How high is a winning cricket score?

By Marcus Wilson 24/03/2015 1

I can't help thinking that the West Indies team got their run chase strategy wrong on Sunday night. They had a tricky task ahead of them. One might say the problem was one of their own making, judging from the rubbish that they served up to Guptill to hit at the end of tne New Zealand innings, but that was in the past. The fact was they had to chase down 393 in 50 overs. How does a team go about doing that?

I would have thought that the obvious answer was 'in the same way that the opposition made it'. In other words, keep the scoreboard turning over nicely in the early stages, but without taking excessive risks, and then, with the wickets in hand (especially Gayle's), hit the accelerator at the end. Rather than taking 50 overs to reach the target, they looked like they were trying to do it in 40. It was never going to work. You can contrast this to the calm manner in which Sri Lanka reached England's 300+ score in the group stages. Nothing flash, no excessive risks, they just ticked the board over just short of the required run-rate, and then with wickets in hand they pushed on at the end to win easily. No fuss, no stupid shots. They knew exactly what was needed, and they achieved it. 

A school of thought says, other things being equal,  it's much better to bat second in a limited overs match, because you know exactly what you have to do. It clarifies your batting strategy: Choose whatever strategy maximizes your chances of reaching the target score within 50 overs. Yes, there are other considerations, our friends Duckworth and Lewis being one, but let's put those aside for the moment.  You have 10 wickets and 50 overs, and a target score. You only need to beat that target by 1 run on the last ball, with one wicket left.  What strategy will maximize your chance of success?

The team batting first has a much less defined problem. They don't know what a winning score is. True, they'll have some feel of what one will be, given the ground, the quality of opposition, the state of the pitch and so on, but fundamentally, they don't know what score is going to be good enough. For example, a team could probably secure a moderate score (let's say 280) with 80% probability, by batting conservatively. Or they could aim for a larger score (say 350) by taking more risks. They might get there with a probability of 25%,  but then there's the possibility (maybe also 25%) they crash out on the way and end up with something more like 250. What  should they do? Which strategy is better? 

It's viewed as a criminal offence if the team batting first fails to bat out its 50 overs. If it's all out beforehand, it has obviously taken too many risks. But also, one could say its a criminal offence if they end up with just a handul of wickets down. In that case they haven't been taking enough risk. Balancing all that up, a typical strategy for a team batting first is to go at a moderate rate to start with, and slowly increase the rate as wickets allow. It seems to have stood the test of time. 

It is possible to make this a bit more mathematical. What a team needs to do is to maximize its score, subject to the constraints that a. it only has 50 overs, and b. it only has 10 wickets. Since the rate of fall of wickets is certainly related to the scoring rate (the higher the runs-per-over, the higher the risk, usually, and the higher the wickets-per-over) we can write down some equations for the situation. [You'll be relieved to hear I won't try to do those here].

It turns out to be is an optimization problem, of which there are many in physics. We can tackle many of them with the Euler-Lagrange equation.  For example, what shape does a chain have when it hangs under its own weight? Take a chain of length 2 metres, secure the ends to posts 1.5 metres apart. The chain clearly sags in the middle, but what shape is the resulting curve?  The chain hangs in such a way as to minimize its gravitational potential energy, subject to the constraint that it has a constant length and that it has to start and end at fixed points. One can put this in equation form and solve for the shape. The resulting curve is called a 'catenary'. 

Certainly, if one could write down the rate of loss of wickets as a function of scoring rate, wickets lost already and other factors, we should be able to have a go at tackling the cricket scoring rate problem. Given the degree of professionalism of the sport, it wouldn't surprise me if that has actually been done by someone. I imagine it would be a closely guarded secret.






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