What is Apple Worth? Part 2: Paradox Resolved?

In part 1 of this series I talked about Daniel Bernoulli's brilliant idea of using a concave utility function to solve the St. Petersburg paradox, a problem originally posed 300 years ago (and a couple of decades before Daniel's solution) by his cousin Nicolas. I explicitly suggested a strong parallel with the way Apple is being undervalued as it gets bigger and bigger, and ended with a discussion of how the overused modern phrase "law of large numbers" (aka lol numbers) as misused by current mainstream financial media talking heads (when referring to the "limits of growth" concept) ironically is in contradiction with the actual fundamental probabilistic concept called the Law of Large Numbers (LLN), first proved by Daniel's and Nicolas' uncle Jacob Bernoulli: If Jacob's LLN applies, the likelihood that Apple will continue to grow would seem higher, given the historical empirical record, not lower (as the lol numbers trite cliche implies). On the other hand, if in fact there are limits to growth being reached, then the actual LLN is violated (independence of trials). In short, the semantic confussion and logical contradiction of invoking a misnamed "law" to justify persistent multiple compression despite continued strong growth in Apple's earnings might get resolved by reinterpreting it not necessarily as limits to growth but instead as an expression of the concept of diminishing marginal utility (although these two concepts might be related after all).

For this part 2 my goal was to explain why and how such a simple solution of a concave utility function works in assessing the pragmatic value placed by most people on such big yet unlikely potential returns as well as why and how the specific logarithmic utility proposed by Bernoulli might not work in every case, and how the theory eventually got refined over the years, decades and centuries. This is no simple task: to wade through 300 years of academic literature on economic theory, and most recently financial theory, from the incipient probability research to modern portfolio theory and CAPM. For an example of the nature of the literature, check out this 850 page book (many pages omitted in that Google Books preview) focused only on a singular very specific thread (the Kelly Criterion) within the broader subject. Even if I could digest and condense all of it into a blog post (which I definitely can't), I wouldn't want to submit my readers to such a dry academic dissertation. Besides, it's mostly way over my head. Hence, after being set back on my promise to post this within a couple of weeks of part 1, I've finally decided to go with a very simplified intuitive exposition, but leaning heavily on the multiple referenced links provided (with the caveat to only rely on Wikipedia for a superficial glance and to conduct your own research through more reliable sources if a deeper understanding is needed). Apologies if this ends up with significant theoretical holes.

What is Apple Worth? Part 1: Regarding Large Numbers

A Big Lottery

“Peter tosses a coin and continues to do so until it should land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation.”

This is how Daniel Bernoulli rephrased a problem nagging a tight circle of mathematicians almost 300 years ago. The issue is that, if you go by the mathematical expected value of the winnings you end up with an infinite value for the lottery: you have 1/2 probability of winning one ducat on the first throw, 1/4 probability of winning two ducats in 2 throws (which adds another 1/2 to the expected value), 1/8 probability of winning 4 ducats in 3 throws (adding another 1/2), and so on adding 1/2 a ducat in value an infinite number of times.

The expected value suggests that a player should be willing to pay any amount asked for a chance to play this lottery. Yet, in real life no one in their right sense would pay more than a few ducats to play this, perhaps somewhere around 5 ducats. Why is it that the mathematical expected value doesn’t reflect the real value placed by real players on this lottery?