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Competitive balance with unbalanced schedules

  • Young Hoon Lee , Yongdai Kim EMAIL logo and Sara Kim
Published/Copyright: May 31, 2019
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Journal of Quantitative Analysis in Sports
From the journal Journal of Quantitative Analysis in Sports Volume 15 Issue 3

Abstract

Many empirical studies on competitive balance (CB) use the ratio of the actual standard deviation to the idealized standard deviation of win percentages (RSD). This paper suggests that empirical studies that use RSD to compare CB among different leagues are invalid, but that RSD may be used for time-series analysis on CB in a league if there are no changes in season length. When schedules are unbalanced and/or include interleague games, the final winning percentage is a biased estimator of the true win probability. This paper takes a mathematical statistical approach to derive an unbiased estimator of within-season CB that can be applied to not only balanced but also unbalanced schedules. Simulations and empirical applications are also presented, which confirm that the debiasing strategy to obtain the unbiased estimator of within-season CB is still effective for unbalanced schedules.

Keywords: competitive balance; unbalanced schedule; unbiased estimation

Acknowledgment

This work was partly supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2016S1A2A2912186) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2017R1A2B2006102).

Appendix

A Derivation of bias of ASD in Eq. (7)

Let Xijk be the indicator random variable that represents whether team i wins against team j in the kth match for i,j=1,…,N,i ≠ j and k=1,…,Kij, where N is the number of teams in a league and Kij is the number of games in which team i plays against team j. We denote Gi=∑j≠iKij the number of games played by team i.

We assume that Xijk∼Bernoulli(pij), where pij is the true probability that team i wins against team j. Then, Yij=∑k=1KijXijk∼B(Kij,pij). Denote pi as the true win probability of team i defined as

pi=∑j≠ipijN−1 for i=1,…,N.

We define the competitive balance as the variance of {pi,i=1,…,N}:

σ2=1N∑i=1N(pi−0.5)2,

and we estimate σ2 by using the plug-in estimator:

σ^2=1N∑i=1N(p^i−0.5)2,

where

p^i=∑j≠iYijGi=∑j≠iKijp^ijGi,p^ij=YijKij.

For given i, Yij,j≠i are independent, we can easily find E(p^i) and E(p^i2) to have

Bias(σ^2)=E(σ^2)−σ2=1N∑i=1NE(p^i−0.5)2−1N∑i=1N(pi−0.5)2=1N∑i=1N(E(p^i2)−E(p^i)−pi2+pi)=1N∑i=1N[1Gi2(∑j≠iKijpij(Kijpij+1−pij)+2∑j,k≠ik>jKijpijKikpik)−1Gi∑j≠iKijpij−pi2+pi]=1N∑i=1N[1Gi2(∑j≠iKijpij(Kijpij+1−pij)+2∑j,k≠ik>jKijpijKikpik)−1(N−1)2(∑j≠ipij2+2∑j,k≠ik>jpijpik)−(1Gi∑j≠iKijpij−1N−1∑j≠ipij)].

B Consistency of p^j and p~j

First consider p~i. Note that E(p^ij)=pij for all j≠i. Moreover, p^ij,j≠i are independent for a given i. Thus, Hoeffding’s inequality implies

Pr(|p~i−pi|≥ϵ)≤2exp(−2ϵ2∑j≠i1/(N−1)2)=2exp(−2ϵ2(N−1)).

Therefore, we have

Pr(maxi|p~i−pi|≥ϵ)≤∑i=1NPr(|p~i−pi|≥ϵ)≤2Nexp(−2ϵ2(N−1))→0.

A similar technique can be applied to show that

Pr(maxi|p^i−E(p^i)|≥ϵ)→0

as N→∞.

C Simulation for temporal variation of winning probabilities

Let pij(t) be the winning probability of team i over team j at time t in a given season. We define CB(t), the CB at time t, as CB(t)=∑i=1N(pi(t)−0.5)2/N, where pi(t)=∑j≠ipij(t)/(N−1). The natural definition of CB for the season would be CBT=∑t=1TCB(t)/T, the average of the temporal CBs. In this section, we compare CB and when CBT when pij(t) are not constant by simulation, where CB is calculated with pi=∑t=1Tpi(t)/T for i = 1, …, N.

Table 8:

Comparison of CB and CBT for various values of σCB and σnoise.

σCBσnoiseCBCBTabs(diff)
0.000.010.00050.00250.0021
0.000.050.00180.01250.0107
0.000.100.00510.02540.0203
0.000.150.00500.03740.0324
0.000.200.00470.05020.0456
0.050.010.05000.05000.0001
0.050.050.05060.05190.0013
0.050.100.05030.05580.0055
0.050.150.04910.06080.0117
0.050.200.04660.06700.0204
0.150.010.15000.15000.0000
0.150.050.14930.14970.0004
0.150.100.15040.15180.0014
0.150.150.14880.15190.0031
0.150.200.14760.15290.0053
0.250.010.25000.25000.0000
0.250.050.24980.24990.0001
0.250.100.24940.24970.0003
0.250.150.24930.25000.0008
0.250.200.24860.24980.0012

Let Si(t)=Si+ϵi(t), where ϵi(t),t=1,…,T are independent Gaussian random variables with mean 0 and variance σnoise2. Then, we define pij(t)=[1+exp{Sj(t)−Si(t)}]−1. For various values of {Si,i=1,…,N}, we compare the CB and CBT by simulation, whose results are summarized in Table 8. In the table, σCB is the CB obtained based on pij=exp(Si)/(exp(Si)+exp(Sj)). Note that CBT is always larger than CB, which is reasonable since temporal variation increases the value of CB at each time point. However, the difference of CB and CBT disappears fast as the baseline CB σCB increases.

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Published Online: 2019-05-31
Published in Print: 2019-08-27

©2019 Walter de Gruyter GmbH, Berlin/Boston

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  3. Relative age effects in American professional football
  4. Optimal shot selection strategies for the NBA
  5. The relative wages of offense and defense in the NBA: a setting for win-maximization arbitrage?
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  7. Competitive balance with unbalanced schedules
https://doi.org/10.1515/jqas-2017-0100
Keywords for this article
competitive balance; unbalanced schedule; unbiased estimation

Articles in the same Issue

  1. Frontmatter
  2. nflWAR: a reproducible method for offensive player evaluation in football
  3. Relative age effects in American professional football
  4. Optimal shot selection strategies for the NBA
  5. The relative wages of offense and defense in the NBA: a setting for win-maximization arbitrage?
  6. Ludometrics: luck, and how to measure it
  7. Competitive balance with unbalanced schedules
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