The 8th Australasian Conference on Mathematics and Computers in
Sport, 3-5 July 2006, Queensland, Australia
STOCHASTIC DOMINANCE AND ANALYSIS OF ODI BATTING PERFORMANCE:
THE INDIAN CRICKET TEAM, 1989-2005
XLRI Jamshedpur School of Management, Jamshedpur, Jharkand, India.
Journal of Sports Science and Medicine (2006) 5, 503 - 508
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|Relative to other team games, the contribution of individual team
members to the overall team performance is more easily quantifiable
in cricket. Viewing players as securities and the team as a portfolio,
cricket thus lends itself better to the use of analytical methods
usually employed in the analysis of securities and portfolios. This
paper demonstrates the use of stochastic dominance rules, normally
used in investment management, to analyze the One Day International
(ODI) batting performance of Indian cricketers. The data used span
the years 1989 to 2005. In dealing with cricketing data the existence
of 'not out' scores poses a problem while processing the data. In
this paper, using a Bayesian approach, the 'not-out' scores are first
replaced with a conditional average. The conditional average that
is used represents an estimate of the score that the player would
have gone on to score, if the 'not out' innings had been completed.
The data thus treated are then used in the stochastic dominance analysis.
To use stochastic dominance rules we need to characterize the 'utility'
of a batsman. The first derivative of the utility function, with respect
to runs scored, of an ODI batsman can safely be assumed to be positive
(more runs scored are preferred to less). However, the second derivative
needs not be negative (no diminishing marginal utility for runs scored).
This means that we cannot clearly specify whether the value attached
to an additional run scored is lesser at higher levels of scores.
Because of this, only first-order stochastic dominance is used to
analyze the performance of the players under consideration. While
this has its limitation (specifically, we cannot arrive at a complete
utility value for each batsman), the approach does well in describing
player performance. Moreover, the results have intuitive appeal.
WORDS: Bayesian, utility function, batting average, conditional
average, geometric distribution.
a game, cricket is a statistician's delight. Each game of cricket
throws up a huge amount of performance related statistics. As other
games have evolved and developed, they too have become richer in
the use of performance statistics. For example, use of statistics
like 'unforced errors' in lawn tennis or 'assists' in basketball
is increasingly becoming popular. However in cricket these statistics
have always been part and parcel of the game. Cricket is one of
the few games in which a 'scorer' is required to continuously maintain
statistical data on key game/player-specific performance statistics.
It is one of the few games that have detailed 'scoring sheets'.
These scoring sheets were maintained manually in the pre-digital
age and are maintained electronically today.
In spite of this legacy and long history of maintaining statistical
data, two aspects associated with cricketing data are striking.
The first is the idiosyncrasy that has persisted in the treatment
of the 'not out' scores of a player. The second is the lack of effort
in exploiting the richness of data to improve the representation
of player performance.
The batting average of player i, Ri, is computed as:
Rit is the number of runs scored by the i th player in the t th
innings; n is the total number of innings in which the i th player
has batted and k is the number of innings in which the i th player
has remained 'not out'.
(1) introduces an upward bias in the average. This bias is caused
because the numerator is the total runs scored over all innings
while the denominator excludes the innings in which the player has
remained 'not out'. This bias cannot seemingly be avoided. Taking
the denominator to be n instead of n-k would instead introduce a
downward bias in the average. A similar problem arises while preparing
the input data required for the stochastic dominance rules developed
later in the paper. The input data that is required is the innings-by-innings
runs scored by the player. What should be done with the scores for
the innings in which the player has remained 'not out'? This paper
first proposes a method to deal with this problem.
The second aspect of cricketing data is the scant attention that
has been focused by researchers on certain aspects of cricket. A
substantial portion of the work has focused on devising optimal
playing strategies. The strategies studied have either focused on
batting strategies (Clarke, 1988; Clarke and Norman, 1999; Preston
and Thomas, 2000;
Swartz et al., 2006)
or on bowling strategies (Rajadhyaksha and Arapostathis, 2004).
A fair amount of work has also focused on the problem of arriving
at a fair result when a game has to be prematurely terminated due
to weather conditions or other disturbances (Duckworth and Lewis,
Preston and Thomas, 2002;
Carter and Guthrie, 2004).
The third stream of work, on the understanding and development of
player-specific performance statistics, (Kimber and Hansford, 1993;
has received little attention. Cricket, with its slow pace and non-continuous
nature, is a very television-friendly game. It allows viewers the
leisure of watching replays without impinging on real-time action.
It thus allows for the presentation of a vast amount of descriptive
statistics during the course of a game. In spite of this feature
of the game and the long history of the game, cricket commentators
sometimes seem to feel constrained by the inability of performance
statistics to really describe player performance. Comments like
"Statistics don't say everything" are very commonly heard.
The attention devoted by researchers to this aspect of cricket,
therefore, seems surprisingly scant in relation to its importance
This paper seeks to develop methods to assess the performance of
batsmen in cricket that (i) makes use of more information than current
methods do and (ii) can be converted into visually appealing graphics
for the television medium. The method is demonstrated using player
statistics for the some of the key members of the Indian One Day
International (ODI) cricket team between 1989 and 2005. The names
of the players included in the study are listed in Table
primary measure of a batsman's performance in cricket today is the
player's batting average defined as in Equation (1). This measure
suffers from the shortcoming that it is a one-dimensional number
and does not capture the richness of the underlying data. Though
cited very often, this measure fails to capture the various facets
of a batsman. It does not provide answers to many questions that
arise during the course of a game. These un-addressed concerns or
questions feature often in the comments of cricket commentators.
For example, commentators of the game are found to say "Player
X is a dangerous player once he is set". Or "Player X
has the ability to convert a good start to a big score". Or
"Though his average does not reflect it, Player X is a more
consistent performer than Player Y".
the raw data
The raw data used in the development of any method for representing
a batsman's performance are the innings-by-innings runs scored by
the player. However, using this raw data poses a problem. In some
of the innings the batsman would not have been dismissed. In such
cases the score would not reflect the number of runs the player
could potentially have gone on to score. The scores for these innings
(the 'not out' situations) have thus to be replaced by a number
that is a good estimate of the number of runs the player would have
scored had he batted on.
In an early work Wood, 1945
had provided empirical support to support the claim that a batsman's
scores follow a geometric distribution. Under this assumption, because
of the memoryless property of the geometric distribution, a batsman's
chance of getting out is independent of the score he is on. This
assumption can be used to arrive at an estimate of the number of
runs a 'not- out' player would have scored had he batted on. However,
the assumption of a geometric distribution for a batsman's scores
might not hold for all players. There may be some players who are
'slow starters' and who therefore do better as they progress. There
may be other players who become more adventurous as their score
increases. For such adventurous players their chances of getting
out might increase as their score increases.
Kimber and Hansford, 1993
did consider deviations from the geometric distribution, but their
focus was on arriving at an optimal estimator for the population
mean. On the other hand, we need a method to arrive at an estimate
of the number of runs a 'not out' batsman would have gone on to
score. A Bayesian approach has been adopted in this paper to arrive
at this estimate. This is achieved in the following manner.
Assume that in his j th innings player i remains 'not out'
on a score of Rij. Define a binary variable Grik
= 0 if Rik< Rij and = 1 if Rik>=Rij
for k = 1, 2,….j-1
Cik = 0 if Rik< Rij and
= Rik if Rik>=Rij for
k = 1, 2,….j-1
estimate of the number of runs that the 'not-out' batsman would
have gone on to score is then given by:
other words, the estimator used for the runs that the 'not out'
batsman would have gone on to score is the conditional average of
the batsman at that point of time, given that he has already scored
a certain number of runs. In every instance of a 'not out', the
batsman's score in that innings j is replaced by the estimate Eij.
This approach has the advantage of handling deviations from the
geometric distribution assumption. It is also information efficient,
with the posterior values of the conditional average incorporating
more information on the batsman's performance. Table
2 gives an example of the computational procedure used for finding
the replacement values for the first two 'not outs' in the career
of one member of the Indian ODI team, Sachin Tendulkar.
The adjusted raw data is now used to arrive at an analytical representation
of the player's batting performance. The approach adopted draws
from methods normally used for the analysis of securities and portfolios
in investment management.
The focus in investment management is on wealth creation. The problem
of portfolio choice is that of selecting a portfolio that maximizes
the utility for the investor. The utility function for the investor
attaches a utility to various levels of wealth. The utility function
can be constrained to have certain properties like non-satiation
(more wealth is always preferred to less) or risk aversion (diminishing
marginal utility for incremental units of wealth). In mathematical
terms the first constraint requires the first derivative of the
utility function to be positive. Again, in mathematical terms the
second constraint requires the second derivative of the utility
function to be negative.
Consistent with some of the above-listed features of utility functions,
the traditional approach to the portfolio selection problem has
been the mean-variance approach. Amongst the alternative approaches
to the portfolio selection problem suggested in the investment management
literature is the set of stochastic dominance rules (Ali, 1975;
To use stochastic dominance rules we need to characterize the utility
function of the investor. According to the first-order stochastic
dominance rules a portfolio A is preferred to another portfolio
B if, for any level of return, the cumulative probability of portfolio
A giving a return lesser than the given level of return is never
greater, and sometimes less, than the cumulative probability of
portfolio B giving a return lesser than that given level of return.
This rule is consistent with the assumption that in the investors'
utility function more wealth is preferred to less. (Elton and Gruber,
Analogous to the portfolio selection problem, a similar approach
is adopted in this paper to represent the batting performance of
cricketers. Using this approach we can say that a batsman A's performance
is better than another batsman B's if, for any level of score, the
probability of batsman A getting a score greater than the given
score is never lesser, and sometimes greater, than the probability
of batsman B getting a score greater than that given score. This
rule corresponds to the first-order stochastic dominance rules and
assumes that more runs are always preferred to less.
The cumulative probability charts of various batsmen can now be
charted with runs on the X-axis (with the origin as zero) and the
probability of scoring more runs than the X-axis value of the score
(that is one minus the cumulative probabilities of scoring lesser
than the X-axis value of score) on the Y-axis. Visually this would
mean that a batsman whose stochastic dominance curve envelops another's
curve stochastically dominates the other batsman.
method is demonstrated using data for the Indian ODI cricket team
spanning the years 1989 (the year one of India's most highly rated
players, Sachin Tendulkar, made his debut) to 2005. This period
was chosen because this was a period during which the compositional
changes in the Indian ODI team were very few. A sample batting performance
stochastic dominance chart output for five Indian players is given
in Figure 1.
Four of the five players represented are essentially specialist
batsmen (Tendulkar, Dravid, Sehwag and Laxman) and one a specialist
bowler (Khan). The results are interesting and have intuitive appeal.
They are consistent with popular notions regarding the batsmen whose
performances were studied. For example, the curve for Sachin Tendulkar,
who is considered an icon of Indian cricket, almost completely envelops
the curves for other players. And the curve for Rahul Dravid, who
is referred to as 'the wall' because of his perceived consistency,
does indeed dominate the curves for other players till the 20 run
point. In other words, the chances of Rahul Dravid getting a score
less than 20 is lesser than the chances for any other player in
the Indian team getting a score lesser than 20. Finally, the curves
for the specialist batsmen very clearly dominate the curves for
the specialist bowlers, as should be the case.
method that has been developed only provides an alternative approach
to represent the batting performance of cricket players. This alternative
approach is visually and intuitively appealing. The attempt in this
paper is not to arrive at a model to rank the utility of players.
Nor is the goal to develop a model to assist in team selection.
The utility of a player goes far beyond the runs scored by him.
Factors like tactical skills, passive support to the partner batsmen,
etc. cannot be gauged by looking at the runs scored. Even if we
use runs scored as the sole measure of utility, first-order stochastic
dominance rules alone cannot be used to rank players in terms of
their utility. And if we go on to second-order stochastic dominance
rules the utility function might not have a negative second derivative.
In other words, there could be potentially match winning situations
in which a batsman who is batting on a very high score (say, 108)
has to score one more run in order for the team to win the match.
In this situation the incremental one run (from 108 to 109) might
be much more valuable than the incremental one run the batsman scored
while he was on a lower score (say, 23) during the same innings.
limits of this study, the paper seeks to highlight the tremendous
scope that exists to improve and develop on the measures currently
used to describe the performances of cricket players in general,
and batsmen in particular. The measures used today do not adequately
capture the richness of the underlying data. Similar approaches
can be adopted to represent the performances of bowlers too.
The problem of dealing with 'not out' scores in cricket is tackled
using a Bayesian approach.
dominance rules are used to characterize the utility of a batsman.
the marginal utility of runs scored is not diminishing in nature,
only first order stochastic dominance rules are used.
results, demonstrated using data for the Indian cricket team are
limitation of the approach is that it cannot arrive at a complete
utility value for the batsman.
Employment: Professor of Finance, XLRI School of Business
and Human Resources, Jamshedpur, Jharkand, India.
Degree: Fellow (IIM Bangalore), M.Stat., B.Sc. (Physics).
Research interests: Investments, Corporate Finance, Cricket.