Harry Gurney is widely regarded as one of the best T20 death bowlers currently around, picking up trophies across the globe in the Pakistan Super League, Big Bash League and T20 Blast (1). The left-armer practises hard on his variations, landing slower balls, cutters and yorkers as part of a rigid routine where he won’t move on until being happy with where the ball is landing (2). It may be his work ethic that has got him this far, but perhaps his Economics degree is giving him that vital extra edge over his opponents through utilising what is known as a ‘mixed strategy’. This article explains the economic theory of a mixed strategy, how it could be used to gain a vital edge over the opposition and might give players insight that previously took years of experience to develop.
If you have ever faced a bowling machine, then you will likely have enjoyed the comfort of knowing where the ball is going to come and the shot that you are going to play before the ball is bowled. It removes the element of surprise and the worry of where the ball might come next, making batting a lot easier. A mixed strategy is used to overcome this problem by randomising between choices in order to avoid predictability and being ‘exploited’ by the other player (3). It puts a question mark in the batter’s mind over the next ball and reduces the ability to just pick where they want to hit it in advance of the ball being bowled. In short, it is a deliberate or planned strategy for how often to bowl variations.
Traditionally, batters have been taught to “play each ball on its merit”, but T20 has shifted this dynamic with Gurney admitting that he’s “constantly thinking about where a batsman’s trying to hit [him] and trying to stay one step ahead of them”(4). This represents a classic ‘game theory’ situation, when the choices a player must make in a game are affected by the actions of the other player/s. Variation is a key strategy in cricket, especially for those who have the ability to consistently bowl the ball where they please. With this level of skill and control, the next step is mental, to select the variation that will work best – this is where the ‘mixed strategy’ comes in.
A mixed strategy is a probability distribution that assigns a likelihood of being selected to each available action (5). The concept applies to a situation where each player has several strategies that they could choose to go for at any one time, known as a ‘profile’ of strategies. The player will then choose each of these options with a certain probability to create their mixed strategy. This can then be used to calculate the expected payoff of the player’s approach and analysis can be done to find the best approach a player could take (by maximising the expected payoff).
Translating this into cricket, a bowler at the start of their run up has a choice to make about the type of ball that they would like to bowl, let’s say that they can either bowl a yorker, bouncer or length ball. These delivery options are the ‘profile’ of strategies. To form the mixed strategy, the bowler would then assign probabilities to each of these options, so they might choose to bowl a yorker 5% of the time, a bouncer for 10% of the deliveries and a length ball the remaining 85%. It is essentially putting in place a formal approach to choose how often to bowl variations.
Whether he picked up this knowledge at Leeds University as an Economics undergrad or not, Harry Gurney seems to hold this concept as a fundamental way to optimise his bowling strategy with one particular interview sticking in my mind to illustrate this (6). One of his “main philosophies” is to be able to bowl at least two types of ball according to the field that he has set: it is vital that there is an element of doubt in the batter’s mind and this creates it, with Gurney then being able to bowl each of the two deliveries with a certain probability.
In the same interview, Gurney then goes on to say:
“Actually, what I’ve realised is that I can bowl far more than two variations to that field and also, what’s becoming increasingly popular in T20 cricket at the moment is bluffing. So, a lot of the bowlers will bring fine leg and third man up, and still bowl a bouncer because they are playing mind games with the batsman. So, in addition to me being able to bowl three or four deliveries to the same field … you could throw in a couple of bluffs as well.”
The bluff captures the essence of using a mixed strategy in T20 bowling, it would be as foolish to bowl it regularly (assigning a high probability and likely go for lots of runs) as it would be to not bowl it at all (assign it a 0% probability and become too predictable).
A professional bowler’s experience and ‘feel’ for the game will arguably guide them to bluffing with a probability close to an optimum mixed strategy. This has been shown in other elite level sporting settings before (7). Penalty takers and goalkeepers in football have been shown to shoot either left, right or down the middle with optimal probabilities assigned to each. It is unlikely that the footballers have sat down to work out their mixed strategy but are likely to have analysed previous data on the opponents they face and, combined with their experience for the game, are then able to perform at an optimal level.
It is possible however to calculate the exact probabilities that will give the best expected payoff and perhaps that vital edge for the team. To illustrate this, imagine that Gurney has the three mixed strategies he can choose from for the field set and variations as described above:
Strategy 1: No bluffing (bouncer 0%), bowl a yorker 75% of deliveries and a slower ball 25%
Strategy 2: Lots of bluffing (bouncer 30%), bowl a yorker 40% of deliveries and a slower ball 30%
Strategy 3: Bit of bluffing (bouncer 5%), bowl a yorker 65% and a slower ball 30%
The best strategy is the one which gives away the fewest runs.
So, for Strategy 1, without the worry of a bouncer, the batter can hit Gurney’s yorkers for an average of 2 runs per ball and is also prepared for the slower ball and hits these at the same rate. This gives an expected payoff of 2 runs per ball (0.75×2+0.25×2) for Strategy 1.
For Strategy 2, the bluff is deployed far more regularly by bowling a bouncer 30% of the time, even though the field is set up for yorkers or slower balls. This creates a big element of doubt in the batter’s mind, reduces their ability to hit the yorkers that now go for 1.25 runs per ball on average and also the slower balls at 1.75. However, the bluff offers the chance for the batter to cash in at an average of 3 runs per ball. This gives an expected payoff of 1.925 runs per ball (0.4×1.25+0.3×3+0.3×1.75) and so although more expensive itself, the sacrifice of bowling the bluff regularly creates enough doubt for the other deliveries that it is just about worth it compared to Strategy 1.
Perhaps though, there is a way of introducing the element of doubt in the batter’s mind without having to bowl the costly bluff delivery as often. Strategy 3 considers this by bowling a bouncer/bluffing 5% of the time, this means the batter isn’t as ready for the bouncer to come, only going at 2.75 runs per ball instead of the previous 3 runs per ball. They also know that the yorker and slower balls are more likely, so the runs per ball increase compared to Strategy 2 but are not as high as in Strategy 1. This strategy would give an expected payoff of 1.65 runs per ball (0.65×1.5+0.05×2.75+0.3*1.8) and so would be the best strategy of the three options.
These examples are of course arbitrary, but I hope that they illustrate the point. With the level of detail that teams now have access to with ball tracking data it should be possible to carry out these sorts of calculations. Working out a mixed strategy would be wise for T20 bowlers in advance of a game and could help to bring their economy rates down to win games for their team.
This is reliant though on a bowler being skilful enough to be able to have the ability to bowl these variations and land the ball where they please. Even most pros are not able to do this, there is only the barest of margins between landing a yorker or bowling a full toss.
Even without this level of control, it doesn’t change the ability to make an optimal decision before the delivery. It could even take into account the likelihood of the bowler actually being able to deliver the ball where intended, this would require an honest appraisal of where the bowler had previously intended to bowl to gain the necessary information. A less skilful player, for example, will struggle to regularly land a yorker and end up with full tosses and half volleys for the batter to tuck into, perhaps resulting on average in a more expensive delivery than say a length slower ball. The mixed strategy can be tailored for this, considering each bowler’s skill level as long as the historic data is there.
There is perhaps less of an application to the longer, red ball forms of the game. Here the batter is far more likely to be ‘playing each ball on its merit’ and the bowler will be bowling fewer variations. It brings the skill of consistently controlling where the ball lands into focus, as batters will tend to cash in on anything that isn’t a ‘good’ length. There is very little room for bluffing.
Some of the best bowlers of recent times are those that can bowl a good length most consistently. James Anderson, Vernon Philander, Bhuvneshwar Kumar and Mohammad Abbas stand out from the rest in terms of their ability to land the ball regularly on a good length and their low test averages are the result of this skill.
On the face of it, it seems as though the best strategy in tests, when selecting the length to bowl, may be just to stick with trying to bowl a ‘good’ length every ball. This would be suitable as a strategy if there was no further information. Although, a batter might have a weakness against the short ball, in which case it would probably make sense to increase the proportion of shorter balls. A pitch might be offering more seam movement, giving the chance for more wickets with the batters playing forward to a full length ball, here an adjustment to pitch it up might be worth it. These nuances are a huge part of the game and should not be overlooked.
James Anderson is perhaps the master of mixed strategy in the test arena, as well as bowling a good length and coming out well in the article mentioned above, he has also honed his selection of in-swing vs. away-swing over time. At the skill level of James Anderson and with the ability to also swing it both ways it is also of course not as simple as selecting the length.
A superb piece of analysis sticks out in my mind by CricViz (8) that illustrates it is Anderson’s change in strategy, rather than action, which has seen his average – over the last couple of years – fall to a level lower than it ever had previously been. Rather than trying to bowl aggressive swinging deliveries, Anderson has shifted to bowling a consistent line outside off-stump with more potential for the ball to go either way and to make the batsmen play. In mixed strategy terms, he has increased the probability assigned to bowling outside off-stump, with an increase from 19% to 26% of balls that actually land there.
He has also significantly reduced the percentage of balls that are in-swinging to the batter, down from 29% to 9% for left-handers and 32% to 11% for right-handers. This has seen his average from in-swingers fall from 26.66 to 7.5 for left-handers and 60.50 to 8.80 for right-handers. This is a phenomenal shift that has hugely contributed to his lower average towards the tail-end of his career and highlights the benefits that a mixed strategy calculation could bring.
It has been honed over years of test cricket experience, being able to watch other bowlers at the top level for over a decade has provided a phenomenal resource for learning and improving. It may be though, that calculating a mixed strategy could help to provide a shortcut for this level of experience. With the detail of data that allows companies, like CricViz, to conduct such in depth analysis, perhaps a young bowler now could be shown the benefit that bowling fewer in-swingers would have on their average and adjust their variations accordingly.
Whether they can then consistently bowl the delivery is another matter. James Anderson’s hard graft, practice and sheer volume of deliveries over the years has undoubtably fashioned his ridiculous level of control. Harry Gurney’s desire to practice his variations and to keep going in until he is landing the ball where he wants is what has driven him to becoming an elite level T20 bowler.
Yet, calculating a mixed strategy using the data that is now available, regardless of the level of skill and format, could help all bowlers to reduce their averages in a far shorter time than the previous years of experience that were needed to gain this knowledge. There is arguably more scope for this in the shorter formats of the game, where a greater number of variations and even bluffing offers the chance for big gains to be made. With the money being spent on white ball cricket nowadays it would be wise, in my opinion, for teams to spend a fraction of it calculating mixed strategies in advance of games. Those that do may gain a vital edge over their opponents.