Richard Garfield, the designer of Magic the Gathering, defines luck in his ITU Copenhagen talk as “uncertainty in outcome”. I think by modeling human reasoning as Bayesian, we can come up with another fruitful, if not a more fruitful, definition.

Suppose I were to take out a quarter from my pocket and ask you to guess my next twenty flips. You perceive the heads/tails probability to be 50/50 with high confidence, and so your accuracy on my first three flips, which turned out to be all heads, is very heavily luck based.

Now suppose that my next fourteen flips are all heads. At this point, you will be quite certain that the game was rigged, and your heads/tails probability becomes 100/0. Indeed, your next three guesses are correct. Curiously, at this point in the game we no longer perceive luck as being involved.

In Bayesian speak, our 50/50 distribution is our prior belief, and 100/0 distribution posterior belief. Note that we consider the prior (no pun intended) to be highly random, whereas the latter not so much. Wikipedia defines random as “the lack of pattern or predictability in events”, which suits our current purpose. We will roll (pun intended) with it.

Now another example.

Suppose I leave right now to catch the next bus to work. The bus arrives just as I get to the bus stop. How lucky!

Now suppose I tell you that I have memorized the bus table and had been stealing glances from the wall clock while talking to you, cutting our conversation short when it’s time to leave. The bus arrives just as I get to the bus stop. Not very lucky, but pretty calculating.

This example is insightful in two ways. Firstly, to be perceived as lucky I don’t have to intentionally make a choice. As long as the outcome benefits me, I seem lucky. Secondly, we can make things seem less luck-based via additional information. My new prior (which is my posterior after memorizing the timetable) was good enough to reduce randomness. What appears to you as random may be fairly predictable for me. Randomness can be subjective. In fact, it often is. The stock market, weather patterns, and even the search for a good romantic partner can be predictable for one and random for another. Thus, a definition of luck necessarily takes subjectivity into account.

So I think that perhaps a better definition for luck is “when apparently unpredictable outcome offers high utility”. In Bayesian speak, when an outcome of good utility occurred despite a prior that doesn’t favor it. We should note too that utility, which is subjective too, also affects our perception of luckiness. In a bet with 90% chance to win one million and 10% to loss one million, the winner is still declared pretty lucky. Our perception of utility is biased. In the case of loss-aversion, sometimes the bias is advantageous.

Prior and utility are both extremely malleable, and a variety of cool insights arise (a.k.a. this is an insight dump where I stop being good at explaining things):

1. Extremely complex conditions gives rise to priors of similar quality to everyone. In guessing the 567,890th digit of pi, everyone starts on the same prior.
2. In fact, if the universe is deterministic, the reason that statistics and probability exists in the first place is because events are too difficult to predict. In that case, there would be no true prior other than “X certainly will happen”. Our predictions will necessarily always be approximations and guesses (a guess that can only be exact if it also states that “X certainly will happen). If the world is deterministic, probability is a summary of what we don’t know.
3. Depths of gameplay may arise from a sufficiently complex condition that allows continual optimization. A first card drawn from a nicely-shuffled deck is complex because shuffling is complex, but shallow because we don’t hold any information that can visibly optimize our prior. A card drawn during the middle of the game is complex not only due to shuffling but also due to a significant number of cards already in play; however, this draw is less shallow because we can use the cards in play to optimize our prior if we wish. When we are down to the last dozen or so cards, the draws are actually deeper because we now have very concrete information to optimize from.
4. When you meet people who to you seems perpetually lucky or unlucky, you should strongly suspect your priors.
5. Knowledge on probability, statistics is very valuable.
6. Knowledge about how to improve your priors is extremely valuable. It’s a prior on how to change your prior when you see rules. See factors of correctness. A hidden value in Bayesian models is actually a value or a prior on confidence (the resilience your priors are to change). The phenomenon illustrated in factors of correctness might point to variability of the of underlying prior on confidence. It seems like more research should be done in that area.
7. Disguising depth through luck allows new players an excuse to feel better. It’s a desirable feature of design.
8. The best estimation for probability is set in stone, so our versatility (once we have a good prior) comes from selecting our situation such that the most likely outcome yields optimal utility. Something like betting on heads on a rigged coin or timing our leave for the bus.
9. Complexity is a gradient scale but has very different design properties on different logarithmic scales.
10. To artificially add luck to anything, create a situation where everyone has the same starting prior. (i.e. dice roll, coin flip, atmospheric pressure).
11. For fun (not for profit), to make yourself more “lucky”, put yourself in situations where unsuspecting high utility situations may occur, without much probability of low utility situations. This might explain why people who are more curious finds more pleasant surprises.