There’s a perennial question about how much achievement something depends on talent, and how much on hard work. Perhaps genius (or even garden variety exceptional performance) is written into someone’s genes, or perhaps what separated Einstein from his peers had more to do with his work ethic than his IQ.

Evidence points in both directions. On the one hand, most high performers, whatever their field, emphasize how important hard work – rather than ‘just talent’ – is to their achievements (e.g. Terrence Tao, Will Smith, Ira Glass, Thomas Edison). Some, like Malcolm Gladwell, talk about a ‘10000 hour rule‘ as the required hard work before one can truly excel. Perhaps the main proponent of the ‘Arbeit uber alles’ approach is Erikson’s work on deliberate practice. On the other hand, there are lots of instances where innate physical or mental characteristics play an important role: the average height of NBA players is 6’7″, Intelligence (albeit imperfectly measured by IQ) seems to predict lots of things (including various intellectual achievements) – and it appears to remain predictive even into the very high range.

So perhaps it is a mix. But the precise mechanism of the mix could be important; how do innate talents and amount of training relate to one another when it comes to achievement? Could some maths help?

A Growth-mindset model

Here’s one suggestion, implied by Uri Baum:

Performance = Talent + Practice intensity x Time practising[ref]Perhaps even better would be to use a time integral here, as likely practice intensity will vary over time. But multiplication is simpler, and simplicity is better than precision for toy models.[/ref]

On this sort of model, talent counts, but as time passes, practice matters more. Unlike talent – a static given – one can grow a stock of practice over time, and time invested in practice and hard work has a rich return on performance (c.f. Hamming’s remarks). An attractive corollary is that if one can improve one’s practice intensity, be that through more focused training, deliberate practice, better learning styles, etc. this acts as a multiplier – working smarter, as well as working harder may be a stronger determinant of success than talent.

If so, extraordinary talent may be a curse – it could let us coast. Bram suggests there might be a mechanism where if we select for exceptional achievement, we select for people with varying mixes of raw talent and hard work. The group which skew more towards the latter may overtake those skewing to the former former over time: those who skew towards more practice time and intensity will be able to grow faster, whilst those who mainly got to where they were ‘just’ on their talent may find they are hitting a wall unless they can improve how they develop.

Empirical trouble

There seem a few holes in the model as-is:

First, this model suggests that investing more time gives linear gains to performance. It is plausible there are diminishing or even saturating returns to practice (I haven’t done sports physiology in a while, but I recall some conditioning does saturate), and so perhaps there should be levelling off.

Second, the relationship doesn’t even appear to be monotonic with respect to time. Elite sportsmen commonly decline in ability as they age past a certain point, and most (no matter the sport) are retired from top level competition in middle age. Yet on the ‘performance equals talent plus practice times time’ model, they should have an advantage over their younger peers as having had more time available for them to have devoted to practice. There is a similar feature in more ‘intellectual’ pursuits: although some (judges, writers) seem to improve steadily with age, it is thought that others tend to ‘peak’ (mathematicans, lyrical poets, chess grandmasters) at varying ages, declining afterwards (c.f. similar work showing a modest decline in labour productivity with advanced age).[ref]I owe this to Richard Batty[/ref]

Third, if hard work generally wins out over practice, you should expect the best performers to be those who have worked the hardest – yet elite performers are marked by dramatic selection on ‘innate’ characteristics: certain ethnic groups outperform others in athletic skills (e.g. virtually everyone who has run a sub-10 second 100m has been of African descent), age and height have already been mentioned, and we could also throw in gender (e.g. the qualifying time for the men’s 100m in the last Olympics was 10.24 seconds; the women’s 100m world record is 10.49 seconds). It seems implausible to say that young men of African descent have a near monopoly on gaining the practice necessary to reach the pinnacle of sprinting. It is harder to demonstrate the same features for intellectual performance, as most ways of standardizing mental ability are unable to resolve into the extreme right tail, but some circumstantial evidence would be the lack of ceiling effect for IQ’s positive relationship with things like PhDs, patents, and job performance, the fact there might be a floor for developing some abilities, and the fact IQ-achievement correlations tend to strengthen with age.

Fourth, practice intensity is not independent of all talents. Being smarter could help you practice smarter, for example (and this is borne out in sundry occupational studies: smarter people tend to pick up skills and knowledge faster than the rest). There may also be a secular ‘success spiral’ effect where initial success (thanks to innate talent, perhaps) encourages further development. Thus talent may not just benefit performance directly, but also exert a beneficial effect on how effective practice is.

Toy model 2: Growth circumscribed by ability

A lot of these features can be approximated by making the model multiplicative, and sticking a concave function on the ‘practice’ component. We need to say that both of these components are pretty important: I would guess the precise mix will vary depending on what we’re interested in.

Performance = Talent x log(practice intensity x Time)[ref]You might argue that there should be saturation rather than a concave but increasing function, although this might vary by field - knowing more stuff might have diminishing returns (but not asymptotically so), maybe athletic conditioning could be 'maxed out'. In any case, logs are simpler, and mutatis mutandis my prior justification for using multiplication rather than time integrals.[/ref]

Suppose I start learning the violin. You can imagine some features I have as talents: pitch discrimination, manual dexterity, the precise dimensions of my hands, creativity. Yet even if I am gifted in all the ‘violinist’ talents, I still need to start practising. Suppose I do so at a steady rate.

On this model my future as a violinist shows fairly clearly diminishing returns. I start with dramatic improvements as I spend time practising, even if I continue practicing steadily my rate of growth levels off: getting from ‘novice’ to ‘pretty good’ may be much easier than getting from ‘pretty good’ to ‘world leading’:[ref]Where relative abilities lie on a cardinal scale is fuel for all sorts of interesting conjecture: is a world leading violinist actually that much better, in objective terms, than someone filling half a desk at a local orchestra? How might it vary by field? Are some ‘easy to master’, and beyond a certain point everyone lies on a par, or might some have a convex relationship?[/ref]

This would still suggest, that if I keep practising, I’ll continue to improve without bound – I could get to being at the level of performance of a world leading violinist if only I had enough time. This seems unlikely: even if I practice steadily throughout my life, there will likely be a peak, and I will not be so good when I get really old. This could be because my violin-playing innate talent will not be static, but a function of time – and unfortunately our physical and mental abilities all deteriorate to at least some degree at we age. So long as the decay is not sub-logarithmic, then our logarithmically increasing reserves of practice will not save us.[ref]This motivates making the benefits of talent and practice multiply rather than add, otherwise it is hard to see why performance will decline without demanding that almost all the variance in performance is due to talent alone[/ref][ref]One wonders whether investigating the dynamics of performance over time might yield helpful clues as to its substructure: one could look at how closely the performance-by-age graph matches talent-by-age for candidate talents thought to be important – perhaps the reason there are many more child prodigies in music, chess, and mathematics than creative writing or poetry, is the latter demand things like emotional development which few are precocious in. Even more interesting, if one already has a good handle on what talents matter for performance, one could compare the graphs by age to look at the likely effect of training, as this will manifest (approximately) as an integral factor of talent, and so introduce a phase lag. With less jargon: if some performance depends almost entirely on talent, it should rise and fall in-step with that talent by age, if other fields of performance rely on having accrued lots of practice, changes in the underlying talent take a long time to manifest in performance. So maybe the reason we so few child prodigies as novelists is not that novel writing demands uniquely ‘adult’ skills and practice, but that novel writing just demands vastly more practice than ‘merely’ playing a violin or learning mathematics.[/ref]

The other key wrinkle is that because of the concave function relating to practice, differences in talent become more important than differences in practice when you have a lot of practice: 11000 units of practice is only 1% better than 10000 units of practice, and so could be swamped by modest differences in ability. Even a fixed multiplier on the amount of practice gets significantly less important as lots of practice is accrued. If we both put in the allegedly requisite 10000 hours of practice to master the violin, and I practice twice as effectively as you, my (effectively) 20000 hours only gives me a <10% advantage over your 10000 hours – putting human ability on a cardinal scale is difficult, but plausibly talent may vary by significantly more than that. If there’s a ceiling effect to practice, then this feature becomes even more marked – effectively our talent sets an envelope of our abilities, and so no amount of increased practice on my part can compensate for your greater talent with a sufficient amount of practice:

(Note also that the more talented you might well grow faster in ability than I do, as talent might help accrue practice better. Further, note our performance curves diverge over time with increasing practice. This might fit with how differences in talent ‘filter to the top’ when looking at elite performance: when everyone has practised so much, they are approximately all on a par on the nearly flat part of the practice curve, and so only variance in talent remains to discriminate.)

Further, my limitations as a violinist may prove objective instead of just positional. Even if I can accrue practice without bound, I may not be able to cram in enough before the vicissitudes of age take their toll: maybe no matter how hard I try from now on, I could never accurately play virtuoso pieces like a Paganini Caprice. This is all before contemplating there could be ‘hard thresholds’ to achievement that no amount of practice could overcome: in the same way I could not play the violin with only one arm, perhaps I could not play certain advanced passages without requisite manual dexterity or hand size.

Aside: exceptional practice (but not talent) as a cause for child prodigy regression?

One plank of support for a model like Bram’s would be if child prodigies tended not to turn into adult elite performers to the degree their early performance would predict. One would need to interpret this with care, as one could see this with regression to the mean, rather than a directional effect. If performance at T and performance at T + several years are positively correlated, it is nonetheless the case that the very best at T will tend to be not as exceptional relative to their peers at T + several years; the effect size of regression to the mean grows as you move out to the tail. A higher performing child is more likely to be a higher performing adult than a lower performing child, but there tend to be many more of the latter, and some of them will ‘get lucky’.

But if it were the case that child prodigies ‘burn out’ before they shine as adults, there could be an adverse selection explanation lying anti-parallel to Bram’s. Instead of a child prodigy never learning to develop but starting to coast, maybe instead what is going on that early achievement loads more strongly on practice accrued than talent, and so ‘child prodigies’ are disproportionately selected from those who have managed to cram as much effective practice into their young lives as possible. Later in life, when practice hits diminishing returns, differences in talent come to the fore, and so those selected less strongly on this factor will begin to loose ground as a group. (In most cases, child prodigies are admired for age-adjusted achievement – for having adult or adult-like abilities despite being so young, but seldom are they outperforming adult elite performers.)[ref]Of course, many child prodigies start practising so much because of their extraordinary talents, but on a population you would see early selection load more to ‘more practice’ over ‘more talent’. Data showing that these people do not practice much more than their peers would refute this hypothesis.[/ref]

A cold shower for the growth mindset

I’m biased, but this model seems provide a fairly good fit for what we see around us. Even if there isn’t a hard and fast rule about 10000 hours, exceptional achievement requires a lot of work, no matter how gifted you are.

Yet life isn’t Arbeit uber alles; the closer we get to the heights of human excellence, ‘giftedness’, ‘talent’, ‘genius’ become more, not less, germane. Although the truly exceptional did work incredibly hard, what distinguished them from their equally-hard working but less exceptional peers were rare and exceptional qualities of their minds and bodies. Qualities that, sadly, tend to be lost with age.[ref]An economic corollary is to wonder how this squares with fairly hierarchical ‘one way’ progression in salary in most employment: if mathematics is a ‘young man’s game’, why do academic salaries steady rise with age? If there is a move from ‘knowledge work’ requiring a large amount of information (and so loading on ‘crystallized g’, positively correlated with age up until middle age, and fairly flat thereafter) to ‘reasoning work’ relying less on prior knowledge and more on mental ability (and so loading ‘fluid g’, which tends to peak in young adulthood and steadily decline thereafter), will ‘peak and decay’ supplant ‘steady progression’ as the normal career trajectory; will 20 year olds supervising 60 year olds, and the former anticipating to be demoted in time? (Is this already happening?)

A more futurist corollary is to wonder what would happen to achievement if we all lived longer. A common case for life extension goes along the lines of, “Think what Newton or Shakespeare or Beethoven could have accomplished if they had centuries of work!” But maybe a 400 year old Newton wouldn’t be that great: he would have been ‘past it’ cognitively for a while, and even in the case where one lives longer with an unaging brain, maybe there aren’t any great returns to practice after a given point in physics. In contrast, even with an aged brain we might be excited to see what would percolate through a 400 year old Shakespeare or Beethoven, given their creative prime was nearer the end of their lives.[/ref]

All of this is bad news to the growth mindset. To a first (moment) approximation, people are average rather than exceptional, and most who aspire to being the best will fail; no matter how carefully they nurture their talent, no matter their determination or dedication or grit (or focus on deliberate practice) they will reach an horizon they cannot escape, its dimensions set by a natural lottery beyond their control or responsibility. The lucky few who drew better tickets can surpass them (c.f., accord). Not only can the seeds of lives be scattered on barren ground to never grow, or among weeds to strangle us, but we can be outgrown and overshadowed by our neighbours.

Regenerating the growth mindset

This pessimism should only take us so far. Even if I am right in the above, this pessimism only really manifests if you want to climb right to the summit of a field. In many things, we are content to be on the foothills, and for many of these things diligent work can get us there. I don’t need to be a particularly cunning linguist to become a stimulating conversation partner in French, nor a top percentile VO2 max to join a cycling club, nor an elite musician if I want to play the violin for my own pleasure. Perhaps some things are only accessible to those few who drew a one-in-a-million ticket on the natural lottery (theoretical physics?) but these are rare almost by tautology; most your peers aren’t super-elite, so if you want to emulate things they can do, you can do to (albeit possibly not quite as well).

If there are really diminishing returns, then this is good news for dilettantes (much less work than a true expert, but a large fraction of the benefit!), and underscores how helpful ‘learning how to learn’ and deliberate practice are, especially to develop renaissance-person affectations, as it lets you race up more linear-regions of the practice curve in less time.

Depressing though it may be for our potential to be circumscribed, perhaps that is better than blaming ourselves simply thinking we didn’t try hard enough. Losing the natural lottery on talent seems less damning than whatever personal flaws meant we didn’t work hard.[ref]Further, why not consider the abilities that engender hard work and conscientiousness talents too? Maybe holding someone who ‘wasn’t that talented but worked hard to make the most of it’ in higher esteem than someone who ‘failed to live up to their potential’ our desire to moralize some character traits and not others. Maybe the slacker ‘can’t help it‘.[/ref]

It also suggests some heuristics for career choice. It reinforces the idea that passion is not enough: no matter how passionate I am about the violin, trying to become a professional violinist at my stage is almost certainly a bad idea. It also suggests looking soberly at your talents and how well they apply to a given field, and also how the impact of people in that field is distributed by performance: Some things look like tournament games with ‘winner takes all’ dynamics and power law distributions – a few big winners, but the median case gets very little: think academia, the arts, or professional sports. If talent really becomes decisive on the right tail (no matter how you work), these fields are best avoided without signals of extremely exceptional ability.[ref]Although people shouldn’t ‘write themselves off’ prematurely. Talent in a given field is multifactorial, and so being ‘sub-incredible’ on one feature is not a disqualification. Kasparov:

I soon realized that my answers were disappointing. I didn’t eat anything special. I worked hard because my mother had taught me to. My memory was good, but hardly photographic…

There is little doubt that different people are blessed with different amounts of cognitive gifts such as long-term memory and the visuospatial skills chess players are said to employ. One of the reasons chess is an “unparalleled laboratory” and a “unique nexus” is that it demands high performance from so many of the brain’s functions. Where so many of these investigations fail on a practical level is by not recognizing the importance of the process of learning and playing chess. The ability to work hard for days on end without losing focus is a talent. The ability to keep absorbing new information after many hours of study is a talent. Programming yourself by analyzing your decision-making outcomes and processes can improve results much the way that a smarter chess algorithm will play better than another running on the same computer. We might not be able to change our hardware, but we can definitely upgrade our software.[/ref] [ref]It is unclear what is generating these power-law distributions. Maybe the underlying talent is power law, so one has to go a lot further along – say – physics ability to get from the average professor to Einstein, than from me to the physics professor (seems unlikely, given mental and physical component performance is usually normal). Maybe – as I imply – the translation from ability to performance: the ‘objective’ difference between Einstein and the average physics prof is not that great, but this small advantage gives a much higher output. Or possibly although achievements are normal, recognition or praise becomes the locus of a Matthew effect: if we look at aggregate measure of ‘speed of physics progress’, the counterfactual impact of Einstein over another average professor was not that marked, but it happens that the recognition of this progress distributes to the biggest names. In the latter case, it may be worth going into physics even if one is near certain one won’t be exceptional, because your real impact may be significant but systematically under-recognized. I’d actually guess the latter case is common (but the subject for another post), but maybe not in cases where the ‘direct’ impact is recognition based – not much point being median novelist if the readership for the median novel is 0![/ref] For the less exceptional (or risk averse, as success in these things depends on luck and other stuff besides ‘raw talent’) fields that distribute normally by performance may be a better bet.

But who knows?