Mathritocracy

Substack absolutely sucks for technical articles so I would highly recommend checking the Notion version out here. The formatting here is terribly off, I tried my best but got lazy.

Disclaimer: All thoughts and opinions are mine. As much as possible, all sources cited have been thoroughly vetted for fair use and accuracy. Very minimal AI (mostly deriving katex equations and proofs) has been used in the writing of this article, hence, there may be mistakes here and there but hey, at least it’s more human than most things out there nowadays.

I am also in no way an expert on politics, economics, sociology or really anything. I’m just a 23 year old Singaporean who has some opinions he can’t keep to himself.

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The idea behind the Singaporean society is maddeningly simple if you look past the smoke and mirrors. To our core, meritocracy runs it all — the idea that our work determines our rewards in life. To be hardworking, smart or cunning, that is the only way forward in our society. Either that or you won the birth lottery and can sit comfortably atop the toil of generations before you (which actually if you think about it for more than two seconds, already betrays the very meaning of what it is to be meritocratic). To many of us, that is what meritocracy is and continues to be: a system that has structured Singapore since her birth and brought us much reward globally and locally.

Competition is great for social progress! Or rather, that is what we’ve been told by our parents, our grandparents and probably some random Grab driver uncles too. “*Work hard, study hard, then ok already! Go to good school, get good job then dun need worry in life one.*” It’s probably way easier to believe them, especially if they appear to have reaped the benefits of our systems. The conclusion about meritocracy can be easily drawn up for those who weren’t handed the short end of the stick after all.

As someone who doesn’t accept wholesale what our elders might sometimes ramble about, I thought I’d write an analysis of the system I grew up in. This may of course, be very different for kids growing up now as our system has evolved, and I’ll do my best to be cognizant of this! This article is not an attempt to glorify or discredit meritocracy but to understand the cogs and gears of it. To add some depth to my (very amateur-ish) analysis, I wanted to borrow some ideas from mathematics, particularly from the field of game theory — which always lends itself handy to the study of social computing and economics. Meritocracy is, as most other socioeconomic systems, very dynamic with certain strengths, contradictions and certain natural thresholds that emerge, which vastly change the outcomes of it.

To begin, we have to go back to the very start of it all. The fight for our first wins.

Why the First Win always matters

Every Singaporean kid learns early that major exams are the biggest indicators of the paths we will take in life. Primary School Leaving Examination (PSLE) determines secondary school. Your secondary school determines your streaming and your O/N-level odds of success. O-levels determines your junior college or polytechnic and so on. This effective chaining of one win to another (or conversely, one loss to another) is what pressurises many kids and even their parents to make the very best out of the first touch point — PSLE. When I was growing up, I remember that my parents, like many of my friends’ parents, did not f*** around when it came to my PSLE. So, naturally I didn’t f*** around too.

Officially, this academic rite of passage to chart our own success story, where the best at each stage and test advance, is the one true Singaporean arc of life. It appears fair and potentially the most socially optimal way to promote some healthy competition in our society.

Let’s branch away for a bit from the boring societal analysis to something even worse, math. In 1976, Derek J. de Solla Price was studying citation networks that formed in research papers. He noticed an emerging pattern, which might seem stupidly obvious to most of us: papers that already had many citations were far more likely to get new citations than papers that had only a few initial citations.

The core idea is immediately obvious: *success breeds success.* Price formalised his observations (referred to as [Preferential Attachment]) by imagining a growing system where new entities enter over time. Each new entity brings with them a certain number of initial connections m. They connect to existing entities, but not randomly. The probablility that they connect to another entity i depends on how many connections k_i that entity already has. Simply put, this can be defined by:

The numerator is entity i’s current connections plus a small baseline k_0, which ensures that even newcomers to this system have some chance. The denominator is a sum across all entities in the system.

Well, so what? The rich get richer, we all already kinda know that. This Price dude must not be very smart I guess. Ok, I’m joking. We can go deeper with this and Price obviously knew that too. He found that after many rounds, the distribution of connections is proportional to:

p_k denotes the fraction of entities with k connections and the negative exponent indicates that this is a power law distribution.

The key insight we can draw here is that after enough time, you get a system where a tiny handful of entities have the most connections. A handful of entities will have a moderate amount and shockingly, the vast majority will have very few. Crucially, *this outcome emerges even if everyone started equally*. It’s pretty beautiful to see the idea of competition and cumulative advantage be expressed mathematically.

We can break away from the numbers now and translate this back into our Singaporean context. Let’s replace our “entities” with stressed-out Singaporean Primary School kids and our “connections” with educational advantages. To provide some solid, real-world examples:

• k_i is obviously not just your exam scores but everything that compounds with it. Think alumni networks, access to tuition, parental involvement in a child’s education, elite school reputation, internship connections etc.
• m is the initial advantages a child starts with, blessed by the divine fate of the universe and the birth lottery as mentioned earlier
• k_0 are the baseline opportunities available to all regardless of background. These are your universal access, social equalisers like SkillsFuture for continuous education and Student Care Centres for after-school care needs across all primary schools.

The model above predicts that the initial small differences in k_i will compound over time. The child who gets into a top primary school has slightly more access to more advantages, increasing their k_i, making more advantages ever so closer for them. By secondary school, the gap has grown. By JC, it’s substantial. By the time that kid hits uni, that gap is pretty much a solid determinant of how their life will go and potentially, how their kids’ lives will go.

It might help to say that I am a benefactor of this system and hence, I feel confident enough to state this. My family is truthfully financially average but I was fortunate to get into a relatively good secondary school and subsequently, one of the top JCs in Singapore. I can confidently say that I wouldn’t have ended up in the university course I’m in today without those prior advantages in life. Am I among the smartest people in my course? Hell no, very very painfully far from it. The point is, my own cumulative advantages have, very fortunately, paid off.

I am also very fortunate to have interacted with many people who have expanded my horizons regarding meritocracy. I have met people who have been disadvantaged by our system early on in life and have had to fight a very noble uphill battle to course-correct the trajectory of their Singaporean arc. I have also met people who have succumbed to the brutality of a system that is designed to breed insurmountable inequality for them. Not only does this spell gloom for them, but I have also seen how this creates immense troubles for their future generations. In a brutal twist of irony, I have also met people who have benefited immensely from a network of nepotism (or “connections” as they would rather have it be called) in the corporate world. Somehow, in an economy as rough as it is now, opportunities like internships and job offers have magically found their way easily into their grasp (yea, I’m looking at you finance sector).

I would argue that this is a symptom of a broken system, one that calls itself meritocratic but is really just nothing more than a bad simulacrum of what meritocracy is. Or, meritocracy truly is engineered to play out this way, so the system isn’t really broken. The question is then, do we want to keep it running like this?

Optimal Competition and Reward

Here’s the next thing on our agenda: is finding the best entities in a system and rewarding them the ideal way to promote innovation and progress?

Let’s imagine now we’re the Ministry of Competition with a fixed budget of rewards that we can distribute to n contestants in our Annual Competition for the Sake of Competition. Each of those n contestants can exert effort e_i at some personal cost of resources. You want to ensure that the overall effort exerted in this competition is maximised (in order to boost productivity and learning).

You have one key parameter that you intend to monitor: how precisely you should link effort to rewards. Let’s model this as r, our grand meritocracy parameter if you could call it that. Let’s kick this competition off!

We can model the probability that a contestant i wins this competition using:

• When r = 0, the formula effectively becomes P_i= \frac{1}{n} . This means that literally everyone has an equal chance of winning the competition. Legit as good as playing TOTO.
• When r = 1, the chances of winning the competition is directly proportional to your effort relative to others. In other words, if you work twice as hard, your chances double. This should sound fair on paper.
• When r is very large, the contestant with the highest effort invested wins with almost guaranteed certainty.

You might be inclined to believe that making the competition more meritocratic would always hence, increase the total effort. If effort determines rewards, then people will obviously try harder, right?

You’ll come to be surprised that there exists a threshold that, beyond that, increasing $r$ actually *reduces total effort*. Let’s say each contestant chooses effort $e_i$ to maximise their expected payoff from the competition:

Here, we represent $V$ as the value of winning and $c(e_i)$ is the cost of the effort. When contestants are not identical in terms of their abilities, some having higher abilities than others, the equilibrium effort levels will end up shifting. For example, a contestant with ability $a_i$, the optimal effort is:

Let’s break it down a little with some of the main insights we can derive from how people respond to meritocratic systems, mathematically.

1. Ability Gaps Get Amplified

Small natural advantages become enormous effort differences as $r$ increases. The system doesn’t just reward talent, it also ironically demands the talented work exponentially harder, while making it pointless for everyone else to even try.

• At $r = 1$: twice the ability → $2^2 = 4$ times the effort
• At $r = 1.5$: twice the ability → $2^5 = 32$ times the effort
• At $r = 1.8$: twice the ability → $2^{14} = 16,384$ times the effort

2. Your Competition Determines Your Effort

The term S in the denominator is everyone’s total performance. If you’re surrounded by people who are damn freaking smart, S is large, potentially pushing your effort down. If you’re the smart fella, S is smaller, and you work harder.

We also see ultimately that context matters. Put the same student in different environments, and they’ll exert completely different effort levels.

3. The Breaking Point at r = 2

Look at the exponent $1/(2-r)$. As $r$ approaches 2, it blows up. Beyond $r = 2$, low-ability contestants stop trying entirely and shut themselves off from the competition. You could think of this as essentially what burnout is in a society that enforces a rat race to survive.

You can think of $2-r$ as a “stretch factor” for effort.

- When $r$ is small, $2-r$ is large, so effort responds gradually to ability.

- As $r$ increases, $2-r$ shrinks, so effort becomes more and more responsive so small ability differences get magnified.

- When $r$ hits 2, the stretch factor becomes zero, meaning effort would need to be infinite to compensate, which is mathematically stupid.

- Beyond $r=2$, the stretch factor becomes negative, which mathematically signals that the low-ability contestants drop out entirely.

So what then is our ideal “level of meritocracy”? I believe that somewhere between 1.5 and 3 lies the sweet spot, where our best get their chance to shine, while also ensuring that they don’t burn out too fast or be overly complacent. Those who may not be as talented still get a slice of the pie and are also not incentivised to disengage with the system. This actually ensures some degree of healthy competition, as I would argue.

It’s hard to gauge truly where we’re at as a country but I’d say that: with our fine-grained scoring system of PSLE, GEP being opened at age 9 and our love of streaming and elitist, branded educational institutions, we’re probably high up on the $r$ scale (probably $r > 3$).

Risk

So far we’ve seen that previous models assume that if you work hard, you get *something* I guess. Probably not really what you deserve, but definitely something! Ultimately, we understand that real life is never tidy and that effort doesn’t always guarantee success. Luck is definitely a key factor in some successes in our lives, whether big or small. Picking option C on a question you don’t know and getting it right. Landing a job you bombed the interview for. Talking to the right person when they were in the right mood. These small random events can change our life trajectories, much like the butterfly effect.

In a low $r$ (where $r$ is the measure of how meritocratic a system is) system, that’s actually perfectly fine. One bad day or bad outcome doesn’t ruin your life permanently. There are second chances for everything. The high entropy of our lives becomes a smooth randomness that we can deal with over time.

Let’s consider however, in a relatively high $r$ system, sort of like our Little Red Dot, randomness is often a terrifying concept for us, as we love to optimise our lives around success and the most optimal path to it. When every contest matters and the difference between winners and losers can be completely different life paths, rational people will obviously not take risks. Why risk everything when you can play it safe and guarantee yourself *at least* *some degree of success*? **We can’t all be winners but at least we can minimise how much we lose out to others.** Typical kiasu mindset type shit.

Let’s revisit that competition we discussed earlier. We want to add something to spice it up a little. Let’s add a dash of luck, a hint of randomness to our system. Let’s redefine performance of our entities to:

where $\varepsilon_i$ is random noise, good luck or bad luck on the day. It could be positive or negative, and its size depends on how much randomness exists in the system. We now see that the probability of winning the competition is now dependent on both effort and luck. Here’s where it gets interesting!

When $r$ is low, luck actually matters more. Your chance of success in the system actually depends on random factors that are beyond your control. To us, beings who are used to trying to unrandomise the random, this might be a travesty. This system however, does have a surprising benefit: people are very much more willing to experiment with different “life arcs”. Think about it like this: if success isn’t guaranteed anyway, you might as well try something different. Something you’re passionate about. Something scary. The downside of failure is actually more limited and less scary because the system is so full of noise and entropy.

When your $r$ is high however, the opposite is true. The contest for success is intended to minimise luck and ensure that the “best” person (i.e. as we discussed earlier, the person who has the best abilities and also puts in the most effort, by right lah) wins. This means that if you are not actually sure where you stand in the pack, the rational choice for you is to actually play it safe because putting in effort into tried and tested routes guarantee you some degree of success. Take the subjects you know you can score in. Do the sport you’re good at for CCA. Stick to the proven methods. Try not to experiment, because experimentation might fail and in a high $r$ system, failure could actually spell your downfall.

We can explore this concept a little more in depth by looking at how people choose different strategies for their life arcs:

- Safe strat: guaranteed performance of at least $a_s$ with moderate effort $e_s$

- Risky strat: potentially higher performance $a_r$ if it pays off, higher effort $e_r$ and uncertainty

In a high $r$ system, the expected payoff of the risky strategy is:

But $P(\text{success} | \text{risk})$ depends on the entire distribution of outcomes. If the downside case means you fall far behind the safe players, your chance of winning drops to near zero. And since $r$ is high, even a small disadvantage in expected performance translates into a huge disadvantage in winning probability. The resultant outcome is obvious: the risky strategy only makes sense if its upside is enormous, significantly larger than the safe strategy’s performance. For most Singaporeans, from what I’ve seen, their risk appetite isn’t large enough to go against the grain. So, we play it safe generally.

We see this manifest all around us actually. In school, students more often than not, choose subjects based on scoring potential and what career those subjects can bring them. Rarely are they picked out of interest (or even if they are it’s because we probably forced ourselves to have an interest in them so we can rationalise it in our heads). The safe combination of STEM subjects is heavily favoured over the arts, even for students who excel insanely in the arts and other creative arenas. I know many Computer Science majors who are insanely talented in the arts, but pursued something as “mainstream” as CS due to career prospects and traditional expectations. In terms of entrepreneurship, it’s seen as something you do only after you have a stable and secure job. It’s more of a hobby that we dabble in sometimes, not a serious commitment because the risk is far too great. This obviously manifests in our startup scene that is, lackluster to say the least (it seems to be getting better with time, however). This is definitely not because of a lack of resources or support by our institutions but really because our population of risk-takers is disappointingly meagre. Singapore isn’t bad for startups contrary to what many might think, **Singaporeans are.**

Ultimately, these are just rational responses to a system where the stakes are high and the margin for error is extremely low and very punishing. A hyper-meritocratic system selects for safe, boring and exam-smart behaviour and not creativity or entrepreneurship. We often term ourselves as a population of exam-takers, which isn’t wrong at all.

Quite dangerously, this disincentivises the willingness to fail and learn, which I believe is highly detrimental. I believe that to fail **productively**, is to chart a new daring path to success. In my classes, people are always terrified of answering questions, in fear that they will be struck down by the hyper-brutal violence of a “*hmm, I’m not sure you’re quite right*” by the teaching assistant. Inspite of all that, I am grateful to see more and more of my generation face this risk head on and try new things. It’s extremely refreshing and I would say “f*** it and try it” for anyone faced with this dilemma of risk aversion.

I do see this as a loss on the whole, for Singapore. Think of all the startups that never got founded because their founders thought their idea was too risky to ever see the light of day. Think of all the deliciously creative works of art that were never birthed by an artist too worried about falling behind. There are often many hidden costs like these, that I suppose we have to bear in our over-optimised meritocracy.

Thoughts

I think the biggest question that remains is: what kind of society do we want to be? Meritocracy has definitely brought us a long way from the underdeveloped, third world country we were post-colonialism. My thoughts are that there are two things that we should pair with our meritocratic system: humility and compassion.

I believe with humility comes a recognition that the successful among us did not get there by effort alone. Conversely, the unsuccessful didn’t fail because of a lack of it. Ultimately, luck and a gazillion other factors might play a role much larger than we cynically would like to admit.

I believe with compassion comes an understanding that in a system designed to produce winners and losers, we owe the losers much more than apathy or a lecture about working harder and how they could have optimised their path to success better. We owe them dignity, support and genuine second chances.

There are many moments I see in my day to day which showcase the beauty and ugliness of meritocracy. The amazing city I see around is the product of infinite generations of hardworking men and women, both born here and those who travelled the seas to contribute to where I call home today. The everyday draining office politics and terrible scandals we see periodically are too, a product of a system that incentivises us to step over one another to climb the ladder of success.

The systems we build today will determine the advice we tell our kids and their kids. Will it be something of tired refrain or something a bit more honest, hopeful and true?

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If you made it here, thank you for reading! I really appreciate it :) Have a lovely day!