# Startup Discrimination Bias It is comforting to believe that intrinsic skill plays a primordial role in success. Sadly, not only is it false; but worse: people that have initial luck (and thus those most likely to succeed and express survivorship bias) have more reasons to falsely believe it to be true. Let’s destroy the idea through maths. ## Prior Luck & Player Skill Let’s pretend that startup success is a game where people have varying skill. It is a fair approximation, as most startups have direct competitors in the same space: some eat the market, and some die. Some startups will inevitably have initial biases in their chance at success, either going for or against them. For instance, the startup may be in a poor country plagued by war, or have a smaller pool of potential hires, or suffer from arbitrary discrimination. Maybe the founders are old black women from an impoverished background, and maybe a lot of dealmakers and VCs believe silly things like “Genetics always influence intelligence” or “Women are unable to take level-headed, wise action” or “The poor are terrible at finance management”. We can make statistics about how these fundamental —and often unmodifiable— priors modify the likelihood of a startup with this baggage to thrive. I call this **prior luck**. Independent from this nature, the players in the startup game have **intrinsic skill.** They make moves, and those can be superior to that of their peers and competitors, increasing their chances of surviving. Let’s start with a trivial modelization and build from there. ### The Two Doors You might have heard about the puzzle story of the two doors and two guards, where one always tells the truth while the other lies. One door yields treasures beyond measure, the other kills you. A more generic variant: you have a binary choice to make, and one branch reaches success. There are no guards; instead, your skill is defined as the probability that you pick the right door. After all, **skill** is about picking up environmental clues to approximate perfect play. - A skill of 0% means that you always pick the wrong door. - 75% means that you pick the wrong door 25% of the time. - 100% means that you always pick the right door: it is perfect play. So far, it is not about luck: you just have an innate quality, skill. It results from the **policy** that you set for your actions. Your understanding of a problem’s territory defines your policy as the actions that maximize your gain based on your situational awareness, according to your best prediction of how the world will evolve from your choices.

A leads to failure, B to success — The optimal policy is to pick B.

Maybe one door has a radioactive symbol on it, and your knowledge of that symbol helps you avoid lethal doses of radiation. Maybe there is a chasm behind the other, and you are pretty sure you can jump it. That challenge is not equivalent to another. Cultural baggage may mistakenly make the average person pick the wrong door. The right move may be counter-intuitive to most people, just like the [Fosbury Flop][]. And the two doors are not equivalent to each other. Unequal odds, knowledge asymmetry, and specific abilities can twist a door’s likelihood of being picked. The average person from a given environment has a given probability of making it through the good door. The resulting probability of success is the **prior luck** we mentioned before. A given individual will have a specific probability of picking the right path. One for each choice. It may be higher than average, and it may be lower. That represents its **skill**. For simplicity, we model it as a single, average probability across all choices. [Fosbury Flop]: https://en.wikipedia.org/wiki/Fosbury_Flop ### Graph Generalization Often, you don’t reach the end of your venture after a single choice. There are multiple steps, and every right choice adds up to success.

A leads to failure, B leads to a C/D choice where C fails and D succeeds. — The optimal policy is to pick B and D.

### Randomness Is All That You Cannot Control The next model improvement is to add the notion of random events. Picking a door does not always result in success with either probability 1 or 0: it can be somewhere in the middle, just like the Casino example. So you pick a door. And behind it, you lose with probability P, and win with probability 1-P.

A leads to failure with probability .6, B leads to failure with probability .7 — The optimal policy is to pick A.

That probability does not necessarily result from true randomness (after all, maybe the world is deterministic). It comes from undecided actions beyond your control. It could be the uncertainty in what your chess adversary plays, or be the consequence of a politician voting a law that has adverse effects to your startup. That politician may not even have thought about you at all, just like the Hitchhikers’ Guide aliens build a hyperspace bypass without thinking about the humans they annihilate with Earth. Some entities are Lovecraftian, and make choices that devastate ecosystems out of sheer disregard for negative externalities. (Humans are like that with animals and flora all the time: we brought [rabbits to Australia][], [tumbleweed to America][], and we [emptied the Aral Sea].) [rabbits to Australia]: https://en.wikipedia.org/wiki/Rabbits_in_Australia [tumbleweed to America]: https://www.youtube.com/watch?v=hsWr_JWTZss [emptied the Aral Sea]: https://en.wikipedia.org/wiki/Aral_Sea ### Graph Generalization We can improve the model one last time. Often, the first choice you make is not the last. After each choice, a dice is rolled, and you change state based on that; then you have another choice, another dice roll: rince, repeat. Eventually, you reach an end state, which is either a success or a failure. This is a rough draft of what a **Markov Decision Process** (MDP) is. In the example below, you first have to make a choice between A and B. If you pick B, you have a 20% chance of reaching a failure state; otherwise, you must pick between C and D.

A leads to failure with probability .7, B leads to failure with probability .2, and B leads to another decision, where C leads to failure with probability .9, and D leads to failure with probability .6. — The optimal policy is to pick D and B.

The definition of **Skill** remains the same: it measures the statistical probability that your policy picks the best option (the perfect play) out of a given decision in the graph. If your understanding of the world holds perfect accuracy, you know precisely the probability that any given choice eventually lands you on a success state. That lets you know which of the two options to choose: that with highest success likelihood. But of course, in the real world, you don’t know that choice A makes you fail in 7 possible futures out of 10. You learn that 30% estimate of success through training, experience, and analysis. In other words, skill. And that is also how you can plan more than one move ahead. If you correctly estimate that move B has a 20% chance of failure, and incorrectly estimate that choice D has a 30% chance of success, then the success rate of chosing this path is 0.8×0.3 = 0.24. Rationally, the better move would then be to pick A. As we know, that mis-estimation makes you traverse a sub-optimal path; you are not skilled enough to recognize that →B→D has better odds. Your skill thus obviously improves your odds of achieving success. To be clear, increasing your skill does not always improve your odds. ### Free Lunch The example above was carefully constructed to break that common misconception. Let’s plot the success likelihood as a function of skill:

Surprise! When you play worse than randomly, there comes a point where you play so bad that you get better results than highly skilled players. I emphasize that this is only valid *for the specific example MDP we saw*. The point of this side-note: > For any learning process, > there exists a game where it fails to reach perfect play. The curve of success rate as a function of skill can have pretty much any shape. The only constant is that its maximum is reached at perfect skill. This weirdness is a consequence of the **no free lunch theorem**: all optimization processes are equivalent when averaged over all possible problems. Thankfully, a large portion of unrealistic or meaningless problems are weighing things down; most useful problems are gradual. For instance, in a game made of an infinite sequence of choices where the probability of success does not change after each choice, there is nothing that can be theoretically learnt. Sorry, Russian Roulette: I claim you are a useless game. We will focus on monotonous problems, which are more well-formed. Those MDPs Modeling uncertainty probabilistically, smears a large aspect of high-level skill acquisition in a deterministic world. Some estimations can be learnt beyond what is rationally achievable. At a human level, that includes detecting bluff, or seeing a coin flip in mid-air to catch it on a chosen side. Tooling can help go further, such as [predict casinoes’ bad PRNG values][PRNG] (yes, that is a real thing), leveraging the advertising infrastructure in a political election game, playing a message customized for each voter to play on their opinions, beliefs and lack of knowledge — learnt through their social media interaction. We could imagine machines that reach skill levels today considered truly superhuman. For instance, aggregating and simulating the [connectome][MindMice] of all humans, lets you predict many events that today would be estimated statistically: from the purchase behaviors to the research breakthroughs, from the political chenanigans to the private commercial plans… [PRNG]: https://www.wired.com/story/meet-alex-the-russian-casino-hacker-who-makes-millions-targeting-slot-machines/ [MindMice]: https://www.sciencedirect.com/science/article/abs/pii/S0092867420310011 ### Meaningful Fitness Landscapes On the flip side, your **prior luck** is the probability of success of a “coin flip” policy: choosing randomly at every turn. Assuming your skill is 50%, you will statistically have the same chance of winning as your prior luck. As it goes up, your success rate rises to that of perfect play (shown as 1 to normalize the scale); as it goes down, your success rate plummets to worst play (normalized to 0). Your skill is like a luck improvement machine. If you have a 10% prior luck, 90% of the sequence of choices you can make lead to a loss. However, thanks to an 80% skill, you will avoid taking a failure path 80% of the time. Your skill still has false positives and false negatives, though: your skill might wrongly discard a successful move (from the small pool of 10% good picks) as a failure. Conversely, it will mislabel 20% of bad moves (from the big pool of 90% failure paths) as leading to success. If `L` is your prior luck, and `S` is your skill level, your chance of success looks like this: S×L ÷ (S×L + (1-S)×(1-L))

Prior luck: .

## Average Prior Luck From an average background, skill has a stable relevance to success. This is actually the most encouraging learning experience. At any point in time, all improvements in skill get satisfactorily rewarded. You start off with low skill: you make a lot of mistakes. Every time you identify one, you can immediately experience an improvement: you win more often. You can easily identify which techniques pay off, and which are counterproductive: you sensibly notice whether your success rate goes up or down. Once you reach average skill among your peers, you succeed as often as the overall population succeeds. However, that doesn’t take into account the fact that you will often face competition from people with much higher prior luck than you do, which will both feel unfair, and cause you losses from which it is extremely hard to learn, because the subtle choices that could have given a win would only have been discovered by people with tremendously higher skill. ## The Nepotistic Tycoon Heir: 99% Prior Luck For them, a marginal improvement in ability always gives them a tremendous competitive advantage. It results in an enticing learning process: everyone starts with low skills, but their improvements quickly yield great results. However, at a high level of play, they can barely tell when they are improving in skill. They must consciously compete exclusively with the very best, and with those of higher prior luck, if they want to get stronger. Otherwise, they cannot verify that their techniques pay off. But they are not particularly incentivized to do so. Competing with the average Jane will yield greater returns. I call this **concave learning**. It is the most rewarding learning experience: you get a lot from improving your skills, and once you feel like your skills are above average, of course you expect to win nearly all of the time! ## From Rags To Riches: 1% Prior Luck The situation is reversed compared to the tycoon heir: prior luck has an outsized effect on the outcome, while skill increase barely has any effect. Only those with outstanding skill see it be rewarded with success. But nobody starts with outstanding skill: the only ones to get there are those ready to fight through endless losses. ## Unfair Face-Off Worse: for someone with 10% prior luck to beat someone with 90% prior luck of average skill more than half the time, they need an insane **99% skill**: they must not make more than a single inferior move in a hundred moves. Their opponent has an invisible safety net: mistakes are OK, as long as they play right half the time. It hurts, because most markets have a magnate at play.

Prior luck: .

Competitor prior luck: .

Competitor skill: .

Conversely, for someone with 90% prior luck to beat someone with 10% prior luck of average skill, they barely need more than a meager 1% skill. They could literally play 98 awful moves out of a hundred… and still win. This counterintuitive reality of our society hurts: the unfortunate ones play with convex learning, starting the game with a feeling that skill does not help, and only seeing results —albeit tremendous ones— once they outshine the very best; while the lucky get not just an immediate boost, but also a reason to sharpen their abilities. ## Notes The assumptions made are fair. Nonetheless, their subtleties bear pointing out. First, **skill** can easily be a fuzzy notion. Here, we define it as the ability to distinguish a good move from a bad one. In a board game, you have a finite number of possible moves, some definitely better than others. As a weaker player, however, two moves may seem equivalent, while a stronger master may know that one leads to a harder fight. In real life, the number of moves is significantly larger, but the principles scale indefinitely. (Aside: equivalently, either you define skill as the detection rate of better moves, and then the prior luck is the average success rate of people with the same background when playing randomly; or you define skill as the percentage of moves that are better than a random person, and then the prior luck is the average success of people with the same background. The former makes skill easy to understand, the latter makes prior luck easy to measure. The relationship between the two stays the same.) One interesting edge-case is 100% skill, aka. perfect play. This means that you can precisely order all possible moves from best to worst with no error. Counter-intuitively, it does not yield guaranteed success: there may be no possible set of actions that end in victory. Which brings us to **chance of success**, the least well-defined concept in this article. Many games have an ambiguous win condition. Some are indecisive (draws), some are leveled (difficulty; point score), some are qualitative (story endings). To assess your own definition of success, you must build your personal stochastic variable and draw its general distribution. That distribution will be affected by what we call the chance of success. With a better-than-average chance of success, applying the general distribution to YOU will improve the probability of your desirable end states. With a 100% chance of success, you get a guaranteed best state (barring randomness in the game, although even that can be included in skill, as we have seen expert players memorize game PRNG outputs to win). Finally, let’s talk about **extreme prior luck**. A 100% skill was extremely improbable; a 100% prior luck is impossible. A 100% prior luck means that literally nothing you do can fail. You would need to be the tyrannic despot of the strongest country. Then, you would have the power to force any endeavor to become a success, even if there is no market for it. But there is always the chance that you suddenly die of a stroke. A 0% prior luck is more common. Someone that is scheduled to die before they can plant the seeds, cannot make their efforts flourish. That is true of life imprisonment, death row, and heavy persecution such as genocide. You could argue that there is always a slim chance that some *deus ex machina* will save you, and that is reasonable. Absolute zero is only truly reached with an irreversible death. ## Acknowledgement I wish to thank Hannah Fry for her insights presented at the [Royal Institution’s 2019 Christmas Lectures][Fry]. [Fry]: https://youtu.be/_q4DrUHKC0Q?t=1041