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How often does the best team win? A unified approach to understanding randomness in North American sport


Statistical applications in sports have long centered on how to best separate signal, such as team talent, from random noise. However, most of this work has concentrated on a single sport, and the development of meaningful cross-sport comparisons has been impeded by the difficulty of translating luck from one sport to another. In this manuscript, we use betting market data to develop a Bayesian state-space model that can be uniformly applied across sporting leagues to better understand the role of randomness in game outcomes. Our model can be used to extract estimates of team strength, the between-season, within-season, and game-to-game variability of team strengths, as well each team's home advantage. We implement our approach across a decade of play in each of the National Football League (NFL), National Hockey League (NHL), National Basketball Association (NBA), and Major League Baseball (MLB), finding that the NBA demonstrates both the largest dispersion in talent and the largest home advantage. Additionally, the NHL and MLB stand out for their relative randomness in game outcomes. We conclude by proposing a new metric for judging league competitiveness that works in absence of factors outside of team control.
How often does the best team win? A unified approach to
understanding randomness in North American sport
Michael J. Lopez
Skidmore College
Gregory J. Matthews
Loyola University Chicago
Benjamin S. Baumer
Smith College
January 24, 2017
arXiv:1701.05976v1 [stat.AP] 21 Jan 2017
Statistical applications in sports have long centered on how to best separate signal, such as team
talent, from random noise. However, most of this work has concentrated on a single sport, and the
development of meaningful cross-sport comparisons has been impeded by the difficulty of translating
luck from one sport to another. In this manuscript, we use betting market data to develop a Bayesian
state-space model that can be uniformly applied across sporting leagues to better understand the
role of randomness in game outcomes. Our model can be used to extract estimates of team strength,
the between-season, within-season, and game-to-game variability of team strengths, as well each
team’s home advantage. We implement our approach across a decade of play in each of the National
Football League (NFL), National Hockey League (NHL), National Basketball Association (NBA),
and Major League Baseball (MLB), finding that the NBA demonstrates both the largest dispersion
in talent and the largest home advantage. Additionally, the NHL and MLB stand out for their
relative randomness in game outcomes. We conclude by proposing a new metric for judging league
competitiveness that works in absence of factors outside of team control. Keywords: sports analytics,
Bayesian modeling, competitive balance, MCMC
1 Introduction
Most observers of sport can agree that game outcomes are to some extent subject to chance. The
line drive that miraculously finds the fielder’s glove, the fumble that bounces harmlessly out-of-
bounds, the puck that ricochets into the net off of an opponent’s skate, or the referee’s whistle on
a clean block can all mean the difference between winning and losing. Yet game outcomes are not
completely random—there are teams that consistently play better and worse. To what extent does
luck influence our perceptions of team strength over time?
One way in which statistics can lead this discussion lies in the untangling of signal and noise when
comparing the caliber of each league’s teams. For example, is team ibetter than team j? And if so,
how confident are we in making this claim? Central to such an understanding of sporting outcomes
is that if we know each team’s relative strength, then, a priori, game outcomes—including wins and
losses—can be viewed as unobserved realizations of random variables. As a simple example, a 75%
probability of team ibeating team jat time kimplies that in a hypothetical infinite number of
games between the two teams at time k,iwins three times as often as j.
Given both national public interest and an academic curiosity that has extended across disciplines,
many innovative techniques have been developed to estimate team strength. These approaches typ-
ically blend past game scores with game, team, and player characteristics in a statistical model.
Corresponding estimates of talent are often checked or calibrated by comparing out-of-sample es-
timated probabilities of wins and losses to observed outcomes. Such exercises do more than drive
water-cooler conversation as to which team may be better. Indeed, estimating team rankings has
driven the development of advanced statistical models (Bradley and Terry, 1952; Glickman and
Stern, 1998) and occasionally played a role in the decision of which teams are eligible for continued
postseason play (CFP, 2014).
However, because randomness manifests differently in different sports, a limitation of sport-specific
models is that inferences cannot generally be applied to other competitions. As a result, researchers
who hope to contrast one league to another often focus on the one outcome common to all sports:
won-loss ratio. Among other flaws, measuring team strength using wins and losses performs poorly
in a small sample size, ignores the game’s final score (which is known to be more predictive of future
performance than won-loss ratio (Boulier and Stekler, 2003)), and is unduly impacted by, among
other sources, fluctuations in league scheduling, season length, injury to key players, and the general
advantage of playing at home. As a result, until now, analysts and fans have never quite been able
to quantify inherent differences between sports with respect to randomness and the dispersion and
evolution of team strength. We aim to fill this void.
In the sections that follow, we present a novel approach for estimating team ability that accounts for
league scheduling, measures each team’s home advantage, and uncovers inherent differences in North
American sport. First, we validate an assumption that game-level probabilities provided by betting
markets provide unbiased and low-variance estimates of the true probabilities of wins and losses in
each professional contest. Second, we implement a modified Bayesian state-space model that uses
these probabilities to capture implied team strength and variability. Next, by examining posterior
estimates of within and between season variability, as well as the overall dispersion in team strength
estimates, we present unique league-level contrasts that to this point have been difficult to capture.
Finally, we conclude by showing that our estimates of team strength improve upon both won-loss
ratio and point differential with respect to correlating to future performance, and use our posterior
draws propose a novel metric of assessing parity. We find that, on account of both narrower talent
distributions and smaller home advantages, a typical contest in the NHL or MLB is much closer to
a coin-flip than one in the NBA or NFL. Additionally, the NHL (from one season to the next) and
NBA (from one week to the next) boast the least consistency in team strength estimates over time.
1.1 Literature review
The importance of quantifying team strength in sport extends across disciplines. This includes
contrasting league-level characteristics in economics (Leeds and Von Allmen, 2004), estimating game-
level probabilities in statistics (Glickman and Stern, 1998), and classifying future game winners in
forecasting (Boulier and Stekler, 2003). We discuss and synthesize below.
1.1.1 Competitive balance
Assessing the competitive balance of sports leagues is particularly important in economics and
management (Leeds and Von Allmen, 2004). While competitive balance can purportedly measure
several different quantities, in general it refers to levels of equivalence between teams. This could be
equivalence within one time frame (e.g. how similar was the distribution of talent within a season?),
between time frames (e.g. year-to-year variations in talent), or from the beginning of a time frame
until the end (e.g. the likelihood of each team winning a championship at the start of a season).
The most widely accepted within-season competitive balance measure is the Noll-Scully (Noll, 1991;
Scully, 1989), computed as the ratio of the observed standard deviation in team win totals to the
idealized standard deviation, defined as that which would have been observed due to chance alone
if each team were equal in talent. Larger Noll-Scully values are believed to reflect greater imbalance
in team strengths.
While Noll-Scully has the positive quality of allowing for interpretable cross-sport comparisons, a
reliance on won-loss outcomes entails undesireable properties as well (Owen, 2010; Owen and King,
2015). For example, Noll-Scully increases, on average, with the number of games played (Owen
and King, 2015), hindering any comparisons of the NFL (16 games) to MLB (162). Additionally,
each of the leagues employ some form of an unbalanced schedule. Teams in each of MLB, the
NBA, NFL, and NHL play intradivisional opponents more often than interdivisional ones, and
intraconference opponents more often than interconference ones, meaning that one team’s won-loss
record may not be comparable to another team’s due to differences in the respective strengths
of their opponents (Lenten, 2015). Moreover, the NFL structures each season’s schedule so that
teams play interdivisional games against opponents that finished in the same spot in the standings
in the prior year. In expectation, this punishes teams that finish atop standings with tougher
games, potentially driving winning percentages toward 0.500. Unsurprisingly, unbalanced scheduling
and interconference play can lead to imprecise competitive balance metrics derived from winning
percentages (Utt and Fort, 2002). As one final weakness, varying home advantages between sports
leagues, as shown in Moskowitz and Wertheim (2011), could also impact comparisons of relative
team quality that are predicated on wins and losses.
Although metrics for league-level comparisons have been frequently debated, the importance of com-
petitive balance in sports is more uniformly accepted, in large part due to the uncertainty of outcome
hypothesis (Rottenberg, 1956; Knowles et al., 1992; Lee and Fort, 2008). Under the uncertainty of
outcome hypothesis, league success—as judged by attendance, engagement, and television revenue—
correlates positively with teams having equal chances. Outcome uncertainty is generally considered
on a game-level basis, but can also extend to season-level success (i.e, teams having equivalent
chances at making the postseason). As a result, it is in each league’s best interest to promote some
level of parity—in short, a narrower distribution of team quality—to maximize revenue (Crooker
and Fenn, 2007). Related, the Hirfindahl-Hirschman Index (Owen et al., 2007) and Competitive
Balance Ratio (Humphreys, 2002) are two metrics attempting to quantify the relative chances of
success that teams have within or between certain time frames.
1.1.2 Approaches to estimating team strength
Competitive balance and outcome uncertainty are rough proxies for understanding the distribution
of talent among teams. For example, when two teams of equal talent play a game without a home
advantage, outcome uncertainty is maximized; e.g., the outcome of the game is equivalent to a coin
flip. These relative comparisons of team talent began in statistics with paired comparison models,
which are generally defined as those designed to calibrate the equivalence of two entities. In the case
of sports, the entities are teams or individual athletes.
The Bradley-Terry model (BTM, Bradley and Terry (1952)) is considered to be the first detailed
paired comparison model, and the rough equivalent of the soon thereafter developed Elo rankings
(Elo, 1978; Glickman, 1995). Consider an experiment with ttreatment levels, compared in pairs.
BTM assumes that there is some true ordering of the probabilities of efficacy, π1, . . . , πt, with the
constraints that πi0 and Pπi= 1. When comparing treatment ito treatment j, the probability
that treatment iis preferable to j(i.e. a win in a sports setting) is computed as πi
Glickman and Stern (1998) and Glickman and Stern (2016) build on the BTM by allowing team-
strength estimates to vary over time through the modeling of point differential in the NFL, which
is assumed to follow an approximately normal distribution. Let y(s,k)ij be the point differential of
a game during week kof season sbetween teams iand j. In this specification, iand jtake on
values between 1 and t, where tis the number of teams in the league. Let θ(s,k)iand θ(s,k)jbe the
strengths of teams iand j, respectively, in season sduring week k, and let αibe the home advantage
parameter for team i, for i= 1, . . . , t. Glickman and Stern (1998) assume that for a game played at
the home of team iduring week kin season s,
E[y(s,k)ij |θ(s,k)i, θ(s,k)j, αi] = θ(s,k)iθ(s,k )j+αi,
where E[y(s,k)ij |θ(s,k)i, θ(s,k)j, αi] is the expected point differential given iand j’s team strengths
and the home advantage of team i.
The model of Glickman and Stern (1998) allows for team strength parameters to vary stochastically
in two distinct ways: from the last week of season sto the first week of season s+ 1, and from week
kof season sto week k+ 1 of season s. As such, it is termed a ‘state-space’ model, whereby the
data is a function of an underlying time-varying process plus additional noise.
Glickman and Stern (1998) propose an autoregressive process to team strengths, whereby over time,
these parameters are pulled toward the league average. One attractive property of this specification
is that past and future season performances are incorporated into season-specific estimates of team
quality. Perhaps as a result, Koopmeiners (2012) identifies stronger fits when comparing state-space
models to BTM’s fit separately within each season. Additionally, unlike BTM’s, state-space models
would not typically suffer from identifiability problems were a team to win or lose all of its games in
a single season (a rare, but extant possibility in the NFL).1For additional and related state-space
resources, see Knorr-Held (2000), Cattelan et al. (2013), Baker and McHale (2015), and Manner
(2015). Additionally, Matthews (2005), Owen (2011), Koopmeiners (2012), Tutz and Schauberger
(2015), and Wolfson and Koopmeiners (2015) implement related versions of the original BTM.
Although the state-space model summarized above appears to work well in the NFL, a few issues
arise when extending it to other leagues. First, with point differential as a game-level outcome,
parameter estimates would be sensitive to the relative amount of scoring in each sport. Thus,
comparisons of the NHL and MLB (where games, on average, are decided by a few goals or runs)
to the NBA and NFL (where games, on average, are decided by about 10 points) would require
further scaling. Second, it is unclear if a Normal model of goal or run differential is appropriate
in low scoring sports like the NHL and MLB. Finally, NHL game outcomes would entail an extra
complication, as roughly 25% of regular season games are decided in overtime or a shootout.
In place of paired comparison models, alternative measures for estimating team strength have also
been developed. Massey (1997) used maximum likelihood estimation and American football out-
comes to develop an eponymous rating system. A more general summary of other rating systems for
forecasting use is explored by Boulier and Stekler (2003). In addition, support vector machines and
simulation models have been proposed in hockey (Demers, 2015; Buttrey, 2016), neural networks
and na¨ıve Bayes implemented in basketball (Loeffelholz et al., 2009; Miljkovi´c et al., 2010), linear
models and probit regressions in football (Harville, 1980; Boulier and Stekler, 2003), and two stage
Bayesian models in baseball (Yang and Swartz, 2004). While this is a non-exhaustive list, it speaks
to the depth and variety of coverage that sports prediction models have generated.
1In the NFL, the 2007 New England Patriots won all of their regular season games, while the 2008 Detroit Lions
lost all of their regular season games.
1.2 Betting market probabilities
In many instances, researchers derive estimates of team strength in order to predict game-level
probabilities. Betting market information has long been recommended to judge the accuracy of
these probabilities (Harville, 1980; Stern, 1991). Before each contest, sports books—including those
in Las Vegas and in overseas markets—provide a price for each team, more commonly known as the
money line.
Mathematically, if team i’s money line is `iagainst team j(with corresponding money line `j),
where |`i| ≥ 100, then the boundary win probability for that team, pi(`i), is given by:
pi(`i) =
100+`iif `i100
100+|`i|if `i≤ −100
The boundary win probability represents the threshold at which point betting on team iwould be
profitable in the long run.
As an example, suppose the Chicago Cubs were favored (`i=127 on the money line) to beat
the Arizona Diamondbacks (`j= 117). The boundary win probability for the Cubs would be
pi(127) = 0.559; for the Diamondbacks, pj(117) = 0.461. Boundary win probabilities sum to
greater than one by an amount collected by the sportsbook as profit (known colloquially as the
“vig” or “vigorish”). However, it is straightforward to normalize boundary probabilities to sum to
unity to estimate pij, the implied probability of idefeating j:
pij =pi(`i)
pi(`i) + pj(`j).(1)
In our example, dividing each boundary probability by 1.02 = (0.559 + 0.461) implies win probabil-
ities of 54.8% for the Cubs and 45.2% for the Diamondbacks.
In principle, money line prices account for all determinants of game outcomes known to the public
prior to the game, including team strength, location, and injuries. Across time and sporting leagues,
researchers have identified that it is difficult to estimate win probabilities that are more accurate
than the market; i.e, that the betting markets are efficient. As an incomplete list, see Harville (1980);
Gandar et al. (1988); Lacey (1990); Stern (1991); Carlin (1996); Colquitt et al. (2001); Spann and
Skiera (2009); Nichols (2012); Paul and Weinbach (2014); Lopez and Matthews (2015). Interestingly,
Colquitt et al. (2001) suggested that the efficiency of college basketball markets was proportional to
the amount of pre-game information available—with the amount known about professional sports
teams, this would suggest that markets in the NFL, NBA, NHL and MLB are as efficient as they
come. Manner (2015) merged predictions from a state-space model with those from betting markets,
finding that the combination of both predictions only occasionally outperformed betting markets
We are not aware of any published findings that have compared leagues using market probabili-
ties. Given the varying within-sport metrics of judging team quality and the limited between-sport
approaches that rely on wins and losses alone, we aim to extend paired comparison models using
money line information to better capture relative team equivalence in a method that can be applied
2 Validation of betting market data
We begin by confirming the accuracy of betting market data with respect to game outcomes. Regular
season game result and betting line data in the four major North American professional sports leagues
(MLB, NBA, NFL, and NHL) were obtained for a nominal fee from Sports Insights. Although these
game results are not official, they are accurate and widely-used. The 2006–2016 seasons were included
in our models, except for the NFL, which used only the 2006–2015 seasons.
These data were more than 99.3% complete in each league, in the sense that there existed a valid
betting line for nearly all games in these four sports across this time period. Betting lines provided by
Sports Insights are expressed as payouts, which we subsequently convert into implied probabilities.
The average vig in our data set is 1.93%, but is always positive, resulting in revenue for the sportsbook
over a long run of games. In circumstances where more than one betting line was available for a
particular game, we included only the line closest to the start time of the game. A summary of our
data is shown in Table 1.
We also compared the observed probabilities of a home win to the corresponding probabilities implied
by our betting market data (Figure 1). In each of the four sports, the efficient market hypothesis
cannot be rejected for any range of implied home win probabilities, based on visual inspection of
Sport (q)tqngames ¯pgames nbets ¯pbets Coverage
MLB 30 26728 0.541 26710 0.548 0.999
NBA 30 13290 0.595 13245 0.615 0.997
NFL 32 2560 0.563 2542 0.589 0.993
NHL 30 13020 0.548 12990 0.565 0.998
Table 1: Summary of cross-sport data. tqis the number of unique teams in each sport q.ngames
records the number of actual games played, while nbets records the number of those games for which
we have a betting line. ¯pgames is the mean observed probability of a win for the home team, while
¯pbets is the mean implied probability of a home win based on the betting line. Note that we have
near total coverage (betting odds for almost every game) across all four major sports.
a LOESS regression model. Thus, we find no evidence to suggest that the probabilities implied by
our betting market data are biased or inaccurate—a conclusion that is supported by the body of
academic literature referenced above. Accordingly, we interpret these probabilities as “true.”
3 Bayesian state-space model
Our model below expands the state-space specification provided by Glickman and Stern (1998) to
provide a unified framework for contrasting the four major North American sports leagues.
Let p(q,s,k)ij be the probability that team iwill beat team jin season sduring week kof sports
league q, for q∈ {M LB, N BA, N FL, N HL}. The p(q,s,k)ij ’s are assumed to be known, calculated
using sportsbook odds via Equation (1). In using game probabilities, we have a cross-sport outcome
that provides more information than only knowing which team won the game or what the score was.
In our notation, i, j = 1, . . . , tq, where tqis the number of teams such that tMLB =tN BA =tN H L =
30 and tNF L = 32. Additionally, s= 1, . . . , Sqand k= 1, . . . , Kq, where Sqand Kqare the
number of seasons and weeks, respectively in league q. In our data, KN F L = 17, KN B A = 25,
KMLB =KN H L = 28, with SN F L = 10 and SM LB =SN BA =SN H L = 11.
Our next step in building a model specifies the home advantage, and one immediate hurdle is that
in addition to having different numbers of teams in each league, certain franchises may relocate from
one city to another over time. In our data set, there were two relocations, Seattle to Oklahoma City
(NBA, 2008) and Atlanta to Winnipeg (NHL, 2011). Let αq0be the league-wide home advantage
(HA) in q, and let α(q)i?be the extra effect (positive or negative) for team iamong game’s played
in city i?, for i?= 1, . . . , t?
q. Here, t?
qis the total number of home cities; in our data, t?
MLB = 30,
NB A =t?
NH L = 31, and t?
NF L = 32.
Figure 1: Accuracy of probabilities implied by betting markets. Each dot represents a bin of implied
probabilities rounded to the nearest hundredth. The size of each dot (N) is proportional to the
number of games that lie in that bin. We note that across all four major sports, the observed winning
percentages accord with those implied by the betting markets. The dotted diagonal line indicates a
completely fair market where probabilities from the betting markets correspond exactly to observed
outcomes. In each sport, this diagonal line lies entirely within the standard error surrounding a
LOESS regression line, suggesting that an efficient market hypothesis cannot be rejected.
Letting θ(q,s,k)iand θ(q,s,k )jbe season-week team strength parameters for teams iand j, respectively,
we assume that
E[logit(p(q,s,k)ij )|θ(q,s,k )i, θ(q,s,k)j, αq0, α(q)i?] = θ(q,s,k )iθ(q,s,k)j+αq0+α(q)i?,
where logit(.) is the log-odds transform.
Let p(q,s,k)represent the vector of length g(q,s,k)containing all of league q’s probabilities in week
kof season s. Our first model of game outcomes, henceforth referred to as the individual home
advantage model (Model IHA), assumes that
logit(p(q,s,k))N(θ(q,s,k )X(q,s,k)+αq0Jg(q,s,k)+α
αqZ(q,s,k), σ2
q,game Ig(q,s,k)),
where θ(q,s,k)is a vector of length tqcontaining the team strength parameters in season sduring
week kand α
αq=nα(q)1,· · · α(q)t?
qo. Note that α
αqdoes not vary over time (i.e. HA is assumed to
be constant for a team over weeks and seasons). X(q,s,k)and Z(q ,s,k)contain g(q,s,k)—the number
of games in league qduring week kof season s—rows and tqand t?
qcolumns, respectively. The
matrix X(q,s,k)contains the values {1,0,1}where for a given row (i.e. one game) the value of ith
column in that row is a 1/-1 if the ith team played at home/away in the given game and 0 otherwise.
Z(q,s,k)is a matrix containing a 1 in column i?if the corresponding game was played in city i?, and
0 otherwise. Finally, σ2
q,game is the game-level variance, Jg(q,s,k)is a column vector of length g(q,s,k)
containing all 1’s, and Ig(q,s,k)is an identity matrix with dimension g(q,s,k)×g(q,s,k).
In addition, we propose a simplified version of Model IHA, labelled as Model CHA (constant home
advantage), which assumes that the HA within each sport is identical for each franchise, such that
logit(p(q,s,k))N(θ(q,s,k )X(q,s,k)+αq0Jg(q,s,k), σ 2
q,game Ig(q,s,k)).
In Model CHA, matrices p(q,s,k),X(q ,s,k),Jg(q,s,k), and Ig(q ,s,k)are specified identically to Model
IHA. As a result, for a game between home team iand away team jduring week kof season s,
E[logit(p(q,s,k)ij )] = θ(q,s,k )iθ(q,s,k)j+αq0under Model CHA.
Similar to Glickman and Stern (1998), we allow the strength parameters of the teams to vary auto-
regressively from season-to-season and from week-to-week. In general, this entails that team strength
parameters are shrunk towards the league average over time in expectation. Formally,
θ(q,s+1,1)|θq ,s,Kq, γq,season , σ2
q,season N(γq,season θ(q,s,Kq), σ2
q,seasonItq) for all s2, . . . , Sq,
θ(q,s,k+1) |θ(q,s,k), γq,w eek, σ2
q,week N(γq,w eekθ(q ,s,k), σ2
q,week Itq) for all s1, . . . , Sq,k2, . . . , Kq.
In this specification, γq ,week is the autoregressive parameter from week-to-week, γq,season is the
autoregressive parameter from season-to-season, and Itqis the identity matrix of dimension tq×tq.
Given the time-varying nature of our specification, we use a Bayesian approach to obtain model
estimates. For sport q, the team strength parameters for week k= 1 and season s= 1 have a prior
distribution of
θ(q,1,1)iN(0, σ2
q,season),for all i1, . . . , tq.
Team specific home advantage parameters have a similar prior, namely,
α(q)i?N(0, σ2
q,α),for i1, . . . , t?
Finally, letting τ2
q,game = 12
q,game ,τ2
q,season = 12
q,week = 12
q,week , and τ2
q,α = 12
we assume the following prior distributions:
q,game Γ(0.0001,0.0001) αq0N(0,10000)
q,season Γ(0.0001,0.0001) γq,season U niform(0,2)
q,week Γ(0.0001,0.0001) γq,w eek Unif orm(0,2)
q,α Γ(0.0001,0.0001)
Our primary interest lies in three levels of variability with respect to the θ(q,s,k)’s. First, there is
variability at any fixed time sand kacross i. This reflects the between-team variability in team
strength; in other words, how equivalent are the teams to one another at a given snapshot in time?
Second, there is variability across k, reflected in the week-to-week autoregressive parameter, γq,week.
This generalizes to how teams can improve or worsen over the course of a season. Third, there is
variability across s, corresponding to the season-to-season autoregressive parameter, γq,season . This
accounts for larger changes to team ability that can occur between seasons.
Posterior distributions of each parameter are estimated using Markov Chain Monte Carlo (MCMC)
methods. We used Gibbs sampling via the rjags package (Plummer, 2016) in the Rstatistical
computing environment to obtain posterior distributions, done separately for each q. Three chains—
using 20,000 iterations after a burn-in of 2,000 draws, fit with a thin of 5 to reduce the autocorrelation
within chains—yielded 4,000 posterior samples in each q. Visual inspection of trace plots with parallel
chains are used to confirm convergence. Comparisons of Models IHA and CHA are made using the
Deviance Information Criterion (DIC, Spiegelhalter et al. (2002)).
While we are unable to share the exact betting market data due to licensing restrictions, a simplified
version of our game-level data, the data wrangling code, Gibbs sampling code, posterior draws, and
the code used to obtain posterior estimates and figures are all posted to a GitHub repository, available
4 Results
In this section we present our results. We begin by validating and comparing the fits of Models IHA
and CHA. We discuss the implications of our estimates of team strength and home advantage, as well
as the interpretation of our variance and autoregressive parameters. We conclude by evaluating our
team strength parameters and illustrating how they could be used to build a league parity metric.
4.1 Model fit
We identified no concerns with the fit of Models IHA and CHA. Trace plots of αq0,γq,season,
γq,week ,σq,game,σq,season , and σq,w eek are shown for each qin Figures 8–11 in the Appendix. Visual
inspection of these plots does not provide evidence of a lack of convergence or of autocorrelation
between draws. These trace plots stem from Model IHA; conclusions are similar when plotting draws
from Model CHA.
Table 2 shows the deviance information criterion (DIC) for each fit in each league, along with the
difference in DIC values and the associated standard error (SE). In each of the leagues, fits with
a team-specific HA (Model IHA) yielded lower DIC’s (lower is better) by a statistically significant
margin, with the most noticeable difference in fit improvement in the NBA.
Model IHA Model CHA Difference (SE)
MLB -8548 -8522 -25.7 (10.8)
NBA 6886 7224 -337.4 (26.6)
NFL -1230 -1216 -13.6 (3.3)
NHL -18335 -18148 -187.6 (20.7)
Table 2: Deviance information criterion (DIC) by sport and model, along with the difference in DIC
and the associated standard errors (SE, in parentheses). IHA: individual home advantage, CHA:
constant home advantage
These results suggest that chance alone likely does not account for observed differences in the home
advantage among teams in each league, with the NBA showing the largest team-to-team differences
in home advantage. As such, results that follow use model estimates from Model IHA.
4.2 Team strength
Table 3 shows summary statistics of the team strength estimates, approximated using posterior
mean draws for all weeks kand seasons sacross all four sports leagues. Overall, there tends to
be a larger variability in team strength at any given point in time in both the NFL and NBA,
with posterior coefficient estimates tending to vary between -1.3 and 1.2 in the NBA and -1.1 and
1.0 in the NFL (on the logit scale). For reference, a team-strength of 1.0 on the log-odds scale
implies a e1.0
1+e1.0= 73.1% chance of beating a league average team in a game played at a neutral site.
The standard deviation of team strength is smallest in MLB, suggesting that—relative to the other
leagues—team talent is more tightly packed. Relative to MLB, spread of team strengths are about
1.3, 3.0, and 3.6 times wider in the NHL, NFL, and NBA, respectively.
League (q) N* min 2.5th Q1 mean Q3 97.5th max sd
MLB 9240 -0.561 -0.384 -0.145 -0.012 0.116 0.324 0.470 0.183
NBA 8250 -2.177 -1.267 -0.485 0.001 0.478 1.209 1.864 0.660
NFL 5440 -1.570 -1.084 -0.390 0.010 0.427 1.034 1.909 0.558
NHL 9240 -1.035 -0.532 -0.169 -0.007 0.173 0.429 0.869 0.246
Table 3: Summary of average week-level team strength parameters, taken on the log-odds scale. N*:
number of unique team strength draws (teams ×seasons ×weeks)
Figure 2 shows estimated team strength coefficients over time. Figures 12–15 (shown in the Ap-
pendix) provide an individual plot for each sport, which include divisional facets to allow easier
identification of individual teams. Teams in Figures 2 and 12–15 are depicted using their two pri-
mary colors, scraped from via the teamcolors package
( in R.
As in Table 3, these figures suggest that the NBA and NFL boast larger between-team gaps in
talent than the NHL and MLB, implying more competitive balance in the latter pair of leagues.
On one level, this stands somewhat in contrast to competitive balance as measured using Noll-
Scully, which alternatively argues that the NFL is more competitively balanced than MLB (Berri,
2014). One likely explanation for this difference is Null-Scully’s link to number of games played,
which artificially makes MLB (162 games) appear less balanced than it actually is and the NFL
(16) appear more balanced. Like Noll-Scully, we conclude that the NBA does not show competitive
balance relative to other leagues.
Our figures also illustrate several other observations. For example, the New England Patriots of the
NFL stand out as having the top performance in the last decade, with an average team strength of
1.91 on the log-odds scale, observed during Week 11 of 2007. In that season, New England finished
the regular season 16-0 before eventually losing in the Super Bowl. The worst performance belongs
to the NBA’s Miami Heat, who during week 23 of the 2007–08 season had a posterior mean team
strength of -2.18. That Heat team finished with an overall record of 15-67, at one point losing
15 consecutive games. Related, it is interesting that the team strength estimates of bad teams in
the NBA (e.g. the Heat in 2007–08) lie further from 0 than the estimates for good teams. This
possibly reveals the tendency for teams in this league to “tank”—a strategy of fielding a weak team
intentionally to improve the chances of having better selection preference in the upcoming player
draft (Soebbing and Humphreys, 2013).
Another observation is that in the NHL, top teams appear less dominant than a decade ago. For
example, there are seven NHL team-seasons in which at least one team reached an average posterior
strength estimate of 0.55 or greater; each of these came during or prior to the 2008–09 season. In
addition to increased parity, the league’s point system change in 2005–06—which unintentionally
encouraged teams to play more overtime games (Lopez, 2013)—could be responsible. More overtime
contests could lead to different perceptions in how betting markets view team strengths, as overtime
sessions and the resulting shootouts are roughly equivalent to coin flips (Lopez and Schuckers, 2016).
Weakest team (Miami Heat)
Strongest team (New England Patriots)
2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016
Team Strength (log−odds scale)
Figure 2: Mean team strength parameters over time for all four sports leagues. MLB and NFL
seasons follow each yearly tick mark on the x-axis, while NBA and NHL seasons begin during years
labeled by the preceding tick marks.
As a final point of clarification in Figures 2, 13, and 15, the periods of time with straight lines of
team strength estimates during the 2012–13 season (NHL) and 2011–12 season (NBA) reflect time
lost due to lockouts.
4.3 Variance and autoregressive parameters
Table 4 shows the mean and standard deviation of posterior draws for γq,season,γq ,week ,σq,game ,
σq,season, and σq,week for each q.
League (q)γq,season γq,week σq ,game σq,season σq ,week
MLB 0.621 (0.031) 1.002 (0.002) 0.201 (0.001) 0.094 (0.005) 0.027 (0.001)
NBA 0.618 (0.041) 0.977 (0.003) 0.274 (0.002) 0.443 (0.02) 0.166 (0.003)
NFL 0.688 (0.041) 0.978 (0.005) 0.232 (0.008) 0.333 (0.02) 0.148 (0.006)
NHL 0.542 (0.028) 0.993 (0.003) 0.105 (0.001) 0.122 (0.006) 0.053 (0.001)
Table 4: Mean posterior draw (standard deviation) by league.
Posterior draws of σq,game suggest that the highest game-level errors in our log-odds probability
estimates occur in the NBA (median posterior draw of σN BA,game = 0.273), followed in order by the
NFL, MLB, and the NHL. Interestingly, although Figure 2 identifies that the talent gap between
teams is smallest in MLB, σMLB,game 2×σN H L,game in our posterior draws. We posit that this
additional game-level error in MLB is a function of the league’s pitching match-ups, in which teams
rotate through a handful of starting pitchers of varying calibers.
We also examine the joint distribution of the variability in team strength on a season-to-season
(σq,season) and week-to-week (σq,week) basis via the contour plot in Figure 3, using separate colors
for each q. Figure 3 reveals that the highest uncertainty with respect to team strength occurs in the
NBA, followed in order by the NFL, NHL, and MLB.
There are a couple of plausible explanations regarding the increased uncertainty in NBA team
strength on a weekly basis. Injuries, the resting of starters, and in-season trades would seemingly
have a larger impact in a sport like basketball where fewer players are participating at a single point
in time. In particular, our model cannot precisely gauge team strength when star players, who could
play, are rested in favor of inferior players. Relative to the other professional leagues, star players
take on a more important role in the NBA (Berri and Schmidt, 2006), an observation undoubtedly
known in betting markets. That said, while there is increased variability in our estimate of NBA
team strengths, when considering differences in team talent to begin with, these absolute differences
Figure 3: Contour plot of the estimated season-to-season and week-to-week variability across all four
major sports leagues. By both measures, uncertainty is lowest in MLB and highest in the NBA.
are not as extreme (e.g., a difference in team strength of 0.05 means less in the NBA than in the
Similarly, Figure 4 displays the joint posterior distribution of γq,season and γq,week via contour plots
for each q. On a season-to-season basis, team strengths in each of the leagues tend to revert towards
the league average of zero as all draws of γq,season <1 for all q. Reversion towards the mean is largest
in the NHL (estimated γNH L,season = 0.54, implying 46% reversion), followed by the NBA (38%),
MLB (38% reversion), and the NFL (31%). However, the only pair of leagues with non-overlapping
credible intervals are the NFL and NHL.
For each of the NHL, NBA, and NFL, posterior estimates of γq,week (as well as 95% credible intervals)
imply an autoregressive nature to team strength within each season. Interestingly, the NBA and
NFL are the least consistent leagues on a week-to-week basis. In MLB, however, team strength
estimates quite possibly follow a random walk (i.e., γMLB ,week = 1), in which the succession of team
strength is unpredictable.
Finally, it is worth noting that our estimates for γN F L,week and γNF L,season—0.98 and 0.69, respectively—
do not substantially diverge from the estimates observed by Glickman and Stern (1998) (0.99
and 0.82). Further, our estimates are more precise. For example, our 95% credible interval for
γNF L,season of (0.61, 0.77) is entirely contained within the interval of (0.52,1.28) reported by Glick-
0.96 0.97 0.98 0.99 1.00 1.01
Figure 4: Contour plot of the estimated season-to-season and week-to-week autoregressive parame-
ters across all four major sports leagues.
man and Stern (1998). In fairness, it is unclear if this increased precision is a function of our
model specification (using log-odds of the probability of a win as the outcome, as opposed to point
differential) or because we used a larger sample(10 seasons, compared to 5).
Like Glickman and Stern (1998), we also observe an inverse link in posterior draws of γNF L,week
and γNF L,season . Given that total shrinkage across time is the composite of within- and between-
season shrinkage, such an association is not surprising (Glickman and Stern, 1998). If one source of
reversion towards the average were to increase, the other would likely compensate by decreasing.
4.4 The home advantage
Figure 5 shows the 2.5th percentile, median, and 97.5th percentile draws of each team’s estimated
home advantage parameter, presented on the probability scale. These are calculated by summing
draws of αq0and α(q)i?for all i?. HAs are shown in descending order to provide a sense of the
magnitude of differences between the home advantage provided in MLB (league-wide, a 54.0% prob-
ability of beating a team of equal strength at home), NHL (55.5%), NFL (58.8%), and NBA (62.0%).
The two franchises that have relocated in the last decade, the Atlanta Thrashers (NHL) and Seattle
Supersonics (NBA), are also included for the games played in those respective cities.
Figure 5 depicts substantial between-franchise differences within both the NBA and NHL. Con-
Philadelphia Phillies
San Diego Padres
Cleveland Indians
Los Angeles Dodgers
Cincinnati Reds
Seattle Mariners
Miami Marlins
Kansas City Royals
Baltimore Orioles
New York Mets
Los Angeles Angels
Atlanta Braves
Washington Nationals
Chicago Cubs
San Francisco Giants
Detroit Tigers
Toronto Blue Jays
Pittsburgh Pirates
Chicago White Sox
St Louis Cardinals
Arizona Diamondbacks
Houston Astros
Minnesota Twins
Tampa Bay Rays
Oakland Athletics
Texas Rangers
New York Yankees
Boston Red Sox
Milwaukee Brewers
Winnipeg Jets
New York Islanders
Ottawa Senators
New Jersey Devils
Colorado Rockies
New York Rangers
Montreal Canadiens
Chicago Blackhawks
Philadelphia Flyers
Toronto Maple Leafs
Pittsburgh Penguins
Boston Bruins
Buffalo Sabres
Arizona Coyotes
Dallas Stars
St. Louis Blues
Carolina Hurricanes
Edmonton Oilers
Tampa Bay Lightning
San Jose Sharks
Vancouver Canucks
Florida Panthers
Washington Capitals
Detroit Red Wings
Colorado Avalanche
Los Angeles Kings
Columbus Blue Jackets
Minnesota Wild
Atlanta Thrashers
Anaheim Ducks
Nashville Predators
Calgary Flames
Los Angeles Rams
Detroit Lions
Miami Dolphins
Philadelphia Eagles
Pittsburgh Steelers
Cincinnati Bengals
Tennessee Titans
New York Jets
Carolina Panthers
Indianapolis Colts
Oakland Raiders
Dallas Cowboys
Green Bay Packers
Cleveland Browns
Tampa Bay Buccaneers
Atlanta Falcons
Washington Redskins
San Diego Chargers
Minnesota Vikings
San Francisco 49ers
Jacksonville Jaguars
Buffalo Bills
Baltimore Ravens
New Orleans Saints
New York Giants
Houston Texans
Arizona Cardinals
Chicago Bears
Kansas City Chiefs
Seattle Seahawks
New England Patriots
Denver Broncos
Brooklyn Nets
Detroit Pistons
New York Knicks
Philadelphia 76ers
Boston Celtics
Miami Heat
Toronto Raptors
Houston Rockets
Chicago Bulls
Los Angeles Lakers
Los Angeles Clippers
Orlando Magic
Minnesota Timberwolves
Oklahoma City Thunder
Memphis Grizzlies
Dallas Mavericks
Washington Wizards
New Orleans Pelicans
Cleveland Cavaliers
Milwaukee Bucks
Indiana Pacers
Atlanta Hawks
Phoenix Suns
Charlotte Hornets
San Antonio Spurs
Seattle Supersonics
Portland Trail Blazers
Sacramento Kings
Golden State Warriors
Utah Jazz
Denver Nuggets
0.54 0.620.5890.555
Probability of beating an equal caliber opponent at home
Estimated Home Advantage by Franchise
Figure 5: Median posterior draw (with 2.5th, 97.5th quantiles) of each franchise’s home advantage
intercept, on the probability scale. We note that the magnitude of home advantages are strongly
segregated by sport, with only one exception (the Colorado Rockies). We also note that no NFL
team, nor any MLB team other than the Rockies, has a home advantage whose 95% credible interval
does not contain the league median.
versely, HA estimates within the NFL and MLB are, with the exception of the Colorado Rockies,
indistinguishable across franchises. Interestingly, the draws of the home advantage parameters for of
a few NFL franchises are skewed (see Denver and Seattle, relative to Detroit), potentially the result
of a shorter regular season. Alternatively, the NFL’s HA may vary by season, game time, or the day
of the game. Anecdotally, night games (Thursday, Sunday, or Monday) conceivably offer a larger
HA than those played during the day (Crabtree, 2014). Informally, NFL team-level HA estimates
are similar in effect size to those depicted by Koopmeiners (2012).
In the NBA, Denver (first) and Utah (second) post the best home advantages, with Brooklyn showing
the worst. This matches the results of Paine (2013), who found significantly better performances
when comparing Denver and Utah to the rest of the league with respect to home and road point
differential. In MLB, the Colorado Rockies stand out for having the highest home advantage, while
the remaining 29 teams boast overlapping credible intervals. We note that teams playing at home
in Denver have the largest home advantages in MLB, the NBA, and the NFL, and the 8th-highest
in the NHL. We speculate that this consistent advantage across sports is related to the home team’s
acclimation to the city’s notably high altitude.
These distinctions have plausible impacts on league standings. An NBA team with a typical home
advantage can expect to win 62.0% of home games against a like-caliber opponent. Yet for Brooklyn,
the corresponding figure is 59.9%, while for Denver, it is 66.1%. Across 41 games (the number each
team plays at home), this implies that Denver’s home advantage is worth an extra 1.68 wins in a
single season, relative to a league average team. Compared to Brooklyn, Denver’s home advantage
is worth an estimated 2.54 wins per year. As one important caveat, our model estimates do not
account for varying line-up and injury information. If opposing teams were to rest their star players
at Denver, for example, our model would artificially inflate Denver’s home advantage.
4.5 Evaluation of team strength estimates
Ultimately, estimates from Model IHA are designed to estimate team quality at any given point in a
season in the absence of factors such as the home advantage and opponent caliber. If these estimates
more properly assess team quality than other metrics of team success (e.g., won-loss percentage or
point differential), they should more accurately link to future performance, such as how well teams
will perform over the remainder of the season.
4 8 12 16 0 20 40 60 80
0 50 100 150 0 20 40 60 80
Game of season
Our estimates
Point differential
Win %
Coefficient of determination with future in−season win %
Figure 6: Coefficient of determination with future in-season win percentage. We note the improve-
ment our team strength estimates offer over season-to-date win percentage and season-to-date point
differential in all sports, especially early in the season.
Figure 6 shows the coefficient of determination (R2) between each team’s future won-loss percentage
in a season and each team’s (i) average team strength estimate from Model IHA, (ii) season-to-date
cumulative point differential, and (iii) season-to-date won-loss percentage. Within each sport, this
is computed by game number, which helps to account for league-level differences in season length.
For purposes of using our team strength estimates, we took the mean posterior draw for each team
in each week a particular contest was played. The lockout-shortened seasons in the NBA (2012) and
NHL (2013) were dropped.
Across each sport, our estimates of team strength consistently outperform past team win percentage
and point differential in predicting future win percentage. This gap is most pronounced earlier in
each season, which is not surprising given the instability of won-loss percentage and point differential
in a small number of games. Differences in predictive accuracy remain throughout most of the regular
season in MLB, the NHL, and the NFL. However, by the NBA’s mid-season, won-loss ratio and point
differential are similar to our estimates of team strength in assessing future performance. By and
large, this confirms the findings of Wolfson and Koopmeiners (2015), who identified that most of the
information needed to predict the remainder of the NBA season is contained within the first third
of the year.
Altogether, results suggest that across seasons and sports, team strength estimates from our state-
space model more accurately assess team caliber than won-loss percentage and point differential.
4.6 How often does the best team win? A new measure of league parity
We conclude by addressing our initial question about the inherent randomness of game outcomes.
One simple way to compare league randomness would be to contrast the observed distribution of
p(q,s,k)ij ’s between each q. However, while sportsbook odds can be used to infer the probability of
each team winning, these odds are only provided for scheduled games. As a result, any between-
league comparisons using sportsbook odds alone would be contingent upon each league’s actual
schedule, and they may not accurately reflect differences that would be observed if all teams were
to play one another.
A second option would be to contrast our posterior draws of θ(q,s,k)ifor all i, either across time
periods or at a fixed point in time, as these estimates account for league particulars such as strength
of schedule. While possible with our team strength estimates, which are presented on identical
scales, such a procedure would not generalize to other sports or leagues where betting market data
may not be unavailable.
Instead, to assess the equivalence of all teams in each league, we consider the likelihood that—given
any pair of teams chosen at random—the better team wins, by simulating estimates of p(q,s,k)ij
using posterior draws of team strength, home advantage, and game level error. For our purposes,
we define the better team to be the one, a priori, with a higher probability of winning that game.
If a contest has no inherent randomness (consider the Harlem Globetrotters), then the better team
always wins.2Conversely, if game-level variability is large relative to the difference in team strength,
then even the inferior team might win nearly half the time.
Using our posterior draws, we approximate the distribution of game-level probabilities between two
randomly chosen teams using the following steps.
Given sport qwith season length Kq, number of seasons Sq, and number of teams tq,
2The Harlem Globetrotters are an exhibition basketball team that plays hundreds of games in a year, rarely losing
1. Draw season ˜sfrom {1, . . . , Sq}, and week ˜
kfrom {1, . . . , Kq}.
2. Draw teams ˜
iand ˜
jfrom {1, . . . , tq}without replacement.
3. Sample one posterior draw of team strength for ˜
iand ˜
iand ˜
j, respectively, from
the posterior distributions of ˜
iand ˜
j’s team strength estimates during season ˜sat week ˜
4. Sample one posterior draw of the HA, ˜αq0, from the posterior distribution of αq0.
5. Sample one posterior draw of the game-level variance parameter, ˜σ2
q,game , and draw a game-
level error, ˜q,game , from ˜q,game N(0,˜σq,game )
6. Impute the simulated log-odds of the better team winning between ˜
iand ˜
j, logit(˜p(q, ˜s,˜
j) =
j+ ˜q,game |, where the better team’s log-odds are based on ˜
j, and ˜q,g ame.
7. Transform logit(˜p(q, ˜s,˜
j) into a probability to obtain a simulated estimate, ˜pq,sim, where
˜pq,sim = ˜p(q,˜s,˜
8. Repeat the above steps nsim times to obtain ˜pq={˜pq,1,...,˜pq,nsim }.
For each q, we simulated with nsim = 1000. Additionally, to remove the effect of each league’s HA
on simulated probabilities, we repeated the process fixing ˜αq0= 0 for each league to reflect game
probabilities played at neutral sites.
Figure 7 shows the cumulative distribution functions (CDFs) for each set of probabilities in each
league. The median probability of the best team winning a neutral site game is highest in the NBA
(67%), followed in order by the NFL (65%), NHL (57%), and MLB (56%). The spread of these
probabilities are of great interest. Nearly every simulated MLB and NHL game played at neutral
site is less than a 3:1 proposition with respect to the best team winning (75%). Meanwhile, roughly
28% of NBA and 20% of NFL neutral site match-ups are greater than this 3:1 threshold.
Factoring in each league’s home advantage works to exaggerate league-level differences. When the
best team plays at home in the NBA, it is always favored to win at least 60% of the time, with the
middle 50% of games ranging from a 68% probability to an 84% probability. Meanwhile, even with
a home advantage, it is rare that the best MLB team is ever given a 70% probability of winning,
with the middle 50% of games ranging from 57% to 63%.
All games
coin flips
All games
0.5 0.6 0.7 0.8 0.9 1.0
Simulated win probability
Solid: neutral site, Dashed: home game for better team
How often does the best team win?
Figure 7: Cumulative distribution function (CDF) of 1000 simulated game-level probabilities in
each league, for both neutral site and home games, with the better team (on average) used as the
reference and given the home advantage.
Finally, we use the CDFs displayed in Figure 7 to quantify the cumulative difference between each
league’s game-level probabilities and a league of coin flips by estimating the approximate area under
each curve. Let P ar ityqbe our parity measure, such that
P arityq= 2 Z1
P(˜pqx)dx ,
where we multiply by 2 in order to scale so that 0 P arityq1, where 1 represents complete
parity (every game a coin flip) and 0 represents no parity (every game outcome pre-determined).
For games with no home advantage, P arityMLB = 0.86, followed by the NHL (0.83), NFL (0.69),
and NBA (0.66). When the best team has a home advantage, parity is again the greatest in the
MLB (0.79), followed by the NHL (0.73), NFL (0.54), and NBA (0.46). These results suggest that
when the best team is playing at home, the NBA is closer to a world where every game outcome is
predetermined than to one where every game outcome is a coin flip. Meanwhile, even when giving
the best team a HA, MLB game outcomes remain lightly-weighted coin flips.
5 Conclusion
5.1 Summary
Using a modified Bayesian state-space model, we estimate both time-varying team strength and
league-level variance parameters in order to better understand the underlying randomness in the
four major North American professional sporting leagues, the NBA, NFL, NHL, and MLB.
Our first finding relates to the relative equivalence of the four leagues. At a single point in time,
team strength estimates diverge substantially more in the NBA and NFL than in the NHL and MLB.
In the latter two leagues, contests between two randomly chosen teams are closer to a coin-flip, in
which each team has a reasonable shot at winning. Understanding this underlying randomness would
appear to be crucial for decision makers in these leagues. At critical moments in a team’s evolution,
such as the a trade deadline, free agency period, or the decision to fire a coach, we recommend
that team officials look past wins and losses to better understand team strength in the context of
their league. As one easy example, it is insufficient to evaluate a baseball or hockey team based on
their performance in the postseason alone, given that so many of those contests are nearly 50-50
Our second set of findings relates to the autoregressive nature of team strengths. Within a season,
posterior estimates suggest that teams in each of the NBA (largest reversion), NFL, and NHL tend
to revert towards the league average in the long term on a week-to-week basis, while trends of team
strength in MLB are indistinguishable from a random walk. On a season-to-season basis, NHL teams
exhibit the largest reversion (nearly 50%) towards the league average, with the other three leagues
falling somewhere between roughly 25% and 40%.
Our next finding relates to the relative equivalence of the home advantage in each league, with the
NBA well ahead of the pack, with teams averaging a 62.0% chance of winning versus a like-caliber
opponent. We also show that the home advantage varies most significantly between venues within
each of the NBA and the NHL. In the NBA, for example, the league’s best team home advantage is
worth a few wins per year, in expectation, over the league’s worst home advantage. Moreover, with
the exception of the Colorado Rockies, it is not clear that any MLB or NFL team has a statistically
significant home effect.
Finally, we identify that incorporating information from betting markets can help to more accurately
gauge the caliber of each league’s teams, as shown by an improved ability to predict future team
performance. Unlike wins and losses or point differential, our estimates of team strength account for
league characteristics such as unbalanced schedules and season length. We conclude by using these
team strength draws to propose a parity metric that can compare team equivalence without being
affected by league-level characteristics like unbalanced schedules.
5.2 Future work
Opportunities to extend our model are plentiful. One approach would use our team strength es-
timates to examine how each league’s scheduling quirks impact resulting won-loss standings. For
example, what is the impact of the unbalanced schedule used in the NFL? A second question con-
cerns the relationship of our estimates of team strength to performance in the postseason. How likely
is it for the best team to win each league’s title? Conversely, how likely is it that the team that
won the postseason tournament was actually the strongest team at the end of the regular season?
Finally, one could use time-varying estimates of team strength to consider the existence of tanking,
in which teams—in order to secure a better draft position—are better off losing games later in the
season. While this has been demonstrated in basketball using betting market data (Soebbing and
Humphreys, 2013), it would also be worth looking at tanking in other leagues, or if team interest in
tanking corresponds to the perceived talent available in the upcoming draft.
To maintain consistency with the NFL’s calendar, we considered time on a weekly basis. More refined
approaches may be appropriate in other sports. As an example, investigation into starting pitchers in
baseball—who change daily—could lead to novel findings. Additionally, another model specification
could consider the possibility that time-varying estimates of team strength follow something other
than an autoregressive structure. One alternative specification, for example, is a stochastic volatility
process (Glickman, 2001). In this respect, our model can be considered a starting point for those
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Supplementary Materials for
“A unified approach to understanding randomness in sport”
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0 1000 2000 3000 4000
Chain index
Figure 8: Trace plots of MLB parameters
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0 1000 2000 3000 4000
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Figure 9: Trace plots of NBA parameters
0 1000 2000 3000 4000 0 1000 2000 3000 4000 0 1000 2000 3000 4000
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Figure 10: Trace plots of NFL parameters
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Figure 11: Trace plots of NHL parameters
AL West
NL West
AL East
NL East
AL Central
NL Central
Team Strength (log−odds scale)
Team strength parameters over time, MLB
Figure 12: Team strength coefficients over time for Major League Baseball.
Weakest team (Miami Heat)
Team Strength (log−odds scale)
Team strength parameters over time, NBA
Figure 13: Team strength coefficients over time for the National Basketball Association.
Strongest team (New England Patriots)
AFC West
NFC West
AFC South
NFC South
AFC North
NFC North
AFC East
NFC East
Team Strength (log−odds scale)
Team strength parameters over time, NFL
Figure 14: Team strength coefficients over time for the National Football League.
Team Strength (log−odds scale)
Team strength parameters over time, NHL
Figure 15: Team strength coefficients over time for the National Hockey League.
... There are many methods to measure these variations, and no single method should be regarded as most appropriate [8]. The withinseason, competitive balance has been evaluated for a number of individual or team sports, including baseball, basketball, football, ice hockey, soccer, speed skating, table tennis, and tennis using the Herfindahl-Hirschman Index (HHI), Competitive Balance Ratio (CBR), the ratio of standard deviations (RSD), Gini coefficient and related Lorenz curve, relative entropy, and other, specific-sport-based criteria, such as points, scores, or time [10][11][12][13][14][15][16][17][18][19][20][21][22]. Between-seasons variation can be evaluated for an individual team using the team-specific variation in standing in different seasons (turnover) [11], or by measuring the concentration of championships in a given time period [8]. ...
... Szymanski [19] evaluated the number of teams accounting for top positions in 1977 -1998 and reported relatively stable competitive balance. Other authors, however, described decline in competitive balance when analyzing the period of 1888 -2007 [20], 1947 -2004 [21], 1948 -2008 [22], 1963 -2005 [23] and 1992 -2010 [24]. Most of the authors noted that the significant decline in competitive balance started between 1987 to mid-1990s. ...
... The value of -0.25 indicated a similar, though nonsignificant trend. Though it was not an objective of the current study to identify factors leading to these changes in competitive balance, previous authors suggested that the change may be related to the back-pass rule [25], the Bosman ruling [20], and/or the increased inequalities in resources between clubs [26]. ...
The concept of competitive balance is considered to be an essential aspect in the field of sport economics. This work describes a novel approach for measuring and evaluating competitive balance through concentration of championships. The concentration of championships was assessed using a sliding window approach with the length of five consecutive competitions (years) and a single competition (year) increment over the whole evaluated period. Because the sliding window was relatively short, the newly developed index, termed ‘We Are the Champions’ (WAC5), is sensitive to rapid changes in competitive balance. The WAC5 index, average WAC5, expected WAC5, and ΔWAC5 were applied to data from 68 competitions of several individual and team sports collected for the 1960 – 2020 period. The significance of indices was tested by resampling (bootstrapping and permutation). The results of the study show a growth in competitive balance of several ice hockey competitions (national and international), but decreasing competitive balance in Formula 1 racing and in several European soccer competitions. In soccer competitions, there was a substantially lower competitive balance in a league than in a domestic cup competition within each country/federation. The difference between the overall competitive balance in the most popular, North American, professional leagues and the top European soccer leagues is growing. A significant grouping of champions was determined for all sports involving individual athletes, but also for several team competitions.
... When the performance of a sports team or sports person deviates from their expected performance, the deviation is attributed to luck (Mauboussin, 2012;Sobkowicz et al., 2020), which is often defined as the randomness of performance (Ben-Naim et al., 2013;Gauriot & Page, 2019) and is difficult to predict (Aoki et al., 2017;Weissbock, 2014). The underlying idea is that performance is a function of skill and luck (Bailey et al., 2020;Connolly & Rendleman Jr, 2008;Levitt & Miles, 2014;Lopez et al., 2018;Yang et al., 2014). ...
... Arguably, these five leagues are not equally competitive, and consequently, randomness of outcome may vary across leagues. Randomness increases in leagues that are competitively more balanced (Lopez et al., 2018). Conducting separate analysis for top of the table and for the bottom of the table reduces the impact of such variability among the leagues. ...
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It is common wisdom that luck plays a role in sports, along with skill. However, there is no consensus among researchers on what constitutes luck. One strand of the literature studied randomness in sports, most of which did the analysis at the levels of pitch actions, or at the match level. There is no empirical study to assess the role of luck in the determination of rank positions in football (soccer) leagues. In this paper, we define X-factor as unforeseen and unaccounted factors and quantify it as the difference between actual and predicted values of performance or outcome variables. For league football, we have perceived the difference between actual and expected goal difference as the X-factor effect in performance, and the difference between actual and expected points as the X-factor effect in outcome. Further, we have ideated that a plausible role of luck cannot be ruled out if the X-factor effect on outcome is significant while that on performance is not. Conducting analyses of variance on observations from seven seasons (2014–15 to 2020–21) in the top tier leagues of England, Spain, Germany, Italy, and France, we detected the presence of a significant and systematic X-factor effect. We have studied the role of luck using Tukey’s HSD test. In general, luck does not play any significant role in determining the rank positions in league football.
... While various dynamic models that use score or score-related information have been proposed for head-to-head games (Harville, 1977;Glickman and Stern, 1998;Lopez, Matthews and Baumer, 2018;Ingram, 2019;Kovalchik, 2020), we are unaware of similar work for multi-competitor games. In this paper, we extend the normal dynamic linear model (DLM) proposed by Harville (1977) and Glickman and Stern (1998) to rate athletes who compete in multi-competitor games. ...
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Sports organizations often want to estimate athlete strengths. For games with scored outcomes, a common approach is to assume observed game scores follow a normal distribution conditional on athletes' latent abilities, which may change over time. In many games, however, this assumption of conditional normality does not hold. To estimate athletes' time-varying latent abilities using non-normal game score data, we propose a Bayesian dynamic linear model with flexible monotone response transformations. Our model learns nonlinear monotone transformations to address non-normality in athlete scores and can be easily fit using standard regression and optimization routines. We demonstrate our method on data from several Olympic sports, including biathlon, diving, rugby, and fencing.
... The basketball technical and tactical analysis system is another ball analysis system of the Software Architecture Laboratory after the research of volleyball and table tennis game information collection and analysis system. The reason the Software Architecture Lab is dedicated to the development of ball information systems is that the information technology in the sports industry has not yet reached a level that matches the degree of information technology [18]. Although the number of game data in the database system is increasing, the technology for analyzing games still uses traditional statistical methods, and most coaches adjust their training and game strategies based on experience and statistics, without any substantial change in technology. ...
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The Apriori algorithm is used to conduct an in-depth analysis and research on the relationship between data mining and penalty decision of multiattribute data in the basketball game scene. The technical and tactical features are analyzed using an improved Apriori algorithm for association rule analysis of basketball game data. The algorithm generates association rules based on mining the set of frequent items among basketball technical actions. The improved algorithm can mine the technical moves that are more connected in the game data, and the analysis results are highly instructive. The technical and tactical directed analysis is divided into two parts: technical and tactical directed action analysis and technical and tactical directed cooperation analysis. The key action analysis uses Markov process-based data mining algorithm to analyze the basketball game data for key score transfer steps and key score loss transfer steps. The algorithm can find the key actions of scoring and key actions of conceding points in the game process, and the analysis results can guide basketball training and games, which has high practical value. Using the collated game data as the independent variable and the number of games won and lost as the dependent variable, logistic regression analysis is applied to derive the characteristics that affect winning. Again, the decision tree algorithm is used to select the significant features that affect winning and to make predictions of team performance. Finally, the technical statistics of the main players in the last three seasons are selected, and the association rule algorithm is applied to derive the degree of influence of player performance on the outcome of the game.
... This dynamic setting has received significantly less attention than its static counterpart and has mostly been studied with a focus on applications, such as sports tournaments [5,11,21]. It has rarely been analysed theoretically in the past however, although some results exist for a state-space generalization of the BTL model [9,10,19] and for a Bayesian framework [10,16]. ...
Many applications such as recommendation systems or sports tournaments involve pairwise comparisons within a collection of n items, the goal being to aggregate the binary outcomes of the comparisons in order to recover the latent strength and/or global ranking of the items. In recent years, this problem has received significant interest from a theoretical perspective with a number of methods being proposed, along with associated statistical guarantees under the assumption of a suitable generative model. While these results typically collect the pairwise comparisons as one comparison graph G, however in many applications-such as the outcomes of soccer matches during a tournament-the nature of pairwise outcomes can evolve with time. Theoretical results for such a dynamic setting are relatively limited compared to the aforementioned static setting. We study in this paper an extension of the classic BTL (Bradley-Terry-Luce) model for the static setting to our dynamic setup under the assumption that the probabilities of the pairwise outcomes evolve smoothly over the time domain [0, 1]. Given a sequence of comparison graphs (G_t)_{t ∈T} on a regular grid T ⊂ [0, 1], we aim at recovering the latent strengths of the items w^*_t ∈ R^n at any time t ∈ [0, 1]. To this end, we adapt the Rank Centrality method-a popular spectral approach for ranking in the static case-by locally averaging the available data on a suitable neighborhood of t. When (G_t)_{t ∈T} is a sequence of Erdös-Renyi graphs, we provide non-asymptotic l_2 and l_∞ error bounds for estimating w^*_t which in particular establishes the consistency of this method in terms of n, and the grid size |T |. We also complement our theoretical analysis with experiments on real and synthetic data.
... In this way, the implied probability and corresponding uncertainty of parameter estimates are still rigorously defined while being directly measurable and more intuitive to understand than traditional Frequentist methods of confidence intervals and p-values. Third, with advancements in computational Bayesian statistics, such as Probabilistic Programming languages 25 and Hamiltonian Monte Carlo (HMC) 26 , we are able to easily define and compute flexible and complex models using various likelihood functions with ease instead of being limited to traditional methods like Normal and Poisson regressions more traditionally used in sports modelling 9,10,17,18,27 . ...
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Home advantage in professional sports is a widely accepted phenomenon despite the lack of any controlled experiments at the professional level. The return to play of professional sports during the COVID-19 pandemic presents a unique opportunity to analyze the hypothesized effect of home advantage in neutral settings. While recent work has examined the effect of COVID-19 restrictions on home advantage in European football, comparatively few studies have examined the effect of restrictions in the North American professional sports leagues. In this work, we infer the effect of and changes in home advantage prior to and during COVID-19 in the professional North American leagues for hockey, basketball, baseball, and American football. We propose a Bayesian multi-level regression model that infers the effect of home advantage while accounting for relative team strengths. We also demonstrate that the Negative Binomial distribution is the most appropriate likelihood to use in modelling North American sports leagues as they are prone to overdispersion in their points scored. Our model gives strong evidence that home advantage was negatively impacted in the NHL and NBA during their strongly restricted COVID-19 playoffs, while the MLB and NFL showed little to no change during their weakly restricted COVID-19 seasons.
... ) seems quite reasonable, since it is expected a that team with a higher point scores more goals rather than lower point team. Estimating the relative strength of teams is an old and large literature, one arm of which is the Bayesian state-space literature (see [5] as a canonical example, and [7] for a recent applied example). Equivalently, we can assume the waiting time to see team A scores r 1 goals to team B and team B scores r 2 goals to team A independently follow ...
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This paper proposes a Bayesian predictive density estimator of time to goal in a hockey game, using ancillary information such as performance in the past, points, and specialists’ opinions about teams. To be more specific, we model time to r-th goal as a gamma distribution. The proposed Bayesian predictive density estimator using the ancillary information belongs to an interesting new version of a weighted beta prime distribution and it outperforms the other estimators in the literature such as the one that does not incorporate this information as well as the plug-in estimator. The efficiency of our estimator is evaluated using frequentist risk along with measuring the prediction error from the old dataset, 2016–2017, to the season 2018–2019 of the National Hockey League.
In wake of the Covid-19 pandemic, 2019–2020 soccer seasons across the world were postponed and eventually made up during the summer months of 2020. Researchers from a variety of disciplines jumped at the opportunity to compare the rescheduled games, played in front of empty stadia, to previous games, played in front of fans. To date, most of this post-Covid soccer research has used linear regression models, or versions thereof, to estimate potential changes to the home advantage. However, we argue that leveraging the Poisson distribution would be more appropriate and use simulations to show that bivariate Poisson regression (Karlis and Ntzoufras in J R Stat Soc Ser D Stat 52(3):381–393, 2003) reduces absolute bias when estimating the home advantage benefit in a single season of soccer games, relative to linear regression, by almost 85%. Next, with data from 17 professional soccer leagues, we extend bivariate Poisson models estimate the change in home advantage due to games being played without fans. In contrast to current research that suggests a drop in the home advantage, our findings are mixed; in some leagues, evidence points to a decrease, while in others, the home advantage may have risen. Altogether, this suggests a more complex causal mechanism for the impact of fans on sporting events.
In wake of the Covid-19 pandemic, 2019-2020 soccer seasons across the world were postponed and eventually made up during the summer months of 2020. Researchers from a variety of disciplines jumped at the opportunity to compare the rescheduled games, played in front of empty stadia, to previous games, played in front of fans. To date, most of this post-Covid soccer research has used linear regression models, or versions thereof, to estimate potential changes to the home advantage. But because soccer outcomes are non-linear, we argue that leveraging the Poisson distribution would be more appropriate. We begin by using simulations to show that bivariate Poisson regression reduces absolute bias when estimating the home advantage benefit in a single season of soccer games, relative to linear regression, by almost 85 percent. Next, with data from 17 professional soccer leagues, we extend bivariate Poisson models estimate the change in home advantage due to games being played without fans. In contrast to current research that overwhelmingly suggests a drop in the home advantage, our findings are mixed; in some leagues, evidence points to a decrease, while in others, the home advantage may have risen. Altogether, this suggests a more complex causal mechanism for the impact of fans on sporting events.
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The betting market for the Women’s National Basketball Association (WNBA) is a thin financial market, which does not attract much interest from sports bettors. Given these characteristics, it is possible that profitable wagering strategies could exist for informed bettors of the WNBA. Using betting data on the WNBA from 2007–2012, we find that simple betting strategies do not earn statistically significant returns. WNBA bettors are like NBA bettors; however, in that they strongly prefer the best teams, particularly when they are on the road. Despite this clear bias, betting against the most popular public wagers is not found to earn statistically significant profits.
League-winning percentage Gini coefficients have seen recent use as measurements of within-season competitive balance in Major League Baseball. The authors demonstrate that the zero-sum nature of league play renders past estimates inappropriate. Adjusted for league play, Gini coefficients reveal a much larger competitive balance problem than shown in previous estimates. However, additional complexities involving unbalanced schedules, interdivisional play, and now interleague play must be overcome before winning percentage Gini coefficients can give precise estimates of competitive balance. The authors suggest using the traditional measures of winning percentage standard deviations and their idealized values to analyze within-season competitive balance over time until these issues are overcome.
Roughly 14% of regular season National Hockey League games since the 2005–06 season have been decided by a shoot-out, and the resulting allocation of points has impacted play-off races each season. But despite interest from fans, players and league officials, there is little in the way of published research on team or individual shoot-out performance. This manuscript attempts to fill that void. We present both generalised linear mixed model and Bayesian hierarchical model frameworks to model shoot-out outcomes, with results suggesting that there are (i) small but statistically significant talent gaps between shooters, (ii) marginal differences in performance among netminders and (iii) few, if any, predictors of player success after accounting for individual talent. We also provide a resampling strategy to highlight a selection bias with respect to shooter assignment, in which coaches choose their most skilled offensive players early in shoot-out rounds and are less likely to select players with poor past performances. Finally, given that per-shot data for shoot-outs do not currently exist in a single location for public use, we provide both our data and source code for other researchers interested in studying shoot-out outcomes.
This paper treats the problem of modeling and forecasting the outcomes of NBA basketball games. First, it is shown how the benchmark model in the literature can be extended to allow for heteroscedasticity and estimation and testing in this framework is treated. Second, time-variation is introduced into the model by introducing a dynamic state space model for team strengths. The in-sample results based on eight seasons of NBA data provide weak evidence for heteroscedasticity, which can lead to notable differences in estimated win probabilities. However, persistent time variation is only found when combining the data of several seasons, but not when looking at individual seasons. The models are used for forecasting a large number of regular season and playoff games and the common finding in the literature that it is difficult to outperform the betting market is confirmed. Nevertheless, a forecast combination of model based forecasts with betting odds can lead to some slight improvements.
This article describes a method for predicting the outcome of National Hockey League (NHL) games. We combine a model for goal scoring and yielding, and one for penalty commission, in a Markov-type computation and a simulation model that produce predicted probabilities of victory for each team. Where these differ substantially from the market probabilities, we make “bets” according to a simple strategy. Our return on investment is both positive and statistically significant.
The relationship between attendance at major league baseball games and the uncertainty of the outcome of each game is examined. We use an a priori measure of uncertainty in estimating the attendance equation. The variable is developed from the betting lines for individual games and measures the probability of a home team victory during the 1988 major league baseball season. The results indicate that uncertainty of outcome is a significant determinant of attendance for major league baseball. In addition, the results are used to determine the probability of a home team victory at which attendance will be maximized.
The predictive performance of various team metrics is compared in the context of 105 best-of-seven national hockey league (NHL) playoff series that took place between 2008 and 2014 inclusively. This analysis provides renewed support for traditional box score statistics such as goal differential, especially in the form of Pythagorean expectations. A parsimonious relevance vector machine (RVM) learning approach is compared with the more common support vector machine (SVM) algorithm. Despite the potential of the RVM approach, the SVM algorithm proved to be superior in the context of hockey playoffs. The probabilistic SVM results are used to derive playoff performance expectations for NHL teams and identify playoff under-achievers and over-achievers. The results suggest that the Arizona Coyotes and the Carolina Hurricanes can both be considered Round 2 over-achievers while the Nashville Predators would be Round 2 under-achievers, even after accounting for several observable team performance metrics and playoff predictors. The Vancouver Canucks came the closest to qualify as Stanley Cup Finals under-achievers after they lost against the Boston Bruins in 2011. Overall, the results tend to support the idea that the NHL fields extremely competitive playoff teams, that chance or other intangible factors play a significant role in NHL playoff outcomes and that playoff upsets will continue to occur regularly.
This article develops a predictive model for National Football League (NFL) game scores using data from the period 1988-1993. The parameters of primary interest - measures of team strength - are expected to vary over time. Our model accounts for this source of variability by modeling football outcomes using a state-space model that assumes team strength parameters follow a first-order autoregressive process. Two sources of variation in team strengths are addressed in our model; week-to-week changes in team strength due to injuries and other random factors, and season-to-season changes resulting from changes in personnel and other longer-term factors. Our model also incorporates a home-field advantage while allowing for the possibility that the magnitude of the advantage may vary across teams. The aim of the analysis is to obtain plausible inferences concerning team strengths and other model parameters, and to predict future game outcomes. Iterative simulation is used to obtain samples from the joint posterior distribution of all model parameters. Our model appears to outperform the Las Vegas "betting line" on a small test set consisting of the last 110 games of the 1993 NFL season.