Would you prefer a batsman who scores 50 every innings, or one who scores 100 half the time and 0 the rest? The ‘expected value’ (batting average) each player offers their team is the same. The decision appears to rest on whether consistency or the potential for a match-defining innings contributes more to your team.
Jarrod Kimber and Andy Zaltzman discussed this conundrum on a recent episode of the ‘Cricket Sadist Hour,’ acknowledging @analytics_jonas as the source of the riddle. The discussion suggests that there is an answer to the question, if sufficient mathematical heft is applied to it. Indeed there may be, but how often in the real world of cricket, or any other occupation, is a difficult decision subject to rational analysis and resolution?
Suprisingly rarely, according to Daniel Kahneman, Nobel prize winner in Economics and author of ‘Thinking, Fast and Slow’. Kahneman challenged the orthodoxy in the social sciences that, “people are generally rational and their thinking is normally sound.” Kahneman and others put judgement to the test by running experiments. What they consistently found were “heuristics and biases.. the simplifying shortcuts of intuitive thinking.”
Following Kahneman’s lead, I won’t try the tricky statistical enterprise of devising an answer to the question. Instead, I will draw on his book to understand why and in which conditions team-mates, fans or even selectors may choose one of these theoretical batsmen over the other.
The numbers (50, 50, 50, 50… and 0, 100, 0, 0, 100, 100…etc) will play a part in these answers, but to begin with, let’s acknowledge that for most of us, much of the time, decisions are not based on numerical analysis, which Kahneman has shown is too effortful and our brains tend to shun.
This is the essence of intuitive heuristics: when faced with a difficult question, we often answer an easier one instead, usually without noticing the substitution.” Kahneman, p12
Instead of attempting the difficult question (which player, based on their pattern of scoring, and the complex structure of a cricket match, will over time contribute more to the team?), we substitute an easier one. In this case, it might be: “Who scored more runs when I last saw them on TV? or, which player has the purer cover-drive? or, which player is being lauded by the ex-player whose opinion I respect?” Kahneman gives examples of how choices are made over political candidates, investment opportunities and charitable donations, but could easily have added cricket team selection to his list.
I will now engage more directly with @analytics_jonas’s original question but introduce some small aspects of context in order to understand how each of the batsmen might, in different circumstances be our preference.
Experience and memory
Our experience of the two batsmen, aggregated over time, will be very similar (if not identical) in terms of the number of runs we see them score. If decision-making were based on lived experience, we may struggle choosing between the consistent batsman and the success/fail player.
However, studies carried out by Kahneman and other psychologists have shown that decisions are not driven by the lived experience, but by the retained memory of the experience.
The memory that the remembering self keeps.. is a representative moment strongly influenced by the peak and the end. (Kahneman, p383)
Taking the ‘peak’ effect first, the success/fail player will clearly have provided greater peaks to plant in the memory than the steady accumulator of half-centuries. The second aspect, the recency effect works equally well for each as around half the time each of the batsmen will have recorded the higher score in their last innings. Combining the ‘peak effect’ and the ‘end effect’, the success/fail player will have created the more favourable impression.
Put now in the position of selector opting for one of our two players for the next match, rationally there is nothing to choose between them, as over two innings they are, on average, expected to contribute the same number of runs.
Daniel Bernoulli, Swiss scientist of the eighteenth century, refined this understanding by observing that most people dislike risk. So while the expected value of the two batsmen is the same, there is a premium on the player guaranteed to score 50 each innings (or a discount on his alter ego). For the selector this may make sense. There may be only one innings in the match, so better bank 50 runs than risk 0. Even with two innings, we cannot be certain that our success/fail player won’t make two ducks, before bouncing back to form in a later game.
Kahneman added further nuance to Bernoulli’s theory from the empirically observed standpoint that people’s decisions are context dependent. The pattern found, and stated in Prospect theory, is that the decision-maker whose prospects are poor is more likely to gamble. On the other hand, when faced with a situation that offers only an upside, decision-makers tend to be averse to risk and seize the sure thing.
One-nil down in the series with one match to play, the opportunity of having the century-maker is more appealing. One-up, at the same stage of the series, and the cautious choice is preferred. As explored in a previous post on declaration decisions, when a skipper is on top in a Test match, losing feels worse than winning feels good.
It is important to emphasise that Kahneman’s work is descriptive of the decisions people (Test selectors?) make and not prescriptive of the decisions they should make. Indeed, Kahneman emphasises that these systematic biases, based in this case on decision makers’ reference points (i.e. their context), can create unfavourable outcomes, with good opportunities passed over and poor choices pursued.
While we wait for advanced cricket analytics to tell us whether absolute consistency trumps its opposite, selectors will have to pick players, pundits will continue to push the interests of their proteges and we followers of the game will invest great hopes in our favourites. By making some effort to examine our preferences we might reveal biases and so move towards more informed selections of, lobbying for and championing of players.