\displaystyle=\frac{\exp(\psi_{i,s})}{\exp(\psi_{i,s})+\exp(\psi_{1-i,s})}$
(2) $\displaystyle\zeta_{i,s}$
$\displaystyle=\frac{\sum_{s^{\prime}<s}y_{i,s^{\prime}}}{\sum_{s^{\prime}<s}(y_{i,s^{\prime}}+y_{1-i,s^{\prime}})}-\frac{\exp(\eta_{i})}{\exp(\eta_{i})+\exp(\eta_{1-i})}$
(3)

Based on the model for the election process given by equations (1)-(3), the
probability for each candidate to win the primaries conditional on the outcome
of a single state’s result is determined. Because the number of states and
territories is too large to allow an exact numerical calculation, the winning
probabilities are approximated by a Monte Carlo simulation using 5’000
simulation runs. As suggested in Section 3.2.5, we choose the difference in
the winning probabilities as the distance function for this contest with
binary reward structure, i. e. to be nominated as a presidential candidate or
not. This distance function leads to symmetric EI estimates for both
candidates. To eliminate the dependency on a particular set of states’
preferences, we simulate 1’000 random draws of state preferences.

The three schedules we evaluate are defined as follows: The regular schedule
is according to the actual election dates555Because of the Covid-19 pandemic,
several states have postponed their election such that the actual dates can
differ from the initially planned schedule. and the allocated number of
delegates by state. In the random schedule, we randomly permute the ordering
of the states keeping the framework of the schedule fixed. The rank increase
scheme ranks the states increasing by their number of delegate votes and
applies the ordering to the actual schedule framework.

###  4.3 Results

(a) Regular

(b) Random

(c) Rank increase

Figure 1: Average event importance estimates over 1’000 states’ (preferences)
samples with linear fit.

Figure 1 shows the average EI estimates over all samples for the three
schedule types. The regular schedule of the 2020 democratic party primaries
(1(a)) displays the increased importance of the early elections, as the
respective states have a higher average EI estimate than the number of
delegates allocated to them would suggest. The randomised schedule (1(b))
reveals a linear relationship between the ability of a state to change the
outcome of the primaries and its number of delegates. Since all states will
eventually benefit from an early position in the schedule, the positive
spillover effects are spread across all states and territories and balance
each other out.

Ordering the states by their number of delegates (1(c)) cannot entirely
eliminate the first-winner effect but substantially alleviates it. The
increased importance of states due to the early election date can be
compensated by a smaller number of delegates, which is an option already
considered in the allocation of delegates. Because an ex-ante randomisation
requires many repetitions to balance the positional effects, the ordered
schedule seems to be a good compromise between practicability and fairness.

Figure 2: Estimated event importance values for Iowa and California at
different hypothetical positions in the election schedule for 400 random
samples of states’ preferences.

To illustrate the importance of both the first-winner effect and the state’s
size we show in Figure 2 the estimated EI values for two states, Iowa (41
delegates) and California (415 delegates), at different hypothetical positions
in the election schedule. For the very early positions (1 and 4), both states
are of considerable importance for the final nomination of the presidential
candidates. From position 5, the "Super Tuesday" on which numerous states hold
their elections, the capability of the elections to influence the final
nomination decreases considerably.

The importance of elections in small states thrives on the fact that there are
spillover effects through the first-winner effect. Because of the substantial
amount of delegates, California remains considerably important for the
nomination result in late stages of the schedule, while the low number of
Iowa’s delegates become irrelevant in many realisations. For Iowa in
particular, if the election would be held in the middle (20) or at the end
(50) of the schedule, the importance of the election would be determined only
by the possibility that the state’s delegates could act as tiebreakers in the
nomination if the election race is close.

##  5 Application: European football leagues

###  5.1 Introduction

As with many analyses of contests, sports data provide a suitable and well-
structured framework for applications because it features accurate
observational data Kahn (2000); Bar-Eli . (2020). We apply the EI framework to
the seven major European football leagues. Those contests have a non-trivial
schedule of multiple events, held sequentially or in parallel, between
pairings of the participants and a non-linear reward structure that can vary
individually or change throughout the season – all of which can be handled
naturally with the proposed framework. With, for example, postponed games
leading to changes in the schedule, or supplementary rewards achieved by
national cup tournaments changing the reward structure individually, this
application is a good showcase to demonstrate the flexibility of the
framework.

Quantifying EI in this context is interesting for several reasons. Contest
designers should avoid match-ups that pit contestants with unequal incentive
levels against each other. Such matches are potentially more susceptible to
bribery and a lower engagement of certain participants could give an unfair
(dis)advantage to participants not even involved in the event itself. Other
valuable use cases of the EI in football tournaments include (a) selecting
intense or interesting matches for prime-time broadcast, (b) improving the
prediction of winning probabilities, and (c) detecting or avoiding unfair
match schedules.666E. g. pairings with unequal EI levels may result in unfair
schedules for involved or non-involved teams.

###  5.2 Setup

####  5.2.1 Data

We analyse data from the 2006/07 through 2018/19 seasons of seven major
European football leagues, namely the German ’Bundesliga 1’, the Dutch
’Eredivisie’, the Spanish ’La Liga’, the French ’Ligue 1’, the English
’Premier League’, the Portuguese ’Primeira Liga’, and the Italian ’Serie A’.
These leagues were the major leagues in Europe in terms of sporting and
financial success throughout the studied period. All leagues are designed as
double round-robin tournaments, which means that each team plays each other
twice - once at each home venue. The rewards are distributed after the season.
With the seven analysed leagues, we cover a variety of different reward
structures. A detailed description of the tournament design, league format,
and reward structure of the considered European football leagues can be found
in Appendix C.1.

For each individual match, we record a long list of characteristics:
describing the match setting, such as the time or day of the week,
characterising the participating teams, as their success in recent matches,
whether they play in international competitions and metrics of the squad
players, e. g. age, height, estimated market value, and preferred foot. The
full set of all 133 covariates is listed in Appendix C.2.

####  5.2.2 Specific application framework

In this section, we describe how we implement the general framework from
Section 3 and elaborate on all generic functions outlined in Algorithm 1.
Algorithm 2 in Appendix C.3 presents the pseudo-code of the specific framework
tailored to this application. At the end of a football season, rewards are
allocated to the teams based on their final rank. The areas in the ranking
which denote the championship title, qualification for international
competitions, and relegation are stated by strict thresholds. We use those
boundaries to group all the ranks between two thresholds as a single
reward.777Financial rewards, i. e. money from broadcasting rights, are
determined by the final league table. However, this gradation is less relevant
for the team, the coaches, and the players. Becoming champions, qualifying for
next year’s European Cup or not being relegated to a lower league is what we
assume is more in the focus of the involved entities. Strict thresholds
implicitly assume that it’s not particularly relevant for teams to finish 10th
or 11th, but that the potential implications around crucial positions in the
ranking – which guarantee participation in next year’s Champions League, for
example – are considerably greater. For a discussion on the financial
dimensions consult GK2020. Although the framework allows for a weighting
scheme, for the sake of simplicity, we consider all thresholds to be equally
relevant to each team. More detailed information on the reward structures per
league and season appear in Appendix C.3.2. Individual updating patterns of
the reward scheme, e. g. because the UEFA Europa League place allocated to the
national cup winner is transferred and included in the league’s season
rewards, are explained in Appendix C.3.3.

In Section 3, we have outlined how the general framework can be employed for
applications with unknown outcome functions and those incorporating the EI
itself. Outcomes of football matches do not follow a deterministic rule and
can only approximately be described by a probabilistic model. We follow the
approach of Goller2018, using an ordered choice model with three outcome
probabilities estimated by the Ordered Forest Lechner Okasa (2019), hereafter
abbreviated as ORF.888The general framework is not restricted to this specific
method and the choice of the underlying outcome model is of second-order (see
Appendix C.4.4). To restrict the number of covariates in the ORF model we
perform a LASSO-based model selection step. Starting from the second
iteration, this set of covariates is extended with the previously estimated EI
values (as outlined in Section 3.2.4). In addition, we also simulate the exact
score of the match, drawn from two independent Poisson distributions, as the
goal difference often serves as a tie-breaker in determining the final
ranking.

The choice of the Jensen-Shannon divergence as the distance function,
specified in equation (4), follows the argumentation in Section 3.2.5 for
settings with multiple event outcomes and rewards. We use a scaling factor of
$\ln(3)^{-1}$ to constrain the EI to the [0,1] interval and weight the
probability distributions $P_{i}$ by the match outcome probabilities
$\\{\pi_{H},\pi_{D},\pi_{A}\\}$ to account for the likelihood of the three
outcome scenarios.

$\displaystyle\text{JSD}_{\pi_{H},\pi_{D},\pi_{A}}(P_{H},P_{D},P_{A}))=\frac{1}{\ln(3)}\left(\mathbf{H}\left(\sum_{i\in\\{H,D,A\\}}\pi_{i}P_{i}\right)-\sum_{i\in\\{H,D,A\\}}\pi_{i}\mathbf{H}(P_{i})\right)$
(4) $\displaystyle\textit{where}\
\mathbf{H}(P)=-\sum_{j=1}^{m}P(x_{j})\ln(x_{j})$

###  5.3 Results

####  5.3.1 Distribution of the estimated values

Figure 3: Home and away team’s event importance estimates grouped by match day
enumerated in relation to the last match day, all seasons, and all leagues.
Square-root transformation to y-axis applied

Figure 3 shows the distribution of estimated EI values by match days. For the
majority of the season, the estimated EI values are concentrated around a
value of about 0.01. In other words, most matches are similarly (un)important
for the first parts of the season. Deviations can be observed in pairings
between teams that are expected to be close in the final end-of-season
standings, as in these matches a positive result implies a negative result for
the opponent. This behaviour changes towards the end of the season, with non-
relevant matches and frequent outliers of particularly important matches. The
uncertainty about the outcome of the rest of the season is reduced with fewer
unknown future results, and the results of individual matches can become more
decisive for the end-of-season rewards. This results in more pronounced values
of the EI towards the end of the season.

As an illustrative example, we show the estimated EI values for the last match
day in the 2017/18 German Bundesliga 1 season in Appendix C.4.1.

####  5.3.2 Predicting match outcomes

To shed light on whether the quantified EI has an impact on outcome
prediction, we compare the estimates of a ’baseline’ ordered forest model that
does not use the EI information with a ’richer’ ORF model that includes the
estimated EI of both, the home and away team, as additional input.

We fit both ORF models on half of the data and predict the outcome
probabilities with the two models on the other half. Based on the outcome
probabilities we construct the expected points (ExpP) measure by awarding
points to the outcomes according to modern football rules – 3 points for a
win, 1 for a draw, and 0 for a loss. This procedure is repeated with swapped
training and prediction samples.

Figure 4: The difference in expected points (ExpP) between the model including
EI variables (Rich) and the baseline model (Base) by the difference in event
importance (EI) between the team and its opponent. Values are rounded to the
nearest grid point. Frequency indicates the number of points on a grid point.
The red line denotes a GAM with a 95% confidence interval. Expected points are
averaged over 100 estimates with different sample splits. Square-root
transformation to x-axis applied.

Figure 4 displays the difference in expected points between the rich and the
baseline model by the difference in the EI values between the two competing
teams. The generalised additive model (GAM) fit on the data confirms that
teams with a higher absolute difference in EI are attributed a higher absolute
prediction of ExpP with the richer model verifying that the inclusion of the
EI variable is relevant for outcome prediction. The variable importance
measures of the EI variables in the rich model are shown in Appendix C.4.3 and
provide evidence for the notable contribution of the EI in the outcome model.

####  5.3.3 Prediction power improvement

In Section 5.3.2 we have shown, that the estimated EI values are picked up by
an enriched outcome model. This raises the question of whether using EI values
in an outcome model improves predictive performance.

We compare seven different prediction models to margin-free betting odds of
the online betting platform B365.999The betting odds serve as a benchmark and
are collected from the website www.football-data.co.uk. To ensure
comparability with the model predictions, we linearly scale the odds to remove
the bookmaker’s margin. The baseline ORF model as described in Section 5.3.2
(ORF), the richer model including the EI values (ORF+EI), and additionally
including the difference of EI estimates (ORF+EI+diff), an ORF model with a
binary importance measure101010A binary variable indicating if the match is
still relevant for attaining a better/worse reward or if the match cannot
change the reward anymore. Several empirical works use such indicators
Fornwagner (2019); Feddersen . (2021) (ORF+naive), an ORF model (ORF+add3)
that adds three covariates,111111To investigate if a potential improvement is
just induced mechanically by the larger set of covariates. We include travel
distance, days since the last match of the home team & days to the next match
of the away team. an ordered logit model with the baseline variables (Logit),
and an ordered logit including the EI estimates (Logit+EI).

To evaluate the out-of-sample prediction accuracy we randomly split the data
into two samples.121212The split is performed on the full-season level to not
give the proposed models an unfair advantage over the betting odds. On one-
half of the seasons, the models are fitted, on the other half, the prediction
accuracy of the models is measured by the log-likelihood and the Brier score
(results for the Brier score can be found in Appendix C.4.6). In each
repetition, we index the accuracy measures by the results of the benchmark
betting-odds model to balance any particular characteristics of the chosen
sample.

Figure 5: Out-of-sample prediction accuracy of different models over 1000
repetitions. Values are quantified in log-Likelihood and indexed in each
repetition by the performance of betting odds.

Figure 5 shows the out-of-sample prediction accuracy. For the logit model, the
addition of the EI results in only a slight improvement, which is probably due
to the linearity constraint. Including EI values in the ORF model
substantially increases the predictive power, as the additional information
contained in the EI variables can be fully utilised, resulting in better
performance than the margin-free betting odds. Recording the event importance
in a binary variable does not improve the accuracy of the prediction. The
model with three added covariates indicates that the increase in prediction
power is not just induced by the larger set of covariates.

Figure 6: The difference in realised and expected points according to betting
odds by the difference in event importance between the team and its opponent.
Values are rounded to the nearest grid point. Frequency indicates the number
of points on a grid point. The red line denotes GAM with a 95% confidence
interval. Square-root transformation to x-axis applied.

To break down the improvement by the EI information on the betting odds, we
present in Figure 6 the difference between the achieved points and the
expected points according to the betting odds in relation to the difference in
the EI values between the teams and their opponents. A GAM fit on all data
points indicates, that in particular across matches where the differences are
small, the EI can partly explain the mismatch in the betting odds. For larger
EI differences, the EI does not provide additional information to the
bookmaker’s model. We deduce that the betting odds already cover the unequal
incentives when they are particularly pronounced but do not fully account for
more subtle disparities in the importance of a match to the competing teams.
This is generally in line with and extends the results of Feddersen2021, which
show that bookmakers are aware of the impact of different incentives on the
outcome of matches on the final match days. We provide additional evidence on
the complimentary informational content of the EI measure to the betting odds
in Appendix C.4.5.

####  5.3.4 Team performance

Besides the usefulness of the EI measure in predictions, we investigate
whether the differences in incentives for teams is reflected in-match
statistics that record a team’s on-field behaviour and performance. For this,
we investigate in-match statistics (data source:
Opta131313https://www.statsperform.com/opta/) for the 2010/11 through 2018/19
German Bundesliga 1 seasons with regards to our EI estimates. The team
performance data is collected individually for both teams and pooled for the
home and away teams. Outcome variables are totals per match, except for ’Duel
win’ and ’Tackles win’ which are shares. For ease of interpretation, the EI
estimates for the home and away teams are each divided into three groups -
’zero’ (EI = 0), ’low’, and ’high’ (above 0.035) EI.1414145% of the EI values
are zero. We therefore choose the high EI threshold at the 95 % quantile of
the EI values to obtain balanced groups.

Figure 7 shows the results for four outcomes: duels per game, number of
completed passes, number of goals scored, and number of goals
conceded.151515In a first step, we run a fixed effect regression for every
outcome individually using the combinations ’Team x home/away x season’, as
well as ’Opponent x home/away x season’ fixed effects’. The resulting
residuals are centred and standardized by ’Team x home/away x season’. On
those scaled residuals we run a regression using again the grid on the EI
categories ’zero’, ’low’, and ’high’ for both competing teams. Complementary
results using other outcomes are shown in Appendix C.4.7.

(a) Duels

(b) Passes

(c) Goals scored

(d) Goals conceded

Figure 7: Linear regression estimates of the centred and standardized
residuals of different outcomes on the Event importance categories. 95%
confidence intervals are in parentheses. The baseline is low by low category.

The results can be summarised as follows: Teams for which a match is
particularly important (i) play more aggressive entering more duels on the
pitch, (ii) play more directly towards goal with fewer passes, fewer touches,
and more entries into the final third and penalty area, (iii) score more
goals. In contrast, teams with zero importance exhibit a more passive style of
play and concede more goals.

####  5.3.5 Public perception

Sport is entertainment and thrives on public perception. If the calculated EI
can represent the (later realised) public interest in a specific match, it
could be useful for several purposes – marketing, ticket pricing, or prime-
time broadcasting selection. With this in mind, we relate the EI to the
stadium attendance turnout, as well as social media attention. While
attendances are officially reported by clubs, social media attention is
captured through club mentions and match hashtags within the 24 hours before
kickoff on Twitter.161616Gathered using Twitter API v2. The analysis yields
almost identical results when the time period is extended to 48h before the
match. Match hashtags are a short abbreviation of the names of the two teams
which are regularly used on Twitter, i. e. #BVBS04 relates to the match of the
teams Borussia Dortmund against Schalke 04. Mentions of club accounts are
Twitter tweets containing the account name of a team, e. g. @LFC - the
official Twitter account of the English team Liverpool FC. Due to the
inconsistency and lack of use of the aforementioned proxies in the early years
and across the leagues, we can only perform this analysis beginning with the
2014/15 season and must exclude the Spanish and Portuguese leagues.

As different clubs have put different emphasis on social media and this has
changed over time, we control for the team and season-specific usage of social
media.171717The procedure of the analysis is similar to the residual analysis
in Section 5.3.4. In a first step we calculate a linear fixed-effect model
containing a FE for every ’team x home/away x season’ interaction and for
every calendar month and using the Twitter or attendance data, both in logs,
as the outcome. In a second step, we centre and standardize the residuals from
the linear fixed-effect models by ’league x season’ pairs. This is the most
natural procedure in our view. By taking nominal outcomes, other FE variants,
or standardising over different groups interpretation of results does not
change. In the analysis, we explain the standardised and centred attention
measures by linear regression on the EI of the two competing teams. Figure 8
presents the point estimates and 95 % confidence intervals of the linear
regressions. Stadium attendance is modestly associated with the home team’s
importance in the match. Here, restrictions on stadium capacity and (pre-sold)
season tickets could mitigate the effect. Thus, social media attention might
give a more clear picture of realised interest. We find team account mentions
are strongly associated with the respective team’s EI measure. Similarly, the
match-tag mentions increase with both teams’ EI. This is consistent with and
complementary to Dobson1992 and Lei2013 reporting higher stadium attendance
for more important sporting events, and recent findings by buraimo2022armchair
that Premier League television audiences are larger for more important
matches.

(a) Home team’s Twitter account mentions

(b) Away team’s Twitter account mentions

(c) Match-tag Twitter mentions

(d) Stadium attendance

Figure 8: Linear regression estimates of the centred and standardized
residuals of different outcomes on the event importance (EI) categories. 95%
confidence intervals are in parentheses. The baseline is low by low category.

##  6 Conclusion

Public perception and academic research analyse incentives in simple
situations where there is a direct link between performance and reward. More
complex situations with indirect rewards and therefore unclear implicit
incentive structures have received little attention.

In this article, we propose a statistical method to quantify the importance of
single events in multi-event contests with end-of-contest reward structures.
Thanks to its flexibility and generality the procedure covers a multitude of
potential applications and can be valuable for various fields, including sales
and marketing, human resources, or operations management. Our event importance
framework can be adapted to different contest structures seen in society and
opens a variety of potential research topics such as behavioural responses
involving implicit incentives or operational fairness concerns in contest and
reward structures. These include, for example, different valuations due to the
order of actions or asymmetric incentives that lead to distorted probabilities
of winning in a contest.

In an application to European football leagues, we show the association of the
quantified importance of a match to in-match behaviour and the performance of
the teams. As discrepancies in the EI can lead to altered outcome
probabilities, the quantification of the event importance can help to ensure
fair tournaments. The event importance measure also addresses other
stakeholders in the football industry. As we show that the EI measure is
consistent with the public interest in terms of social media and stadium
attendance it can be useful for dynamic ticket pricing or TV stations that
want to broadcast the most attractive match. Lastly, we illustrate the value
of the EI measure for predicting match outcomes and point out under which
circumstances the bookmakers do not yet account for the event’s importance.

For the application to the US presidential primaries, we quantify the higher
relevance of early election dates induced by the first-winner effect. For
small states with a low number of delegates, this can substantially boost
their influence on the nomination outcome as otherwise, their votes become
irrelevant in many of the primaries. We show that the two investigated
hypothetical schedules lead to more equitable distribution in the ratio of
event importance values to the number of delegates rewarded by the election.

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lifelike orange-picking task. Groups of four subjects picked oranges under
three payoff conditions: personal reward, collective reward, and intergroup
competition with a reward to the most efficient team. On average, the
collective reward rule resulted in a 30 percent loss in production compared to
the personal payoff rule. The intergroup competition eliminated this loss of
productivity and was more effective the more similar the competing teams were
with respect to overall abilities. The implications of these findings are
discussed in light of a new approach to game-theoretic equilibrium solutions.
© 1993 by Academic Press, Inc., author = Erev, Ido and Bornstein, Gary and
Galili, Rachely, issn = 10960465, journal = Journal of Experimental Social
Psychology, number = 6, pages = 463–478, title = Constructive intergroup
competition as a solution to the free rider problem: A field experiment,
volume = 29, year = 1993 @articleGoddard2005, abstract = In the previous
literature, two approaches have been used to model match outcomes in
association football (soccer): first, modelling the goals scored and conceded
by each team; and second, modelling win-draw-lose match results directly.
There have been no previous attempts to compare the forecasting performance of
these two types of model. This paper aims to fill this gap. Bivariate Poisson
regression is used to estimate forecasting models for goals scored and
conceded. Ordered probit regression is used to estimate forecasting models for
match results. Both types of models are estimated using the same 25-year data
set on English league football match outcomes. The best forecasting
performance is achieved using a ’hybrid’ specification, in which goals-based
team performance covariates are used to forecast win-draw-lose match results.
However, the ifferences between the forecasting performance of models based on
goals data and models based on results data appear to be relatively small. ©
2004 International Institute of Forecasters. Published by Elsevier B.V. All
rights reserved., author = Goddard, John, journal = International Journal of
Forecasting, keywords = Bivariate Poisson,Football match results,Ordered
probit, number = 2, pages = 331–340, title = Regression models for forecasting
goals and match results in association football, volume = 21, year = 2005
@articleGoller2021, author = Goller, Daniel, doi = 10.1007/s10479-022-04563-0,
journal = Annals of Operations Research, title = Analysing a built-in
advantage in asymmetric darts contests using causal machine learning, volume =
forthcom., pages = 1–31, year = 2022 @articleGoller2020, abstract = Balancing
the allocation of games in sports competitions is an important organizational
task that can have serious financial consequences. In this paper, we examine
data from 10,142 soccer games played in the top German, Spanish, French, and
English soccer leagues between 2007/2008 and 2016/2017. Using a machine
learning technique for variable selection and applying a semi-parametric
analysis of radius matching on the propensity score, we find that all four
leagues have a lower attendance in games that take place on four non-
frequently played days than those on three frequently played days. We also
find that, in all leagues, there is a significantly lower home advantage for
the underdog teams on non-frequent days. Our findings suggest that the current
schedule favors underdog teams with fewer home games on non-frequent days.
Therefore, to increase the fairness of the competitions, it is necessary to
adjust the allocation of the home games on non-frequent days in a way that
eliminates any advantage driven by the schedule. These findings have
implications for the stakeholders of the leagues, referees’ and calendar
committees as well as for coaches and players., author = Goller, Daniel and
Krumer, Alex, issn = 03772217, journal = European Journal of Operational
Research, keywords = Performance,Schedule effects,Soccer, number = 2, pages =
740–754, publisher = Elsevier B.V., title = Let’s meet as usual: Do games
played on non-frequent days differ? Evidence from top European soccer leagues,
volume = 286, year = 2020 @articleGonzalez-Diaz2012, abstract = Stakes affect
aggregate performance in a wide variety of settings. At the individual level,
we define the critical ability as an agent’s ability to adapt performance to
the importance of the situation. We identify individual critical abilities of
professional tennis players, relying on point-level data from twelve years of
the US Open tournament. We establish persistent heterogeneity in critical
abilities. We find a significant statistical relationship between identified
critical abilities and overall career success, which validates the
identification procedure and suggests that response to pressure is a
significant factor for success. © 2012 Elsevier B.V., author = González-Díaz,
Julio and Gossner, Olivier and Rogers, Brian W., issn = 01672681, journal =
Journal of Economic Behavior and Organization, keywords = Career
success,Critical ability,Heterogeneity,Performance,Pressure, number = 3, pages
= 767–781, publisher = Elsevier B.V., title = Performing best when it matters
most: Evidence from professional tennis, volume = 84, year = 2012
@articleHarb-Wu2019, abstract = Performing in front of a supportive audience
increases motivation. However, it also creates psychological pressure, which
may impair performance, especially in precision tasks. In this paper, we
exploit a unique setting in which professionals compete in a real-life contest
with large monetary rewards in order to assess how they perform in front of a
supportive audience. Using the task of shooting in the sprint competitions of
professional biathlon events over a period of 16 years, we find that for both
genders, biathletes from the top quartile of the ability distribution miss
significantly more shots when competing in their home country compared to
competing abroad. Our results are in line with the hypothesis that high
expectations to perform well in front of a friendly audience prompt
individuals to choke when performing skill-based tasks., author = Harb-Wu, Ken
and Krumer, Alex, issn = 01672681, journal = Journal of Economic Behavior and
Organization, keywords = Biathlon,Choking under pressure,Home
advantage,Paradoxical performance effects of incentives,Social pressure, pages
= 246–262, publisher = Elsevier B.V., title = Choking under pressure in front
of a supportive audience: Evidence from professional biathlon, volume = 166,
year = 2019 @bookHastie2009, address = 2nd. ed. New York, author = Hastie,
Trevor and Tibshirani, Robert and Friedman, Jerome, isbn = 9781479932115,
publisher = Springer, title = The Elements of Statistical Learning - Data
mining, inference, and prediction, year = 2009 @articleJennett1984, author =
Jennett, Nicholas, journal = Scottish Journal of Political Economy, number =
1, pages = 176–198, title = Attendances, Uncertainty of Outcome and Policy in
Scottish League Football, volume = 31, year = 1984 @articleKlaassen2001,
abstract = This article tests whether points in tennis are independent and
identically distributed (iid). We model the probability of winning a point on
service and show that points are neither independent nor identically
distributed: winning the previous point has a positive effect on winning the
current point, and at “important” points it is more difficult for the server
to win the point than at less important points. Furthermore, the weaker a
player, the stronger are these effects. Deviations from iid are small,
however, and hence the iid hypothesis will still provide a good approximation
in many cases. The results are based on a large panel of matches played at
Wimbledon 1992–1995, in total almost 90,000 points. Our panel data model takes
into account the binary character of the dependent variable, uses random
effects to capture the unobserved part of a player’s quality, and includes
dynamic explanatory variables. © 2001 American Statistical Association.,
author = Klaassen, Franc J.G.M. and Magnus, Jan R., isbn = 0162145017531, issn
= 1537274X, journal = Journal of the American Statistical Association,
keywords = Binary choice,Dependence,Dynamic panel data,Linear probability
model,Nonidentical distribution,Random effects,Tennis, number = 454, pages =
500–509, title = Are points in tennis independent and identically distributed?
Evidence from a dynamic binary panel data model, volume = 96, year = 2001
@articleKrumer2017, abstract = The order of actions in contests may have a
significant effect on performance. In this study, we examine the role of the
schedule in round-robin tournaments with sequential games between three and
four contestants. Our empirical analysis, based on soccer FIFA World Cups and
UEFA European Championships, and on two Olympic wrestling events, reveals that
there is a substantial advantage to the contestant who competes in the first
and third matches. Our findings are in line with the hypothesis that winning
probabilities in multi-stage contests with sequential games are endogenously
depending on the schedule of contests as predicted by game-theoretical models.
We also discuss possible ways to reduce the effect of the schedule., author =
Krumer, Alex and Lechner, Michael, issn = 00142921, journal = European
Economic Review, keywords = First-mover advantage,Performance,Schedule
effects,Soccer,Wrestling, pages = 412–427, publisher = Elsevier B.V., title =
First in first win: Evidence on schedule effects in round-robin tournaments in
mega-events, volume = 100, year = 2017 @articleKrumer2017a, abstract = We
study round-robin tournaments with either three or four symmetric players
whose values of winning are common knowledge. With three players there are
three rounds, each of which includes one pair-wise game such that each player
competes in two rounds only. The player who wins two games wins the
tournament. We characterize the subgame perfect equilibrium and show that each
player’s expected payoff and probability of winning is maximized when he
competes in the first and the last rounds. With four players there are three
rounds, each of which includes two sequential pair-wise games where each
player plays against a different opponent in every round. We again
characterize the subgame perfect equilibrium and show that a player who plays
in the first game of each of the first two rounds has a first-mover advantage
as reflected by a significantly higher winning probability as well as by a
significantly higher expected payoff than his opponents., author = Krumer,
Alex and Megidish, Reut and Sela, Aner, issn = 01761714, journal = Social
Choice and Welfare, number = 3, pages = 633–658, publisher = Springer Berlin
Heidelberg, title = First-mover advantage in round-robin tournaments, volume =
48, year = 2017 @articleLechner2019, abstract = In econometrics so-called
ordered choice models are popular when interest is in the estimation of the
probabilities of particular values of categorical outcome variables with an
inherent ordering, conditional on covariates. In this paper we develop a new
machine learning estimator based on the random forest algorithm for such
models without imposing any distributional assumptions. The proposed Ordered
Forest estimator provides a flexible estimation method of the conditional
choice probabilities that can naturally deal with nonlinearities in the data,
while taking the ordering information explicitly into account. In addition to
common machine learning estimators, it enables the estimation of marginal
effects as well as conducting inference thereof and thus providing the same
output as classical econometric estimators based on ordered logit or probit
models. An extensive simulation study examines the finite sample properties of
the Ordered Forest and reveals its good predictive performance, particularly
in settings with multicollinearity among the predictors and nonlinear
functional forms. An empirical application further illustrates the estimation
of the marginal effects and their standard errors and demonstrates the
advantages of the flexible estimation compared to a parametric benchmark
model. A software implementation of the Ordered Forest is provided in GAUSS as
well as in the R-package orf available on CRAN., archivePrefix = arXiv,
arxivId = 1907.02436, author = Lechner, Michael and Okasa, Gabriel, eprint =
1907.02436, issn = 2331-8422, journal = arXiv:1907.02436, title = Random
Forest Estimation of the Ordered Choice Model, year = 2019 @articleLei2013,
abstract = Sports teams have incentives to put more effort into games with an
immediate effect on standings compared to games that do not, possibly
affecting outcome uncertainty. We develop a measure of game outcome
uncertainty, game importance (GI), that captures how each game affects a team
’ s standing and can be calculated for individual games. Results show that
observed variation in GI explains observed variation in attendance, game
outcomes, and margin of victory at MLB games over the 1994 through 2010
seasons, suggesting that GI is an influential indicator in fans ’ attendance
decisions, consistent with the uncertainty of outcome hypothesis., author =
Lei, Xinrong and Humphreys, Brad R., issn = 15590410, journal = Journal of
Quantitative Analysis in Sports, keywords = Game Attendance,Game Importance,
Game Outcome,Major League Baseball,Uncertainty Of Outcome, number = 1, pages =
25–36, title = Game importance as a dimension of uncertainty of outcome,
volume = 9, year = 2013 @articleLink2016, abstract = This research explores
the influence of match importance on player activity in professional soccer.
Therefore, we used an observational approach and analyzed 1,211 matches of
German Bundesliga and 2nd Bundesliga. The importance measurement employed is
based on post season consequences of teams involved in a match. This means, if
a match result could potentially influence the final rank, and this rank would
lead to different consequences for a team, such as qualification for Champions
League opposed to qualification for Europe League, then this match is
classified as important; otherwise not. Activity was quantified by TOTAL
DISTANCE COVERED, SPRINTS, FAST RUNS, DUELS, FOULS and ATTEMPTS. Running
parameters were recorded using a semi-automatic optical tracking system, while
technical variables were collected by professional data loggers. Based on our
importance classification, low important matches occurred at the beginning of
round 29. A two-way ANOVA indicates significantly increased FAST RUNS (+4
percent, d = 0.3), DUELS (+16 percent, d = 1.0) and FOULS (+36 percent, d =
1.2) in important matches compared to low important ones. For FAST RUNS and
FOULS, this effect only exists in Bundesliga. A comparison of the two leagues
show that TOTAL DISTANCE COVERED (+3 percent, d = 0.9), SPRINTS (+25 percent,
d = 1.4) and FAST RUNS (+15 percent, d = 1.4) are higher compared to 2nd
Bundesliga, whilst FOULS is less in Bundesliga (-7 percent, d = 0.3). No
difference in player activity was found between matches at the beginning of a
season (round 1-6) and at the end of a season (round 29-34). We conclude that
match importance influences player activity in German professional soccer. The
most reasonable explanation is a conscious or unconscious pacing strategy,
motivated by preserving abilities or preventing injury. Since this tendency
mainly exists in Bundesliga, this may suggest that more skilled players show a
higher awareness for the need of pacing., author = Link, Daniel and De
Lorenzo, Michael F., issn = 19326203, journal = PLoS ONE, number = 6, pages =
1–10, pmid = 27281051, title = Seasonal pacing - Match importance affects
activity in professional soccer, volume = 11, year = 2016 @articleMoreira2013,
author = Moreira, Alexandre and Freitas, Camila and Nakamura, Fabio and Drago,
Gustavo and Drago, Murilo and Aoki, Marcelo, journal = The Journal of Strength
and Conditioning Research, number = 1, pages = 202–207, title = Effect of
Match Importance on Salivary Cortisol and Immunoglobulin A Responses in Elite
Young Volleyball Players, volume = 27, year = 2013 @articlePaserman2010,
author = Paserman, Daniele M., journal = mimeo, title = Gender Differences in
Performance in Competitive Environments? Evidence from Professional Tennis
Players, year = 2010 @articleStefani2007, abstract = Comprehensive rankings of
football teams have become an important, and occasionally controversial,
feature of many football codes. The rationale behind these systems needs to be
understood. The historical evolution of the current eight codes is briefly
traced from just two: association football and rugby football, played under
various forms in the mid 19th century. Based on current rules, the eight codes
fall into four groups: first, Australian Rules football and Gaelic football;
second, American college football, American professional football and Canadian
football; third, rugby union and rugby league; and fourth, soccer.
Comprehensive rating systems exist for three codes. For American college
football, the Bowl Championship Series or BCS system places top US college
football teams into a national championship game and other important "bowl
games". That system combines two normally incompatible components, an
objective adjustive computer component and a subjective human-poll component.
The composite has been controversial in four of the nine years of service,
when the computer component differed from the human component resulting in
major changes that favored the human component each time. For rugby union, the
International Rugby Board or IRB system employs a predictor/corrector
adjustment in which defeating a weak team provides less gain than defeating a
strong team while losing to a weak team elicits a much larger negative
adjustment than losing to a strong team, arguably a fair and efficient methods
for rating competitors. For soccer, FIFA have improved the previous rating
systems with a new and simpler system which takes into account strength of
opponents and game importance; however, all losses are treated as equal
regardless of the opponents, and home advantage is ignored. An Elo based
system, employing many of features of the IRB system, appears to have
advantages over the FIFA system., author = Stefani, Ray and Pollard, Richard,
issn = 2194-6388, journal = Journal of Quantitative Analysis in Sports,
keywords = football, American football, rugby union, FIFA, so, number = 3,
title = Football Rating Systems for Top-Level Competition: A Critical Survey,
volume = 3, year = 2007 @articleWager2018, abstract = Many scientific and
engineering challenges—ranging from personalized medicine to customized
marketing recommendations—require an understanding of treatment effect
heterogeneity. In this article, we develop a nonparametric causal forest for
estimating heterogeneous treatment effects that extends Breiman’s widely used
random forest algorithm. In the potential outcomes framework with
unconfoundedness, we show that causal forests are pointwise consistent for the
true treatment effect and have an asymptotically Gaussian and centered
sampling distribution. We also discuss a practical method for constructing
asymptotic confidence intervals for the true treatment effect that are
centered at the causal forest estimates. Our theoretical results rely on a
generic Gaussian theory for a large family of random forest algorithms. To our
knowledge, this is the first set of results that allows any type of random
forest, including classification and regression forests, to be used for
provably valid statistical inference. In experiments, we find causal forests
to be substantially more powerful than classical methods based on nearest-
neighbor matching, especially in the presence of irrelevant covariates.,
archivePrefix = arXiv, arxivId = 1510.04342, author = Wager, Stefan and Athey,
Susan, eprint = 1510.04342, issn = 1537274X, journal = Journal of the American
Statistical Association, keywords = Adaptive nearest neighbors
matching,Asymptotic normality,Potential outcomes,Unconfoundedness, number =
523, pages = 1228–1242, publisher = Taylor & Francis, title = Estimation and
Inference of Heterogeneous Treatment Effects using Random Forests, volume =
113, year = 2018 @articleWright2014, abstract = This paper surveys the
academic OR/analytics literature describing research into the laws and rules
of sports and sporting competitions. The literature is divided into post hoc
analyses and proposals for future changes, and is also divided into laws/rules
of sports themselves and rules/organisation of tournaments or competitions.
The survey outlines a large number of studies covering 21 sports in many parts
of the world. The analytical approaches most commonly used are found to be
various forms of regression analysis and simulation. Issues highlighted by
this survey include the different views of what constitutes fairness and the
frequency with which changes produce unintended consequences. © 2013 Elsevier
B.V. All rights reserved., author = Wright, Mike, issn = 03772217, journal =
European Journal of Operational Research, keywords = Analytics,OR in
Sport,Review,Rules,Survey,Tournaments, number = 1, pages = 1–8, publisher =
Elsevier B.V., title = OR analysis of sporting rules - A survey, volume = 232,
year = 2014 @articleLahvicka2015, abstract = This article presents a new
method of calculating match importance. Match importance is defined as
strength of relationship between the match result and a given season outcome.
Probabilities of all necessary match result-season outcome combinations are
estimated by Monte Carlo simulation. Using actual results of 12 seasons of
English Premier League and betting odds, it is shown that both match result
and season outcome predictions are realistic. The method provides results that
are close to Jennett’s approach; however, it does not require ex post
information and can be used for any type of season outcome., author =
Lahvička, Jiří, journal = Journal of Sports Economics, keywords = Monte
Carlo,match importance,seasonal uncertainty, number = 4, pages = 390–409,
title = Using Monte Carlo Simulation to Calculate Match Importance: The Case
of English Premier League, volume = 16, year = 2015 @articleSoebbing2013,
abstract = A growing body of literature indicates that sports teams face
incentives to lose games at the end of the season. This incentive arises from
a league’s entry draft policy. We use data from betting markets to confirm the
existence of tanking, or the perception of tanking, in the National Basketball
Association (NBA). Results from a Seemingly Unrelated Regression (SUR) model
of point spreads and point differences in NBA games indicate that betting
markets believe that tanking takes place in the NBA, even though the evidence
that tanking actually exists is mixed. Other NBA policy changes also affect
betting market outcomes. © 2011 Western Economic Association International.,
author = Soebbing, Brian P. and Humphreys, Brad R., isbn = 7804923406, issn =
10743529, journal = Contemporary Economic Policy, number = 2, pages = 301–313,
title = Do gamblers think that teams tank? Evidence from the NBA, volume = 31,
year = 2013 @articleRohde2017, abstract = Europe’s professional football clubs
engage in intraseasonal races to win the league, qualify for Union of European
Football Associations (UEFA) competitions, and avoid relegation. In these
competitions, playing talent is a scarce good as players invest effort in the
form of fitness and the risk of injuries. Thus, managers face incentives to
adjust effort levels by means of changing the value of starting squads. This
paper analyzes whether managers save efforts in the absence of financial
incentives or ahead of more important games. The theoretical model extends
previous papers from the contest theory and shirking literature to a match-day
panel regression analysis of strategic behavior. We build on two unbalanced
panels covering 10 consecutive seasons in the German Bundesliga (n=6,120) and
UEFA Champions League (n=1,920). The insights generated in this article are
important for national league administrators as well as the UEFA in aligning
their competitions towards each other and increasing incentives towards
participating teams., author = Rohde, Marc and Breuer, Christoph, issn =
1930076X, journal = International Journal of Sport Finance, keywords = Contest
theory,Financial incentives,League design,Managerial effort,Soccer,Sport
economics, number = 2, pages = 160–182, title = Financial incentives and
strategic behavior in european professional football: A match day analysis of
starting squads in the German Bundesliga and UEFA Competitions, volume = 12,
year = 2017 @articleKendall2017, abstract = Wright (2014) recently presented a
survey of sporting rules from an Operational Research (OR) perspective. He
surveyed 21 sports, which consider the rules of sports and tournaments and
whether changes have led to unintended consequences. The paper concludes:
“Overall, it would seem that this is just a taster and there may be plenty
more such studies to come”. In this paper we present one such study. This is
an interdisciplinary paper, which cuts across economics, sport and Operational
Research. We recognize that the paper could have been published in any of
these disciplines but for the sake of continuity with the paper that motivated
this study, we wanted to publish this paper in an OR journal. We look at
specific examples where the rules of sports have led to unforeseen and/or
unwanted consequences. We hope that the paper will be especially useful to
sports administrators, helping them to review what has not previously worked
and also encouraging them to engage with the scientific community when
considering making changes. We believe that this is the first time that such a
comprehensive review of sporting rules, which have led to unexpected
consequences, has been published in the scientific literature., author =
Kendall, Graham and Lenten, Liam J.A., issn = 03772217, journal = European
Journal of Operational Research, keywords = Incentives,Sport,Strategies,
number = 2, pages = 377–394, publisher = Elsevier B.V., title = When sports
rules go awry, volume = 257, year = 2017 @articlePaul2003, abstract = In
recent years the National Hockey League (NHL) has put policies in place to
boost attendance. Specifically, these changes have been to curb violence,
increase scoring, and move to an unbalanced schedule featuring more games
against regional rivals. This research looks at variations in game-to-game
attendance in the NHL, focusing on these policy changes. It is found that
violence, specifically fighting, tends to attract fans in large numbers across
the United States and Canada. Surprisingly, increases in scoring, ceteris
paribus, tend to depress attendance. The change in scheduling by the NHL,
however, has been a success, with divisional rivals increasing attendance in
U.S. cities and additional contests against other Canadian teams increasing
attendance in Canada., author = Paul, Rodney J., journal = American Journal of
Economics and Sociology, number = 2, pages = 345–364, title = Variations in
NHL attendance: The impact of violence, scoring, and regional rivalries,
volume = 62, year = 2003 @articleGoddard2004, abstract = An ordered probit
regression model estimated using 10 years’ data is used to forecast English
league football match results. As well as past match results data, the
significance of the match for end-of-season league outcomes, the involvement
of the teams in cup competition and the geographical distance between the two
teams’ home towns all contribute to the forecasting model’s performance. The
model is used to test the weak-form efficiency of prices in the fixed-odds
betting market. A strategy of selecting end-of-season bets with a favourable
expected return according to the model appears capable of generating a
positive return. Copyright © 2004 John Wiley & Sons, Ltd., author = Goddard,
John and Asimakopoulos, Ioannis, issn = 02776693, journal = Journal of
Forecasting, keywords = Efficiency, ordered probit,Sports betting, number = 1,
pages = 51–66, title = Forecasting Football Results and the Efficiency of
Fixed-odds Betting, volume = 23, year = 2004 @articleFeddersen2021, abstract =
Research Question: : The study asks whether bookmakers alter the closing
betting odds on European football matches due to potential variation in team
incentives to exert effort to win games based on position in the standings.
Effort is a multidimensional concept and teams have many ways to alter effort
supplied in a match. For example, the players included or left out of the
starting 11, the tactics used in a specific match, substitutions made during a
match, and other factors under team control, can be interpreted as changes in
effort supplied. Research Methods: OLS models explaining observed variation in
betting odds are estimated for roughly 25,000 regular season football games in
8 top European domestic leagues from 2000–2001 through 2018–2019. Results:
Regression results indicate bookmakers alter betting odds on matches involving
clubs in the middle of the domestic league standings that have no chance of
qualifying for the Champions/Europa Leagues or being relegated. In other
words, bookmakers and bettors believe these teams will not put forth maximum
effort to compete in and win these matches. We find similar results for teams
already assured of relegation and teams that already clinched the league
championship. Implications: Overall, the research contributes to literature
analyzing the impact of incentives in sporting contests. As the world of
professional sport and sports gambling become increasingly interconnected and
integrated, the results of the study present opportunities not only for
collaboration between the parties but also an increased understanding on the
role league design has on individual match outcomes., author = Feddersen, Arne
and Humphreys, Brad R. and Soebbing, Brian P., issn = 1746031X, journal =
European Sport Management Quarterly, keywords = Tournament
theory,football,incentives,sports betting, number = forthcom., pages = 1–17,
publisher = Taylor & Francis, title = Contest incentives, team effort, and
betting market outcomes in European football,

year = 2021 @articleFornwagner2019, abstract = This paper analyzes data from a
tournament, namely the National Hockey League regular scheduled season of
games, which provides incentives to increase effort in order to reach the
playoffs and incentives to decrease effort once a team has been eliminated
from playoff considerations because of the entry draft. Our results show that
teams react to these dual incentives - they win more games when there is still
a chance to reach the playoffs and lose more after being eliminated from
playoff considerations. One can argue that losing more games after having no
more chance to reach the playoffs could be the result of lower motivation or
disappointment. This is the first study to show that this is not the only
explanation for a higher amount of lost games. Instead, we find that there is
a concrete strategy behind losing., author = Fornwagner, Helena, issn =
01674870, journal = Journal of Economic Psychology, keywords = 3720,Dual
incentives,National Hockey League,Natural field experiment,Sport
data,Strategic losing,Tournaments, pages = 1–12, title = Incentives to lose
revisited: The NHL and its tournament incentives, volume = 75, year = 2019
@articleScarf2009, abstract = Operational Research may be used to compare
different designs for a sporting contest or tournament. This paper considers a
methodology for this purpose. We propose a number of tournament metrics that
can be used to measure the success of a sporting contest or tournament, and
describe how these metrics may be evaluated for a particular tournament
design. Knowledge of these measures can then be used to compare competing
designs, such as round-robin, pure knockout and hybrids of these designs. We
show, for example, how the design of the tournament influences the outcome
uncertainty of the tournament and the number of unimportant matches within the
tournament. In this way, where new designs are proposed, the implications of
these designs may be explored within a modelling paradigm. In football
(soccer), the UEFA Champions League has adopted a number of designs over its
50 year history; the design of the tournament has been modified principally in
response to the changing demands of national league football and television -
the paper uses this particular tournament to illustrate the methodology. ©
2008 Elsevier B.V. All rights reserved., author = Scarf, Philip and Yusof,
Muhammad Mat and Bilbao, Mark, issn = 03772217, journal = European Journal of
Operational Research, keywords = Competitive balance,OR in
sport,Simulation,Sports tournaments, number = 1, pages = 190–198, publisher =
Elsevier B.V., title = A numerical study of designs for sporting contests,
volume = 198, year = 2009 @articleScarf2008, abstract = A quantitative measure
of "match importance" is useful in a number of decision problems, for example:
as a metric in tournament design; for selecting matches for broadcasting; for
scheduling matches in a tournament; and for assigning referees. To date
measures of match importance used in such analyses have been relatively naïve.
We discuss a general measure that considers the effect of a particular match
on the end of tournament position, given the results of all other matches,
some played, some predicted. We use logistic regression to predict matches and
Monte Carlo simulation to compute the match importance measure, and apply
these to soccer matches in the English Football Association Premier League. ©
2006 Elsevier Ltd. All rights reserved., author = Scarf, Philip A. and Shi,
Xin, issn = 03050548, journal = Computers and Operations Research, keywords =
Championship significance,Contest,Outcome uncertainty,Sport, number = 7, pages
= 2406–2418, title = The importance of a match in a tournament, volume = 35,
year = 2008 @articleCorona2017, abstract = Identifying the decisive matches in
international football tournaments is of great relevance for a variety of
decision makers such as organizers, team coaches and/or media managers. This
paper addresses this issue by analyzing the role of the statistical approach
used to estimate the outcome of the game on the identification of decisive
matches on international tournaments for national football teams. We extend
the measure of decisiveness proposed by Geenens (2014) in order to allow us to
predict or evaluate the decisive matches before, during and after a particular
game on the tournament. Using information from the 2014 FIFA World Cup, our
results suggest that Poisson and kernel regressions significantly outperform
the forecasts of ordered probit models. Moreover, we find that although the
identification of the most decisive matches is independent of the model
considered, the identification of other key matches is model dependent. We
also apply this methodology to identify the favorite teams and to predict the
most decisive matches in 2015 Copa America before the start of the
competition. Furthermore, we compare our forecast approach with respect to the
original measure during the knockout stage., author = Corona, Francisco and De
Dios Tena Horrillo, Juan and Wiper, Michael Peter, issn = 15590410, journal =
Journal of Quantitative Analysis in Sports, keywords = Kernel
regression,Poisson model,entropy,game importance,ordered probit model, number
= 1, pages = 11–23, title = On the importance of the probabilistic model in
identifying the most decisive games in a tournament, volume = 13, year = 2017
@articleSchilling1994, author = Schilling, Mark F ., journal = Mathematics
Magazine, number = 4, pages = 282–288, title = The Importance of a Game,
volume = 67, year = 1994 @articleGeenens2014, abstract = In sport tournaments
in which teams are matched two at a time, it is useful for a variety of
reasons to be able to quantify how important a particular game is. The need
for such quantitative information has been addressed in the literature by
several more or less simple measures of game importance. In this paper, we
point out some of the drawbacks of those measures and we propose a different
approach, which rather targets how decisive a game is with respect to the
final victory. We give a definition of this idea of game decisiveness in terms
of the uncertainty about the eventual winner prevailing in the tournament at
the time of the game. As this uncertainty is strongly related to the notion of
entropy of a probability distribution, our decisiveness measure is based on
entropy-related concepts. We study the suggested decisiveness measure on two
real tournaments, the 1988 NBA Championship Series and the UEFA 2012 European
Football Championship (Euro 2012), and we show how well it agrees with what
intuition suggests. Finally, we also use our decisiveness measure to
objectively analyse the recent UEFA decision to expand the European Football
Championship from 16 to 24 nations in the future, in terms of the overall
attractiveness of the competition. © 2013 Elsevier B.V. All rights reserved.,
author = Geenens, Gery, issn = 03772217, journal = European Journal of
Operational Research, keywords = Applied probability,Entropy,Game
importance,OR in sports,Uncertainty modeling, number = 1, pages = 156–168,
publisher = Elsevier B.V., title = On the decisiveness of a game in a
tournament, volume = 232, year = 2014 @articleGoossens2012, abstract =
Recently, most clubs in the highest Belgian football division have become
convinced that the format of their league should be changed. Moreover, the TV
station that broadcasts the league is pleading for a more attractive
competition. However, the clubs have not been able to agree on a new league
format, mainly because they have conflicting interests. In this paper, we
compare the current league format, and three other formats that have been
considered by the Royal Belgian Football Association. We simulate the course
of each of these league formats, based on historical match results. We assume
that the attractiveness of a format is determined by the importance of its
games; our importance measure for a game is based on the number of teams for
which this game can be decisive to reach a given goal. Furthermore, we provide
an overview of how each league format aligns with the expectations and
interests of each type of club. © 2010 Springer Science+Business Media, LLC.,
author = Goossens, Dries R. and Beliën, Jeroen and Spieksma, Frits C.R., issn
= 02545330, journal = Annals of Operations Research, keywords = Football,Match
importance,Optimization,Simulation,Tournament design, number = 1, pages =
223–240, title = Comparing league formats with respect to match importance in
Belgian football, volume = 194, year = 2012 @articleVanDijk2001, abstract = In
the reported experiment different payment schemes are examined on their
incentive effects. Payments based on individual, team and relative performance
are compared. Subjects conducted computerized tasks that required substantial
effort. The results show that individual and team payment induced the same
effort levels. In team production free-riding occurred, but it was compensated
by many subjects providing more effort than in case of individual pay. Effort
was higher, but more variable in tournaments, while in case of varying
abilities workers with relatively low ability worked very hard and drove up
effort of the others. Finally, attitudes towards work and other workers
differed strongly between conditions. © 2001 Elsevier Science B.V. All rights
reserved., author = Van Dijk, Frans and Sonnemans, Joep and Van Winden, Frans,
isbn = 3102052541, issn = 00142921, journal = European Economic Review,
keywords = Experiment,Payment schemes, number = 2, pages = 187–214, title =
Incentive systems in a real effort experiment, volume = 45, year = 2001

@articleSzymanski2003, author = Szymanski, Stefan, isbn = 9780230274273,
journal = Journal of Economic Literature, pages = 1137–1187, title = The
economic design of sporting contests, volume = XLI, year = 2003

@articleSheremeta2018, abstract = Group contests are ubiquitous. Some examples
include warfare between countries, competition between political parties,
team-incentives within firms, and rent-seeking. In order to succeed, members
of the same group have incentives to cooperate with each other by expending
effort. However, since effort is costly, each member also has an incentive to
abstain from expending any effort and instead free ride on the efforts of
other members. Contest theory predicts that the intensity of competition
between groups and the amount of free riding within groups depend on the group
size, sharing rule, group impact function, contest success function, and
heterogeneity of players. We review experimental studies testing these
theoretical predictions. Most studies find significant over-expenditure of
effort relative to the theory and significant heterogeneity of behavior within
and between groups. Also, most studies find support for the comparative
statics predictions of the theory (with the exception of the “group size
paradox”). Finally, studies show that there are effective mechanisms that can
promote within-group cooperation and conflict resolution mechanisms that can
de-escalate and potentially eliminate between-group conflict., author =
Sheremeta, Roman M., issn = 14676419, journal = Journal of Economic Surveys,
keywords = Contests,Experiments,Groups, number = 3, pages = 683–704, title =
Behavior in Group Contests: a Review of Experimental Research, volume = 32,
year = 2018 @articleRosen1985, abstract = The role of rewards for maintaining
performance incentives in multi-stage, sequential games of survival is
studied. The sequential structure is a statistical design-of-experiments for
selecting and ranking contestants. It, author = Rosen, Sherwin, journal =
National Bureau of Economic Research Working Paper Series, number = 1668,
title = Prizes and Incentives in Elimination Tournaments, year = 1985
@articlePrendergast1999, author = Prendergast, Canice, issn = 00220515,
journal = Journal of Economic Literature, number = 1, pages = 7–63, title =
The provision of incentives in firms, volume = 37, year = 1999
@articleLazear2000, author = Lazear, Edward, journal = The American Economic
Review, Papers and Proceedings, number = 2, pages = 410–414, title = The Power
of Incentives, volume = 90, year = 2000 @articleKleinTeeselink2020, abstract =
This paper examines how within-match variation in incentives affects the
performance of darts players. The game of darts offers an attractive naturally
occurring research setting, because performance can be observed at the
individual level and without obscuring effects of risk considerations and
behavior of others. We analyze four data sets covering a total of 29,381 darts
matches of professional, amateur, and youth players. We find that amateur and
youth players display a sizable performance decrease at decisive moments.
Professional players appear less susceptible of such choking under pressure.
Our results speak to a growing literature on the limits of increasing
incentives as a recipe for better performance., author = Klein Teeselink,
Bouke and Potter van Loon, Rogier J.D. and van den Assem, Martijn J. and van
Dolder, Dennie, issn = 01672681, journal = Journal of Economic Behavior and
Organization, keywords = Choking under pressure,Darts,Incentives,Performance
pressure, number = August, pages = 38–52, title = Incentives, performance and
choking in darts, volume = 169, year = 2020 @articleIqbal2019, abstract = The
discouragement effect of being the lagging player in multi-stage contests is a
well-documented phenomenon. In this study, we utilize data from 2447 Davis Cup
matches in team tennis tournaments to test the effect of being behind or ahead
on individuals’ performance with and without intermediate prizes. Using
several different strategies to disentangle the effect of being ahead in the
interim score from the effect of selection, we find the usual discouragement
effect. However, the discouragement effect disappears after the introduction
of intermediate prizes in the form of ranking points. The lagging favorite had
close to a 20-percentage point greater probability of winning compared to
matches without such a prize. We show that this result is not driven by the
selection of better players into tournaments with intermediate prizes. As
predicted by previous theoretical studies, our empirical findings suggest that
intermediate prizes may mitigate or even eliminate the ahead–behind effects
that arise in multi-stage contests., author = Iqbal, Hamzah and Krumer, Alex,
issn = 00142921, journal = European Economic Review, keywords = Collective
decision-making,Discouragement,Multi-stage contests,Tennis, pages = 364–381,
publisher = Elsevier B.V., title = Discouragement effect and intermediate
prizes in multi-stage contests: Evidence from Davis Cup, volume = 118, year =
2019 @incollectionGoller2018, author = Goller, Daniel and Knaus, Michael C and
Lechner, Michael and Okasa, Gabriel, booktitle = A Modern Guide to Sports
Economics, chapter = 22, editor = Koning, Ruud H. and Kesenne, Stefan, pages =
335–355, publisher = Edward Elgar Publishing, title = Predicting Match
Outcomes in Football by an Ordered Forest Estimator, year = 2021
@articleGinsburgh2003, author = Ginsburgh, V. and van Ours, J., journal = The
American Economic Review, number = 1, pages = 289–296, title = Expert Opinion
and Compensation : Evidence from a Musical Competition, volume = 93, year =
2003 @articleLin1991, title=Divergence measures based on the Shannon entropy,
author=Lin, Jianhua, journal=IEEE Transactions on Information theory,
volume=37, number=1, pages=145–151, year=1991, publisher=IEEE

@bookLaffont2002, author = Laffont, Jean-Jacques and Martimort, David,
mendeley-groups = MI_project, publisher = Princeton University Press, title =
The Theory of Incentives: The Principal-Agent Model, year = 2002

@articleCohen-Zada2017, abstract = We exploit a unique setting in which two
professionals compete in a real-life tennis contest with high monetary rewards
in order to assess how men and women respond to competitive pressure.
Comparing their performance in low-stakes versus high-stakes situations, we
find that men consistently choke under competitive pressure, but with regard
to women the results are mixed. Furthermore, even if women show a drop in
performance in the more crucial stages of the match, it is in any event about
50 percent smaller than that of men. These findings are robust to different
specifications and estimation strategies., author = Cohen-Zada, Danny and
Krumer, Alex and Rosenboim, Mosi and Shapir, Offer Moshe, issn = 01674870,
journal = Journal of Economic Psychology, keywords = Choking,Competitive
pressure,Gender,Performance,Tennis, pages = 176–190, publisher = Elsevier
B.V., title = Choking under pressure and gender: Evidence from professional
tennis, volume = 61, year = 2017

@articleGK2020, title = Let’s meet as usual: Do games played on non-frequent
days differ? Evidence from top European soccer leagues, journal = European
Journal of Operational Research, volume = 286, number = 2, pages = 740-754,
year = 2020, issn = 0377-2217, author = Daniel Goller and Alex Krumer,
keywords = Performance, Schedule effects, Soccer, abstract = Balancing the
allocation of games in sports competitions is an important organizational task
that can have serious financial consequences. In this paper, we examine data
from 10,142 soccer games played in the top German, Spanish, French, and
English soccer leagues between 2007/2008 and 2016/2017. Using a machine
learning technique for variable selection and applying a semi-parametric
analysis of radius matching on the propensity score, we find that all four
leagues have a lower attendance in games that take place on four non-
frequently played days than those on three frequently played days. We also
find that, in all leagues, there is a significantly lower home advantage for
the underdog teams on non-frequent days. Our findings suggest that the current
schedule favors underdog teams with fewer home games on non-frequent days.
Therefore, to increase the fairness of the competitions, it is necessary to
adjust the allocation of the home games on non-frequent days in a way that
eliminates any advantage driven by the schedule. These findings have
implications for the stakeholders of the leagues, referees’ and calendar
committees as well as for coaches and players.

@articleAlchian1972, archivePrefix = arXiv, arxivId =
https://knowledgehub.unsse.org/es/knowledge-hub/relaciones-entre-los-ods-el-
plan-para-una-decada-cooperativa-y-el-balance-social-en-cooperativas-2/,
author = Alchian, Armen and Demsetz, Harold, eprint =
/knowledgehub.unsse.org/es/knowledge-hub/relaciones-entre-los-ods-el-plan-
para-una-decada-cooperativa-y-el-balance-social-en-cooperativas-2/, journal =
The American Economic Review, number = 5, pages = 777–795, primaryClass =
https:, title = Production, Information Costs, and Economic Organization,
volume = 62, year = 1972

@articlekahn2000sports, title=The sports business as a labor market
laboratory, author=Kahn, Lawrence M, journal=Journal of economic perspectives,
volume=14, number=3, pages=75–94, year=2000

@articlebar2020ask, title=Ask not what economics can do for sports-Ask what
sports can do for economics, author=Bar-Eli, Michael and Krumer, Alex and
Morgulev, Elia, journal=Journal of Behavioral and Experimental Economics,
volume=89, pages=101597, year=2020, publisher=Elsevier

@articlelechner2020orf, title=orf: Ordered Random Forests, author=Lechner,
Michael and Okasa, Gabriel, journal=R package version 0.1.3, year=2020

@articledanziger2011extraneous, title=Extraneous factors in judicial
decisions, author=Danziger, Shai and Levav, Jonathan and Avnaim-Pesso, Liora,
journal=Proceedings of the National Academy of Sciences, volume=108,
number=17, pages=6889–6892, year=2011, publisher=National Acad Sciences

@articlede2005save, title=Save the last dance for me: Unwanted serial position
effects in jury evaluations, author=de Bruin, Wändi Bruine, journal=Acta
psychologica, volume=118, number=3, pages=245–260, year=2005,
publisher=Elsevier

@articleklumpp2006primaries, title=Primaries and the New Hampshire effect,
author=Klumpp, Tilman and Polborn, Mattias K, journal=Journal of Public
Economics, volume=90, number=6-7, pages=1073–1114, year=2006,
publisher=Elsevier

@articlepreston2003cheating, title=Cheating in contests, author=Preston, Ian
and Szymanski, Stefan, journal=Oxford review of economic policy, volume=19,
number=4, pages=612–624, year=2003, publisher=Oxford University Press

@articletaylor2002losing, title=Losing to win: Tournament incentives in the
National Basketball Association, author=Taylor, Beck A and Trogdon, Justin G,
journal=Journal of Labor Economics, volume=20, number=1, pages=23–41,
year=2002, publisher=The University of Chicago Press

@articleknight2010momentum, title=Momentum and social learning in presidential
primaries, author=Knight, Brian and Schiff, Nathan, journal=Journal of
political economy, volume=118, number=6, pages=1110–1150, year=2010,
publisher=University of Chicago Press Chicago, IL

@bookmayer2003front, title=The front-loading problem in presidential
nominations, author=Mayer, William G and Busch, Andrew E, year=2003,
publisher=Brookings Institution Press

@articleridout2008importance, title=The importance of being early:
Presidential primary front-loading and the impact of the proposed western
regional primary, author=Ridout, Travis N and Rottinghaus, Brandon,
journal=PS: Political Science & Politics, volume=41, number=1, pages=123–128,
year=2008, publisher=Cambridge University Press

@articleharris1987racing, title=Racing with uncertainty, author=Harris,
Christopher and Vickers, John, journal=The Review of Economic Studies,
volume=54, number=1, pages=1–21, year=1987, publisher=Wiley-Blackwell

@articleapesteguia2010psychological, title=Psychological pressure in
competitive environments: Evidence from a randomized natural experiment,
author=Apesteguia, Jose and Palacios-Huerta, Ignacio, journal=American
Economic Review, volume=100, number=5, pages=2548–64, year=2010

@articleburaimo2022armchair, title=Armchair fans: Modelling audience size for
televised football matches, author=Buraimo, Babatunde and Forrest, David and
McHale, Ian G and Tena, J d D, journal=European Journal of Operational
Research, volume=298, number=2, pages=644–655, year=2022, publisher=Elsevier

@ArticleNielsen2021, AUTHOR = Nielsen, Frank, TITLE = On a Generalization of
the Jensen–Shannon Divergence and the Jensen–Shannon Centroid, JOURNAL =
Entropy, VOLUME = 22, YEAR = 2020, NUMBER = 2, ARTICLE-NUMBER = 221, ISSN =
1099-4300, ABSTRACT = The Jensen–Shannon divergence is a renown bounded
symmetrization of the Kullbackndash;Leibler divergence which does not require
probability densities to have matching supports. In this paper, we introduce a
vector-skew generalization of the scalar alpha; -Jensenndash;Bregman
divergences and derive thereof the vector-skew alpha; -Jensenndash;Shannon
divergences. We prove that the vector-skew alpha; -Jensenndash;Shannon
divergences are f-divergences and study the properties of these novel
divergences. Finally, we report an iterative algorithm to numerically compute
the Jensenndash;Shannon-type centroids for a set of probability densities
belonging to a mixture family: This includes the case of the
Jensenndash;Shannon centroid of a set of categorical distributions or
normalized histograms.

@incollectionmcfadden_conditional_1974, address = New York, title =
Conditional logit analysis of qualitative choice behavior, language = en,
booktitle = Frontiers in Econometrics, publisher = Academic press, author =
McFadden, Daniel, editor = Zarembka, Paul, year = 1974, pages = 105–142

@dataDataverse, author = Goller, Daniel and Heiniger, Sandro, publisher =
Harvard Dataverse, title = Replication code and results for: ’A general
framework to quantify the event importance in multi-event contests’, UNF =
UNF:6:LktvG47zGdoPNd+HgJeKLA==, year = 2022, version = V1, doi =
https://doi.org/10.7910/DVN/F3QA9N

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