In the previous post I discussed the value of economical vs strike bowling in T20 cricket, using my simple match simulation model to attempt to quantify the value of wickets in hand vs additional runs. This time I will use the same model to find the impact that a single opening batsman can have on a team’s average run total. Once again I am going with the same lineups as before, with a grid varying the strike rate of a single batsman from 70% of his usual strike rate to 127% in one direction, and the same for his balls faced per dismissal in the other. You can read the first post for an explanation of the model – here I present the results:

The key thing to note is that the boundaries between two colors in this plot, indicating the lines of constant score for a given strikerate-balls faced pair, are almost completely horizontal. In other words, and I don’t this would surprise anyone, the value of batting in this game is strongly correlated to strike rates, with the ability to stay at the crease having only secondary value – there is a very large difference between batsmen at the same batting average, which can be seen by following the numbered lines on the plot. The lines do get more steep as strike rate increases, but this is simply due to the added value of having one’s best hitter at the crease longer. The graph is normalized against a baseline batsman with an average of 22 and strike rate of 135, which is probably a little low in the average and high in the strike rate, but let’s assume that the openers are taking the correct aggressive strategy.

In fact the worst thing one can have in this form of the game is not an opener that tends to get out early, but instead one that plays a lot of dot deliveries. Even a batsman averaging in the upper teens can keep the average score on par with a slightly increased strike rate, because there’s not the threat of being bowled out in this game. But for batsmen with strike rates even 90% of this baseline, the run totals will suffer regardless of how long the player stays in.

Another thing to note is the difference in value between the absolute best and worst batsmen in this game. The best batsman in twenty20 cricket so far has been Andrew Symonds. With an average of 40 and strike rate of 161, he’s reached a level beyond what this plot can account for. It’s hard to believe that he could average more in T20’s than in ODI’s, and so maybe it’s just variance, but running this same simulation for a player with Symonds’ current stats yields an average total of 149.2, an incredible 10 runs higher than that of the “average” opener. A difference of 10 runs in a T20 match is simply amazing – enough to account for one and a half to two additional wins on average over a single IPL season. Having him come in at his usual number 4 instead of opening, my model has the total slightly reduced to 148.8, still massive.

To think about the “worst” opener is a little more difficult since the worst opener would probably be a bowler. But the player who opens regularly, or at least bats at the top of the order, that comes to mind first is the dreadful Sourav Ganguly. Ganguly was solid, if not overrated, in the longer form of the game, but his strike rate of 103 is simply not going to get it done in this form. In fact a team with Ganguly opening the batting would average 135.2, four runs lower than the standard total. Consider this – in a Kolkata vs. Deccan game, if all the other players are average, Deccan will start with a 13 run head start simply on the basis of having Symonds vs. Ganguly. Despite taking up a large portion of Deccan’s salary cap, Symonds is worth the money.

I just blew about 100 bucks to put this site back up for another year, so hopefully I will write on it. I am not going to be doing any game prediction stuff since it takes way too much time and doesn’t work anyway – instead the type of things I will write will be more like this one, although I hope to connect more to actual cricket in the future.

Having been inspired by a recent article on baseballprospectus, I thought it would be interesting to look at one of the classic parts of cricket, slipping in the yorker. Everyone knows that the yorker is most effective when the batsman isn’t expecting it, and that it’s best at the end of an innings since it’s hard to deal with a perfect yorker unless one “cheats” out of the crease. But if the bowlers resort to it every time, the batsman will simply charge down the track, play scoop shots or other crazy things, and the yorker loses a lot of its effectiveness. The same kind of thing could be said about bouncers, slower balls, and even a lot of other aspects of cricket like team or pitch selection (and I plan to think about some of those as well with this type of framework).

The question is this – how often should the bowler slip in a yorker? It of course depends on how often a batsman will expect one. Obviously if the batsman is looking for the yorker the bowler wants to bowl length and vice versa, but neither can truly know for sure when that is the case. But suppose the batsman is very unlikely to anticipate a yorker – this would be the best time to bowl one and the bowler should know this and would adjust. The batsman would then adjust and go back to looking for standard length, and so on. In general if the batsman anticipates yorkers too much, the bowler can benefit by going to an all-length strategy and vice versa.

However, somewhere in between a 0 and 100% chance of the batsman looking for a yorker, there is an point where the bowler will not be able to improve his outcome by switching to all yorkers or all length balls, and similarly for the batsman and his anticipation. This is called the mixed strategy Nash equilibrium. Obviously it is always best to “guess right” so to speak, but this is impossible, although some batsmen are said to have instincts on these things. But the next best thing is to pick a strategy where probabilisticaly you cannot be exploited by your opponent either way, which is what the mixed strategy is.

Furthermore, in cricket this situation is more interesting, because in each delivery there are 0-6 runs, or 0 or 1 wickets lost. But the value of taking a wicket changes over time – in the death overs it may only be worth a dot ball, whereas removing the last specialist batsman in a tight chase might be worth 20 runs or more. The result is that yorkers, which tend to be somewhat less likely to take a wicket against good batsmen but allow slightly fewer runs, tend to be a better strategy at the end of the innings where the value of taking a wicket is usually lower.

To find an example where we can see this, I create a simple model between the bowler and batsman. In real cricket the bowler can bowl at any length, but let’s assume that the bowler can only bowl a yorker or length ball, and the batsman can anticipate either one. Furthermore let’s assume that probability of taking a wicket and the runs scored (on average) per each delivery follows the following table:

Each of the boxes represents the outcome where the batsman anticipates either a length ball or yorker, and the bowler bowls either a length ball or yorker, respectively. The first number in each column is the probability of the batsman getting out and the second number is the number of runs scored – note that this is a little different than a standard payout chart for those of you familiar with these things. So for example, in the top left box, the batsman is looking to hit against length and the bowler bowls length. The chance of taking a wicket is 5% and the batsman will score 1.4 runs per ball in this scenario. Note the more favorable set of outcomes for the bowler when the batsman “guesses wrong”.

There are several ways to find the equilibrium strategy for the bowler, but the easiest is to look at the problem from the following perspective: What percentage of the time should the bowler bowl a yorker such that the batsman could not benefit by automatically choosing one strategy over the other? To find this, one has to calculate the value (in runs) for the batsman in each of the four scenarios, assuming the bowler bowls a  length delivery with probability q (and hence a yorker with probability 1-q). When the value for the two scenarios involving anticipating length is equal to those where the batsman looks for a yorker, the equilibrium has been reached – the batsman can not benefit by choosing one over the other.

The value in runs is obvious for the run per ball situation, but the value of taking the batsman’s wicket will vary heavily depending on the match situation. Let’s call this value “a” – note that this will have a negative in front of it for the purposes of this, since less runs will be given up overall as a result of taking a wicket. For each term, the equation is then:

(-(value of wicket)*(probability of taking a wicket in this scenario) + (runs given up in this scenario)) * (probability of scenario occuring)

Looking at this from the perspective of a batsman, facing a bowler who will bowl  length with probability q, the equation is then:

(-.05a + 1.4)q + (-.07a + 0.8)(1-q) = (-.07a + 1.0)q + (-.03a + 1.2) (1-q)

The left side is the batsman’s expectation, in runs, if he anticipates a length delivery, and the right side is his expectation in runs if he anticipates a yorker.

Solving for q one obtains

q = (.04*a + .4)/(.06*a + .8)

which looks like this:

As expected, the length ball is a better strategy the more important it is to take a wicket. Just starting off by plugging in some numbers, suppose it is worthless to take a wicket (this is impossible as a wicket is worth a dot ball at the least). Then the bowler should bowl half yorkers and half length balls. Looking at the table this makes sense, as ignoring the fall of a wicket, the average number of runs scored by the batsman is 1.1 regardless of which he chooses if the bowler mixes it up 50/50 either way. Otherwise one choice becomes superior to the other. Adding wickets falling makes things more complicated, but basically the length ball is more likely to take a wicket with the numbers I chose and so in general the more valuable a wicket the more favorable it is to bowl length balls, and the numbers reflect this. For a wicket being infinitely important, the bowler should bowl length with probability 2/3.

The numbers I picked are completely arbitrary, although I tried to capture the idea of cricket with the numbers I chose. Actually I think the difference between the batsman guessing right and wrong might be far more drastic than what I chose, which might make the deviation in strategy far larger. But this is not the point – the idea is to see in numbers the value of mixing up deliveries, and how this changes throughout an innings. The most important thing to take from this is that even when one strategy is in general worse than the other (here the yorker is worse than length unless the batsman is surprised), there is value in mixing up your strategy simply to force the opposition to respect it.