# Talent, luck and success: simulating meritocracy and inequality with stochasticity

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I remember when I was fresh out of academia, looking for industry jobs to work as a data scientist. Some of my friends in a situation like mine got a job easily. That was not my case. It took me a long time to finally land on a job. Whenever I got a rejection email after an interview, I was harshly critical of myself. What did I do wrong? How can I do better next time? Am I not smart enough? Shouldn’t I have said that? And so on. Difficult times.

One day, to encourage myself, I watched a video about self compassion. An important lesson from the video was not to underestimate luck. This idea reassured me, but also made me think often about how to counterbalance luck to build a fairer society. That’s why this paper got me really excited. It asks why such asymmetry in wealth in our society exists whereas our talent (such as intelligence) is normally distributed. If we live in a truly meritocratic society, it’s not supposed to happen, is it? Based on model simulations, the authors concluded that randomness plays a fundamental role in selecting the most successful individuals.

However, as some people have already criticized, the authors only did a basic job on their model simulation and somewhat exaggerated the results. Regardless of the shortcomings, I still found it interesting because it showed an example of quantified luck in socioeconomic settings. Since the model was quite simple, I decided to implement the model in Python with some modifications to manipulate the model parameters. I also simulated various socioeconomic scenarios to find insights on the relationship between talent, luck, and success.

# The paper

The authors used a mathematical model to simulate luck and to quantify its effect. In the model, some events happen to people and their resources (capital) change over time. Here the amount of capital is the measure of one’s success.

## People

1000 people are randomly placed in a 2D space (Fig. 1). They neither reproduce nor die. At the beginning of a simulation, everyone has the same amount of capital (C=10). Everyone has a talent (T) parameter, which is drawn from a normal distribution with 𝜇=0.6 and 𝜎=0.1. The talent does not change over time. Since the talent follows a normal distribution, technically it can be negative or larger than 1, but for convenience we assume T~[0, 1].

## Events and simulation

A single run of the simulation is equal to 40 years of working life (from age 20 to 60). Each time step is 6 months, so we have 80 time steps in total. At a given time step, there are 500 lucky and 500 unlucky events, green and red dots respectively (Fig. 1). In the beginning of a run, the dots are randomly placed. Once the simulation starts, at every time step the dots move around the space as if they are gas particles. If a person is hit by a green dot, a lucky event happens; red, unlucky event.

When a lucky event happens, your capital is doubled but only if you are talented enough to get profit out of it (i.e., only if a random draw between 0 and 1 is smaller than your talent, T):

When an unlucky event happens, your capital is always halved:

## Key results

Even though the talent is normally distributed among the population, after a single run of the simulation (40 years or 80 time steps), the capital distribution becomes highly asymmetric: many are poor and few are rich (Fig. 2).

In this run, the poorest person (the leftmost dot in Fig. 3) has talent that is +1𝜎 above average (T>0.7), but the richest person has average talent, which makes this world un-meritocratic. The authors ran multiple simulations and found similar results. They concluded that

Randomness plays a fundamental role in selecting the most successful individuals.

## Criticism

First, the paper lacks explanation on their methodology. Why did they simulate the randomness and luck in a 2D space? The authors also didn’t provide explanation on their parameter value selection. Studies like this (model simulations on existing phenomena) often require careful parameter selection using meta analysis. Second, they didn’t even attempt to vary the parameter values and jumped to a conclusion too soon. Plus, it would’ve been nice if they tried to validate their theory by fitting the model to the real-world data. Finally, making a big statement like the quote above, without careful evalutation of the model, can be misleading.

## Can we do better?

I implemented the model in Python with some modifications to play with various parameters in the model. I found interesting effects of the parameters on talent and success. I also modified the model to introduce other socioeconomic scenarios.

# Implementing the model

Instead of using a 2D simulation, I decided to simulate the random events by simply using a Bernoulli process:

• At every time step, flip a coin: there is a probability, P_event, that an event happens. It can be either a lucky or an unlucky event.
• If an event occurs, there’s 50–50 chance of it being lucky or unlucky.

This way, it is easy to quantify and understand the stochastic nature of luck. For instance, if P_event = 0.1, a lucky or an unlucky event occurs every 5 years on average. Everything else is the same as the original model in the paper (equal start of capital, normally distributed talent, etc.).

The new model can create similar results from the paper (Fig.4): the capital distribution is highly skewed even though we started from normally distributed talent. We can also visualize the changes in capital over time (Fig. 5). Notice that this particular individual is very talented (T=0.8) but due to a series of bad luck (red dots), her capital decreased.

# Why it looks like average people are more successful

Fig. 5 shows an example of a talented person being very unsuccessful because of bad luck. However, is it really true that luck plays the key role to select most successful individuals as the authors claimed?

Similar to the author’s results (Fig. 3), a simulation from the new model shows an average person being the most successful (having the highest capital) but not the most talented (the highest talent).

Why is this the case? The answer is actually quite simple. It is because we have too few samples for the highly talented. In this model, capital is only increased by lucky events. Although talent helps get profit out of a lucky event, since the probability of having multiple lucky events is small, the expected number of average-talented people getting lucky multiple times is higher than the high-talented people simply because there are more average-talented people. We can confirm this idea by using uniformly distributed talent.

Now, the average-talented are less likely to outperform the high-talented. In fact, there is a positive trend, showing that highly talented people tend to be wealthier than others. Hence, it is the interplay between stochasticity (luck) and normally distributed talent that makes it look like average people are doing better than the talented, not just the randomness itself.

# Now the fun part!

Another reason I found the paper interesting is that it tackles the idea of meritocracy that affects many aspects of our lives and society. Assuming that talent is an inherent trait, it would be interesting to see how luck affects the relationship between talent and success. The talent-capital scatter plot can visualize meritocracy, economic inequality, and stochasticity.

• If talent and capital show positive correlation, we can say that the world is meritocratic; the more talented you are, the more successful you are likely to be.
• Due to luck, inevitably there is a degree of stochasticity, which creates a cloud of dots, not a straight line. If there is a wide spread of dots at a given talent value, you may say the world is more stochastic (more luck-driven), which in turn, makes it less meritocratic because then the talented are no more successful than the average.
• The width of capital distribution is a measure of economic inequality. The wider it becomes, the more inequality exists.

From now on, we will see many talent-capital relationship plots from different parameter values and scenarios. Keep Fig. 8 in mind to help you understand those plots in terms of meritocracy, inequality and stochasticity.

# The effect of stochasticity

In the new model, we defined the P_event parameter as the probability of having a lucky or unlucky event at a given time step. In fact, P_event is probably the most interesting parameter because it shows how we will be affected by the prevalence of luck in a society. In the simulation above (Figs. 4–7), we fixed P_event=0.1, as if an event happens every 5 years. What if we vary the parameter? The P_event parameter can be interpreted in the following way:

• A small P_event value means the world is deterministic and predictable.
• A large P_event value means a highly stochastic and unpredictable world.

Fig. 9 shows how stochasticity is manifested in talent-capital relationship. P_event=0.001 means an event can happen at every 500 years and P_event=0.9 means an event can happen at about every 6 months.

• When P_event is small, nothing really happens and everyone’s capital is more or less the same as the start amount (C=10).
• However, the more stochastic the world becomes (wider cloud of dots) , the more often our lives are bombarded by unpredictable lucky and unlucky events, which creates large economic inequality.
• With high stochasticity (P_event>0.1), talent becomes more useful because people have to be smart enough to profit from the lucky events. Therefore, we start seeing positive correlation in the high stochasticity settings, which means talent does matter in this setting.
• Also, with high stochasticity, more people fall into poverty because of the imbalance between the return of the lucky and unlucky events; unlucky events always punish individuals
• but the lucky events only take effect if you are smart enough to get profit out of them:

# Scenario 1: Balanced unlucky events

The first thing I wanted to try was to fix this imbalance. Instead of always being punished by unlucky events, we can let people escape from bad luck if they are smart enough, which is a pretty reasonable assumption. Here, the new function for unlucky events becomes:

Notice that we have the conditional term as we had in the lucky event function.

After fixing the imbalance, now people have more capital than before (Fig. 10) because now people face unlucky events less frequently, especially if they are talented. We also see talent playing a stronger role (steeper slope) because now talent affects both lucky and unlucky events.

# Scenario 2: Talent directly affects the return of luck

Now let’s assume that talent can directly affect how much we can profit or lose from the events. We can remove the stochastic process in the lucky and unlucky function (rand[0,1]) and come up with the following:

One benefit of this scenario is that now the only parameter governing the stochasticity is P_event. This is a reasonable assumption too. For instance, A’s talent is 0.2 and B’s is 0.8. When a lucky event occurs, A’s capital becomes ×1.2 but B’s becomes ×1.8 because B is more talented. When an unlucky event occurs, A’s capital gets ×0.2 (losing 80%) but B’s is now ×0.8 (losing only 20%). Thus, the high talent is helpful in both situations.

Scenarios 1 and 2 look similar but when P_event gets larger, we have steeper slope in Scenario 2, mainly due to the less talented getting poorer (Fig. 11). This is because Scenario 1’s coefficients are fixed (2 and 0.5) but here they are not (1+T and T). Since T~[0, 1], it is possible that T < 0.5, meaning the less talented can have harsher consequences from bad luck than Scenario 1 where bad luck halves capital at most.

# Scenario 3: Paycheck (“Hard work pays off”)

So far, if nothing happens at a given time step, capital remains the same. This is not realistic because in general people have regular income from their jobs. We can implement this idea by adding a term to capital when neither lucky nor unlucky events occur:

The new term, wT is the amount of one’s paycheck. w represents the importance of the paycheck. The higher it is, the more pay you get. w is multiplied by the talent parameter T to implement a meritocratic idea: the more talented you are, the fatter the paycheck you get.

## If you have work, you can bounce back from bad luck!

The main effect of the paycheck is to cause a stable increase in capital. This greatly helps people recover from the worst situations (e.g., a series of unlucky events). For instance, Fig. 12 shows how a very unlucky person’s capital changes over time. She had many unlucky events (many red dots) but thanks to her paycheck, she was able to bounce back and keep earning money. Since she kept making money even after unlucky events, then even though another unlucky event hit her, her capital did not plummet too dramatically.

## Paycheck and inequality

To understand Fig. 13, you might need to recall that everyone starts with the same amount of capital (C=10), and this determines the magnitude of the paycheck in relation to starting capital. For instance, for a person whose talent is 0.8, if w=10, her paycheck is 8, which is quite large. On the contrary, if w=0.01, her paycheck is now 0.008, negligible compared to 10 (the initial amount).

This is why as w gets larger, the band of dots get tighter in Fig. 13; capital is dominated by paycheck, which is deterministic (no randomness). This also means less inequality because we have a narrower wealth distribution.

## With vs. without paycheck

We see very interesting results when we compare models with and without a paycheck (Fig. 14). First, because we have additional income, the paycheck increases capital in general. In the high-stochasticity settings (P_event > 0.1) in Fig. 14, instead of showing linear talent-capital relationship, the yellow dots (paycheck scenario) form a curved shape where the capital stays higher for less talented people than the scenario without a paycheck. This is because the paycheck shields them from bad luck to a certain degree because they still make money from working (paycheck) and they can still increase their capital even after unlucky events.

# Scenario 4: Interest rate

Another way to increase capital when nothing happens is to introduce an interest rate (r). So far, the capital does not increase by itself. In this scenario, at every time step (every 6 months), your capital increases based on an interest rate (all the other rules remain the same). Interest rate is applied in the beginning of each time step before other events (e.g., lucky or unlucky events) happen.

## Interest rate without paycheck

Obviously, higher interest rates increase capital further. In Fig. 15, how much the dots are spread seem similar across different interest rates. But remember that the y axis is on a log scale. Although the spread of dots looks similar between the leftmost plot (r=0.001) and the rightmost (r=0.25), the actual capital difference is much larger among the population when r=0.25 (compound growth). Thus, when the interest rate increases, even if the capital of all individuals increases, there can be more economic inequality.

## Interest rate with paycheck

When the interest rate and paycheck both exist, the plots look slightly different (Fig. 16). When the interest rate is low (leftmost), the spread of dots is narrower (see the leftmost one in Fig. 15). This is because in this scenario, when the interest rate is low, your capital is mostly dominated by your paycheck.

Let’s compare scenarios with and without an interest rate when a paycheck (w=1) exists. When the interest rate is applied (yellow dots in Fig. 17), if the world is more stochastic (P_event > 0.1), we get a wider distribution of wealth compared to the no-interest scenario (blue dots in Fig. 17). Note that we see the J-shape in the distribution when P_event > 0.1, as we did in Fig. 14. This is because of the paycheck which prevents the less talented from going into extreme poverty.

# Scenario 5: “The rich get luckier, the poor get unluckier”

One might say this is probably the most realistic scenario so far. When one gets rich, she is more likely to have more influence and power than others, and thus she might get luckier than others. On the other hand, if one gets poor, she might be likely to face unlucky events more often than others because she just doesn’t have enough resources to cope with those events. Unfortunately, this actually happens often in our society.

So far, in our model, when an event happens, there’s an equal chance (0.5) of it either being lucky or unlucky. To introduce our new unfortunate scenario, we need to change the probability of an event being lucky or unlucky by recalibrating the probability at every time step based on how much capital a person owns:

• At every time step, we compute a z-score for every individual based on the capital distribution of the population of that time step.
• Then we compute cumulative probability for each z-score, which becomes the probability of an event being lucky for the next time step.
• For instance, at a given time step, if your capital is within the top 1% of the population, at the next time step, if an event happens, there’s 99% chance that it is a lucky event and 1% chance that it’s an unlucky event.
• On the other hand, if your capital is within the bottom 1% of the population, at the next time step, if an event happens, there’s 1% chance of it being lucky and 99% chance of it being unlucky.

When this scenario is applied, we see the talented can get extremely rich (Fig.18, the rightmost) under high stochasticity, creating huge inequality. In fact, now the maximum of the y-axis is nearly 10²⁴, almost 10¹³ times larger than before. What is interesting though is that the less talented do not get too poor. You guessed it: they still have paychecks!

## Can it be worse?

What happens if we remove the paycheck from the model?

Not surprisingly, when the paycheck is gone, less talented people fall into extreme poverty because they cannot bounce back from unlucky events, which creates an extreme version of inequality.

# Scenario 6: Income tax

One of the measures to reduce inequality is to impose an income tax. Although in the U.S. there are discrete tax brackets, to simplify the matter, I used a method similar to the one in Scenario 5 to implement income-based tax rates:

• First, compute z-score for every individual based on their capital among the population.
• Compute a rate as we did to compute the probability of an event being lucky but we now scale this number so that they fall in a specific range defined by minimum and maximum rates.
• For instance, let’s assume we have [0.1 (10%), 0.4 (40%)] range of tax rate. If your capital is within the top 1% of the population, your tax rate is 99% × (0.4–0.1) + 0.1 = 39.7%. If you’re at the bottom 1%, your tax rate is 1% × (0.4–0.1) + 0.1 = 10.03%.
• Tax is calculated after a paycheck is earned, and deducted from the total capital.

Obviously, the more tax we pay, the less capital we own. Tax can reduce the variance of the capital (compare the leftmost and the middle plots in Fig. 20) and create less economic inequality. However, a large tax rate (Fig. 20, the rightmost) can also make the slope very flat, playing an anti-meritocratic role.

# Scenario 7: Social safety net

Finally, what if we have a hard lower bound for one’s capital as a social safety net? For example, we can have a scenario where one’s capital cannot go lower than the starting amount.

Fig. 21 shows that the social safety net has more effect when P_event is large (> 0.1). This implies that even if we live in a world with high randomness, this type of social safety net can prevent us from getting too poor.

# Other scenarios

• Uneven starting amount of capital: the paper and the scenarios above assume everyone starts with the same amount of capital. What will happen if the amount of capital is a random number drawn from a distribution? Uniform, normal or log-normal?
• Redistributing wealth (the total amount of capital): we never tracked the total amount of capital but if the rich make a lot more money than the poor, what is going to happen if we distribute the rich’s high income to the poor? What kind of redistribution strategy can we choose?
• Capitalism (a fraction of the poor’s labor goes to the rich): in a capitalistic society, employees work for employers and some fraction of the value of the employees’ labor goes to employers. This is definitely going to worsen the inequality, but how much?
• The rich pay to get away with unlucky events: we also see this happening in our society. If the rich pay a certain price to change their fate, how would the talent-capital relationship look?
• A group of people are inflicted by bad events all at the same time: this is similar to what natural disasters do. In this scenario, 2D simulation can be actually useful although we will have to redefine lucky and unlucky events.
• The rich live longer, the poor live shorter: so far, we have assumed that everyone has the same number of time steps. Unfortunately, in real world, the rich are likely to have more access to better medical service and they might live longer because of that.
• Inheritance: speaking of lifespan, what happens if individuals can reproduce and pass down the wealth to their children?
• Basic income: we saw how helpful a paycheck is to prevent people from falling into poverty. However, in our model, a paycheck depends on talent. What if we have a term that is constant and independent of talent? What will its effect be compared to a paycheck?
• Stochastic pay: the paycheck in our model is deterministic and proportional to one’s talent. We all know that this is not the case in the real world. Talented people can have a hard time getting a decent job and they might end up having menial jobs with low pay. This means luck can affect our paycheck as well.
• Time-variable pay: people’s paycheck may vary over time as well. They can get promotions over time and make more money, or the opposite can occur.
• The poor pay more to buy things than the rich: last year, while listening to WNYC’s On the Media podcast, I learned interesting facts about poverty such as that being poor is very expensive. For instance, you can’t buy things in bulk and so you end up paying more. You can’t buy healthy food, you become sick, and then you pay money for medical expenses. When this ironic fact is implemented in our model, how would the talent-capital relationship look?
• Debt: in our models, the capital can go as low as possible but is never negative. In reality, people may borrow money and accumulate debt. What’s going to happen if we allow debt in a society? What if the interest rate is applied to debt as well as capital?

# Future ideas

## Fitting parameters

A great benefit of building quantitative models is that we can fit parameters to real data to estimate the parameters. For instance, we can estimate P_event for a given society, which can tell us how often a random event occurs in the society. We can even go further and estimate the parameter for different societies or countries. Then we can quantitatively answer questions:

• How large is the effect of luck in a society compared to others?
• How often do random events happen in a country, and how this affects people’s lives?

## Timing of events

We haven’t discussed much about the timing of events because in some scenarios the timing does not matter at all. However, in more complicated scenarios, timing may make a visible difference. For instance, is it easier to recover from bad luck if unlucky events happen in one’s 20s than in one’s 60s? This can be extended to questions such as:

• How should people prepare for their future?
• What kind of economic measures can we take to protect senior citizens?

# Conclusions

## Stochasticity disrupts meritocracy

If lucky events can bring large amounts of profit and unlucky events can bring huge losses to individuals, due to the stochastic nature of their lives, this will inevitably disrupt meritocracy. Even without other socioeconomic situations that can aggravate this phenomenon, it appears that people with average talent often do better than the more talented simply because there are many people of average talent and a few will be lucky.

## The steepness of the talent-success slope is related to fairness and inequality

In a meritocratic society, there is positive correlation between talent and success. But how steep should this relationship be? If the talent-success slope is very steep (i.e., a small talent difference causes a large capital gap), you might end up having a wide distribution of wealth (severe economic inequality). On the other hand, if the slope is quite flat (i.e., a big talent difference makes little difference in capital), the society has weak meritocracy, which may disincentivize people from achievements. The right degree of steepness depends on how one defines fairness and how much inequality a society can handle.

## Paychecks protect people from poverty even in the worst case

One of the most interesting results I found was the effect of a paycheck (stable income). Obvious in retrospect! Even in the worst case where someone faced many unlucky events, if they had a paycheck or stable income, they can always bounce back and recover from bad luck. Plus, we saw that economic inequality can be diminished when the paycheck is a large fraction of one’s capital. Personally, I already knew stable income is useful, especially for the poor, but simulating it using a simple mathematical model was an interesting and meaningful experience.

## Data visualization can provide insights on socioeconomic problem

Although we simplified complex topics like meritocracy and economic inequality, we were still able to visualize how various socioeconomic parameters influence the talent-success dynamics. Data visualization helps understand how the terms in a quantitative model interact with each other. It can also inspire us to explore new ideas and variations on an old one. When visualizing data, it’s important to start off simple, to run sanity checks like summary statistics or scaling, and to ask “Why?” whenever you find interesting patterns.

# Acknowledgment

Many interesting discussions about the paper helped me develop these ideas and write this post. The biggest help came from my brilliant husband, Gabe Bodeen, who brainstormed with me, challenged my ideas, and came up with interesting scenarios that I was able to implement. Roni Kobrosly, a data scientist and one of my best friends, contributed ideas about how to counterbalance luck in our society. In a reading group at Arm Research, I presented the paper to my coworkers and they shared ideas and encouraged me to write this post. I genuinely appreciate all the help I’ve received. Finally, I would like to thank Pluchino, Biondo, and Rapisarda for their interesting and inspiring research.

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