Tie-breaker

In this week's column in our Insights newsletter, I wonder a bit about whether T20 matches really need tie-breakers. 

I’m not convinced there’s anything wrong with a tie. Why are we always trying to break them?

We left Friday’s T20 match between the Black Caps and India after the 16th over. New Zealand needed only 26 runs from 24 balls with plenty of wickets in hand. The WASP had New Zealand almost certain to win. It was well past 11 pm, and our 9-year-old was dozing off.

I read CricInfo’s commentary aloud to my rather more awake son and his friend as we walked to the car. Back at the stadium, the roars we heard from a crowd heavy with India’s supporters during the final over probably meant wickets rather than boundaries, as CricInfo eventually confirmed.

So they were off to another Super Over that would take the game to a too-familiar outcome, well past midnight. And we were off to get the kids to bed.

The drive home had me wondering about tiebreakers.

If both teams end a match with an identical score, is there any fair way of determining which side deserved to win?

Deciding a match on the number of boundaries tends to reward the flashier team over a patient one grinding forward on ones and twos. Is the former really better than the latter?

Equally, handing a win to the team with more wickets in hand says that it’s worse to run out of wickets than to run out of overs in a limited-overs game. Both are surely valuable, so why set the one above the other?

But going to a Super Over is plainly a mistake – not simply because of any recent and repeated unpleasantness. You might think that, because an extra over in a twenty-over game gives us 5% more information about which team is really the better one, it is a fair way of resolving a tie. But this format privileges the team with top-heavy talent over the side with talent spread across its order.

Ties are more likely to happen when both teams have comparable skill, so it is no surprise that picking a winner between them involves some arbitrariness. Worse, every method of choosing can skew the pitch.

Yet nothing in cricket demands every match have a winner or loser. We could just accept that both teams were equally decent on the night.

At least that's better than being forced to consider New Zealand’s performance in the Super Overs. 

I tend to run cricket stuff past Scott Brooker to make sure I'm not beclowning myself too badly as a relatively recent convert to the game. 

Scott had an excellent suggestion for a way of avoiding ties, should one wish to avoid ties. 

At the start of the second innings, flip a coin. The coin flip determines whether the chasing team needs to meet the defending team's score to win, or exceed that score. Both teams then have the full inning to chase or defend a known target. No chance of a tie. And nothing that skews play. I rather like it. If you don't want to allow ties.

And this idea also has merit:

Maybe the side that wins the toss should always has to have one more run than the opponent to win. The side that loses must only be equal to the opposition to win, thereby balancing the winner’s advantage in being able to decide whether to bat first or not.

— Russell Hutchinson (@russellhutchins) February 6, 2020

Cricket and Collaboration

Matt Lowe's job market paper coming out of MIT looks excellent. And I love that he markets it with a tweetstream.

🚨 Revised JMP on intergroup contact 🚨

Some (me) wondered whether this day would ever come. To celebrate I'd like to tell a story about cricket, caste, and different types of intergroup contact. Are you sitting comfortably?https://t.co/VwMC3ulRj3

1/N pic.twitter.com/GELj6yRjgM

— Matt Lowe (@hmmlowe) August 15, 2019

Sri Lanka has finally finished their first innings in the test against New Zealand, so you've time to give the paper a read:

Types of Contact: A Field Experiment on Collaborative and Adversarial Caste Integration

Matt Lowe

Abstract

I estimate the effects of two types of intergroup contact: collaborative and adversarial. I randomly as-signed Indian men from different castes to participate in cricket leagues or to serve as a control group. League players faced variation in collaborative contact, through random assignment to homogeneous-caste or mixed-caste teams, and adversarial contact, through random assignment of opponents. Collaborative contact increases cross-caste friendships and efficiency in trade, and reduces own-caste favoritism. In contrast, adversarial contact generally reduces cross-caste interaction and efficiency.League participation reduces intergroup differences, suggesting that the positive aspects of intergroup contact in the leagues more than offset the negative aspects in this setting.

What a fun design, and great excuse to get to watch a lot of cricket:

From a sample of 1,261 men, I randomized 800 to play in eight month-long cricket leagues, and assigned the others to a control group.  Of those assigned to play, I assigned 35% to homogeneous-caste teams, and the others to mixed-caste teams.  This randomization gave the first type of cross-caste contact: collaborative – those on the same team shared the common goal of winning matches. Once teams formed, I chose opponents randomly to create the second type of cross-caste contact: adversarial– those on opposing teams had opposing goals. I measured intergroup behavioral outcomes one to three weeks after each league ended.

Morning Roundup

The closing of a big set of browser tabs brings a few gems.

• Rohan Pearce at Computerworld notes a warning from the Law Council of Australia that Oz's encryption law breaks a few international privacy standards, and could then come into conflict with GDPR. 
• Jacob Sullum at Reason goes through some shonky anti-vaping research that claims vaping causes heart attacks, but that ignores that the heart attacks happened before the smokers switched to vaping.
• Shoeshine over at the NBR warns that woke investment strategies do not always pay off. I particularly wonder about NZX here. NZX's report on Corporate Social Responsibility suggests that boards with stronger gender diversity show higher returns - but the academic literature says there is no effect. And there's now a Kiwisaver Fund that makes sure not to invest in firms that have inadequate boardroom gender diversity. It all makes for fun pondering. If an investor picked up that Fund based on NZX's assertions about the excess returns to be had through that investment strategy, and was disappointed, could there be recourse?
• A sad end for Capital Liquor, which seems set to be hounded out of existence by the Police and Medical Officer of Health through the DLC process. They've not done anything wrong - they just want to operate during the hours allowed for under the Local Alcohol Policy. But the District Licencing Commission look to be trying to set a 6pm closing time for downtown bottleshops, licence by licence. 
• In this week's coming NBR column, I make passing reference to underarm bowling. One of the younglings here didn't know what I was talking about. If you don't know what that means, watch the documentary or get ready for the re-education camps. Discussion of the docco here. 

Is 400 the new 300 in ODIs?

The current ODI series between England and NZ has been quite extraordinary. England has scored more than 300 in every game, but has lost twice. Yesterday morning, New Zealand’s total of 349 was not only chased down by England, it was chased down with ease, with England losing only three wickets and having 6 overs to spare. 

My sense from my Twitter feed is that the conventional wisdom is as follows:

1. The rule changes dating from October 2012 (that saw two new balls in each innings and a reduction in the number of fielders allowed outside the circle in non-powerplays) allowed for larger scores, as the outfield gaps and still-hard balls allow batsmen to score at will in the final overs.
2. These rule changes coincided with new batting skills honed in 20-20 competitions like the IPL.
3.  New Zealand has been leading the way in showcasing an aggressive approach to cricket; England  prior to now has continued to play with an outdated conservative style, but has now belatedly accepted the new approach, in which “400 is the new 300”.

There is probably much right with this version of events, but I’m not fully convinced. Here are some raw numbers. Since the October 2012 rule changes prior to the current series between England and New Zealand, there were 177 ODI games played involving two teams from the top 8 (defined as the test-playing nations excluding Zimbabwe and Bangladesh), excluding games with a Duckworth-Lewis-Stern reduction in overs. Of those, 56 (or 31%) saw the team batting first score 300 or more. This is certainly a higher rate than we would have seen in past eras, but not as high as conventional wisdom seems to be suggesting. Moreover, getting to 300 still made the team batting first the overwhelming favourite. Of those 56 games with a first-innings score in excess of 300, the team batting first won 48 (86%). More pertinently, of the 18 games where the first innings score only just got to 300 (defined as a score between 300 and 310), the team batting first won 14, which is still a %77 success rate. If 400 is the new 300, it really should be easier to chase down 300 than these data suggest.

I wrote last year about how, once you adjust for mismatches between teams and where the game has been played, there wasn’t much evidence in the data for a general trend towards increasing first-innings scores. Taking all games from the start of the English 2002 season through to the end of the World Cup, controlling for team ability, home-field advantage, and the ground being used, first innings scores since Oct 2012 are only 12 runs higher on average than in the 10 years before Oct 2012. (For data geeks, I describe the exact model at the bottom.)

The following graph illustrates the lack of a trend. The small red dots are the difference between the first-innings score and a prediction based on the team batting, the team bowling, which team (if any) was playing at home, the ground at which the game was played, and allowing for a 12-run premium for the current rules. The solid red dots are a 25-game smoothed moving average, to take out some of the random variation and make any trends clearer. Although there appears to be a bit of an upward trend over the period since October 2012, scores by the end of this period were still only 28 runs higher than in the 2002-2012 period, suggesting that 328 is the new 300!

But now look at the four blue dots. These are the out-of-sample prediction errors for the first four ODIs between England and NZ in the current series. These predictions take into account England’s and New Zealand’s recent (since Oct 2012) batting and bowling form, England’s home-field advantage, and how high scores typically are at those four grounds. The prediction, actual score, and prediction error are as follows:

Ground	Predicted Score	Actual Score	Prediction Error
Edgbaston	235	408	173
The Oval	256	398	142
The Rose Bowl	269	302	33
Trent Bridge	232	349	117
Chester-le-Street	225 (Eng) 228 (NZ)	?	?

The point here is that rather than there having been a world-wide trend in the past few years that England have only now come to grips with; the current series has been extraordinary in every respect even in comparison to recent history. So what is going on? I can think of four hypotheses:

1. I have made a massive coding error in my database.
2. There has been a structural break in conditions: The four English groundsmen have produced very different pitches than in the past, ones much more favourable to high scores.
3. There has been a structural break in team quality: Both NZ and England have better batting and/or worse bowling in this series than they had in the recent past.
4. These four games have been black-swan events; and things will return to normal soon.
5. There has been a strategic mindset shift in both New Zealand and England.

When things look extraordinary, coding errors are always a good bet, and I wouldn’t rule this out, but the raw predictions don’t look too far out from my own intuition, so I  don’t think this is the problem here.

I can’t comment on whether conditions were very different from usual in the four games so far, but I haven’t seen any commentary from England suggesting that the groundsmen have been producing untypical pitches, so I suspect hypothesis 2 is not the right one.

There is probably some truth to hypothesis 3. I don’t think the batting is too much different, but the bowling is quite possibly weaker. New Zealand have lost Vettori, Anderson and Milne from their World Cup bowling line-up, and have had Southee and Boult together for only one of the four matches. England have rested Anderson and Broad. Even so, I would put my money on the final two hypotheses explaining most of the data. 

The idea that teams are not aggressive enough when batting is something Scott Brooker and I have been saying for a long time. Back when he was writing his thesis, Scott experimented with what an average team would be able to achieve if they applied the optimal level of aggression. Based on the strike rates and dismissal rates that we can observe batsmen having in different game situations (e.g. conservative batting in the middle overs versus aggression in the death overs), he constructed a set of frontiers describing the trade-off between risk and return for typical batsmen in positions #1-#11, and then simulated optimal behaviour. He found that scores could be roughly 30 runs higher if batting teams were more aggressive, but that there would be more variance in scores and a higher probability of not batting out the overs. This was based on data from before 2007. It is likely that under the new rules, the value of extra aggression is even higher. 

What I think we are seeing in the current series is two teams who keep pushing the boundaries of this approach, forcing the other team to react in kind, and so all kinds of previously unrealised potential that has existed for a while is now being revealed. Contrary to the conventional wisdom, I don’t think this has been New Zealand’s approach before now. Rather, I think they have emphasised retaining wickets during the middle overs in preparation for an all-out assault in the final 10. New Zealand’s famous aggression in the World Cup was mostly seen in its approach to bowling, putting an emphasis on wicket taking rather than containment.

If I am right, that there has been a mind-set change for both teams in the current series, I am mindful that Scott’s conclusion was that the additional 30 runs on average would come alongside a big increase in variance. This brings me back to the black-swan hypothesis. We have seen scores more than 100 runs in excess of what recent form would have predicted on average. It is likely that in each game, there was a degree of luck. Batsmen got away with taking risks on these occasions, but we could just as easily have seen scores that were quite low. Even allowing for the fact that dropped catches are more likely when batsmen are hitting the ball hard, the catching in the current series does seem to have been below par. Realistically, 370 might be the new 300, but equally 180 the new 200.

Method:

My prediction model was based on a database of all non-rain-affected ODIs involving the top-8 countries since May 2002, using only games played on grounds where there were at least 10 matches played (but also including Chester le Street, as that is the venue for the final ODI between England and NZ).

The model was an OLS regression of first innings score on a dummy variables for the batting-team country, dummy variables for the bowling-team country, a dummy variable for each of the 53 grounds, a dummy variable for when the batting team was playing at home, and for when the bowling team was playing at home. Finally, I added a dummy variable for matches played since Oct 2012, and more dummy variables for this recent-era interacted with the batting and bowling team dummies. 

These interaction teams mean that only post-Oct-2012 data is used to determine the effect of team ability on scores; the only reason for pooling the data with the pre-Oct-2012 era is to provide enough data to estimate ground effects. Essentially, the model is assuming that the relative impact on scores of being at a particular ground and the relative impact of home-field advantage has not changed from before the Oct 2012 to after.

In the 25-game  moving average shown in the chart above, the data is broken around the structural break of Oct 2012, so that the smoothed line before the break is not influenced by games played after the break and vice versa. 

Bleg: Quantifying the value of bowling variety

A common critique of the current English ODI side is that they suffered from having a sameness to their bowling attack--a series of right-handed fast-medium swing bowlers. It's an interesting question. Obviously, diversity in bowling styles can only take you so far: there is a limit to how much aggregate quality a team would be prepared to sacrifice in order to increase diversity, but it is not clear that there is any value to diversity at all. The question was raised recently in the following tweets:

@CricketFanBob Variety is often said to be advantageous, but has this ever been tested with data? May not be true.
— ballsintherightareas (@ballsrightareas) March 3, 2015

There are some obvious ways in which diversity might improve a team's bowling as a unit. First, having the style of bowling change from over to over might make it harder for batsmen to settle into a rhythm. Second, if some bowling types are more effective wicket takers against right-handed batsmen and others against left-handed, then style diversity might help stop one batsman running away with a game. But these are big mights. The twitter thread above led to this request:

@ballsrightareas @CricketFanBob I wonder if we can interest @seamus_hogan to take on this challenge?
— Declaration Game (@chrisps01) March 3, 201

I'm up for the challenge, but I can't see obvious solutions to three conceptual problems:

1. As @CricketFanBob asks, how do you define variety?

If you look at the cricinfo player profiles, you will find that Bill O'Reilly and Clarrie Grimmett, who played in the same Australian test team, are both classified as "legbreak googly". This is true, but this simple classification does not tell you that O'Reilly was an unusually fast spinner who liked to bowl with the wind, while Grimmett was a more-classical flight-into-the-wind legbreak bowler. Similarly, player profiles will tell you that Joel Garner and Malcom Marshall were both "right fast", but the difference between quite fast delivered by a 6'8" bowler and extremely fast delivered by a 5'11" bowler is probably quite substantial. Maybe, however these examples are sufficiently rare that simple cricinfo categorisations are sufficient. But...

2. ...What is the best way of aggregating these bowling-type classifications into a measure of variety.

Is RFM, RFM, LF, RM, SLO more diverse than RFM, LF, RM, SLO, SLO? I think so, but how do you quantify that. And above all,

3.... How do you assess what performance a given set of individual bowler abilities would be expected to produce in order to assess whether variety (or its absence) can explain some of the difference?

In particular, how do you control for the endogeneity that, for example, a spin bowler in spin-friendly conditions will probably a) be in a team with other spin bowlers to take advantage of those conditions, and b) likely to do better than average because of those conditions, making it difficult to infer any value to diversity that might exist.

I have some ideas, but I suspect that the number of variables needed would exhaust the useful degrees of freedom. Any thoughts?

Competitive ODI matches

Before the current cricket world cup started, the International Cricket Council (ICC) announced that the next event (in 2019), would feature only 10 teams, the eight highest-ranked to qualify automatically, and two to be selected by a qualifying tournament to be played in Bangladesh in conditions totally different from the ones that will prevail in the tournament proper, which is to be held in England. 

This is a quite horrible policy. Quite apart from it being totally detrimental to attempts to develop interest in the game in countries outside of the traditional powerhouses, the presence of the so-called "associate nations" in the current tournament has given it so much of its colour, both through their players and their fans. It may be the games at the pointy end of the tournament that matter the most in terms of finding a winner, but the games involving associate nations in the group stages were wonderful celebrations of the game, and a tournament without them would be so much poorer. 

In words that he almost certainly now regrets, ICC chief executive, David Richardson, gave as one of the justifications for the move that 

[T]he World Cup itself, the premium event, without exception should be played between teams that are evenly matched and competitive. 

Twitter is having a field day with this, as the games involving two top-eight teams in this world cup have been anything other than competitive, including the four quarter-final matches. This leaves me to wonder if there is anything in the rule changes that were brought in a couple of years ago that are making games less competitive in the sense of seeing fewer games where there is real uncertainty about the outcome until late in the second innings.

The results in the current tournament are exhibit A for the hypothesis that games are becoming less competitive, but it may be that there has simply been a widening in ability between the top and bottom teams in the top 8. I believe, however, that there is good theoretical reasons to expect to see fewer close finishes, even in games between evenly matched teams.

To explain, consider the following fictitious game. I go first and draw 10 random numbers from a distribution, and my score is the sum of the 10 numbers. Before each draw, I can choose the mean and variance of the distribution, according to a menu of choices in which there is an inverted-U shaped relationship between variance (on a horizontal axis), and mean (on the vertical). I maximise my expected score by choosing the variance in the middle aligning with the peak of the upside-down U. To complete the game, my opponent also draws 10 random numbers, choosing from the same menu the mean and variance before each draw, and wins if his total exceeds mine. Now if I luck out and get a very high score, my opponent will be best advised to choose a high variance strategy. Occasionally, he will succeed in chasing down my high score, but most likely he will fail and lose heavily . On the other hand, if I have very poor luck, my opponent should choose a low-variance strategy, which won't maximise his expected score but will maximise his probability of beating mine. In this case, he will likely win easily. On average, the player going second will win more than 50% of the games as he will have the chance to adjust his strategy to the score of the player going first.

To continue with this analogy, now bring in a rule change that sees the inverted-U move such that the score-maximising strategy has a higher variance. If this happens, we would expect to see a bigger range of scores by the players going first, and as a result of this:

1. an increase in the winning percentage of the player going second, and
2. a decrease in the number of games where the result is still uncertain when the player drawing second only has 2-3 numbers left to draw. 

So what is the cricket analogy here? We know that batting teams have a lot of control over the level of risk. Bowling teams also control risk levels, mostly through decisions on how attacking to make their field settings), but we also know that faster scoring happens in the first innings as wickets become less costly to batting teams, suggesting that the batting teams have more strategic control over risk. So in my analogy, the player drawing first and choosing the risk level is the team batting first, and the advantage to the second player is the second-innings advantage that is borne out in the data but not always accepted by captains when they win the toss. The rule change I am thinking of is the restriction to having no more than 4 fielders outside of the circle in non-power play overs, down from 5 previously. Now having more fielders inside the circle makes it harder for a batting team to score if they are playing conservatively, but makes it easier to score when taking risks. The fielding restrictions then gives the team batting first an incentive to take more risk. We saw this when the batting powerplay was first introduced. It only increased the average score of the team batting first by about 4 runs, largely because games where the batters were able to use it to heavily increase their scoring were balanced by others where the powerplay led to a quick loss of wickets. 

I haven't had a chance to look at the data before and after the latest rule change (dating from October 2012), but my prediction is that, once you control for team ability, we will have seen 

1. an increase in the advantage to batting second if you win the toss; and 
2. a decrease in the number of games where the game is still in the balance late into the second innings. 

I probably won't get a chance to crunch the numbers any time soon, but if anyone wants to run with testing these hypotheses with the data, be my guest, I'm happy to be a co-author. 

Batting out your overs

The mantra that "the biggest sin a team batting first in an ODI can commit is to not bat our its overs" has long been a bugbear of mine. As Dan Liebke noted in a rant about net-run-rate the other day, 

We've had Duckworth Lewis for decades now and, even if the mathematics of it is beyond most casual fans, the basic concept that wickets remaining are a resource that need to be considered along with overs remaining is pretty well established. 

Yes, a team has two resources. If it is a sin to not use one of those two resources to the max, why is not also a sin to bat out 50 overs leaving capable batsmen in the pavilion with their pads on? A batting team has to manage both declining resources with no certainty as to the effect that its actions will have on either the rate of scoring or the loss of wickets. 

So I was very happy to see Chris Smith take on this mantra in his Declaration Game blog, and also to see him quote a former player, Geoff Lawson, who was prepared to take a contrarian view. 

`Why?' asked Geoff Lawson, who went on to rationalise that if all the batting side attempted was to survie the 50 overs, they were very unlikely to set a winning total. `Wouldn't it be better', Lawson argued, `to hit out wiht the aim setting a challenging targe, accepting the risk that they could be bowled out, than to crawl to an unsatisfactory total?'

Lawson is right, although maybe not quite. In this quote, he seems to be suggesting that a team that is heading towards a very low score might as well start taking more risks to get to a competitive total. This is a manifestation of a mathematical theorem known as Jensen's inequality, when optimising over a relationship that is not linear, but actually, the relationship between the total score and the probability of winning is pretty much linear over the range of possibilities that can occur on any particular ball. That means, that a batting team should always ignore the current score, accept bygones as bygones, and base their level of aggression on how many balls and wickets they have remaining.

As it happens, we can quantify this decision reasonably precisely. The graph below gives a measure of what I like to call "deathness" for the first innings. The particular metric I use is the payoff to a risky single. Imagine that the batsmen have to choose between trying for a run or not. If they choose not to run, they will score 0 runs but not lose a wicket. If they try for the run, there is some probability that attempt will fail and one batsman will be run out, or they might succeed. What probability of being run out would be too high to make the risk not worth the cost. The graph shows that cross-over probability as a function of the number of overs bowled, for each possible number of wickets lost. The higher is the probability, the greater is the risk that it is worth taking and so the greater is the level of deathness (so called, because the final overs in an innings where batsmen start to take higher levels of risk is often termed "the death"). The actual numbers aren't particularly interesting (most decisions on aggression are about striking the ball, not about whether to attempt a run), but the comparison across different lines in the graph is revealing. So, for example, the graph reveals that if a particular level of aggression is warranted after 40 overs when a team is 5 wickets down, then the same level can be justified at 23 overs if no wickets have been lost.  

Before getting to batting out your overs, a few things to note about this graph:

1. It is based on WASP data that predates the rule change to two new balls and only four outside the circle. That said, the basic story would not change using more recent data or some other estimate of the cost of a wicket such as the Duckworth-Lewis tables. 
2. This table indicates what the expected payoffs are to different levels of risk and return in different game situations; it does not show what different risk-return combinations are possible. So, for 0-7 wickets down, the graphs indicate that the cost of risk is high at the start of the innings (the probability of a run-out has to be very low to justify attempting a run). With the fielding restrictions in the first 10 overs, however, it can be that the return to batsmen from a particular level of risk is much higher than in the middle overs, so that a high-risk strategy is still worthwhile, despite the costs. 
3. The graphs all hit 100% for the final ball of the innings. That makes sense. It is simply saying that as long as there is any probability whatsoever of not being run out, you might as well keep running until you lose your wicket on the final ball.  
4. Interestingly, though, for 1, 3 and 6 wickets lost, the graphs hit 100% before getting to the final ball of the innings. Remember that this is based on average-team versus average-team data. What is going on here is that on average the batters deeper in the batting order, are better at power slugging than those further up the order. So, for example, it is common for a batting order to have two aggressive openers followed by an accumulating #3 to take the team through the middle overs. If a team gets to 43 overs with only one wicket down, it might be better to go for a suicidal run (with the #3 coming to the danger end) and bring in a power hitter than to play out a dot ball. 
5. The graph for 9 wickets down slopes down for most of the graph. This is mostly reflects out-of-sample extrapolation (there is no actual data for games where a team is 9 wickets down after 2 overs), and also the fact that when a team is 9 wickets down very early, there is almost no chance they will bat out their overs, and are likely to lose their last wicket any time so its worthwhile the batters taking risky singles while they are still there to do so. The longer the innings progresses, the less reason there is to think that the next wicket is imminent and so more need for caution. 
6. While there is a general tendency for the graph to be lower the more wickets that have been lost, this tendency is not absolute. This is because, while losing a wicket will reduce the expected number of runs the team will score, the cost of the next wicket is not necessarily greater. For example, after about 46 overs, the incremental cost to a team of losing its 7th wicket is less than losing its 5th or 6th at that stage, so a team being 6 wickets down should be more aggressive than one that has lost only 4 or 5 wickets. 

So let's now think about batting out your overs. In the World Cup game between New Zealand and England, England batting first lost their 6th wicket at 28.1 overs, their 7th later in the same over, and their 8th at 30.4 overs. Looking at the purple, yellow and pink lines, the deathness measures at 28-30 overs, are all pretty much the same. Yes, a lot more caution was called for than if they had only been 2 wickets down at that point (and so Broad's approach at that point was probably not beyond reproach), but also the optimal strategy was not for the team to go into its shell. Rather the situation called for playing in much the same style as any team should do in the middle overs (10-30) but delaying all-out aggression for a bit longer than if they had more wickets in hand. This pretty much describes any situation where the "make sure you bat out your overs" comment is likely to arise. A team should probably delay its all-out assault for a bit if it loses too many wickets, but at no point should it bat more conservatively than in a normal middle-orders situation.  

The trouble with Net Run Rate

In any competition in which there is pool or round-robin play to rank teams before playoff rounds, there needs to be some method of deciding the relative ranking of teams who finish equal on wins and losses. Ideally, this method will reward the teams that have performed best, and also not create any perverse incentives for teams to do anything other than act in a way to maximise their probability of winning.

A nice example of perverse incentives came in the 1999 Cricket World Cup. Only two teams out of New Zealand, Australia, and West Indies were going to carry on from their group into the next round. The rules were such that teams carried through only their results against other teams that made it to the next round. Prior to the match between the West Indies and Australia, New Zealand had beaten Australia but had lost to the West Indies. Australia therefore needed to beat the West Indies, but also wanted WI to be the team that carried through with them so that their loss against NZ didn't matter. As is traditional in the Cricket World Cup,the method used to rank teams with equal numbers of wins and losses, was net-run-rate (NRR)--the difference in a team's average runs scored per over faced and its average runs conceded per over bowled. Batting second, Australia therefore did a deliberate go-slow in order to win, with their 5th wicket partnership taking an extraordinary 127 balls to score the 49 remaining runs needed for a win. This was designed to elevate the West Indies' NRR above New Zealand's. As it turned out, the strategy was not successful, as New Zealand still had a match against the lowly ranked Scotland, and took extraordinary risks to not only win that match but win it by a sufficient margin for their NRR to overtake the West Indies'.

In the current World Cup, there isn't the same "super 6" 2nd stage where teams only carry through some of their points from the first round, but NRR is still used as the tie-breaker. This system is still flawed, as exemplified by Tuesday's match between New Zealand and Scotland. Anyone looking at the two innings scored could be mistaken for thinking that the match was close. It wasn't. What happened was that New Zealand bowled Scotland out for a very low total, and was almost guaranteed a win. When it was New Zealand's turn to bat, they strove to win the match in a few overs as possible, in order to maximise their runs-per-over figure. The fact that they lost 7 wickets in the attempt meant that they did present Scotland with the sniff of a chance of an upset, but the 7 wickets will have no bearing on their eventual NRR.

This exemplifies three problems with NRR:

1. The effect of a large win against a lower-ranked team on NRR depends on which team bats first, since the team batting second only bats until it has overtaken the other team's score, meaning that that innings gets a lesser weight in the runs-per-over calculation than an innings where all 50 overs are faced. 
2. The magnitude of a victory when the team batting second wins is a function not only of how many balls it took the team to amass the winning total but also the number of wickets lost in the process. NRR only takes the former into account. This creates the perverse incentive where New Zealand put their win (slightly) at risk by worrying only about how many overs they used and not how many wickets they lost. 
3. The ranking of two or more teams should not depend on which one beat up the most on a team ranked well below them. If, as could easily happen, three teams (say, Australia, New Zealand and Sri Lanka), finish in a tie for first place in their group, the determination on goes through the quarter finals ranked 1st, 2nd, 3rd, should not come down to which team beat Scotland b the biggest margin.

So with these flaws in mind, here is a sequence of proposals to replace NRR with a different tie-breaking rule.

Adjustment 1: To deal with the first problem above, use the average margin of victory/loss rather than NRR: If the team batting second loses, its margin is its score divided by the score required to tie the match. This will be less than 1. The winning team's margin is the reciprocal of this--the target score divided by the chasing team's score. If the team batting second wins, its margin is the number of balls available to it + 1 divided by the number of balls actually used. The losing team's margin is again the reciprocal of this. In the case of a tie, the margin is 1.0 for both teams.

Adjustment 2: To deal with the problem of teams sacrificing wickets for the sake of fast scoring, amend Adjustment 1 in the case where the team batting second wins, by dividing the predicted score at the end of 50 overs by the score required to tie (the implicit score predictor in Duckworth-Lewis would work for this, although I'd prefer to use WASP due to its adjustment to conditions).

Adjustment 3: Make the calculations iteratively. Let there be n teams in a pool. Construct the table at the end of pool play using points scored, and using Adjustments 1 and 2 to rank teams otherwise tied. Then remove the bottom-ranked team and give them a rank of n. Now reconstruct the table using only games played amongst the remaining n-1 teams, and again find the lowest ranked team. Give it rank n-1, remove it and reconstruct the table with the remaining n-2 teams, etc. As an example of how this could be beneficial, imagine that in the current world cup, Sri Lanka beat Australia, Australia beat NZ, and all three beat England and the other three teams except that the game between Australia and Scotland is rained out. Under the system in place for this competition, Sri Lanka and NZ would finish ahead of Australia simply because Australia were denied to opportunity to play Scotland. Under Adjustment 3, the games against Scotland would be irrelevant for deciding the relative ranking of the top three teams. *

Adjustment 4: O.K. now I am getting well out of the realm of feasible rules into the kind of competition we would have if the ICC comprised exclusively economists, but it is fun to speculate. My adjustment 2 still does not properly align incentives because maximising the expected margin of victory is not the same thing as maximising the probability of victory. So instead, let's define the margin of victory in the following way. Draw the WASP-worm graph of the percentage probability of winning for the second innings as a function of the number of balls bowled. This is a graph  is contained within a rectangle that has a length of 300 and a height of 100. The value for the team batting second would be the area under the graph divided by the area above it. The value for the team batting first would be the reciprocal. Using this method, it would be possible for the winning team to have a lower score than the losing team, but no matter: this scheme means that the way to maximise your team's tie-break variable would be to maximise your probability of winning.

Adjustment 4 tries to align incentives with the only thing that should ever matter in sport--trying to win--but it doesn't deal with the situations like Australia's go-slow against the West Indies in 1999 (or NZ's go slow against South Africa three year's later that shut Australia out of their own tri-series final). The format used in this year's World Cup does not contain the possibility of such strange incentives, but Adjustment 3 would add that. With an obvious nod to the Gibbard-Satherwaite Theorem and Arrow's Impossibility Theorem, then, let me suggest throwing all of these out the window and instead using the following manipulation-proof tie-breaking formula:

Adjustment 5: Rank all teams leading into the tournament based on recent performances. In the event of two or more teams being tied on points at the conclusion of pool play, their relative ranking will be according to their pre-tournament ranking, fully independent of play during the tournament.

* The ICC might argue that they have addressed the problem in a simpler way by restricting the next tournament to only 10 teams. But the results to date in this World Cup suggest that there will still be some very weak teams and non-competitive matches given the non-competitive process for selecting the 10 teams.

A safe and proper tournament

The practice, also called courtsiding (it first emerged in tennis), is not illegal, but the gambling markets it serves, mostly in India, usually are. The International Cricket Council (ICC) prohibits it in the fine print of its match tickets and would rather avoid the unsavoury association.

Police help the ICC make sure of it. Specially-trained plain-clothes officers are patrolling New Zealand matches through their mandate with the ICC and the Government to help deliver a safe and proper tournament. Essentially it is the same powers they have in evicting a drunk person, but it speaks of the seriousness with which pitchsiding is treated. All the men kicked out of Hagley Oval on Saturday were banned from other World Cup matches and face arrest if they try to gain entry. Past this, however, they are not in criminal trouble.

So says The Press.

The fine print on the entry tickets to my house note that, as condition of entry, if the guests are there as we put the kids to bed at 7:30 pm, the kids will likely ask them to sing bedtime songs. Some guests have been reluctant to do so, and we've not pressed the issue. But now that we know that the police are happy to enforce general Terms and Conditions of entry, well, that guests sing for their supper will be better enforced.

Seriously, though: what specifically authorises the New Zealand Police Force to go around looking for breaches of ticket terms and conditions? Why did anybody think this was an appropriate use of police resources?

The police lament their resourcing issues when they argue that all the bars should close at ridiculously early hours: Basically, nobody should be allowed to be out late because it makes rostering too hard for the cops. And yet they've time to send plainclothes detectives out to stop people from telling other people that a bat hit a ball?

If the ICC were running the nightclub-bouncer part of the operation, then asking the Police to enforce trespass orders against those they didn't want at the matches, I could understand it. But sending in cops to watch for breach of ticket terms and conditions?

Aggressive Opening Batmen in ODIs

After five games in the 2015 cricket World Cup, an interesting pattern is emerging: So far, the average score of the team batting first has been 323 runs (well above the historical average for ODIs), and has gone on to win the match in four of the five matches, and yet in three of the four cases where the team batting first won, it was the losing team that won the toss and sent the opposition in to bat. Captains are likely to see this pattern and adjust their strategy, but I think that would be a mistake.

One of the interesting things me to watch out for going into this World Cup was the approach taken by both the batting and bowling teams in the opening overs of the first innings.It has become conventional wisdom that in the modern game it is crucial for batsmen to be aggressive from the outset. This view is seen and heard in media commentary, and is revealed as strategy in team selections: After After playing down the order last year, Brendan McCullum has returned to the opening spot along side Martin Guptil, as New Zealand empahsise a high-risk, high-return approach to batting from the outset. Australia are opening with two high-risk, high-return batsmen, and England recently dropped their relatively conservative opening batsmen and captain, Alistair Cook and also elevated the more-aggressive Moeen Ali to the top of the order.

But is this conventional wisdom correct? Obviously, faster scoring is better for a batting team than slower, given the same likelihood of losing a wicket, and conversely less risk is better than more for a given strike rate, but what is the trade-off? Peter Miller, aka The Cricket Geek, has expressed the trade-off as follows:

In many ways, getting out for a 15-ball 30 is less of a crime than 65 off 90 deliveries. 

This captures the essential difference between test cricket (most of the time) and limited-overs cricket. In the former, time is not a constraint, so the key to a large score is to not be dismissed; 65 contributes more to your team than 30 in most circumstances. In limited overs cricket, every ball faced is a ball that is not available to your teammates. The opportunity cost of balls used up has to be weighed against the runs scored. But is Peter's summary of the trade-off correct? As it happens, there is a measure of the opportunity cost of wickets lost and balls used it up that can answer this question exactly. It is WASP. A player's contribution to his team in the first innings is the amount that WASP advances by on the balls that that batsman faces. Let's imagine that an opening batsman is the first to be dismissed having either scored 30 runs and faced 15 of the first 30 deliveries, or having scored 65 runs and faced 90 of the first 180 deliveries. Which option will have advanced WASP by the larger amount? Well that depends on whether the game is played on a high-scoring or low-scoring pitch. In 250 conditions, the more aggressive opener would have had a net contribution of just under 4 runs compared to just over 11 for the less-aggressive player. In 300 conditions, however, 30 off 15 would give a slightly negative contribution, but 65 off 90 a more-negative one. The cross-over point is when the par score is 278 (roughly).

I reckon that 280 is probably about the average par score for the pitches being played on in this world cup (the succession of first-inning scores over 300 are misleading, as in every game so far, the more-fancied team has batted first), so Peter's example is very finely calibrated but correct.

But there is a seeming inconsistency in the conventional wisdom. As the same time that the consensus is that batters need to be attacking from the outset, media commentary is emphasising the importance of early wickets. And again, this appears to be accepted in team strategies. New Zealand has expressed the intention of attacking from the outset, when bowling as well, being prepared to concede runs in the search for wickets. Australia have their bowling spearheaded by Mitchell Johnson who, it is said, may prove expensive but can also destroy a team with early wickets, and so on. That is, the conventional wisdom is both that opening batsmen have to be aggressive from the start and that it is important for bowling team to secure early wickets. But ifthe risk-return trade-off is such that the benefit of quick runs to the batting team makes it worthwhile taking the risk of early wickets, then the reverse should be true for the bowling team. In the numerical example above, while it is true that 30 off 15 is a better contribution than 65 off 90 on a 300 pitch, both contributions are negative relative to the average opening batsman performance.

Actually, the view that bowling needs to be aggressive at the outset is probably closer to the truth than the view that batting needs to be. At the start of an ODI first innings, the cost of a wicket is between 25 and 30 runs, depending on the conditions. As long as a team has wickets in hand, that cost diminishes steadily as balls are used up, and the trade-off between risk and return favours greater aggression. (Of course, balancing that is the fielding restrictions in the first 10 overs, which lowers the risk from fast scoring.)

If this combination of aggressive batting and aggressive bowling/fielding carries through, in the World Cup, I expect to see a high-variance in first-innings scores: some high scores where the aggressive strategy pays off, and some low ones where rapid wickets impose a large cost. In some cases, when a high first-innings score is achieved, the same strategy will pay off for the chasing team; when a low first-innings score is made, the chasing team should always win by being more conservative. For that reason, notwithstanding the results in the first five games, I still believe that the toss-winner should choose to bat second.* Let's see how it plays out.

* The exception to this rule is when a highly ranked team plays one who is much weaker. In this case, the crazy net-run-rate method for choosing between teams with the same number of wins and losses, implies that the better team should choose to bat first, just to make sure that they bat for the full 50 overs and have more weight on that game in the NRR calculations. But that is the subject for a later post.

More on Courtsiding

Eric is wondering first, what business the NZ police have enforcing ICC ticket terms and conditions regarding "courtsiding" by evicting from NZ cricket grounds spectators who are in violation of the T&C but not breaking any law, and second, why the ICC even wants to crack down on courtsiding. 

Eric, being a public choice guy, considers two reasons why the ICC officials might rationally want to ban courtsiding. I am more inclined to Hanlon's razor, and so here I want to explain why the ICC should be embracing courtsiding. 

Courtsiding, the sending of real-time game information by a live spectator to bookmakers or gamblers overseas quicker than the slightly delayed television feed,  is a kind of insider trading, so let's think about the analogy to financial markets. For all that they occasionally get things seriously wrong with bad consequences, financial markets are on balance a huge force for social good, enabling the efficient matching of savings to productive investment opportunities, and facilitating risk sharing. Less obviously, but still true, speculation is, on balance, beneficial, allowing information to be embedded in prices and letting people make good resource-allocation decisions in their area of expertise without being economic experts as well. This last point may not be self-evidently true to all readers of this blog, but just run with it for now.

Despite this view that speculation is beneficial, we have laws to exclude people with access to privileged information from using that information for financial gain betting on financial markets. The reason for thus is that for speculation to be beneficial, it needs to be competitive. If outsiders feel that they are playing in a rigged game, they will opt out of the market leaving a small group of insiders able to manipulate prices. The case against insider trading is not as clear cut as one might think, but the logic is clear in principal.

Now consider getting on cricket matches. Let's assume that the ICC's objective is to maintain the integrity of the sport (at least as far as games involving India, Australia and England are concerned) not the integrity of gambling markets per se. The analogy to the proscription on insider trading in financial markets then breaks down. The ICC's main concern with gambling on matches, should be the incentive it gives for large payments being made to players to deliberately lose matches or to indulge in spot fixing. If it became well known that the gambling markets were fixed due to things other than player match fixing such as courtsiding, the incentive for outsiders to participate in gambling markets would be reduced, with a corresponding reduction in the financial incentives to bribe players. 

So here is my advice to the ICC: Don't worry about courtsiding; get into the game yourself. And don't do it surreptitiously: Announce to the world loud and clear that you are participating in the betting markets, using real-time information from games, using private information from coaches about hidden player injuries, using information from the professional version of the Duckworth-Lewis formula (which is not in the public domain), etc. Who knows, if you make enough money from this enterprise, you might be able to afford to invite some associate nations to the 2019 World Cup. 

Private Law - Cricket edition

Cops moonlighting, out of uniform, as event security - I can get that.

This one I'm less clear on:

Police have warned people who try to manipulate betting on the ICC Cricket World Cup 2015 that they will be caught and banned from all grounds involved in the tournament.

Police evicted several people from the opening match at North Hagley Park in Christchurch today for breaching the terms and conditions of their tickets, including some who were caught "courtsiding".

So what criminal offences are these courtsiders committing?

Courtsiding refers to the practice of spectators within venues relaying information of incidents during games to people overseas, taking advantage of broadcasting time delays to manipulate betting.

It is different from match-fixing, which is the manipulation of sporting events to achieve a pre-determined outcome.

Courtsiding is not illegal in New Zealand, but it is a breach of the terms and conditions of ICC Cricket World Cup 2015 tickets.

Operation Commander for the ICC Cricket World Cup 2015 policing operation Superintendent Sandy Manderson said Police knew how to idenfity people who were courtsiding and those attempting it would be caught.

"We know what to look for.

We’re aware that people are attempting to operate at venues and they will be detected, evicted and trespassed from all venues. 

So the New Zealand Police are enforcing the terms and conditions of the ICC Cricket World Cup tickets. I could get their enforcing trespass where the venue kicked somebody out for breach of conditions, but why are the Police investigating breach of ticket terms and conditions?

If you were at the venue and didn't know that Sky had gone to commercial, would tweeting a 6 or a wicket get you kicked out of the match by the cops?

Are ODI Scores Increasing? UPDATED

I had a conversation with a sports blogger, John Rogers, on Twitter last week. John Rogers had tweeted a link to a blog post he had written on why the WASP projection being used in BSkyB's coverage of limited overs cricket this English summer is necessarily inaccurate. His point is that ODI cricket is evolving quickly, both in the equipment and the style of batting, so that historical data is a poor guide to how many runs you can expect a team to score.

There is always a tradeoff in statistical work between using only the most recent data to capture trends, and using a longer time period to get more statistical significance. Now, in principle, since WASP is calibrated to a par score set by the broadcast commentators, any trend in scoring that has occurred within the period of the data used to estimate the model could be adjusted for in the par score. The setting of a par score is both a strength and weakness of WASP. The strength is that it allows game-specific information to be factored into the projections such as using local knowledge to assess how the pitch is likely to play. The weakness, however, is that the commentators might suffer from the common human biases of seeing patterns in essentially random data, and I wonder if the view that batting power is increasing is an example of that.

So I was interested to see if John's perception of a recent increase in scoring rates due to teams having more "lower-order hitters", better bats, etc. is borne out in the data. There is no doubt that there has been an increase in scoring over time. For example, all of the 16 ODI matches (all involving top-8 countries) where the team batting second has scored 330 or more have occurred this century. Only 5 of those 16, however, occurred this decade, suggesting that maybe the changes are not so recent.

Extreme scores like these are not necessarily indicative of a general trend, so some regression analysis is called for. John's hypothesis seems to be mainly based on increased rates of scoring by lower-order power hitters near the end of the innings. I don't have the full ball-by-ball database to hand, just a record of scores and results, but if the theory is correct, it should show up in total scores. Now WASP is currently based on ODI data from 2006 involving the top-8 teams, so I had a look at all non-rain-shortened games involving those teams from May 1 2006, using a dummy variable for each year starting May 1. First, I looked at the evolution of first innings scores over that time. To control for different abilities across countries, I ran an OLS regression of first-innings score on dummy variables for the team batting first and for the team bowling first, as well as a dummy variable for each of the 8 years in the database. To further control for differences across grounds, I restricted the data set to games played at grounds where there were at least 10 matches played in this period, and included a dummy variable for each ground. This left me with 245 games. The results are shown in by the blue line in the graph below, with the line showing (left axis) the average first innings score for the average team against the average team at the average ground. There clearly has been very little change over these 8 years.



John's blog post, however, seemed to refer specifically to the ability of teams to chase down large scores, so I separately looked at whether there has been a change in the the probability of the team batting second winning using a probit regression. Because differences in grounds largely affect ease of scoring in both innings, and because probabilistic models require more data to get precise estimates, I used the full dataset without dummy variables for the ground, but again controlled for team ability and included dummy variables for each year. The results are shown in red on the same graph (right axis). Probabilistic models typically require a lot more data, and so I wouldn't put too much faith in the estimates for any one year. But there doesn't seem to be a clear recent trend to it being easier to chase down scores than in previous years, although there was a strange dip in the period 2007-2009 that has since been reversed.

I suspect what is happening is a perception bias. There probably has been a recent increase in power hitting as a result of batsmen taking more risks, but that has been balanced by an increase in the rate of dismissals. And this leads to the reality being different from common perceptions. 20-20 has conditioned us to thinking that it is easy to score 8-9 runs an over on small grounds with flattish pitches. And it is. But it requires aerial shots, unlike 5-6 an over, which can be achieved entirely along the ground with 1s and 2s and the occasional bad ball cut or driven for 4 along the ground too fast for the cover sweeper to collect. With modern bats and batting, it is not difficult to sustain 8-9 an over through regular sixes and lofted 4s, but it is hard to do so without losing regular wickets. But wickets arrive randomly. Now think of the commentators bias. If a batsman hits a clean six, he is lauded for his good shot. If he mistimes it and is caught, as often as not he will be criticised for "taking unnecessary risk" or "not waiting for the right ball". (Have you ever heard a commentator criticise a batsman for taking unnecessary risk after making a clean hit for 6?) This creates an impression that the good shots are normal, and the wickets are just an avoidable failure rather than both being natural consequences of a particular level of aggression. Combine that with our recollections of past matches. Sometimes a team chasing 120 off 72 balls will have a randomly good passage scoring at that rate without any lofted shots going to hand, and it will make it look like such fast scoring is easy. At other times, we will see a procession of wickets and we will be thinking how the batsmen threw the game away. It is the first case that sticks in our mind when we make our own assessment of probabilities, and so we inflate in our own minds what the probabilities of winning are when a team is chasing a large total.

Data (even historical data that may become out of date) is a good antitdote to these perception biases.

UPDATE: Chris Smith, of the wonderful cricket blog, Declaration Game, asks by tweet if the results would have been very different had I not restricted to top-8 countries, and controlled for team ability and ground. That is, would we observe a general increase in scoring, but one attributable to having more games with weak teams, and more smaller grounds. Rerunning the numbers on the first-innings scores, if we include Bangladesh, Zimbabwe, Ireland and Afghanistan in the data (I don't have other countries in my database), and don't control for team ability, and don't control for grounds, we see a 10-run increase between the periods 2002-2007 to 2008-2013. Removing the four weaker teams and controlling for team ability only reduces that change by 2 runs. The big change comes when we restrict the data to grounds with at least 10 games in the dataset  (still 540 games) and control for the ground. This reduces the change down to 2 runs.

Irrational Expectations in Cricket Redux

This post is in part a follow-up post to this one from 2012 about irrational expectations in cricket, but is more a response to some recent twitter activity in the U.K. BskyB have been using WASP in their coverage of the recent ODI and 20-20 series between England and Sri Lanka, and this has provoked some angry twitter comments. Defenders like David Lloyd 

Folk who understand cricket don't understand WASP. Folk who don't understand cricket understand WASP .. "Who's winning?"
— David 'Bumble' Lloyd (@BumbleCricket) May 28, 2014

or Adam Lewis in this post, point out that a metric like WASP can be very useful for newcomers to watching cricket to give a sense of who is winning at any particular time and how comprehensively. The idea behind Adam’s post is that WASP tells cricket newcomers what experienced watchers already know in their gut. But just how good is the gut of experienced watchers? Well that is hard to measure, but I think it is reasonable to assume that highly paid captains of international teams probably have at least as good an intuition from the game from being actively involved. So let’s look at a very simple decision that captains have to make: whether to bat first or second on winning the toss.

I am currently working on a project with a student from India, Pranav Bhargava, to estimate rankings of teams. In the process we came across the following interesting result: A model that estimates the probability that the team batting second would win an ODI as a function of the quality of the two teams playing, fits the data better than one that estimates the probabiliyt that the team who wins the toss wins the game. Looking at the raw data, we find that the team batting second won 53% of the 1294 games played between May 2002 and May 2014, but the team winning the toss won only 51%. This is a small difference but it is masked the fact that the best team over this period, Australia, batted first more often. When controlling for team ability, the difference is more marked.

This makes no sense at all. While the team batting second wins slightly more often than the team batting first, indicating a second-innings advantage on average, the advantage will not apply in every game, depending on the pitch and the abilities of the teams playing. The captain who wins the toss has the option of choosing to always bat second, or to choose to bat first if these game-specific factors suggest that would be better. Accordingly, the team winning the toss should win more often than the team batting second.

O.K. so let’s give the captains the benefit of the doubt. It seems unlikely with such a large sample, but maybe the random toss has, by chance, been won by the weaker team more often than the stronger team. So we investigated this further. We measired separate team ability measures for each of the top 11 countries (the top 8 + Bangladesh, Zimbabwe, and Ireland) for when they won the toss and lost the toss, and found that for some matchups, losing the toss would be preferable to winning it! In particular, three teams—Australia, Pakistan, and Zimbabwe—make the wrong decision according to the data more than 50% of the time, and so would prefer to lose the toss if playing against a clone of themselves. The remaining teams make the right decision more than 50% of the time, but most are sufficiently imperfect that if playing against Australia or Zimbabwe, would be better off losing the toss and relying on the opposition to make the wrong decision! Only Ireland out of the top 11 teams has a decision record that makes it desirable for them to win the toss against any opposition.

So far, these results replicates results in Bhaskar (2007), but with a slightly different method, suggesting that the results are robust. One criticism of both sets of results, however, is that in using the full sample of games to estimate what should be the correct decision, we are using information from matches that would not have been played at the time captains made their decisions. So we divided the data into two eras of 647 matches each. We used the first era to estimate when it would be better to bat first rather than second, and then used this to compare outcomes to predictions in the second era. We find that teams win more than predicted when captains make the right decision and less than predicted when they make the wrong decision. Put another way, the variable on “correct decision”, is strongly and positively significant in a regression modelling the probability of success. And this uses only information on how well teams have played batting first and second in the first 6 years of the data to predict outcomes in the second 6 years. Real-world captains have more up-to-date information about how teams are playing as well as information about ground conditions on the day. 

At this point, I can’t see any comeback. The information available to our model is strictly less than that available to captains, yet our model can outperform international ODI captains quite significantly.

So what is going on? I think there are likely two sources of imperfect understanding by captains at play here. The first is that captains forget that this is a zero-sum game. If you are a team that is better at chasing than setting a score, but are playing against a team that is much better at setting than chasing, the optimal decision is to bat first, holding conditions equal. But teams possibly play to their own strengths rather than also considering their opponents weaknesses. Another possibility, that I suggested in the earlier post, is a misunderstanding of the regression fallacy: on average, the easier the batting conditions, the higher is the first-innings score. And, on average, the higher the first-innings score, the higher is the probability that the team batting first wins the game, since, on average, higher first innings scores indicate a better than average batting performance. But these two facts don’t in themselves imply that the team batting first has a higher chance of winning when batting conditions are easy.

There are other stories one can tell for the source of the errors made by captains, and we are investigating whether we see in the data what the source is. But the bottom line is that careful data analysis with limited information outperforms professional gut opinion with full information, and by a considerable degree!