gareth.jones Section name Department of Economics

Centre for Institutional Performance (CIP)

Economic Analysis Research Group (EARG)

On Fixing International Matches by Sarah Jewell and J. James Reade

Discussion Paper No. 113 October 2014

Department of Economics University of Reading Whiteknights Reading RG6 6AA United Kingdom

www.reading.ac.uk

© Department of Economics, University of Reading 2014

On Fixing International Cricket Matches∗

Sarah Jewell Department of Economics, University of Reading J. James Reade Department of Economics, University of Reading and Programme for Economic Modelling (EMoD), Institute for New Economic Thinking at the Oxford Martin School October 31, 2014

Abstract Corruption is hidden action which distorts allocations of resources away from competitive outcomes. Hence the detection of such actions is both difficult yet important. In many economic contexts, agent actions are unobservable by principals and hence detection is difficult; sport of- fers a well-measured context in which individual actions are documented in great detail. In recent years the sport of cricket, which records a huge volume of statistics, has been beset by a number of corruption scandals surrounding the fixing of matches. We use 18 (ODI) matches that are known to be fixed by one of the teams involved and anal- yse a wide range of observed statistics from all ODI matches since 1971, in order to determine whether corruption manifests itself in recorded out- comes. We find that corruption does affect a number of observed outcomes in anticipated ways, suggesting that both the increased reporting of statis- tics, and the statistical analysis of them may be a useful tool in detecting corruption. JEL Classification: D73, C50, L83. Keywords: Corruption, Econometric Modelling, Sport.

1 Introduction

Corruption is hidden action undertaken by economic agents which distorts al- locations of resources away from competitive outcomes. Being hidden, the problem, thus, is detection of such actions. The more an activity is recorded and documented, the more likely is the detection of actions intended to be hid- den. In this paper we use the context of a copiously documented sport to attempt to detect and understand corruption.

∗We would like to thank the University of Reading for their financial support in the writing of this paper, and Reade would like to thank, in addition, the Open Society Foundations (OSF) and Oxford Martin School (OMS). We also thank Danial Malik and Benjamin Leale-Green for research assistance in the writing of this paper, and Declan Hill and Dorian Owen for helpful suggestions.

1 In recent years the sport of cricket has been beset by a number of corruption scandals surrounding the fixing of matches; Condon (2001), King (2000), and Qayyum (1998) providing three judicial investigations conducted. From the wording of these reports, Condon (2001) in particular, it is clear that what is known about corrupt activity is likely to be a fraction of the total amount of corruption taking place. The problem remains just as pertinent as when these reports were written; de Speville (2011) wrote in another report commissioned by the International Cricket Council (ICC) that “the analysis of corruption in cricket contained in Sir Paul Condon’s report of April 2001 remains as valid today as it was then. Regrettably there is no reason to believe that the risk is any less today than it was then.”. Sport, and cricket in particular, generate vast amounts of well-measured data, but also attracts those seeking to extract rents and fix outcomes. This increases the likelihood that hidden actions such as agreements to fix competitive outcomes, will manifest themselves in observable outcomes and hence statistics. We present a list of 18 one-day international (ODI) matches that are known to have been fixed by one of the teams involved, based on previous public inves- tigations. We analyse a wide range of observed statistics from ODI matches and find that corruption does affect a number of observed outcomes in anticipated ways. The implication is that the better documentation of economic activity, and the rigorous analysis of such collected data, might be an effective tool in detecting corruption. In Section 2 we outline the nature of match fixing and nest it in an economic theory context before outlining our econometric method for understanding more about the methods of match fixing. In Section 3 we introduce our dataset, and in Section 4 we set out the results of our investigation. Section 5 concludes.

2 Theory and Method

Match fixing can be defined as an agreement between interested parties stipulat- ing a particular outcome for an otherwise competitive sporting contest. Preston and Szymanski (2003) provides an economic model of cheating in sports, exam- ining the marginal benefit and marginal cost of engaging in corrupt activity. In their model, sporting participants are approached by ‘fixers’ seeking to ar- range outcomes in order to profit via the placing of bets. These fixers offer the sporting participant a financial package, or bribe, in return for assisting in arranging this outcome.1 They identify the following important dimensions in the subsequent decision of the sporting participant about whether to accept the bribe: • The financial reward from competing honestly, hence the compensation packages of participants, and the reward from winning any given match (in terms of prizes and prestige). • The financial reward from accepting a bribe to fix a match.

1The agreed outcome may not occur as there is a noisy relationship between effort exerted to fix an outcome and the actual outcome. Cricket is an intrinsically uncertain game; book- makers only predict the correct outcome less than 50% of the time (based on a sample of 2,268 bookmaker prices from 139 domestic one-day matches since 2012 in and Australia). It is also possible that an agent exerts zero effort towards fixing an outcome by reneging on the agreement made.

2 • The level of corruption within the bookmaking industry, and the existence of large underground betting markets. • Likelihood of being caught: This clearly depends on the methods chosen to fix games as well as the efforts directed by authorities to detect corruption. • Severity of punishment: The lighter the punishment, the lesser the deter- rent. Both Condon (2001) and Qayyum (1998) note that pay has often been low in cricket, both as a proportion of the overall turnover of the sport and rela- tive to top performers in other popular sports. Furthermore, there are often matches whose significance is trivial; in a series of matches between two teams, often one team has won sufficiently many matches by the final match such that that match cannot influence the final outcome. Additionally, matches are of- ten played in neutral countries where the game is less popular (for example, Morocco, Malaysia and the UAE); such matches could be argued to have less significance to the competitors.2 Prior to the concerted efforts of the late 1990s and early 2000s, it is reasonable to suggest that the likelihood of being caught was quite low, and furthermore the severity of punishment small too as countries sought to minimise the damage both to the credibility of the sport, and to the strength of their national team caused by players being involved in match fixing. Since the spate of allegations and reports around the turn of the century, it is clear that the ICC has acted substantially along many of these di- mensions; during international matches player communications are now strictly monitored, and a code of practice exists that players must sign and adhere to. Player remuneration has also improved, in places dramatically with, for exam- ple, the advent of the , a domestic cricket competition. Indeed, de Speville (2011) ostensibly is a review of the measures put in place since Condon (2001), and as well as warning that corruption is likely still rife (both spot fixing and match fixing), suggests a number of tightenings of these procedures by the ICC. Hill (2010) discusses at length the threat of match fixing to professional sport, but the logic is rooted in the uncertainty of outcome hypothesis of Rottenberg (1956); demand for sport depends on the outcome being uncertain, and fixing outcomes by definition reduces that uncertainty. Hill also details some of the methods by which matches are fixed; participants in the targetted match are approached, commonly through a trusted third party (e.g a former player), and usually groomed for the purpose of fixing matches rather than offered bribes outright, in the first instance, for one-off fixes. Instructions will be passed to players regarding the targetted match. Hill considers a number of football matches known to have been fixed, and compares a set of collected statistics regarding these games to those in ‘clean’ games. By considering many of the measured dimensions of such sporting contests, it ought to be possible to learn more about the methods chosen by match fixers. He found that there was no impact on penalties awarded (unless the referee was involved in fixing the match), significantly more own goals and significantly fewer red cards in fixed matches. The implication, consistent with Preston and Szymanski (2003), is

2Historically such neutral venues were chosen to broaden the appeal of the game, but more recently for security reasons have been playing their ‘home’ matches in the UAE (see http://goo.gl/Pag3tT for a Wikipedia article on this).

3 that fixers will seek less obvious ways to influence match outcomes to minimise the likelihood of detection; for example, regarding red cards, these are salient events in football matches and are usually heavily analysed at the time and afterwards.3 Football, historically, has been a moderately well recorded game, but not to the extent that cricket has been. Since the 18th century vast amounts of information has been recorded about a huge number of cricket matches; Figure 1 shows an example scorecard, record- ing in great detail a match between two villages in Cambridgeshire, England, in 1953.4 For an ODI match, the average scorecard contains well 200 statis- tics on individual and group actions undertaken during a match.5 A batsman has his score recorded, the method he was got out, the number of balls faced, and how many 4s and 6s were scored; also recorded are byes and leg-byes. For the whole team the point at which each batsman was removed is recorded (fall of ). The number of balls bowled by a bowler is recorded along with the number of runs conceded, as are the number of maidens (overs without runs conceded), number of , and wides and no balls.6 Increasingly, websites such as the source of our data, www.cricinfo.com, record even greater amounts of information about matches, for example ball-by-ball commentaries detailing the ends bowled towards, the participants in each action (ball) of the match, and the outcome of each ball is usually described in detail via a short sentence. A sport as comprehensively recorded as cricket must make hidden action more difficult, and as such we try and detect where corruption manifests itself in cricket matches. Reflecting again on the economic model of Preston and Szymanski (2003), clearly fixers will seek methods that reduce the likelihood of being caught, but will also choose methods that have the greatest likelihood of influencing match outcome. Anecdotally, winning the toss and the subsequent decision to bat or field first, plus early wicket partnerships amongst a team’s higher quality bats- men are the most important factors in determining match outcome. Individuals involved in fixing an outcome would be expected to perform below their usual standard, and this might be missed catches and poorly selected shots; these kinds of actions would then be reflected in the scores achieved by individuals and teams, and associated bowling statistics. Channelling the information in a number of reports into cricket corruption (Qayyum, 1998; Condon, 2001; Central Bureau of Investigation, 2000; Radford, 2011), we have a list of 18 ODI cricet matches whose outcome is known to have been fixed (see Section A for the list); this is distinct from another phenonema that has received attention in cricket, that of spot fixing. By match-fixing we refer to the main outcome of a cricket match, namely which team wins the match, being arranged in some way in advance of the match taking place, whereas spot fixing refers to sub-outcomes of matches being arranged.7

3Additionally, a player once excluded from the match is unable to influence its outcome. 4Websites such as www.cricinfo.com and cricketarchive.com have matches back hundreds of years for all levels of cricket. 5Based on all eleven batsmen and five bowlers per team. A team must name a minimum of five bowlers per match, as each can bowl at most 20% of overs available. 6The number of wides and no balls is not recorded at individual level for all of our matches, only for the more recent matches. 7As already meantioned, however, there is a multitude of what we might call sub-outcomes from a cricket match; all of the myriad of outcomes of specific match events that contribute in different ways to the overall outcome. For example, each of the (maximum) 300 balls bowled

4 Figure 1: Example of a cricket scorecard from a 1953 match between two villages in Cambridgeshire.

Our method is to statistically model a wide range of these sub-outcomes recorded for each cricket match via its scorecard, in order to determine whether or not significant differences exist between the 18 matches known to be cor- rupt, and all other ‘similar’ cricket matches.8 We might think, in a slightly perverse manner, about the 18 known fixed matches as our ‘treatment group’, matches treated with corruption, with the remaining matches being our ‘control group’, in order to nest this into the traditional scientific framework of analysis. Naturally, match fixers do not select their target matches at random, and we must build this into our analysis. While we may notice statistically significant differences between our treatment group and control group, this alone does not provide evidence regarding the method of corruption being used. We must firstly control for all other explanations for variation in the variables we model, and secondly we must be sure that one single match in our 18 is not driving the results, rather than a genuine difference in means between the groups. More broadly we aim to avoid false positives; aside from the intellectual failure of com- mitting type I and type II errors, it is clearly unhelpful if any method aimed at detecting criminal activity detects perfectly innocent behaviour, and/or fails to spot the criminal activity. The list of 18 matches helps reduce the likelihood of the former, type I, error, and a thorough investigation of each variable model mitigates the risk of the latter, type II, error. Our object of interest when investigating variables is the moments of their distributions, and in particular their expected value; how many wickets do we expect a bowler to take, for example. All recorded cricket match statistics have by each team can have a range of possible outcomes from a no ball to the batsman scoring a number of runs. As all of these outcomes are recorded, they are often bet upon by keen bettors, and thus are subject to the possibility of being fixed in order that bettors might be able to extract economic rents; such fixing is known as ‘spot fixing’. This is undoubtedly also a significant problem for the sport; whilst the impact of a single ball can be argued to be negligible on the overall outcome of a match, fixing such events naturally raises questions about the integrity of the game. We believe that the methods outlined and utilised in this paper could be applied in the context of spot fixing also. 8By similar we refer to ODI cricket matches only.

5 statistical distributions, and we seek to model a number of them. Specifically, we model scorecard statistic i for match t, denoted as yit, based on a vector of K explanatory variables Xit, and further we add a dummy variable Dt which is 1 if match t is known to have been fixed, and zero otherwise. Hence we a range of models of the form:

yit = β0 + β1Xit + β2Dt + uit. (1)

The K-dimensional coefficient matrix β1 contains the impact of our control variables; the factors that explain the variable we are modelling. The distribu- tion of the error term uit depends on the distributional assumption for statistic i; in all cases in this paper we appeal to the central limit theorem given our large sample size, and assume that uit is identically and independently normally distributed.9 The coefficient β2 captures the difference in expected value between the matches in our control group, and the 18 fixed matches, for the statistic we are modelling. Hence of interest is the significance of this difference in means; it dictates whether this particular statistic is different in known fixed matches relative to all other matches, and hence is indicative of the methods employed by those seeking to fix outcomes. We test whether β2 = 0, which will determine whether in the known fixed matches there was significantly different behaviour in the particular aspect of the game being modelled, such as bowler economy, or runs scored by a batsman. Naturally, an important part of any such regression model is the control group; the matches for which Di = 0, those that, ideally, are not fixed. It could be that many of the matches in our control group are actually fixed, and fixed in similar ways to our treatment group (certainly, Condon (2001) suggests this is likely). If this is the case, the likely effect will be to reduce the significance of Di in our models. This is because it is unlikely that all other matches are also fixed, and it is quite likely that only a small proportion of the matches in our control group are fixed. Hence provided the fixers in the games we identify behave in a reasonably similar manner, we can still expect that we will observe these patterns using our regression methods, even if some of our control group matches do happen to be fixed. Additionally, it might be proposed that each fixed match is different, and hence it is futile looking for patterns in the 18 fixed matches relative to the remaining (presumably) non- fixed matches. However, it is plausible that the same group of match fixers may approach players in a number of different matches, and given the incentives (methods that are particularly effective and less likely to be detected) facing fixers it seems plausible that different groups would follow similar procedures to fix outcomes. Having a treatment group of just 18 matches and a control group of 3492 matches could be seen as rather unbalanced. However, if the mean value of the statistics we model is significantly different for the 18 corrupt matches from the remainder this will still provide a statistically significant statistic. Realistically we might anticipated that we are more likely to find insignificant effects with such a small treatment group, meaning that failing to spot corrupt action is more

9For the models where it might be argued another statistical distribution is more appro- priate, we run that model for reference purposes and comment on any differences observed between the models.

6 likely, rather than erroneous conclusions in favour of corrupt activity. With any statistically significant , we analyse which of the 18 matches, if any can be singled out, are driving the significant statistic found, again in order to guard against one outlier match driving our results. We additionally list a set of a further 48 matches (see Section B on page 24) that are deemed ‘suspicious’ and referred to in Polack (2000); we also run robustness checks on regressions including these matches, finding broadly similar results to those listed in this paper for the 18 matches. We note the possibility of endogeneity; while we may find significant coeffi- cients in matches known to be fixed, it need not be that these correlations are causal; rather it may be that fixers chose these particular matches due to instrin- sic characteristics of the matches. We accept an absence of causal identification in the analysis in this paper as our aim here is to merely identify patterns in outcomes corresponding to matches known to have been influenced.

3 Data

We collect data from all recorded ODI matches from www.cricinfo.com.10 As of August 21 2014, this amounts to 3510 matches. We extract information from all webpages on www.cricinfo.com related to a match; primarily this is scorecard information, but also ball-by-ball information, and information on partnerships and match commentaries. Figure 3 shows the breakdown of matches per year, showing that the format took some time to become popular, with only a few hundred matches taking place in the first ten years. The peak year was 2007 with 190 matches taking place, driven mainly by a particular World Cup format which involved a large number of matches. One-day cricket involves each team batting once, facing at most a stipulated number of overs (and hence balls).11 The match begins with a coin toss, of which the winner decides whether to bat or bowl first. Batsmen from each team then attempt, in turn, to score as many runs as possible in the limited number of balls they face. The is complete when either the entire team is bowled out, or the total number of balls have been used. The winner is the team that scores the most runs. In 121 matches no result is recorded, in 32 the match is tied (each team scores the same number of runs), and in two the team which lost the least number of wickets in reaching equal scores won the match. Two matches were awarded to Pakistan due to

10By means of explaining the focus on what may seem like a small number of countries, cricket is not a global sport in the sense that football (soccer) is. By and large the international cricket community aligns closely with the sphere of influence of the British Empire (Perkin, 1989). The matches in our dataset involve overwhelmingly the major test-playing nations; India have played the most, at 856, followed by Australia (830), Pakistan (817), Sri Lanka (722), the West Indies (713), New Zealand (654) and England (622). The latter is despite England competing in the first ODI match against Australia in 1971. Bangladesh (287) and Kenya (154) form the next group, then Scotland (67), Ireland (83), Canada (77) and the Netherlands (76) form another group, with Afghanistan (37), Namibia (6), Bermuda (35), the UAE (14), Hong Kong (6), East Africa (3), and the USA (2) forming the remainder. A graphical representation of this information can be found in Figure C on page 33 in the Appendix. 11In the early years there was some variation in the number of overs, and even balls per over, but since the late 1990s ODI matches have been exclusively 50 overs per team, hence 300 balls.

7 ODI Matches per Year, 1971−2013 150 100 Number of Matches 50 0

1970 1980 1990 2000 2010

Year

Figure 2: Number of One Day International (ODI) matches per year since 1971. Not including 2014 matches as year incomplete. Source: Cricinfo.com.

their opponents, India (1978) and England (2001), conceded, while in 1996 Sri Lanka were awarded victory over India by default. The remaining matches are plotted in Figure 3, which plots the innings scores of each team in a match, with first innings scores along the vertical axis, and second innings scores along the horizontal axis. Black crosses represent standard match outcomes whilst red squares refer to matches where the Duckworth and Lewis (1998) formula was applied to revise the required totals for the team batting second. As such, apart from Duckworth and Lewis matches, there are very few points to the right of the 45-degree line, as this would represent the team batting second (chasing) scoring many more runs than the team batting first; once the chasing team has reached the target set by the first team, the match is concluded. In order to provide a thorough analysis, two steps are required; firstly we must create a comparable margin of outcome variable for each type of outcome, in order to subsequently create a measure of team quality based on outcomes.12 We create a comparable margin of victory by imputing what a chasing team’s total score would have been had they used their remaining balls (given wickets in hand) using a non-parametric method based on all ODI matches since 2001, and create a ranking for teams using the (Elo, 1978). Details are provided in Appendices D and E.

12We could simply use a binary variable for match outcomes to create a ranking of teams, but this would exclude a lot of information contained in margins of victory.

8 Innings Totals for all recorded ODIs 400 300 200 Total: Team Batting First Team Total: 100

Normal Innings

0 Duckworth−Lewis Affected Innings

0 100 200 300 400

Total: Team Batting Second

Figure 3: Cross plot of innings totals for both teams in all ODIs. Crosses mark innings completed without incident, red filled squres reflect innings affected by rain delays and hence subject to Duckworth-Lewis adjustments.

9 3.1 Control Variables We include a large number of control variables in our regression models; the full list can be found in the Appendix in Table 6 (on page 27). We can summarise them into the following categories: • Match-specific variables: Information specific to the match, such as where it took place, its timing during the day, and whether it was important for the outcome of a tournament. • Innings-specific variables: Information specific to which innings an event took place in, such as what innings was being played, whether the team batting won the toss. • Team-specific variables: Information specific to the teams involved, such as their overall strength, and recent form. • Player-specific variables: Information specific to players such as their age and handedness. • Batsman-specific variables: Information specific to batsmen such as their batting average and frequency of being out to particular methods. • Bowler-specific variables: Information specific to bowlers such as their style of bowling and their bowling performances. • Fixed-effects: We add three types of fixed effects: 1. Teams: We add dummy variables for each team involved in match to capture team strengths. 2. Country: We add dummy variables for countries hosting matches; often conditions in particular countries can be inducive to particular types of play which may affect outcomes. 3. Year: We add a dummy for each year to capture secular trends; for example, innings totals have been increasing throughout our dataset; up until the turn of the century they increased by a run per year from around 190 in the 1970s to 210 in the 1990s, and since then they have increased by just over half a run per year.

4 Results

We first provide some statistical context for the 18 fixed matches. Of the coun- tries involved, Pakistan and India each appear nine times, followed by Australia (4); England, and the West Indies (3), and New Zealand, Zim- babwe, Kenya and Bangladesh once. Of the teams actually fixing, on eight occasions it is India, Pakistan seven times, and Australia, South Africa and Kenya once. Six of the eighteen matches take place in India, three in England, two in each of South Africa, Australia and the UAE, and one in each of New Zealand, Sri Lanka and Zimbabwe.13 Only two of the matches are dead rub- bers, although as a proportion (11%) this is higher than in our overall dataset 13It ought to be noted that, Australia, India and England aside, these proportions are not different to how many ODI matches in total have been played in these countries; Australia has hosted 16% of all ODI matches and 11% of all corrupt matches, England has 10% and 17% respectively, and India has 11% and 33% respectively.

10 (9%). Five of the eighteen take place on neutral venues, which as a proportion is smaller than the number of neutral matches in the dataset overall. Thirteen of the eighteen fixing teams are playing on tour, rather than playing at home. In four of the 18 matches the fixing side eventually wins the match; on each of these four occasions the fixing team is India. As discussed in Section 2, this need not cast doubt on the accuracy of the list, but more points to the intrinsic uncertainty of cricket match outcomes, and the reality that there is a large number of participants in a match, not all of whom can be influenced. In terms of margin of victory, the absolute margin of victory is slightly larger in our fixed matches than in other matches (on average by 7.5 runs), although this is not significant.14 This might be anticipated, as mentioned in Section 2, as fixers will choose methods that have the greatest likelihood of being successfull — taking actions early to ensure a larger margin of victory rather than waiting until late in the game when the outcome remains in doubt. Our analysis now proceeds by formally modelling a number of variables from ODI cricket matches; having set out our control variables in Section 3.1, we then briefly model the coin toss (Section 4.1) before concentrating much more strongly on the subsequent coin toss decision (Section 4.2), the scores achieved in the first and second innings of matches (Sections 4.3.1–4.4), the partnerships in those two innings (Sections 4.5.1–4.5.2), before considering batsman- and bowler-specific statistics in Sections 4.6 and 4.7.

4.1 Coin Toss Although it might be anticipated the coin toss cannot be fixed, this need not be the case. Many of the allegations centre on agreements being formed between players and officials surrounding particular outcomes, and up until recently the toss outcome need only be known between the captains of the two teams and one official, all of whom could agree on a particular outcome.15 We know which team was fixing in each match, and hence we can investigate coin tosses (and subsequent decisions) in those matches. Of the 3510 coin tosses, the team listed first in the match has won 1754, or 50.03% of them. In our 18 matches, the fixing team won the toss 13 times. The likelihood of this occurring at random is about 3.3%, which is slightly below the level of statistical significance we commonly accept. Translated into a simple regression of coin toss on a dummy variable taking the value 1 if the team listed first is known to have fixed the outcome, and a dummy variable taking the value 1 if the team listed second is known to have fixed the outcome, this yields a t-statistic of 1.9; particularly with a sample size of 3510, we would not accept this at conventional significance levels. This illustrates the difficulty in detecting corrupt activity; we have corrobo- rating evidence here that these 18 matches were indeed fixed (and we know the fixing team), and as such we can have more confidence that this is not simply a correlation but the result of corrupt activity. But any statistical method to detect corruption will struggle to detect such marginal results without corrob- orating evidence, particularly in large datasets.

14We compare margins of victory converted into runs, even if the victory margin was actually in terms of wickets and balls (see Appendix D for details on the method employed to convert). 15See Radford (2011, e.g. pp. 84, 232) for an example of such allegations.

11 We can interact the team fixing dummies with team fixed effects in order to try and determine whether any of the fixing teams in particular followed this strategy; if we do so, however, all of the interaction terms are insignificant, suggesting a lack of data in order to determine this. If we interact the team fixing dummies with a dummy for whether the match was played on neutral territory, this has no impact and dilutes the near significance of the team2 dummy. The neutrality dummy in itself is insignificant. If we add dummy variables for the country in which a match took place, we find that two countries in which there is no ODI cricket team and in which a small number of matches took place in our dataset (Morocco and Malaysia), do appear to generate mildly significant coefficients. This again supports the idea that matches played on neutral territory, particularly international non-World Cup tournaments, are more likely to be targetted by match fixers. This, though, is uncorroborated in our dataset.16 As would be expected, team strength has no impact on coin toss, nor do a range of other explanatory variables (not reported).

4.2 Coin Toss Decision Even if the actual toss outcome cannot be fixed, the decision a team makes once the toss is won can clearly be used to influence outcomes. Bhaskar (2009) considers the toss decision, noting its importance for outcomes, and suggests that, controlling for conditions, captains do not act rationally in the choices they make. He considers the suspicious matches noted in Polack (2000) amongst other explainations for the inconsistent coin toss choices by captains, but his results suggest that the most likely explanation is that captains tend to over-estimate their own strengths and under-estimate those of their opponents. In ODIs, 48.6% of the time the team that wins the toss chooses to field, although this is heavily influenced by whether the match is a ‘day-night’ match (one which the second team is likely to bat at night-time); without controlling for any other factors, for daytime matches the winning captain chooses to field almost two thirds of the time, but in day-night matches this reduces to only just over a third of the time, as instead winning captains elect to bat first. As such, Bhaskar also makes this distinction, focussing on daytime matches. Of our 18 fixed matches, twelve times the toss-winning captain elects to field; at two thirds this ratio is distinctly higher than the baseline 48.6% noted above. We present results from a regression model that adds the control variables listed in Table 6 in the first column of Table 1, and shows that fixing teams statistically significantly choose to field more than to bat. Fixing teams are about 45% more likely to choose to bat first.17 This seems plausible; by being the team chasing a target, a fixing team has more control over the outcome, as it can field poorly thus facilitating a large target being set, and subsequently bat poorly, ensuring that target is not met. We do not report control variables in Table 7, but instead in Table 7 on page 28 in the Appendix; we now provide a brief discussion of the salient control variables.18. We find that the last coin toss decision made by teams involved

16Five of our eighteen known fixed matches take place on neutral territory. 17We found similar results when estimating probit and logit models for this binary dependent variable. 18We also included team fixed effects, country/neutral venue fixed effects, and a variable for

12 matters significantly; if the last toss decision was to field, then this decision is 8% more likely to be to field. This implies some persistence in decision-making whereby captains choose to bat or field not based on current conditions but some other factors, and perhaps echoes Bhaskar’s lack of learning explanation for toss decision irrationality. Similarly, if the opposing team’s last decision was to field (bat), there is a similar impact: a captain is 8% more likely to choose to field (bat). Controlling for relative strengths via Elo strength ratings is insignificant, as is controlling for experience and batting and bowling strength via recent scores. A captain with a more experienced batting line-up is more likely to choose to field first when measured by average age, though less likely when measured by average number of innings players have batted in. A captain with a higher scoring batting line-up is (insignificantly) more likely to choose to bat first, suggesting that a more experienced team is seen to be better equipped to chase a target than a younger but higher scoring one. The opposite is true for the opposing team’s batting experience and strength; a captain prefers to let an experienced opponent bat first and a higher scoring one bat second. Considering a captain’s bowling line-up, a more experienced own bowling line- up sees a captain more likely to bat first, while the more economical a bowling line-up, the more likely a captain is to field first. The more fast bowlers in a captain’s (opponents) team, the (more) less likely he is to field first, and the more spin bowlers in a captain’s opponents’ line-up, the less likely he is to choose to field first; the presence of spin bowlers in a captain’s own team has no significant impact. A captain playing at home is no more likely to choose to field, and neither does playing on neutral territory has any significant effect. The series score does appear to matter however; a team ahead in a series is more likely to choose to field, a team behind more likely to choose to bat. Dead rubbers appear to have no impact on the decision made.

Table 1: Condensed regression outputs for post-toss decision and first/second innings totals. See Table 7 on page 28 for control variables.

Dependent variable: toss.team.field innings total.1 innings total.1 innings total.2 innings total.2 (1) (2) (3) (4) (5) Constant 0.530 80.039∗∗∗ 118.104∗∗∗ 32.610∗∗ 8.722 (0.393) (16.852) (43.424) (14.103) (35.953) team.fixing 0.449∗∗∗ −23.433 −25.837 −20.985∗ −18.961 (0.146) (23.262) (22.788) (11.984) (11.835) team.fixing.opp 0.528 27.653∗ 17.476 5.633 3.434 (0.325) (14.406) (14.188) (19.345) (19.009)

Team fixed effects Y N Y N Y Host-country fixed effects Y N Y N Y Year fixed effects Y N Y N Y Observations 3,336 3,339 3,339 3,335 3,335 R2 0.222 0.199 0.266 0.408 0.454 Adjusted R2 0.187 0.187 0.234 0.399 0.429 Residual Std. Error 0.451 (df = 3192) 51.818 (df = 3289) 50.314 (df = 3196) 42.797 (df = 3284) 41.715 (df = 3191) F Statistic 6.368∗∗∗ (df = 143; 3192) 16.705∗∗∗ (df = 49; 3289) 8.175∗∗∗ (df = 142; 3196) 45.315∗∗∗ (df = 50; 3284) 18.534∗∗∗ (df = 143; 3191) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 team strength; these are not reported in Table 7, but were jointly significant and are available using the code and data made available on our workpage http://goo.gl/cNPwt

13 4.3 Innings Totals Following the toss, one team bats first, whether by own choice or that of an opposing captain. We investigate innings totals in turn, controlling for team strengths, levels of experience, whether or not the batting team won the toss (hence elected to be batting at that point), recent scores by both batting and fielding teams (form), whether the match is a day/night match, and whether it is taking place on neutral territory, or in one of the two countries competing (home advantage). Columns (2)–(5) of Table 1 report results from first and second innings totals.

4.3.1 Team Batting First Investigating first innings totals using regression methods, column (2) of Ta- ble 1 reports our model without fixed effects, and column (3) the model with fixed effects included. A fixing team will, if fielding, concede more runs, and if batting will score fewer runs. Of these effects, however, only for a fielding team is this effect significant, and even then only without including fixed ef- fects. The fixed effects are significant, and increase the explanatory power of the model substantially, and hence ought not to be ignored. Nonetheless the fixing coefficients remain in the anticipated direction. Considering the control variables in Table 7, we find that team strength is irrelevant for a first innings score, although experience appears to matter, yielding higher scores for more experienced batting line-ups (both own team and opposition), and lower scores for more experienced bowling line-ups (again for both). A batting line-up composed of more left-handed players scores significantly fewer runs, while left-handers in the bowling line-up have no impact on innings scores. The series score has a small effect; a team one match ahead will score an additional two runs. On batting and bowling strengths, the only variable that appears to matter is the economy rate of the opposition bowlers; for each additional run a bowling team, on average, has conceded per over, a batting team scores 11–12 more runs.19 A team that chose to bat will score, on average four more runs than a team chosen by their opponents to bat. There is persistence in posting innings scores; an extra run on average over a team’s most recent five games translates into 0.132 extra runs in the current match. Actual outcomes of recent matches do not appear to matter, however. There appears to be a home advantage; teams batting first who are playing on home territory will score 13–16 more runs than those playing on foreign territory.

4.4 Second Innings Total Turning to the second innings scores, these are posted by teams chasing the target set by the team that batted first. Columns (4) and (5) of Table 1 report regressions of the second innings total. Considering the impact of fixing teams, if the fixing team is batting second (more common), they score significantly less,

19Additionally, it would appear batsmen are aware of the economy rate of their own bowlers; for each additional run their bowlers are expected to concede, batsmen will score about six more.

14 by around 18–20 runs, while if the fixing team is fielding (less common), the impact is in the expected direction as the batting team scores more, but this impact is insignificant. The control variables are shown in Table 7; the only difference in terms of specification from the regression models for the first innings totals is that we include the first innings total, which enters with a coefficient of about 0.6, suggesting that setting all other factors equal to their mean values, a chasing team will only score about 60% of the total set by the team batting first. There is no significant impact of the toss decision, and day/night matches have a significant negative impact on second innings totals, but only between 5 and 7 runs. Playing on neutral territory does not have any impact, while there is a smaller home advantage for second innings totals than first innings totals; only about 7 runs. As with first innings totals, team experience matters, and adds again close to 0.1 runs per game played historically. Interestingly compared to first innings totals, which were affected by recent scores by the batting team, second innings scores appear more affected by outcomes of recent matches; a team that won their last match is likely to add more runs, while a team facing an opponent with more recent wins is likely to post a lower score. The experience of batting and bowling line-ups appears to matter, and in particular a fielding team with more debutants will concede more runs in a second innings.

4.5 Fall of Wickets/Partnerships Each innings can be decomposed into its constituent parts: partnerships be- tween batsmen. Each scorecard includes information on the ‘fall of wickets’ (FOW), which gives an indication of how the score was built up. While build- ing a score is primarily an individual pursuit by whichever batsmen is ‘on strike’ (facing the bowler), at all times there are two batsmen on the field and these partnerships are generally regarded as important in how scores develop.20 Fig- ure 4 gives an indication of the distribution of these points at which partnerships break down as an innings is built up; the black distribution is the number of runs scored when the first wicket falls, and subsequent fall of wicket distribu- tions become more purple in colour, and can be seen to flatten out. Although the modal fall of first wicket is zero, the mean is 34 and the median is 22. The second wicket falls on average after 68 runs (mean, 55 median), the third at around 100, fourth about 125, fifth 150, and so on — plotted graphically in Figure 6 (on page 34 in the Appendix). It may be that patterns can be detected here which further shed light on match-fixing activity; fixing fielding teams may allow important partnerships to last longer, for example, while fixing batting teams may exhibit shorter part- nership, at least by those batsmen involved in fixing outcomes.21 We consider the two innings separately; we included first innings fall of wicket totals in second innings regressions but found them to be very insignificant and

20An example of such teamwork can be seen often towards the end of an innings, where a stronger batsman will play a shot on the final ball of the over that allows one run to be scored in order that he is ‘on strike’ at the start of the next over (when the end the bowler bowls from is switched). 21Although as batting collapses, where a significant number of wickets are lost for a small number of runs, are reasonably common, it could be that a fixing batsman could influence non-fixing colleagues to similarly perform poorly.

15 Fall of Wicket Scores

FOW 1 FOW 6 200 FOW 2 FOW 7 FOW 3 FOW 8 FOW 4 FOW 9 FOW 5 FOW 10 150 100 Frequency 50 0

0 50 100 150 200 250 300 350

Score

Figure 4: Distribution of fall of wicket scores for all recorded ODI innings. Source: Cricinfo hence they are omitted. We include the fall of wickets for all previous wickets when attempting to explain the fall of wicket score for any given point in the innings, and this enables us to essentially model each partnership; the difference between FOW scores is the length of a given partnership.

4.5.1 First Innings Considering FOWs in the first innings, we present our regression results in Table 2, with each column devoted to each fall of wicket score. As with Table 1, here we also report in the text only the fixing coefficients, and in Table 8 on page 29 in the Appendix we present the same table with all control variables (except the fixed effects) reported. Considering the impact of match fixing, a fixing team scores fewer runs for each fall of wicket, although none of these are significant, all with t-statistics of about one. For a fixing fielding team, the first three wickets fall for significantly more than if the fielding team was not fixing the match. None of these effects are trivial; for the fixing fielding team these coefficients equate to a batting team scoring an additional 50 or more runs. Somewhat unexpectedly, the seventh wicket partnership in matches where a fixing team is batting is statistically very significant and doubles the average size of that partnership; however, rather than indicating a pattern related to corrupt activity, this is dominated by one particularly large seventh wicket partnership.22 Briefly describing the control variable coefficients, the lower order the wicket

22Indeed, the coefficient is driven by a 114-run seventh wicket partnership in the Kenya vs Zimbabwe match, the fifth largest in ODI history, and at the time a record stand.

16 partnership (larger numbers in the table), the more exclusively important the fall of the previous wicket is for explaining the size of the partnership. The experience of batting and fielding teams matters most at first, but this decreases throughout the innings. As such, being at home or on neutral territory, being a stronger team, being in better form (as a team), the series score (or whether it’s a dead rubber), and whether it’s a day/night match all are insignificant for the fall of wicket.

17 Table 2: Regressions of the falls of wicket in first innings

Dependent variable: FOW1.1 FOW2.1 FOW3.1 FOW4.1 FOW5.1 FOW6.1 FOW7.1 FOW8.1 FOW9.1 FOW10.1 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Constant −9.219 10.300 −8.199 17.379 37.681 2.915 4.962 42.553∗∗∗ 15.983 2.459 (32.408) (33.350) (32.238) (30.102) (26.010) (21.161) (19.045) (15.278) (15.135) (14.004)

team.fixing −18.563 −18.562 −13.464 −12.544 −11.751 −8.476 23.868∗∗∗ 0.807 −3.372 8.436 (17.121) (17.613) (17.006) (15.744) (13.378) (10.500) (8.708) (7.321) (6.428) (5.761)

team.fixing.opp 22.382∗∗ 16.239 16.940 −4.082 −2.316 7.497 −5.638 −7.101 −2.022 4.484 (10.696) (11.009) (10.632) (9.847) (9.098) (7.464) (6.512) (6.529) (5.736) (5.581)

Observations 3,339 3,337 3,321 3,259 3,103 2,821 2,393 1,926 1,436 933 Team fixed effects Y Y Y Y Y Y Y Y Y Y Host country fixed effects Y Y Y Y Y Y Y Y Y Y Year fixed effects Y Y Y Y Y Y Y Y Y Y R2 0.079 0.544 0.694 0.769 0.830 0.884 0.908 0.944 0.954 0.972 Adjusted R2 0.039 0.524 0.680 0.758 0.822 0.878 0.902 0.940 0.949 0.966 Residual Std. Error 37.549 (df = 3196) 38.621 (df = 3193) 37.282 (df = 3176) 34.494 (df = 3113) 29.267 (df = 2957) 22.912 (df = 2674) 18.892 (df = 2245) 14.083 (df = 1777) 12.197 (df = 1287) 9.061 (df = 784) F Statistic 1.942∗∗∗ (df = 142; 3196) 26.658∗∗∗ (df = 143; 3193) 50.075∗∗∗ (df = 144; 3176) 71.348∗∗∗ (df = 145; 3113) 99.748∗∗∗ (df = 145; 2957) 139.389∗∗∗ (df = 146; 2674) 150.993∗∗∗ (df = 147; 2245) 203.843∗∗∗ (df = 148; 1777) 181.271∗∗∗ (df = 148; 1287) 181.826∗∗∗ (df = 148; 784) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 18 Table 3: Regressions of the falls of wicket in second innings

Dependent variable: FOW1.2 FOW2.2 FOW3.2 FOW4.2 FOW5.2 FOW6.2 FOW7.2 FOW8.2 FOW9.2 FOW10.2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Constant −26.014 −16.318 11.538 14.224 −6.773 −8.669 −18.126 31.859 12.768 13.897 (30.572) (30.083) (31.308) (29.686) (27.658) (26.514) (23.456) (21.768) (20.453) (19.003)

team.fixing 0.268 −4.871 3.516 −0.132 −11.222 −7.910 −8.400 −1.734 0.633 −6.541 (10.051) (9.446) (9.403) (8.492) (7.577) (7.038) (6.852) (6.292) (5.546) (4.926)

team.fixing.opp −15.226 19.784 5.728 −0.650 −2.131 12.327 17.025 4.827 −3.249 −3.117 (16.090) (16.872) (16.797) (17.542) (18.346) (23.200) (19.235) (16.542) (14.603) (12.080)

Team fixed effects Y Y Y Y Y Y Y Y Y Y Host country fixed effects Y Y Y Y Y Y Y Y Y Y Year fixed effects Y Y Y Y Y Y Y Y Y Y Observations 3,284 3,150 2,927 2,622 2,306 2,039 1,786 1,577 1,369 1,113 R2 0.106 0.547 0.644 0.743 0.810 0.852 0.894 0.921 0.937 0.958 Adjusted R2 0.065 0.525 0.625 0.727 0.797 0.841 0.885 0.913 0.929 0.952 Residual Std. Error 35.269 (df = 3140) 33.105 (df = 3005) 32.887 (df = 2781) 29.597 (df = 2475) 25.229 (df = 2158) 22.350 (df = 1890) 18.397 (df = 1636) 15.695 (df = 1427) 13.737 (df = 1219) 11.156 (df = 962) F Statistic 2.595∗∗∗ (df = 143; 3140) 25.164∗∗∗ (df = 144; 3005) 34.694∗∗∗ (df = 145; 2781) 48.918∗∗∗ (df = 146; 2475) 62.568∗∗∗ (df = 147; 2158) 73.687∗∗∗ (df = 148; 1890) 92.862∗∗∗ (df = 149; 1636) 112.352∗∗∗ (df = 149; 1427) 121.504∗∗∗ (df = 149; 1219) 146.702∗∗∗ (df = 150; 962) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 4.5.2 Second Innings In Table 3 we present similar regression models but for second innings FOWs.23 All fixing coefficients are very insignificant in the second innings. This perhaps reflects that the first innings is the focus for fixing teams, rather than the second innings. This is consistent with Hill (2010) who notes that it is optimal to take early actions to ensure a particular desired outcome, rather than wait until later when it might be too difficult to affect the outcome without increasing the likelihood of being detected. The additional control added is the first innings score, the target that the team batting second must achieve. As with the first innings, we note that constant terms are significant despite all our variables being demeaned (and no fixed effects are added). As with the first innings, experience matters, and perhaps to be expected, day/night matches see significantly shorter partnerships for some partnerships. Outside of these variables, little else appears to be consistently significant throughout the wickets.

4.6 Individual Batting Statistics In addition to considering aggregate batting statistics, as we have thus far, we can consider individual batsmen. Each scorecard lists the score each batsman achieved, in a particular number of balls and minutes, and the method by which they were got out. In Table 4 we present regression results for runs scored by batsmen, the likelihood of them being caught out, bowled out, run out, , out via LBW, and finally ending an innings . As with earlier regressions, in Table 10 on page 31 in the Appendix we present the regressions with control variables (though not fixed effects). If a fielding team is fixing, each batsman scores two extra runs, although this is statistically insignificant. A batsman in a fixing team scores three fewer runs, and this is statistically significant, and is more likely to be run out and less likely to be not out. Although there are no effects of a fixing fielding team that are significant, nonetheless a fixing fielding team is less likely to be caught out (t-statistic of 1.6), reflecting poorer fielding. Considering control variables in the full table, in the first innings (relative to second), batsmen are more likely to score, be caught, bowled and run out, but less likely to be out LBW or not out. The effect on runs is stronger in day/night matches, while the impact on the other variables of day/night matches is mixed. The difference in quality of teams (Elo scores) appears not to matter for individual batsmen, but there appears to be a home advantage in terms of runs scored (1.3 runs), being caught and LBW out (less likely), and being not out (more likely). A batsman’s order matters, and his playing history has an impact also; for example, a player who has been bowled out, or out LBW, is more likely to be out by these methods in the future. The form of a batsman (recent runs) has some impact, as does his quality (batting average), while left- handed players are more likely to be caught and LBW out, and debutants less likely to be caught out and more likely to be not out (likely reflecting they are lower order). Considering the other batsman-specific variables in Table 4, we run linear probability models for the events of a batsman getting out via being caught

23Again, see the Appendix, and Table 9 on page 30, for the control variable coefficients.

19 Table 4: Regressions of the various statistics related to individual batsman performance.

Dependent variable: batsman.runs caught.out bowled.out run.out stumped.out lbw.out not.out (1) (2) (3) (4) (5) (6) (7) Constant 6.503 0.381∗∗∗ 0.393∗∗∗ 0.014 −0.037 0.105∗ 0.127∗ (4.785) (0.097) (0.074) (0.057) (0.030) (0.054) (0.073) team.fixing −3.136∗ 0.024 −0.020 0.042∗ 0.0004 0.011 −0.055∗∗ (1.811) (0.037) (0.028) (0.022) (0.011) (0.020) (0.028) opp.fixing 2.220 −0.065 0.021 0.003 0.001 0.003 0.039 (1.963) (0.040) (0.030) (0.023) (0.012) (0.022) (0.030)

Team fixed effects Y Y Y Y Y Y Y Country fixed effects Y Y Y Y Y Y Y Year fixed effects Y Y Y Y Y Y Y Observations 59,439 59,439 59,439 59,439 59,439 59,439 59,439 R2 0.158 0.071 0.009 0.009 0.008 0.018 0.139 Adjusted R2 0.156 0.069 0.007 0.006 0.006 0.016 0.137 Residual Std. Error (df = 59310) 23.678 0.481 0.368 0.282 0.150 0.268 0.361 F Statistic (df = 128; 59310) 86.757∗∗∗ 35.551∗∗∗ 4.284∗∗∗ 4.034∗∗∗ 3.606∗∗∗ 8.382∗∗∗ 74.846∗∗∗

(column (2)), bowled (3), run out (4), stumped (5), or caught leg before wicket (6). Finally in column (7) we consider the likelihood of being not out at the end of an innings; at least one batsman must be not out by design as a team is composed of 11 men yet only has ten wickets per innings, but two can be not out if a team reaches its full set of overs or reaches its target. Considering specifically the fixing-related variables, we find that batsmen on a fixing team will be significantly more likely to be run out, and significantly less likely to be not out, and if a batsman is facing a fixing fielding side he is significantly less likely to be caught out. All these reflect more careless play by teams — careless team work in determining whether to run, and more careless fielding.

4.7 Individual Bowling Statistics The aim of a fielding team is to bowl out the batting team, and a crucial component of that is the set of bowlers. In this section we model all individual bowler statistics as a large panel. A number of statistics are collected on bowlers from an innings: overs bowled, runs conceded, number of maidens, wickets taken and in later matches the number of wides and no balls bowled. We focus on the bowler’s economy and the number of wickets taken in the innings as we might expect fixing teams to concede more runs and have higher economy rates and take fewer wickets. When the fixing team is batting we may expect them to score few runs and get out more easily so would expect the opposition team to have lower economy rates and take more wickets when they are bowling. In Table 5, as we have done with previous tables in the text, we present regression results for bowler’s economy and wickets taken in the innings, just reporting the fixing variables, with full results (except the fixed effects) in the Table 11 in the appendix on page 32. The signs of the match-fixing coefficients are as expected with economy rates higher when the fixing team is bowling and lower when they are batting; fewer wickets are taken by the fixing team and more taken by the opposition. None of these coefficients are significant, but there is some evidence (at the 10% level) that economy rates are lower when the fixing team is batting.

20 Considering the control variables, we find that bowlers who are bowling at home have a lower economy and take more wickets than when they are bowling away from home. Economy rates are lower and more wickets are taken in the first innings. Bowlers playing in day night matches have lower economy rates and take more wickets, although these results are reduced if they are bowling in the first innings of a day night match; reflecting that the team batting first in a day night is more likely to win. When the batting team is more likely to win i.e. bowlers are facing stronger opposition, bowlers on average have a higher economy rate and take fewer wickets. Economy is higher for dead rubbers which may reflect that teams put in less effort or try out new ideas in these matches. More experienced bowlers have lower economy rates which reflect that bowlers often have higher economy rates early on in their career whilst they are still learning about the international game. Those who have a higher economy either in their career so far or in recent games have higher economy rates, whilst bowlers who have taken more wickets across their careers and in the last five matches are likely to continue to do so in the current innings. Left handed bowlers have lower economy as do fast and medium bowlers, with fast bowlers and left handed bowlers also likely to take more wickets. Economy rates have got higher over time which reflects that average innings scores have gone up over time.

Table 5: Regressions for bowlers

Dependent variable: bowler.econ bowler.wickets (1) (2) Constant 1.208∗∗ 1.358∗∗∗ (0.552) (0.459) team.fixing 0.183 −0.137 (0.195) (0.107) opp.fixing −0.368∗ 0.125 (0.213) (0.096)

Observations 41,702 41,702 Year fixed effects Y Y Team fixed effects Y Y Country fixed effects Y Y R2 0.121 0.061 Adjusted R2 0.118 0.058 F Statistic (df = 134; 41567) 40.27∗∗∗ 20.71∗∗∗ Note: Robust standard errors in brackets Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

21 5 Conclusions

In this paper we investigate corruption in cricket. Match fixing in cricket has become an issue of grave concern in recent years, and it is believed that much fixing goes on undetected. As such, it is of great practical relevance that more tools be provided to assist the detection of match fixing. In this paper we make use of information on known fixed matches, and exploit the vast richness of data that each cricket match produces to generate insights into the methods via which match outcomes might be fixed. We find that the vast majority of statistics in cricket matches are influenced in the expected direction in the presence of known corruption; for example, fixing teams win tosses, choose to field first, and allow early partnerships to flourish in opposing teams, contributing substantially to large targets being set, which the fixing team is then unable to obtain through weaker batting performacnes. Batsmen in fixing teams are more likely to score fewer runs, be run out, and less likely to be not out, while batsmen facing fixing teams are less likely to be caught out. Considering bowling we find that the statistics of bowlers facing fixing batting teams are better, and bowlers in fixing teams are worse. Although the significance of these coefficients tends to be relatively low, this can be attributed in part to the small number of matches we treat as known to be fixed. Naturally those seeking to fix match outcomes gain financially from doing so, and as such even if current methods are detected, will likely attempt to employ different methods in the future; indeed it could be argued that the more recent emphasis on spot fixing exemplifies this. However cricket remains a highly transparent game statistically in that great volumes of statistics are produced for each match; despite modelling a number of the statistics produced by cricket matches in this paper, we have still barely scratched the surface of what is available. Such great documentation makes hidden action more difficult to carry out, and ought to be encouraged within the game: High levels of statistical recording at all levels of the game, and furthermore ought to be best practice across all sports where there are concerns about match fixing and other forms of corruption. Statistical modelling is a powerful tool to detect corrupt activity; by its nature corrupt activity must have a significant impact upon some aspect of a sporting contest, whether recorded or otherwise. Even spot fixing, where players are encouraged to take particular actions at particular points in matches, in principle is detectable given initiatives such as the widespread ball-by-ball commentary on cricket matches that exists now. Caution must nonetheless be applied; in this paper we are able to exploit corroborating evidence which reduces the likelihood of making false positives. We thus conclude by asserting that the goal of detecting match fixing and corrupt behaviour has a valuable aid in statistical analysis, to which end this paper contributes.

A Known Fixed Matches

List of fixed cricket matches (match-fixing team is written in bold):

1. ODI No. 684, India v Pakistan, Wills Trophy, 6th match, 23rd October 1991, Sharjah Cricket Association Stadium, UAE (Qayyum, 1998).

22 2. ODI No. 697, Australia v India, ODI, 15th December 1991, Adelaide, Benson & Hedges World Series, 6th match, (Central Bureau of Investiga- tion, 2000; Radford, 2011). 3. ODI No. 705, Australia v India, ODI, 14th Jan 1992, Sydney, B & H World Series, 11th match, (Radford, 2011). 4. ODI No. 759, England v Pakistan, 20th August 1992, Trent Bridge ODI, Texaco Trophy Nottingham, (Radford, 2011). 5. ODI No. 813, India v England, 6th ODI, Gwalior, Captain Roop Singh Stadium, Charms Cup 92/93, 4th March 1993, (Radford, 2011). 6. ODI No. 814, India v England, 7th ODI, Gwalior, 5th March 1993, (Rad- ford, 2011). 7. ODI No. 895, New Zealand v Pakistan, 5th ODI, 16th March 1994, Bank of New Zealand Trophy 1993/94 Lancaster Park, Christchurch (Radford, 2011). 8. ODI No. 923, Pakistan v Australia, Singer Cup, Colombo, Sinhalese Sports Club Ground, 7th September 1994, p.187, Qayyum (1998). 9. ODI No. 941, India v West Indies, ODI, Wills World Series 1994/1995, 30th October 1994, Modi Stadium, Kanpur. Central Bureau of Investiga- tion (2000). 10. ODI No. 970, South Africa v Pakistan, Mandela Trophy 1st final, ODI, 10th January 1995, Newlands, Cape Town, Qayyum (1998). 11. ODI No. 971, South Africa v Pakistan, Mandela Trophy 2nd final, ODI, 12th Jan 1995, Newlands, Cape Town, Qayyum (1998). 12. ODI No. 1132, India v South Africa, Titan World Series 1996/97, 29th October 1996, Madhavrao Scindia Cricket Stadium, Rajkot (Central Bu- reau of Investigation, 2000; Radford, 2011). 13. ODI No. 1260, Pakistan v West Indies, ODI, 12th Dec 1997, Sharjah Cricket Association Stadium, Akai-Singer Champions Trophy, Qayyum (1998). 14. ODI No. 1417, India v Pakistan, Pepsi Cup 1998/99, 24th March 1999, Sawai Mansingh Stadium, Jaipur, Central Bureau of Investigation (2000). 15. ODI No. 1470, Australia v West Indies, 1999 World Cup, 30th May 1999, Old Trafford, Preston and Szymanski (2003). 16. ODI No. 1471, Pakistan v Bangladesh, 1999 World Cup, 31st May 1999, County Ground, Northampton, (Berry 2004, The Telegraph) [??? CHASE UP]. 17. ODI No. 1576, India v South Africa, 5th ODI, 19th March 2000, SA in India 1999/00 Nagpur Radford (2011). 18. ODI No. 1915, Zimbabwe v Kenya, 2nd ODI, 11th December 2002, Kwekwe, Radford (2011).

23 B Secondary List of ‘Suspicious’ Matches

Here we present a secondary list of matches that are deemed suspicious, and listed in Polack (2000). Where neither team is highlighted, the umpire is alleged to have been the fixing party in the match.

1. ODI No. 475, Pakistan vs Australia, 1987-11-04, Reliance World Cup, Gaddafi Stadium, . 2. ODI No. 685, India vs Pakistan, 1991-10-25, Wills Trophy, Sharjah Cricket Association Stadium.

3. ODI No. 912, India vs Pakistan, 1994-04-22, Pepsi Austral-Asia Cup, Sharjah Cricket Association Stadium. 4. ODI No. 935, Pakistan vs Australia, 1994-10-22, Wills Triangular Series, Rawalpindi Cricket Stadium. 5. ODI No. 976, New Zealand vs India, 1995-02-16, New Zealand Centenary Tournament, McLean Park, Napier. 6. ODI No. 980, Zimbabwe vs Pakistan, 1995-02-22, Pakistan in Zimbabwe ODI Series, Harare Sports Club. 7. ODI No. 982, Zimbabwe vs Pakistan, 1995-02-25, Pakistan in Zimbabwe ODI Series, Harare Sports Club. 8. ODI No. 984, Zimbabwe vs Pakistan, 1995-02-26, Pakistan in Zimbabwe ODI Series, Harare Sports Club. 9. ODI No. 1067, Pakistan vs South Africa, 1996-02-29, Wills World Cup, National Stadium, Karachi.

10. ODI No. 1078, India vs Pakistan, 1996-03-09, Wills World Cup, M Chin- naswamy Stadium, Bangalore. 11. ODI No. 1107, England vs Pakistan, 1996-08-29, Texaco Trophy, Old Trafford, Manchester.

12. ODI No. 1109, England vs Pakistan, 1996-08-31, Texaco Trophy, Edg- baston, Birmingham. 13. ODI No. 1111, England vs Pakistan, 1996-09-01, Texaco Trophy, Trent Bridge, Nottingham.

14. ODI No. 1167, South Africa vs India, 1997-01-23, Standard Bank Inter- national One-Day Series, Springbok Park, Bloemfontein. 15. ODI No. 1169, India vs Zimbabwe, 1997-01-27, Standard Bank Interna- tional One-Day Series, Bank Park, Paarl. 16. ODI No. 1172, South Africa vs India, 1997-02-02, Standard Bank In- ternational One-Day Series, Crusaders Ground, St George’s Park, Port Elizabeth.

24 17. ODI No. 1173, South Africa vs India, 1997-02-04, Standard Bank Inter- national One-Day Series, Buffalo Park, East London.

18. ODI No. 1174, India vs Zimbabwe, 1997-02-07, Standard Bank Interna- tional One-Day Series, Centurion Park. 19. ODI No. 1175, India vs Zimbabwe, 1997-02-09, Standard Bank Interna- tional One-Day Series, Willowmoore Park, Benoni. 20. ODI No. 1200, West Indies vs India, 1997-04-26, India in West Indies ODI Series, Queen’s Park Oval, Port of Spain, Trinidad. 21. ODI No. 1201, West Indies vs India, 1997-04-27, India in West Indies ODI Series, Queen’s Park Oval, Port of Spain, Trinidad. 22. ODI No. 1202, West Indies vs India, 1997-04-30, India in West Indies ODI Series, Arnos Vale Ground, Kingstown, St Vincent. 23. ODI No. 1203, West Indies vs India, 1997-05-03, India in West Indies ODI Series, , Bridgetown, Barbados. 24. ODI No. 1259, England vs India, 1997-12-11, Akai-Singer Champions Trophy, Sharjah Cricket Association Stadium.

25. ODI No. 1326, South Africa vs Pakistan, 1998-04-23, Standard Bank International One-Day Series, Newlands, Cape Town. 26. ODI No. 1365, Pakistan vs Australia, 1998-11-06, Australia in Pakistan ODI Series, National Stadium, Karachi.

27. ODI No. 1368, Pakistan vs Australia, 1998-11-08, Australia in Pakistan ODI Series, Arbab , . 28. ODI No. 1371, Pakistan vs Australia, 1998-11-10, Australia in Pakistan ODI Series, Gaddafi Stadium, Lahore.

29. ODI No. 1467, England vs India, 1999-05-30, ICC World Cup, Edgbaston, Birmingham. 30. ODI No. 1484, Australia vs Pakistan, 1999-06-20, ICC World Cup, Lord’s, London. 31. ODI No. 1495, India vs West Indies, 1999-09-07, Coca-Cola Singapore Challenge, Kallang Ground, Singapore. 32. ODI No. 1496, India vs West Indies, 1999-09-08, Coca-Cola Singapore Challenge, Kallang Ground, Singapore. 33. ODI No. 1522, India vs New Zealand, 1999-11-05, New Zealand in India ODI Series, Municipal Stadium, Rajkot. 34. ODI No. 1523, India vs New Zealand, 1999-11-08, New Zealand in India ODI Series, Lal Bahadur Shastri Stadium, Hyderabad, Deccan. 35. ODI No. 1524, India vs New Zealand, 1999-11-11, New Zealand in India ODI Series, Captain Roop Singh Stadium, Gwalior.

25 36. ODI No. 1525, India vs New Zealand, 1999-11-14, New Zealand in India ODI Series, Nehru Stadium, Guwahati.

37. ODI No. 1526, India vs New Zealand, 1999-11-17, New Zealand in India ODI Series, Feroz Shah Kotla, Delhi. 38. ODI No. 1544, South Africa vs Zimbabwe, 2000-01-21, Standard Bank Triangular Tournament, New Wanderers Stadium, . 39. ODI No. 1546, South Africa vs England, 2000-01-23, Standard Bank Triangular Tournament, Goodyear Park, Bloemfontein. 40. ODI No. 1549, South Africa vs England, 2000-01-26, Standard Bank Triangular Tournament, Newlands, Cape Town. 41. ODI No. 1555, South Africa vs Zimbabwe, 2000-02-02, Standard Bank Triangular Tournament, Kingsmead, Durban. 42. ODI No. 1557, South Africa vs England, 2000-02-04, Standard Bank Triangular Tournament, Buffalo Park, East London. 43. ODI No. 1558, South Africa vs Zimbabwe, 2000-02-06, Standard Bank Triangular Tournament, Crusaders Ground, St George’s Park, Port Eliz- abeth. 44. ODI No. 1560, South Africa vs England, 2000-02-13, Standard Bank Triangular Tournament, New Wanderers Stadium, Johannesburg. 45. ODI No. 1572, India vs South Africa, 2000-03-09, South Africa in India ODI Series, Nehru Stadium, Kochi. 46. ODI No. 1573, India vs South Africa, 2000-03-12, South Africa in India ODI Series, Keenan Stadium, Jamshedpur. 47. ODI No. 1574, India vs South Africa, 2000-03-15, South Africa in India ODI Series, Nahar Singh Stadium, Faridabad.

48. ODI No. 1575, India vs South Africa, 2000-03-17, South Africa in India ODI Series, Indian Petrochemicals Corporation Limited Sports Complex Ground, Vadodara.

C Additional Tables and Figures

26 Table 6: Control variables included in regression models. Included in regression model No. Name Description Toss Decision Innings 1 Total Innings 2 Total Innings 1 FOW Innings 2 FOW Batsman Bowler Fixed effects 1. team.i Fixed effects for team listed first/second YYYYYYY 2. country Fixed effects for country in which match took place YYYYYYY 3. year Fixed effects for year in which match took place YYYYYYY Match-specific variables 4. daynight 1 if match is day/night match YYYYYYY 5. neutral 1 if match is on neutral territory YYYYYYY 6. dead.rubber 1 if match is meaningless (outcome does not affect overall series/tournament outcome) YYYYYYY 7. [bat/toss].team.series.score.i Series score expressed in terms of batting team YYYYYYY Innings-specific variables 8. FOWx.i Fall of wicket x in innings i ∈ [0, 1] NNYYNN 9. innings.1 Dummy = 1 if first innings NNNNNYY 10. innings total.1 Total score from first innings NNYNYNN 11. bat.team.toss.i Team batting won toss (hence decided to bat) NYYYYYY Team-specific variables 12. [bat/toss].team.home.i Batting [fielding]/[non-]toss-decision-making team is at home (in innings i) YYYYYYY 13. [bat/toss].team.last.score.[opp].i Last score by batting [fielding]/[non-]toss-decision-making team NNNYYNN 14. [bat/toss].team.last.5.scores.[opp].i Average of last five scores by batting [fielding]/[non-]toss-decision-making team NNNYYNN 15. [bat/toss].team.won.last.[opp].i 1 if batting [fielding]/[non-]toss-decision-making team won last match (in innings i) NNNYYNN 16. [bat/toss].team.won.last.5.[opp].i Proportion of last five matches won by batting [fielding]/[non-]toss-decision-making team (in innings i) NNNYYNN 17. [bat/toss].team.E.i Elo-generated prediction for match outcome, expressed as probability batting/toss-decision-making team wins YYYYYYY 18. [bat/toss].team.last.f.[opp].i 1 if batting [fielding]/[non-]toss-decision-making team chose to field last time they won toss NNNYYNN 27 19. [bat/toss].team.exp.[opp].i Number of ODIs played by batting [fielding]/[non-]toss-decision-making team prior to this match NNNYYNN 20. [bat/toss].team.[bat/bowl].exp.[opp].i Average number of overs bowled by batsmen/bowlers in batting [fielding] team in innings i NNNYYNN 21. [bat/toss].team.[bat/bowl].age.[opp].i Average age of batting/bowling [fielding]/[non-]toss-decision-making team NNNYYNN 22. [bat/toss].team.[bat/bowl].deb.[opp].i Number of players in batting [fielding]/[non-]toss-decision-making team playing their first ODI match YYYYYNN 23. [bat/toss].team.bat.mean.runs.[opp].i Average of batting averages by batsmen in batting [fielding] team in innings i NNNYYNN 24. [bat/toss].team.bat.left.[opp].i Proportion of left-handed players in batting [fielding]/[non-]toss-decision-making team NNNYYNN 25. [bat/toss].team.bowl.mean.econ.[opp].i Average economy rate of bowlers in batting [fielding] team in innings i NNNYYNN 26. [bat/toss].team.bowl.left.[opp].i Proportion of left-handed bowlers in batting [fielding]/[non-]toss-decision-making team NNNYYNN 27. [bat/toss].team.bowl.spin.[opp].i Proportion of spin bowlers in batting [fielding]/[non-]toss-decision-making team NNNYYNN 28. [bat/toss].team.bowl.fast.[opp].i Proportion of fast bowlers in batting [fielding]/[non-]toss-decision-making team NNNYYNN 29. [bat/toss].team.bowl.med.[opp].i Proportion of medium bowlers in batting [fielding]/[non-]toss-decision-making team NNNYYNN Player-specific variables 30. [batsman/bowler].debut 1 if player making debut as batsman/bowler NNNNNYY 31. [batsman/bowler].experience Number of innings batted in/overs bowled NNNNNYY 32. [batsman/bowler].age Age of bowler/batsman NNNNNYY 33. [batsman/bowler].left =1 if bowler/batsman left handed NNNNNYY Batsman-specific variables 34. batsman.prop.not.out Proportion of matches (career to date) where batsman has been not out NNNNNYN 35. batsman.prop.caught Proportion of times (career to date) player out where batsman has been caught out NNNNNYN 36. batsman.prop.bowled Proportion of times (career to date) player out where batsman has been bowled out NNNNNYN 37. batsman.prop.run.out Proportion of times (career to date) player out where batsman has been run out NNNNNYN 38. batsman.prop.stumped Proportion of times (career to date) player out where batsman has been stumped out NNNNNYN 39. batsman.prop.lbw Proportion of times (career to date) player out where batsman has been lbw out NNNNNYN Bowler-specific variables 40. bowler.careerecon bowler career economy prior to the match NNNNNNY 41. bowler.career.avwkts bowler average wickets per match NNNNNNY 42. bowler.wickets.last number of wickets bowler took in last match NNNNNNY 43. bowler.econ.last economy rate in bowler’s last match NNNNNNY 44. bowler.wickets.last.5 number of wickets bowler took in last 5 matches NNNNNNY 45. bowler.econ.last.5 economy rate across bowlers last 5 matches NNNNNNY 46. bowler.fast =1 if bowler is fast or fast-medium NNNNNNY 47. bowler.medium = 1 if bowler is medium or medium fast or slow medium (used as reference variable) NNNNNNY Table 7: Full regression output (minus fixed effects coefficients) for regressions for post-toss decision and first/second innings totals. Corresponds to condensed outputs in Table 1 on page 13.

Dependent variable: toss.team.field innings total.1 innings total.1 innings total.2 innings total.2 (1) (2) (3) (4) (5) Constant 0.53 80.039∗∗∗ 118.104∗∗∗ 32.610∗∗ 8.722 (0.393) (16.852) (43.424) (14.103) (35.953)

team.fixing 0.493∗∗∗ −28.456 −24.980 −21.557∗ −17.850 (0.146) (23.416) (22.941) (11.980) (11.885)

team.fixing.opp 0.560∗ 31.326∗∗ 20.952 6.339 0.501 (0.324) (14.496) (14.331) (19.343) (19.024)

innings total.1 0.581∗∗∗ 0.602∗∗∗ (0.014) (0.015)

daynight −0.256∗∗∗ 2.745 4.128∗ −4.933∗∗∗ −6.927∗∗∗ (0.021) (2.124) (2.427) (1.754) (2.013)

neutral 0.037 4.033 5.851∗ 2.668 2.045 (0.030) (2.829) (3.185) (2.405) (2.777)

dead.rubber −0.028 3.918 2.736 4.409∗ 4.896∗∗ (0.026) (2.900) (2.868) (2.399) (2.381)

team.toss 4.508∗∗ 4.398∗∗ −2.375 −2.994∗ (1.918) (1.940) (1.587) (1.611)

team.series.score 0.022∗∗∗ 2.202∗∗ 1.520∗ −0.330 −0.522 (0.008) (0.892) (0.883) (0.738) (0.733)

team.home 0.046 13.532∗∗∗ 16.177∗∗∗ 7.306∗∗ 6.952∗∗ (0.035) (3.963) (4.235) (3.271) (3.515)

team.last.score −0.00003 0.049∗∗ 0.032∗ −0.001 −0.006 (0.0002) (0.019) (0.019) (0.016) (0.015)

team.last.score.opp −0.0004∗∗ 0.035∗ 0.029 0.014 0.011 (0.0002) (0.019) (0.019) (0.016) (0.016)

team.last.5.scores −0.0001 0.136∗∗∗ 0.054 0.073∗∗ 0.016 (0.0003) (0.038) (0.038) (0.031) (0.032)

team.last.5.scores.opp 0.0004 −0.025 −0.013 0.017 0.004 (0.0003) (0.038) (0.039) (0.031) (0.031)

team.won.last −0.018 −0.006 −0.357 3.355∗ 3.824∗∗ (0.021) (2.321) (2.275) (1.965) (1.937)

team.won.last.opp 0.017 −0.994 −0.805 1.192 2.066 (0.021) (2.378) (2.336) (1.917) (1.887)

team.won.last.5 −0.068 6.65 2.767 5.51 2.705 (0.042) (4.610) (4.610) (3.902) (3.927)

team.won.last.5.opp −0.022 −8.388∗ −4.924 −14.683∗∗∗ −11.705∗∗∗ (0.041) (4.720) (4.733) (3.815) (3.827)

team.E −0.029 −0.263 −3.269 1.098 2.041 (0.030) (3.345) (3.722) (2.741) (3.074)

team.exp −0.00004 0.074∗∗∗ 0.061∗∗ 0.043∗∗∗ 0.015 (0.0002) (0.007) (0.029) (0.006) (0.023)

team.exp.opp 0.00005 −0.053∗∗∗ −0.012 −0.035∗∗∗ −0.046∗ (0.0002) (0.007) (0.028) (0.006) (0.024)

team.bat.deb 0.003 0.683 0.451 −0.735 −0.414 (0.015) (1.623) (1.639) (1.390) (1.441)

team.bat.deb.opp 0.066∗∗∗ −1.898 −1.247 0.033 0.256 (0.015) (1.682) (1.735) (1.341) (1.361)

team.bat.exp −0.001 0.170∗∗∗ 0.199∗∗∗ 0.158∗∗∗ 0.174∗∗∗ (0.001) (0.054) (0.057) (0.047) (0.050)

team.bat.exp.opp −0.001 0.182∗∗∗ 0.160∗∗∗ 0.043 0.054 (0.001) (0.057) (0.060) (0.045) (0.047)

team.bowl.exp −0.00004 −0.017∗∗∗ −0.019∗∗∗ −0.006 −0.008 (0.0001) (0.006) (0.006) (0.005) (0.006)

team.bowl.exp.opp 0.0002∗∗∗ −0.008 −0.014∗∗ −0.025∗∗∗ −0.026∗∗∗ (0.0001) (0.006) (0.007) (0.005) (0.005)

team.bowl.deb −0.038∗ 2.022 2.378 0.619 0.729 (0.020) (2.285) (2.278) (1.816) (1.829)

team.bowl.deb.opp −0.018 0.988 2 3.911∗∗ 4.501∗∗ (0.020) (2.198) (2.206) (1.896) (1.898)

team.bat.age 0.019∗∗ −1.473∗ −1.283 −2.131∗∗∗ −2.917∗∗∗ (0.009) (0.800) (0.846) (0.693) (0.746)

team.bat.age.opp −0.003 −0.707 −0.945 0.13 −0.022 (0.009) (0.840) (0.901) (0.664) (0.704)

team.bat.age.sq −0.003∗∗ −0.715∗∗∗ −0.792∗∗∗ −0.427∗∗∗ −0.250∗ (0.002) (0.153) (0.169) (0.133) (0.146)

team.bat.age.sq.opp 0.0002∗ −0.239 −0.462∗∗ −0.061 −0.106 (0.0001) (0.196) (0.209) (0.149) (0.163)

team.bowl.age −0.012∗ 1.835∗∗ 1.736∗∗ −0.712 0.092 (0.007) (0.718) (0.769) (0.586) (0.639)

team.bowl.age.opp −0.019∗∗∗ 1.365∗ 1.271∗ 1.119∗ 2.252∗∗∗ (0.007) (0.705) (0.767) (0.593) (0.638)

team.bowl.age.sq −0.001 0.167 0.095 −0.001 −0.013 (0.001) (0.165) (0.168) (0.140) (0.144)

team.bowl.age.sq.opp 0.002 −0.268 −0.033 −0.174 −0.257∗ (0.001) (0.179) (0.188) (0.139) (0.147)

team.bat.mean.runs −0.003 0.229 0.026 0.916∗∗∗ 0.956∗∗∗ (0.002) (0.180) (0.193) (0.152) (0.166)

team.bat.mean.runs.opp 0.009∗∗∗ 0.02 0.06 0.665∗∗∗ 0.579∗∗∗ (0.002) (0.184) (0.200) (0.149) (0.161)

team.bat.left 0.058 −19.229∗∗∗ −9.492 −0.668 0.187 (0.056) (5.798) (6.387) (4.717) (5.415)

team.bat.left.opp −0.049 11.539∗∗ 12.820∗∗ −11.614∗∗ −5.938 (0.056) (5.704) (6.523) (4.806) (5.305)

team.bowl.econ −0.010 5.829∗∗∗ 2.485 −1.865 −1.477 (0.024) (2.259) (2.722) (2.048) (2.396)

team.bowl.econ.opp 0.006 11.505∗∗∗ 9.444∗∗∗ −3.866∗∗ −2.599 (0.026) (2.468) (2.883) (1.868) (2.261)

team.bowl.left −1.238 21.378 −41.705 57.846 90.222 (0.759) (74.284) (77.207) (92.057) (94.998)

team.bowl.left.opp −0.415 108.086 145.841 −75.458 −56.507 (0.840) (111.545) (114.666) (61.365) (64.073)

team.bowl.spin 0.421 37.282 28.01 19.616 4.014 (0.256) (27.567) (29.171) (19.427) (21.197)

team.bowl.spin.opp −0.764∗∗∗ 35.575 30.82 −6.830 −11.827 (0.225) (23.514) (25.545) (22.775) (24.203)

team.bowl.fast −0.985∗ −128.475∗∗ −155.402∗∗∗ −56.542 −114.766∗ (0.551) (53.940) (56.800) (59.713) (61.680)

team.bowl.fast.opp 1.319∗∗ −92.565 −65.803 12.571 −27.670 (0.578) (72.342) (74.334) (44.584) (47.102)

team.bowl.medium 0.365∗ 33.851∗ 53.610∗∗ 3.823 −16.510 (0.192) (20.353) (22.156) (14.427) (15.986)

team.bowl.medium.opp −0.089 0.017 1.724 19.277 5.356 (0.172) (17.465) (19.258) (16.804) (18.375)

team.last.f 0.047∗∗∗ (0.017)

team.last.f.opp 0.056∗∗∗ (0.017)

Team fixed effects Y N Y N Y Host-country fixed effects Y N Y N Y Year fixed effects Y N Y N Y Observations 3,336 3,339 3,339 3,335 3,335 R2 0.222 0.199 0.266 0.408 0.454 Adjusted R2 0.187 0.187 0.234 0.399 0.429 Residual Std. Error 0.451 (df = 3192) 51.818 (df = 3289) 50.314 (df = 3196) 42.797 (df = 3284) 41.715 (df = 3191) F Statistic 6.368∗∗∗ (df = 143; 3192) 16.705∗∗∗ (df = 49; 3289) 8.175∗∗∗ (df = 142; 3196) 45.315∗∗∗ (df = 50; 3284) 18.534∗∗∗ (df = 143; 3191) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

28 Table 8: Regressions of the falls of wicket in first innings (with fixed effects, although fixed effects coefficients are not shown.

Dependent variable: FOW1.1 FOW2.1 FOW3.1 FOW4.1 FOW5.1 FOW6.1 FOW7.1 FOW8.1 FOW9.1 FOW10.1 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Constant −9.219 10.300 −8.199 17.379 37.681 2.915 4.962 42.553∗∗∗ 15.983 2.459 (32.408) (33.350) (32.238) (30.102) (26.010) (21.161) (19.045) (15.278) (15.135) (14.004) bat.team.fixing.1 −18.563 −18.562 −13.464 −12.544 −11.751 −8.476 23.868∗∗∗ 0.807 −3.372 8.436 (17.121) (17.613) (17.006) (15.744) (13.378) (10.500) (8.708) (7.321) (6.428) (5.761) bat.team.fixing.opp.1 22.382∗∗ 16.239 16.940 −4.082 −2.316 7.497 −5.638 −7.101 −2.022 4.484 (10.696) (11.009) (10.632) (9.847) (9.098) (7.464) (6.512) (6.529) (5.736) (5.581)

FOW1.1 0.999∗∗∗ 0.028 −0.011 −0.005 0.020 0.004 0.014 0.007 0.006 (0.018) (0.025) (0.024) (0.022) (0.019) (0.018) (0.016) (0.018) (0.019)

FOW2.1 0.941∗∗∗ 0.015 −0.018 −0.030∗ −0.003 0.011 −0.016 −0.002 (0.018) (0.023) (0.020) (0.018) (0.017) (0.015) (0.016) (0.019)

FOW3.1 0.900∗∗∗ 0.017 0.040∗∗ 0.023 0.016 0.013 −0.008 (0.017) (0.021) (0.018) (0.017) (0.015) (0.016) (0.017)

FOW4.1 0.903∗∗∗ −0.028 −0.028 −0.007 0.006 −0.005 (0.016) (0.020) (0.019) (0.016) (0.017) (0.018)

FOW5.1 0.911∗∗∗ −0.001 −0.009 −0.020 0.008 (0.016) (0.022) (0.020) (0.021) (0.020)

FOW6.1 0.916∗∗∗ −0.006 0.032 0.007 (0.018) (0.023) (0.023) (0.022)

FOW7.1 0.929∗∗∗ −0.032 −0.042 (0.018) (0.029) (0.028)

FOW8.1 0.948∗∗∗ 0.101∗∗∗ (0.024) (0.033)

FOW9.1 0.879∗∗∗ (0.026) daynight 1.444 −0.938 1.142 4.406∗∗∗ −0.146 2.313∗ −0.533 −0.869 −0.006 1.144 (1.811) (1.863) (1.803) (1.678) (1.460) (1.196) (1.071) (0.905) (0.930) (0.874) neutral 3.019 4.524∗ 0.210 −0.796 −0.020 2.455 −0.402 0.698 −0.232 −0.397 (2.377) (2.447) (2.370) (2.208) (1.912) (1.553) (1.381) (1.131) (1.156) (1.082) dead.rubber −0.177 −1.236 3.234 −2.371 0.801 3.183∗∗ 0.870 −1.354 0.935 1.959∗ (2.140) (2.201) (2.132) (1.994) (1.730) (1.438) (1.293) (1.065) (1.081) (1.045) bat.team.toss.1 1.718 2.300 0.789 1.101 0.917 −0.553 −0.086 −0.289 −0.506 −0.171 (1.448) (1.490) (1.443) (1.347) (1.174) (0.966) (0.868) (0.732) (0.742) (0.710) bat.team.series.score.1 0.393 0.793 0.287 0.427 0.802 −0.073 0.303 0.194 −0.211 −0.673∗∗ (0.659) (0.678) (0.656) (0.615) (0.532) (0.438) (0.401) (0.338) (0.348) (0.327) bat.team.home.1 7.760∗∗ 4.725 4.006 2.999 2.619 2.195 1.106 −0.817 1.830 −2.692∗ (3.160) (3.254) (3.153) (2.951) (2.560) (2.098) (1.894) (1.601) (1.614) (1.562) bat.team.last.score.1 −0.006 −0.011 0.017 0.002 0.001 0.013 −0.009 0.004 0.016∗∗ −0.003 (0.014) (0.015) (0.014) (0.013) (0.012) (0.009) (0.009) (0.007) (0.007) (0.007) bat.team.last.score.opp.1 0.020 −0.0001 −0.013 −0.012 0.007 0.011 0.002 0.015∗∗ 0.008 0.002 (0.014) (0.014) (0.014) (0.013) (0.011) (0.009) (0.008) (0.007) (0.007) (0.006) bat.team.last.5.scores.1 −0.021 −0.006 0.015 0.030 0.055∗∗ −0.028 −0.002 0.022 0.004 0.025∗ (0.028) (0.029) (0.028) (0.026) (0.023) (0.019) (0.017) (0.014) (0.014) (0.013) bat.team.last.5.scores.opp.1 −0.046 −0.007 −0.036 0.007 −0.009 0.00004 0.012 −0.037∗∗ 0.011 0.025∗ (0.029) (0.030) (0.029) (0.027) (0.024) (0.019) (0.018) (0.015) (0.015) (0.014) bat.team.won.last.1 0.311 −1.356 1.401 0.264 −2.152 0.699 −0.014 0.130 −0.440 1.225 (1.698) (1.748) (1.690) (1.580) (1.373) (1.122) (1.007) (0.849) (0.882) (0.851) bat.team.won.last.opp.1 −2.512 0.297 0.751 0.217 −1.449 −0.471 0.918 −0.089 −0.069 −0.325 (1.744) (1.795) (1.736) (1.624) (1.408) (1.154) (1.039) (0.863) (0.871) (0.802) bat.team.won.last.5.1 −0.126 2.596 −1.884 −1.316 −0.418 0.971 3.355 −0.255 1.063 −3.462∗∗ (3.441) (3.539) (3.425) (3.200) (2.780) (2.285) (2.050) (1.704) (1.717) (1.630) bat.team.won.last.5.opp.1 2.264 −2.758 −2.248 −0.773 −0.719 −3.572 0.315 1.718 0.461 −3.664∗∗ (3.532) (3.639) (3.518) (3.280) (2.862) (2.348) (2.097) (1.750) (1.769) (1.666) bat.team.E.1 −2.028 −0.719 −2.467 −2.178 1.051 1.363 −0.331 1.314 −0.810 2.393 (2.777) (2.858) (2.769) (2.590) (2.257) (1.862) (1.696) (1.445) (1.470) (1.470) bat.team.exp.1 0.032 0.018 −0.009 0.009 0.023 −0.017 0.003 0.014 0.017 0.004 (0.021) (0.022) (0.021) (0.020) (0.017) (0.014) (0.013) (0.010) (0.011) (0.010) bat.team.exp.opp.1 −0.014 −0.018 0.004 0.007 −0.005 −0.010 −0.009 0.012 0.001 −0.018 (0.021) (0.022) (0.021) (0.020) (0.017) (0.014) (0.013) (0.011) (0.012) (0.011) bat.team.bat.deb.1 0.703 0.224 0.152 −0.068 −1.003 −0.113 −0.033 0.272 0.747 0.248 (1.223) (1.259) (1.228) (1.145) (0.988) (0.810) (0.713) (0.581) (0.597) (0.602) bat.team.bat.deb.opp.1 −0.450 −1.729 1.400 0.256 −2.513∗∗ −0.011 0.199 −0.278 −0.788 0.598 (1.295) (1.332) (1.287) (1.204) (1.042) (0.846) (0.754) (0.610) (0.626) (0.644) bat.team.bat.exp.1 0.103∗∗ 0.163∗∗∗ 0.064 0.071∗ 0.009 0.022 0.047∗ −0.026 −0.008 −0.049∗∗ (0.042) (0.044) (0.042) (0.040) (0.035) (0.029) (0.026) (0.022) (0.022) (0.019) bat.team.bat.exp.opp.1 0.059 0.148∗∗∗ 0.082∗ 0.035 −0.018 0.034 −0.015 0.006 −0.002 −0.003 (0.045) (0.046) (0.045) (0.042) (0.037) (0.031) (0.028) (0.023) (0.023) (0.020) bat.team.bowl.exp.1 −0.010∗∗ −0.006 −0.009∗ −0.007 −0.003 −0.004 −0.001 0.001 −0.002 −0.003 (0.005) (0.005) (0.005) (0.004) (0.004) (0.003) (0.003) (0.002) (0.002) (0.002) bat.team.bowl.exp.opp.1 −0.006 −0.011∗∗ −0.005 −0.006 −0.002 −0.006∗ 0.0003 0.003 −0.001 0.003 (0.005) (0.005) (0.005) (0.005) (0.004) (0.003) (0.003) (0.002) (0.002) (0.002) bat.team.bowl.deb.1 −0.999 1.562 −2.091 1.306 2.444∗ 3.541∗∗∗ 1.185 1.680∗ −0.926 −0.920 (1.700) (1.749) (1.691) (1.599) (1.398) (1.162) (1.062) (0.899) (0.908) (0.894) bat.team.bowl.deb.opp.1 3.439∗∗ 0.476 0.157 −1.803 2.285∗ 0.548 0.181 −0.085 1.400 −0.036 (1.646) (1.694) (1.645) (1.544) (1.354) (1.140) (1.030) (0.849) (0.863) (0.828) bat.team.bat.age.1 −1.592∗∗ 0.206 −0.830 0.056 −0.209 −0.026 −0.260 0.011 0.281 0.836∗∗∗ (0.631) (0.650) (0.629) (0.591) (0.515) (0.427) (0.383) (0.315) (0.316) (0.291) bat.team.bat.age.opp.1 −0.429 −0.180 0.065 −0.009 0.034 −0.332 0.120 −0.289 −0.474 −0.031 (0.672) (0.692) (0.670) (0.629) (0.551) (0.456) (0.414) (0.349) (0.349) (0.318) bat.team.bat.age.sq.1 −0.170 −0.222∗ −0.123 −0.315∗∗∗ −0.238∗∗ −0.048 −0.135∗ −0.021 −0.078 −0.002 (0.126) (0.130) (0.126) (0.118) (0.101) (0.082) (0.073) (0.061) (0.060) (0.054) bat.team.bat.age.sq.opp.1 −0.334∗∗ −0.025 0.057 −0.385∗∗∗ −0.075 0.014 −0.077 −0.113 −0.022 0.153∗∗ (0.156) (0.160) (0.155) (0.145) (0.128) (0.106) (0.092) (0.077) (0.075) (0.072) bat.team.bowl.age.1 0.528 0.216 0.515 −0.285 1.082∗∗ 0.447 −0.029 −0.040 0.418 0.292 (0.574) (0.591) (0.572) (0.537) (0.472) (0.390) (0.358) (0.303) (0.306) (0.279) bat.team.bowl.age.opp.1 1.016∗ −0.253 0.779 −0.010 0.490 0.058 0.199 −0.451 0.019 −0.404 (0.573) (0.590) (0.570) (0.535) (0.464) (0.383) (0.343) (0.286) (0.288) (0.265) bat.team.bowl.age.sq.1 0.014 −0.153 0.173 0.081 0.082 −0.020 0.030 0.150∗∗ −0.100 −0.107 (0.126) (0.129) (0.125) (0.117) (0.105) (0.086) (0.077) (0.064) (0.066) (0.066) bat.team.bowl.age.sq.opp.1 0.156 0.220 −0.011 −0.084 −0.107 −0.134 0.084 −0.096 −0.022 −0.094 (0.140) (0.145) (0.140) (0.130) (0.113) (0.095) (0.083) (0.069) (0.069) (0.064) bat.team.bat.mean.runs.1 0.316∗∗ 0.045 0.100 0.103 0.036 0.092 −0.131 0.046 0.072 −0.005 (0.144) (0.149) (0.144) (0.135) (0.118) (0.098) (0.088) (0.073) (0.072) (0.071) bat.team.bat.mean.runs.opp.1 0.238 −0.255∗ 0.308∗∗ 0.169 0.082 0.069 0.037 −0.106 0.015 0.102 (0.149) (0.154) (0.149) (0.140) (0.123) (0.103) (0.092) (0.076) (0.078) (0.072) bat.team.bat.left.1 −5.498 −1.417 −3.728 2.697 3.047 −2.090 −3.833 −2.291 −5.935∗∗ −2.509 (4.767) (4.905) (4.751) (4.446) (3.876) (3.183) (2.865) (2.389) (2.384) (2.167) bat.team.bat.left.opp.1 6.405 13.898∗∗∗ 10.092∗∗ 5.562 5.359 −2.220 −2.748 −2.476 −0.900 −0.125 (4.868) (5.010) (4.862) (4.556) (3.984) (3.293) (2.981) (2.468) (2.464) (2.242) bat.team.bowl.econ.1 0.770 −0.049 2.521 −0.105 −0.190 0.362 1.429 −0.383 0.418 0.909 (2.031) (2.090) (2.023) (1.889) (1.646) (1.333) (1.190) (0.985) (1.101) (1.016) bat.team.bowl.econ.opp.1 1.032 3.065 1.444 4.268∗∗ −0.322 1.496 2.402∗ −1.204 −0.644 0.587 (2.152) (2.213) (2.143) (2.005) (1.742) (1.426) (1.274) (1.040) (1.042) (1.029) bat.team.bowl.left.1 −85.331 −6.667 −10.173 128.621∗∗ −62.545 52.256 −54.712 7.863 10.583 −19.415 (57.619) (59.290) (57.297) (54.615) (48.426) (39.636) (35.164) (28.405) (30.772) (28.243) bat.team.bowl.left.opp.1 −41.593 19.416 38.881 86.796 −40.961 56.047 −10.727 −16.741 −27.880 47.873 (85.575) (88.024) (85.026) (78.830) (69.479) (61.677) (51.786) (53.599) (49.183) (49.642) bat.team.bowl.spin.1 −5.132 0.241 15.728 6.088 13.646 −6.156 12.115 −5.041 −7.036 −9.677 (21.770) (22.397) (21.702) (20.124) (17.506) (14.586) (12.837) (10.644) (10.457) (9.542) bat.team.bowl.spin.opp.1 5.070 16.888 5.053 10.304 −13.027 16.987 4.099 10.768 −2.302 7.898 (19.064) (19.611) (19.008) (17.816) (15.444) (12.833) (11.860) (10.173) (10.029) (9.098) bat.team.bowl.fast.1 5.187 15.631 −66.541 −62.200 −52.702 −18.592 −55.398∗∗ −6.693 −2.448 −7.223 (42.390) (43.609) (42.129) (39.125) (33.910) (27.241) (24.285) (19.674) (20.934) (19.376) bat.team.bowl.fast.opp.1 −32.503 −16.337 −116.642∗∗ 26.400 −1.097 −8.961 38.752 15.909 −22.839 −14.061 (55.475) (57.063) (55.128) (51.135) (43.674) (34.838) (29.940) (24.858) (25.650) (29.298) bat.team.bowl.medium.1 −5.545 20.703 7.369 13.712 9.066 −11.533 28.586∗∗∗ 6.050 7.946 −5.386 (16.535) (17.036) (16.514) (15.433) (13.505) (11.173) (10.410) (8.629) (8.749) (9.306) bat.team.bowl.medium.opp.1 −2.262 −15.244 16.884 −9.941 −3.865 7.233 −0.805 0.852 1.669 −0.083 (14.372) (14.784) (14.330) (13.382) (11.658) (9.532) (8.542) (7.140) (7.310) (6.961)

Observations 3,339 3,337 3,321 3,259 3,103 2,821 2,393 1,926 1,436 933 R2 0.079 0.544 0.694 0.769 0.830 0.884 0.908 0.944 0.954 0.972 Adjusted R2 0.039 0.524 0.680 0.758 0.822 0.878 0.902 0.940 0.949 0.966 Residual Std. Error 37.549 (df = 3196) 38.621 (df = 3193) 37.282 (df = 3176) 34.494 (df = 3113) 29.267 (df = 2957) 22.912 (df = 2674) 18.892 (df = 2245) 14.083 (df = 1777) 12.197 (df = 1287) 9.061 (df = 784) F Statistic 1.942∗∗∗ (df = 142; 3196) 26.658∗∗∗ (df = 143; 3193) 50.075∗∗∗ (df = 144; 3176) 71.348∗∗∗ (df = 145; 3113) 99.748∗∗∗ (df = 145; 2957) 139.389∗∗∗ (df = 146; 2674) 150.993∗∗∗ (df = 147; 2245) 203.843∗∗∗ (df = 148; 1777) 181.271∗∗∗ (df = 148; 1287) 181.826∗∗∗ (df = 148; 784) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

29 Table 9: Regressions of the falls of wicket in second innings

Dependent variable: FOW1.2 FOW2.2 FOW3.2 FOW4.2 FOW5.2 FOW6.2 FOW7.2 FOW8.2 FOW9.2 FOW10.2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Constant −26.014 −16.318 11.538 14.224 −6.773 −8.669 −18.126 31.859 12.768 13.897 (30.572) (30.083) (31.308) (29.686) (27.658) (26.514) (23.456) (21.768) (20.453) (19.003) bat.team.fixing.2 0.268 −4.871 3.516 −0.132 −11.222 −7.910 −8.400 −1.734 0.633 −6.541 (10.051) (9.446) (9.403) (8.492) (7.577) (7.038) (6.852) (6.292) (5.546) (4.926) bat.team.fixing.opp.2 −15.226 19.784 5.728 −0.650 −2.131 12.327 17.025 4.827 −3.249 −3.117 (16.090) (16.872) (16.797) (17.542) (18.346) (23.200) (19.235) (16.542) (14.603) (12.080) innings total.1 0.049∗∗∗ 0.042∗∗∗ 0.058∗∗∗ 0.054∗∗∗ 0.066∗∗∗ 0.080∗∗∗ 0.075∗∗∗ 0.064∗∗∗ 0.072∗∗∗ 0.052∗∗∗ (0.013) (0.012) (0.013) (0.013) (0.012) (0.012) (0.011) (0.011) (0.010) (0.010)

FOW1.2 0.938∗∗∗ −0.008 −0.002 −0.007 0.002 0.024 0.012 0.006 0.024 (0.018) (0.027) (0.028) (0.027) (0.026) (0.024) (0.023) (0.022) (0.021)

FOW2.2 0.896∗∗∗ −0.014 0.013 0.029 −0.010 −0.036 −0.003 −0.026 (0.020) (0.027) (0.026) (0.025) (0.023) (0.022) (0.022) (0.020)

FOW3.2 0.902∗∗∗ −0.017 0.024 −0.013 0.019 −0.018 0.011 (0.019) (0.026) (0.025) (0.023) (0.022) (0.022) (0.020)

FOW4.2 0.899∗∗∗ −0.048∗ 0.009 −0.019 0.008 0.003 (0.019) (0.027) (0.024) (0.023) (0.022) (0.021)

FOW5.2 0.908∗∗∗ 0.021 0.025 0.004 −0.023 (0.021) (0.027) (0.025) (0.025) (0.022)

FOW6.2 0.875∗∗∗ −0.068∗∗ −0.060∗∗ −0.002 (0.022) (0.028) (0.027) (0.025)

FOW7.2 0.986∗∗∗ 0.053 0.032 (0.023) (0.034) (0.030)

FOW8.2 0.925∗∗∗ 0.005 (0.026) (0.034)

FOW9.2 0.919∗∗∗ (0.026) daynight −5.492∗∗∗ −1.726 −0.212 −3.603∗∗ 0.937 −5.735∗∗∗ 0.499 −1.578 −1.736 0.172 (1.719) (1.655) (1.702) (1.632) (1.481) (1.413) (1.250) (1.147) (1.085) (0.986) neutral 1.268 0.902 −0.811 0.537 0.576 0.223 4.685∗∗∗ −1.507 −3.893∗∗∗ 0.711 (2.366) (2.265) (2.331) (2.209) (2.000) (1.892) (1.663) (1.521) (1.437) (1.313) dead.rubber 1.134 1.491 2.749 1.562 0.837 3.821∗∗ −0.600 −2.292∗ −0.468 2.072∗ (2.028) (1.937) (1.990) (1.937) (1.768) (1.687) (1.493) (1.375) (1.296) (1.164) bat.team.toss.2 −4.913∗∗∗ −0.994 −1.557 −1.161 −0.234 −1.705 0.210 −0.373 −0.322 −0.618 (1.373) (1.322) (1.363) (1.303) (1.194) (1.139) (1.007) (0.925) (0.881) (0.809) bat.team.series.score.2 0.682 −0.298 −0.644 0.203 −0.848 −0.118 0.342 −0.089 0.689∗ 0.402 (0.623) (0.601) (0.620) (0.596) (0.542) (0.513) (0.462) (0.421) (0.394) (0.359) bat.team.home.2 6.367∗∗ 0.810 1.912 −1.857 1.311 2.322 5.551∗∗ −0.423 −2.745 2.571 (2.996) (2.884) (2.984) (2.856) (2.610) (2.476) (2.192) (2.010) (1.912) (1.739) bat.team.last.score.2 0.007 0.015 −0.005 −0.008 −0.006 0.0003 −0.001 −0.016∗ 0.004 0.007 (0.013) (0.013) (0.013) (0.013) (0.011) (0.011) (0.010) (0.009) (0.008) (0.008) bat.team.last.score.opp.2 0.010 −0.013 0.012 0.019 −0.020∗ −0.007 0.008 −0.006 0.011 −0.004 (0.014) (0.013) (0.013) (0.013) (0.012) (0.011) (0.010) (0.009) (0.008) (0.008) bat.team.last.5.scores.2 −0.041 0.007 0.020 −0.031 −0.006 −0.038∗ 0.007 0.048∗∗∗ −0.007 0.012 (0.027) (0.027) (0.027) (0.026) (0.024) (0.023) (0.020) (0.019) (0.018) (0.016) bat.team.last.5.scores.opp.2 −0.020 −0.022 −0.036 −0.027 −0.023 0.047∗∗ 0.002 0.026 −0.0001 −0.014 (0.027) (0.026) (0.027) (0.026) (0.023) (0.022) (0.020) (0.018) (0.017) (0.016) bat.team.won.last.2 0.992 −1.520 0.342 1.116 2.309 −0.216 0.072 0.701 0.577 0.564 (1.651) (1.590) (1.636) (1.565) (1.436) (1.370) (1.232) (1.122) (1.050) (0.957) bat.team.won.last.opp.2 2.859∗ 0.466 −0.725 −1.943 1.469 0.708 1.252 −0.014 −0.502 0.247 (1.609) (1.545) (1.594) (1.502) (1.380) (1.308) (1.158) (1.060) (1.000) (0.927) bat.team.won.last.5.2 −4.668 3.554 −0.802 −1.963 −2.306 3.934 1.183 0.862 −0.934 −2.518 (3.350) (3.231) (3.347) (3.226) (2.938) (2.816) (2.498) (2.275) (2.165) (1.963) bat.team.won.last.5.opp.2 −11.116∗∗∗ −4.266 −5.611∗ −0.913 −1.546 −10.074∗∗∗ −2.508 0.392 −0.725 1.406 (3.265) (3.145) (3.242) (3.098) (2.840) (2.693) (2.384) (2.186) (2.073) (1.877) bat.team.E.2 −0.523 2.168 0.212 2.852 −0.138 −0.480 −2.987 1.177 1.579 −2.434 (2.623) (2.515) (2.604) (2.501) (2.299) (2.190) (1.973) (1.810) (1.732) (1.580) bat.team.exp.2 0.009 −0.032∗ 0.024 −0.008 0.005 0.016 −0.021 0.038∗∗∗ 0.017 −0.004 (0.020) (0.019) (0.020) (0.018) (0.017) (0.016) (0.014) (0.013) (0.012) (0.012) bat.team.exp.opp.2 −0.030 −0.007 0.001 0.002 −0.036∗ 0.003 −0.014 0.013 0.00001 −0.002 (0.020) (0.020) (0.021) (0.020) (0.019) (0.018) (0.016) (0.015) (0.015) (0.014) bat.team.bat.deb.2 1.167 2.027∗ −1.935 1.578 −0.357 0.023 0.287 −0.798 −0.084 −0.140 (1.222) (1.176) (1.214) (1.148) (1.064) (1.002) (0.894) (0.821) (0.762) (0.730) bat.team.bat.deb.opp.2 −0.213 −1.581 1.495 0.171 2.495∗∗ 0.667 0.573 −0.028 0.103 −1.882∗∗∗ (1.159) (1.126) (1.152) (1.083) (0.981) (0.930) (0.809) (0.752) (0.710) (0.643) bat.team.bat.exp.2 0.066 0.024 0.053 0.094∗∗ 0.060 0.011 0.031 −0.076∗∗ −0.042 −0.007 (0.043) (0.043) (0.045) (0.044) (0.041) (0.040) (0.036) (0.033) (0.031) (0.029) bat.team.bat.exp.opp.2 −0.039 0.097∗∗ 0.124∗∗∗ 0.024 0.095∗∗ −0.032 0.022 −0.017 −0.033 −0.015 (0.041) (0.040) (0.042) (0.042) (0.039) (0.039) (0.035) (0.032) (0.030) (0.027) bat.team.bowl.exp.2 0.005 −0.014∗∗∗ −0.011∗∗ −0.004 −0.007∗ −0.006 −0.004 −0.001 −0.001 0.003 (0.005) (0.005) (0.005) (0.005) (0.004) (0.004) (0.004) (0.003) (0.003) (0.003) bat.team.bowl.exp.opp.2 −0.003 −0.006 −0.011∗∗ −0.010∗∗ −0.007 −0.006 −0.005 0.001 0.003 0.001 (0.005) (0.004) (0.005) (0.004) (0.004) (0.004) (0.003) (0.003) (0.003) (0.003) bat.team.bowl.deb.2 −0.439 3.355∗∗ −0.215 1.578 −1.235 0.337 −2.887∗∗ 0.704 0.550 3.592∗∗∗ (1.563) (1.504) (1.550) (1.496) (1.369) (1.312) (1.155) (1.079) (1.025) (0.930) bat.team.bowl.deb.opp.2 0.949 −0.900 2.039 0.379 2.735∗ 1.651 2.681∗∗ −0.087 −0.024 0.152 (1.607) (1.539) (1.612) (1.555) (1.456) (1.404) (1.244) (1.141) (1.078) (1.026) bat.team.bat.age.2 −1.060 −0.166 −1.275∗ −1.051 −1.092∗ −0.504 −0.615 −0.316 0.710 0.244 (0.645) (0.637) (0.681) (0.670) (0.633) (0.615) (0.563) (0.531) (0.510) (0.489) bat.team.bat.age.opp.2 0.275 −1.283∗∗ −1.922∗∗∗ −0.672 −0.032 0.482 0.430 0.619 −0.109 0.096 (0.611) (0.603) (0.642) (0.639) (0.601) (0.594) (0.537) (0.499) (0.477) (0.436) bat.team.bat.age.sq.2 0.312∗∗ 0.203 0.221 −0.104 0.208∗ −0.138 −0.051 −0.030 −0.046 0.070 (0.129) (0.128) (0.137) (0.131) (0.121) (0.115) (0.101) (0.096) (0.090) (0.080) bat.team.bat.age.sq.opp.2 0.190 0.111 0.184 −0.235 0.251 −0.166 −0.098 0.040 −0.004 0.042 (0.145) (0.146) (0.162) (0.164) (0.155) (0.153) (0.136) (0.126) (0.121) (0.112) bat.team.bowl.age.2 −0.661 1.111∗∗ 1.212∗∗ −0.130 0.524 −0.165 −0.103 0.194 0.430 −0.120 (0.550) (0.536) (0.562) (0.553) (0.505) (0.486) (0.445) (0.409) (0.386) (0.352) bat.team.bowl.age.opp.2 0.499 0.279 0.633 0.400 0.981∗ 0.336 0.334 0.765∗ 0.004 −0.224 (0.546) (0.534) (0.565) (0.549) (0.510) (0.489) (0.442) (0.416) (0.398) (0.379) bat.team.bowl.age.sq.2 0.039 −0.074 −0.108 −0.059 −0.124 0.132 0.238∗∗ 0.004 −0.048 −0.054 (0.123) (0.120) (0.126) (0.121) (0.113) (0.107) (0.099) (0.091) (0.089) (0.082) bat.team.bowl.age.sq.opp.2 −0.177 −0.168 −0.315∗∗ 0.105 −0.063 0.090 −0.117 0.063 −0.147 −0.016 (0.127) (0.125) (0.132) (0.129) (0.120) (0.115) (0.102) (0.096) (0.091) (0.082) bat.team.bat.mean.runs.2 0.772∗∗∗ 0.921∗∗∗ 0.769∗∗∗ 0.805∗∗∗ 0.467∗∗∗ 0.468∗∗∗ 0.033 0.160 0.158 0.072 (0.141) (0.141) (0.150) (0.148) (0.139) (0.135) (0.122) (0.113) (0.109) (0.105) bat.team.bat.mean.runs.opp.2 0.955∗∗∗ 0.569∗∗∗ 0.691∗∗∗ 0.759∗∗∗ 0.218 0.508∗∗∗ 0.017 −0.143 0.065 −0.145 (0.138) (0.137) (0.145) (0.143) (0.133) (0.135) (0.123) (0.116) (0.110) (0.104) bat.team.bat.left.2 2.906 3.329 7.754 0.650 1.681 −0.777 7.205∗ 0.788 −3.621 6.836∗∗ (4.698) (4.626) (4.880) (4.744) (4.379) (4.171) (3.737) (3.475) (3.377) (3.073) bat.team.bat.left.opp.2 4.624 −1.792 −6.749 −0.347 4.304 1.724 4.343 −7.087∗∗ 2.020 −0.129 (4.637) (4.605) (4.816) (4.711) (4.363) (4.176) (3.708) (3.412) (3.239) (2.998) bat.team.bowl.econ.2 4.282∗∗ −3.308∗ −2.014 0.714 −0.907 −2.663 1.947 −1.107 −0.297 0.865 (2.040) (2.007) (2.071) (2.014) (1.838) (1.763) (1.574) (1.433) (1.356) (1.252) bat.team.bowl.econ.opp.2 −1.733 0.303 −2.196 −1.385 −0.369 −1.131 −1.068 2.283∗ 1.733 −0.352 (1.923) (1.852) (1.922) (1.822) (1.662) (1.657) (1.453) (1.338) (1.256) (1.162) bat.team.bowl.left.2 69.659 97.144 −5.794 23.868 1.911 120.286∗∗ 3.382 −13.889 −47.953 34.856 (80.502) (77.489) (77.929) (74.048) (64.473) (58.066) (51.180) (45.997) (48.052) (50.950) bat.team.bowl.left.opp.2 29.554 −32.949 −36.574 −3.974 3.170 −16.759 −28.182 −40.646 16.841 0.009 (54.190) (53.380) (53.450) (51.463) (49.432) (46.090) (39.696) (34.354) (34.009) (31.968) bat.team.bowl.spin.2 11.615 −1.645 −25.537 16.708 12.356 10.827 −22.297∗ 7.424 −4.721 −19.489∗∗ (18.041) (17.330) (17.877) (16.848) (15.474) (14.707) (13.003) (11.887) (11.034) (9.645) bat.team.bowl.spin.opp.2 11.093 31.475 3.856 −3.486 4.766 16.571 −15.604 −3.501 −18.864 −25.737∗∗ (20.701) (19.953) (20.708) (19.469) (18.124) (17.477) (15.063) (13.889) (13.569) (12.017) bat.team.bowl.fast.2 −52.755 5.423 −23.377 −58.630 −4.632 −34.277 −77.606∗∗ 3.770 14.339 21.857 (53.340) (50.522) (55.601) (51.007) (48.609) (43.535) (39.463) (34.699) (31.007) (30.303) bat.team.bowl.fast.opp.2 32.667 −49.118 −54.420 −30.730 −5.223 −43.785 −12.122 5.129 10.660 13.912 (40.519) (38.979) (39.904) (37.510) (33.425) (32.566) (28.252) (25.206) (23.466) (20.697) bat.team.bowl.medium.2 −2.257 4.349 −7.908 −11.544 −7.262 −0.401 −3.414 −20.827∗∗ 5.320 4.515 (13.585) (13.017) (13.423) (12.675) (11.550) (10.964) (9.748) (8.893) (8.467) (7.732) bat.team.bowl.medium.opp.2 −21.974 25.858∗ 14.147 −12.734 −12.744 18.762 4.413 −11.289 −5.501 −8.896 (15.581) (15.033) (15.606) (15.042) (13.898) (13.175) (11.844) (10.710) (10.107) (9.263)

Team fixed effects Y Y Y Y Y Y Y Y Y Y Host country fixed effects Y Y Y Y Y Y Y Y Y Y Year fixed effects Y Y Y Y Y Y Y Y Y Y Observations 3,284 3,150 2,927 2,622 2,306 2,039 1,786 1,577 1,369 1,113 R2 0.106 0.547 0.644 0.743 0.810 0.852 0.894 0.921 0.937 0.958 Adjusted R2 0.065 0.525 0.625 0.727 0.797 0.841 0.885 0.913 0.929 0.952 Residual Std. Error 35.269 (df = 3140) 33.105 (df = 3005) 32.887 (df = 2781) 29.597 (df = 2475) 25.229 (df = 2158) 22.350 (df = 1890) 18.397 (df = 1636) 15.695 (df = 1427) 13.737 (df = 1219) 11.156 (df = 962) F Statistic 2.595∗∗∗ (df = 143; 3140) 25.164∗∗∗ (df = 144; 3005) 34.694∗∗∗ (df = 145; 2781) 48.918∗∗∗ (df = 146; 2475) 62.568∗∗∗ (df = 147; 2158) 73.687∗∗∗ (df = 148; 1890) 92.862∗∗∗ (df = 149; 1636) 112.352∗∗∗ (df = 149; 1427) 121.504∗∗∗ (df = 149; 1219) 146.702∗∗∗ (df = 150; 962) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

30 Table 10: Regressions of the various statistics related to individual batsman performance.

Dependent variable: batsman.runs caught.out bowled.out run.out stumped.out lbw.out not.out (1) (2) (3) (4) (5) (6) (7) Constant 6.503 0.381∗∗∗ 0.393∗∗∗ 0.014 −0.037 0.105∗ 0.127∗ (4.785) (0.097) (0.074) (0.057) (0.030) (0.054) (0.073)

team.fixing −3.136∗ 0.024 −0.020 0.042∗ 0.0004 0.011 −0.055∗∗ (1.811) (0.037) (0.028) (0.022) (0.011) (0.020) (0.028)

opp.fixing 2.220 −0.065 0.021 0.003 0.001 0.003 0.039 (1.963) (0.040) (0.030) (0.023) (0.012) (0.022) (0.030)

innings.1 0.709∗∗∗ 0.035∗∗∗ 0.008∗∗ 0.012∗∗∗ −0.001 −0.010∗∗∗ −0.043∗∗∗ (0.251) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

daynight −0.723∗∗ −0.009 0.018∗∗∗ 0.001 0.001 0.012∗∗∗ −0.023∗∗∗ (0.345) (0.007) (0.005) (0.004) (0.002) (0.004) (0.005)

innings.1:daynight 1.249∗∗∗ −0.004 −0.012∗ 0.005 −0.003 −0.005 0.018∗∗∗ (0.425) (0.009) (0.007) (0.005) (0.003) (0.005) (0.006)

E 2.234 −0.018 −0.0001 0.018 −0.005 −0.036∗ 0.035 (1.665) (0.034) (0.026) (0.020) (0.011) (0.019) (0.025)

series.score 0.130 0.002 −0.002 −0.001 −0.001 −0.0004 0.002 (0.097) (0.002) (0.002) (0.001) (0.001) (0.001) (0.001)

dead.rubber 0.642∗∗ 0.003 −0.0002 −0.006 0.001 0.001 0.001 (0.313) (0.006) (0.005) (0.004) (0.002) (0.004) (0.005)

bat.team.home 1.284∗∗∗ −0.011∗∗ −0.003 −0.002 −0.001 −0.010∗∗∗ 0.027∗∗∗ (0.238) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

bat.team.toss −0.155 0.004 −0.0004 −0.001 0.002∗ −0.001 −0.003 (0.205) (0.004) (0.003) (0.002) (0.001) (0.002) (0.003)

bat.team.neutral 0.240 −0.002 −0.005 −0.008∗∗ −0.001 0.004 0.010∗∗ (0.292) (0.006) (0.005) (0.003) (0.002) (0.003) (0.004)

bat.team.won.last 0.227 −0.001 −0.002 −0.002 0.002 0.002 0.002 (0.247) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

bat.team.won.last.opp 0.140 0.0004 −0.001 0.002 0.001 −0.005∗ 0.002 (0.246) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

bat.team.won.last.5 −0.731 −0.003 −0.002 0.003 −0.001 0.004 0.0003 (0.497) (0.010) (0.008) (0.006) (0.003) (0.006) (0.008)

bat.team.won.last.5.opp −1.209∗∗ 0.007 −0.003 0.002 −0.001 0.016∗∗∗ −0.021∗∗∗ (0.492) (0.010) (0.008) (0.006) (0.003) (0.006) (0.008)

batsman.order −5.119∗∗∗ −0.088∗∗∗ 0.003 −0.014∗∗∗ −0.009∗∗∗ −0.011∗∗∗ 0.117∗∗∗ (0.243) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

batsman.order.sqrt 12.546∗∗∗ 0.222∗∗∗ −0.029∗ 0.066∗∗∗ 0.037∗∗∗ 0.016 −0.300∗∗∗ (0.973) (0.020) (0.015) (0.012) (0.006) (0.011) (0.015)

batsman.prop.not.out −3.999∗∗∗ −0.049∗∗∗ −0.001 0.008 −0.006 −0.010 0.065∗∗∗ (0.844) (0.017) (0.013) (0.010) (0.005) (0.010) (0.013)

batsman.prop.caught −2.874∗∗ −0.026 0.012 0.009 −0.009 −0.001 0.017 (1.123) (0.023) (0.017) (0.013) (0.007) (0.013) (0.017)

batsman.prop.run.out −3.828∗∗∗ −0.027 −0.004 0.019 −0.009 −0.003 0.027 (1.236) (0.025) (0.019) (0.015) (0.008) (0.014) (0.019)

batsman.prop.bowled −3.496∗∗∗ −0.031 0.052∗∗∗ −0.007 −0.008 0.009 −0.010 (1.154) (0.023) (0.018) (0.014) (0.007) (0.013) (0.018)

batsman.prop.stumped −2.711 −0.014 0.020 −0.003 −0.0002 0.001 −0.012 (1.895) (0.038) (0.029) (0.023) (0.012) (0.021) (0.029)

batsman.prop.lbw −3.230∗∗ −0.074∗∗∗ −0.041∗ 0.012 −0.002 0.055∗∗∗ 0.053∗∗∗ (1.353) (0.027) (0.021) (0.016) (0.009) (0.015) (0.021)

batsman.experience 0.047∗∗∗ 0.0001 −0.00002 −0.0001∗∗ −0.0001∗∗∗ −0.0001 0.0002∗∗ (0.005) (0.0001) (0.0001) (0.0001) (0.00003) (0.0001) (0.0001)

batsman.experience2 −0.0001∗∗∗ −0.00000 −0.00000 0.00000 0.00000∗ 0.00000 −0.00000 (0.00002) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000) (0.00000)

batsman.runs.last.5 0.008∗∗∗ 0.0001∗ −0.00001 −0.0001∗∗∗ 0.00001 −0.0001∗∗∗ 0.0001∗ (0.002) (0.00004) (0.00003) (0.00002) (0.00001) (0.00002) (0.00003)

batsman.bat.average 0.165∗∗∗ −0.0002 −0.001∗∗∗ 0.00002 −0.0001 −0.0001 0.001∗∗∗ (0.011) (0.0002) (0.0002) (0.0001) (0.0001) (0.0001) (0.0002)

batsman.age 0.422∗ 0.0001 −0.006 −0.0003 0.0004 0.005∗ 0.001 (0.239) (0.005) (0.004) (0.003) (0.002) (0.003) (0.004)

batsman.age2 −0.008∗ −0.00002 0.0001∗ 0.00001 −0.00001 −0.0001∗ −0.00002 (0.004) (0.0001) (0.0001) (0.00005) (0.00003) (0.00005) (0.0001)

batsman.left 0.354 0.016∗∗∗ −0.006 0.001 0.002 −0.018∗∗∗ 0.004 (0.229) (0.005) (0.004) (0.003) (0.001) (0.003) (0.003)

batsman.debut −1.258 −0.078∗∗∗ 0.015 0.015 −0.011 −0.003 0.065∗∗∗ (1.258) (0.026) (0.020) (0.015) (0.008) (0.014) (0.019)

Team fixed effects Y Y Y Y Y Y Country fixed effects Y Y Y Y Y Y Year fixed effects Y Y Y Y Y Y Observations 59,439 59,439 59,439 59,439 59,439 59,439 59,439 R2 0.158 0.071 0.009 0.009 0.008 0.018 0.139 Adjusted R2 0.156 0.069 0.007 0.006 0.006 0.016 0.137 Residual Std. Error (df = 59310) 23.678 0.481 0.368 0.282 0.150 0.268 0.361 F Statistic (df = 128; 59310) 86.757∗∗∗ 35.551∗∗∗ 4.284∗∗∗ 4.034∗∗∗ 3.606∗∗∗ 8.382∗∗∗ 74.846∗∗∗

31 Table 11: Regressions for bowlers

Dependent variable: bowler.econ bowler.wickets (1) (2) Constant 1.208∗∗ 1.358∗∗∗ (0.552) (0.459) team.fixing 0.183 −0.137 (0.195) (0.107) opp.fixing −0.368∗ 0.125 (0.213) (0.096) bat.team.home 0.185∗∗∗ -0.075∗∗∗ (0.023) (0.013) neutral 0.073∗∗∗ -0.013 (0.028) (0.017) innings.1 -0.047∗∗ 0.165∗∗∗ (0.024) (0.014) daynight -0.135∗∗∗ 0.087∗∗∗ (0.033) (0.019) innings.1:daynight 0.138∗∗∗ -0.103∗∗∗ (0.040) (0.024) toss.bat -0.02 0.019∗ (0.020) (0.011)

Bat.team.E 1.856∗∗∗ -0.629∗∗∗ (0.341) (0.194) series.score 0.011 -0.001 (0.007) (0.004) dead.rubber 0.123∗∗∗ 0.018 (0.035) (0.019) bowler.experience -0.001∗∗∗ 0.0001∗∗∗ (0.000) (0.000) bowler.experience2 0.000∗∗∗ -0.000∗ (0.000) (0.000) bowler.careerecon 0.171∗∗∗ -0.009 (0.030) (0.011) bowler.career.avwkts -0.100∗∗∗ 0.254∗∗∗ (0.029) (0.016) bowler.wickets.last 0.01 0.003 (0.009) (0.006) bowler.econ.last 0.044∗∗∗ -0.008∗∗ (0.007) (0.003) bowler.wickets.last5 -0.009∗∗ 0.015∗∗∗ (0.003) (0.002) bowler.econ.last5 0.089∗∗∗ 0.005 (0.017) (0.010) bowler.age -0.022 0.004 (0.022) (0.012) bowler.age2 0.0003 -0.0002 (0.000) (0.000) bowler.left -0.139∗∗∗ 0.042∗∗∗ (0.025) (0.015) bowler.fast -0.252∗∗∗ 0.201∗∗∗ (0.026) (0.015) bowler.spin -0.171∗∗∗ -0.007 (0.029) (0.015) bowler.debut 1.595∗∗∗ 0.168∗∗∗ (0.151) (0.049)

Observations 41,702 41,702 Year fixed effects Y Y Team fixed effects Y Y Country fixed effects Y Y R2 0.121 0.061 Adjusted R2 0.118 0.058 F Statistic (df = 134; 41567) 40.27∗∗∗ 20.71∗∗∗ Note: Robust standard errors in brackets Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

32 tepst rdc eodinnsttl n ec ac ucmsuigadnmcpro- dynamic a using outcomes match hence and totals innings method. second gramming predict to attempts team victory the the runs match, total the the won between first difference the batting which — team (the on runs the second of depending If batting terms differently in expressed declared is match. is cricket the in won Outcome victory team of of Types margin The Different Equating D since country by year per Cricinfo.com. matches Source: (ODI) International 1971. Day One Total 5: Figure vr eann n h ubro ikt hsn emhsi ad based hand, in has team a chasing a provides wickets such of as number interrupted, and the being and team innings remaining scores. each one-day overs the run to a quantifying adjusting available of of (DL) event for resources method (1998) the authorities, the Lewis in cricketing considers team and How the which chasing Duckworth (plus by a wickets for adopted terms. more worth? target incomparable and method, balls are runs, a and these more wickets proposed yet Both are victory, runs larger solitary the many a a runs. Of by denotes 290 is by balls) victory 300. smallest more being allotted the largest to the first, of remaining), batting the balls team balls run, the 36 no by only (and won using wicket matches wicket 1644 single with a side, over a chasing losing Just by the without by winning won winning from use). been have ranging can (1711) team margin record each on a matches the balls ODI target of all cricket its number of one-day reaches maximum half since team balls, fixed chasing (and a remaining the have wickets hand of matches other terms the in declared on is If victory first. batting team Total Matches Played 24 diinly h AP(inn n crn rdco)o roe n oa (2011) Hogan and Brooker of Predictor) Scoring and (Winning WASP the Additionally, 0 200 400 600 800

Afghanistan

Africa XI

Asia XI

Australia

Bangladesh chasing Total number ofODImatches played by country/team Bermuda

Canada

East Africa em aae n h agtstte ythe by them set target the and managed team) England

Hong Kong 24

ICC World XI h Lmto osdr h ubrof number the considers method DL The

33 India

Ireland

Kenya

Namibia

Netherlands

New Zealand

Pakistan

Scotland

South Africa

Sri Lanka

UAE

USA

West Indies

Zimbabwe Fall of Wicket Scores

● Mean ● ● ● Median ●

●

● 150

● 100 ● Frequency

● 50

● 0

0 2 4 6 8 10

Score

Figure 6: Mean and median scores when wickets fall during innings. Source: Cricinfo on fitting curves to historical data, and gives a percentage of total resources remaining. As such it provides a method for converting wickets and balls (resources) into runs. However, although tables do exist, it is nonetheless a difficult method to replicate due to its being protected by commercial interests. We utilise a simple non-parametric approach to equating scores; we collect, using ball-by-ball data since 2001, all the final innings scores for each wickets remaining and balls remaining combination throughout each innings.25 We restrict attention only to innings that were completed, either first or second innings. This equates to around 650,000 observations, and we use this to project how many more runs would have been scored by a team winning with a certain number of wickets and balls remaining. We collect together all scores with 6 or more wickets remaining, as for balls remaining less than 150 these curves are indistinguishable from each other. We plot all the curves in Figure D; these show that, for example, a team that won with 50 balls and one wicket remaining could only have expected to have scored another 13 or 14 runs, whereas a team winning with 50 balls and 5 wickets remaining could be expected to have added a further 55 runs. On the other hand, if a team won with less than an over (six balls) to spare, regardless of how many wickets remained they would be unlikely to add many more than 6–8 runs.

25It is only in 2001 that cricinfo lists ball-by-ball information for matches.

34 Number of runs scored with wickets/balls left combinations

1 wickets left 6 wickets left 2 wickets left 7 wickets left 3 wickets left 8 wickets left 4 wickets left 9 wickets left

200 5 wickets left 10 wickets left 150 100 Wickets remaining Wickets 50 0

0 50 100 150 200 250 300

Balls remaining

Figure 7: Average number of runs scored (vertical axis) for teams with particular number of wickets (each line) and balls (horizontal axis) left. Source: Cricinfo.

E Team Strengths

With margins of victory calculated, we are able to more accurately calculate team strengths. The ICC does provide ODI rankings for teams, and has done since 2002, but their calculation is not revealed explicitly, nor are they available back to the 1970s.26 A common alternative to official rankings is to calculate Elo rankings (Elo, 1978); each team is given a strength from which a match prediction is calculated, and deviations from that prediction are then used to update team strengths. If team A has true strength at time t of RA,t and team B has true strength at time t of RB,t, then the expected score for team A against team B is: 1 Q E = = A , (2) A (R −R )/400 1 + 10 B,t A,t QA + QB and the expected score for team B against team A is: 1 Q E = = B , (3) B (R −R )/400 1 + 10 A,t B,t QA + QB

RA,t/400 RB,t/400 where EA + EB = 1 and QA = 10 and QB = 10 . Naturally, the true strengths of teams is unknown, and hence in practice one must choose a starting value for RA,t and allow it to be updated after each match. If the actual outcome of the match at time t for team A, SA,t, differs from the expected

26See http://www.icc-cricket.com/team-rankings/odi on the ICC’s official rankings.

35 outcome then that team’s score needs updating; if SA,t = EA,t then the existing strength for each team is accurate. While we could simply treat SA,t as binary, scaling the match outcome between 0 and 1 (as we do) allows information on the closeness of the match to be built into the updating process. Updating in the event of SA,t 6= EA,t is done according to the formula:

RA,t+1 = RA,t + K(SA,t − EA,t). (4)

The factor K can be varied and is conventionally set at 32 although it is often argued that other values produce more “accurate” rankings. The setting of K affects both the convergence of RA to its true value and also the variation around that true value. For example, in a well-known soccer variant of the Elo system, used in Reade and Akie (2013), FIFA World Cup Finals matches are weighted three times what international friendly matches are. We simply apply a weight of 40 for all ODI matches on the basis that, unlike sports such as football, there are no full international friendlies.27 The resulting rankings are plotted in Figure E; long periods of dominance by Australia (yellow) can be traced (larger number equals greater team strength), as well as recently South Africa and historically the West Indies. The major test-playing nations appear distinctly stronger than the associate nations. We will use these team strengths to help explain a number of aspects of match outcomes. We will go deeper than overall match outcome variables as although fixing a match must involve influencing the final outcome in terms of scores, our interest is in the methods as well as the outcomes. It is also possible that the match outcome variable has sufficiently high variation that it will be impossible to spot any pattern in our 18 fixed matches. We thus further consider the fall of wicket of batsmen in terms of runs scored (and overs played), and the statistics recorded for each bowler.

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36 Elo Rankings of ODI Nations

New Zealand Sri Lanka Netherlands Australia South Africa Kenya England Bangladesh Canada West Indies Zimbabwe Afghanistan

1100 India Scotland Pakistan Ireland 1050 Elo Ranking 1000 950 900

1970 1980 1990 2000 2010

Date

Figure 8: Elo rankings of most ODI-playing nations since 1971. Source: Cricinfo and authors’ own calculations.

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