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. 

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.