# Batting out your overs

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.

*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.

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

- 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.
- 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. - 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.
- 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. - 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.
- 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.