How much would you pay to play a game in which a coin was flipped, and each time it landed heads the payout was doubled from a starting value of $1? The expected value of this game is not finite, and yet most people are not willing to pay more than a few bucks for it, much less borrow to play.
This is the so-called St. Petersburg Paradox. The earliest solution was proposed by Bernoulli, who suggested that the value people attached to their wealth went as a logarithmic rather than linear function of increasing wealth. Given that assumption, he calculated the amount a person should be willing to wager as log(change in wealth due to payout from lottery) - log (wealth - wager).
But this calculation is not quite correct even though it resolves the paradox by explaining why most people won't wager much: People won't wager much because that logarithm runs off to infinity fast as the wager approaches even a fraction of wealth -- producing a zero in the denominator.
One way to see that this calculation is incorrect is because it treats the wager as a static cost, existing independent of how the payoff increases with increasing flips. Another way to say the same thing is that the wager has time-value in the form of interest. So the expected value and the wager should be treated together in the utility function in successive flips of the coin. What you should wager depends not just on how much, but when the payoff comes.
Ole Peters tells the full story of the St. Petersburg Paradox, including the errors in its analysis later introduced by economist Karl Menger, and propagated by Arrow and Samuelson, in glorious detail here.
I am grateful to Rick Bookstaber for pointing me to the paper.
"So what?" you might ask. Well, if one gets into the habit of treating economic variables as time-series rather than ensemble averages -- as the correction demands -- then one immediately is forced out into the realm of non-equilibrium statistical mechanics and dynamics.
Ole Peters and Alexander Adamou have already proposed a correction to EMH. According to them, it is not prices that are efficient and cannot be arbitraged, but rather the use of leverage to produce larger than logarithmic gains.
I'm completely drowning in patent litigation right now, but eventually I would like to offer some thoughts on what these papers mean for competition and innovation policy. What should be considered analogous to leverage in intellectual property markets?
The potential implications for investors are also interesting and need to be thought through. Do we now have a quantitative insight into how and why so many of the investors who have consistently beat the market over the past few decades have advised against leverage. Buffett, in particular, is famous from eschewing leverage and advising others to do the same.