Broken Symmetry

Resolving the St. Petersburg Paradox and the Flaws in the Efficient Market Hypothesis Too

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.

RIP Mandelbrot

A man after my own heart. Here's a short autobiographical summary of his intellectual odyssey. And here's a recent paper (via Marginal Revolution) applying some of his insights. I particularly liked this bit:

Economic systems tend to be around the critical point in general. Open markets are typical examples. When there are more buyers than sellers, the market price goes up causing decrease in the number of buyers and increase in sellers. Such well-known pulling back mechanism obviously keeps the system around the critical point. The important fact is that this criticality does not stabilize the absolute value of the market price but only stabilizes the statistics of market price fluctuations to follow the critical fluctuations. As the dealers in open markets tend to care about only the relative market price, whether it goes up or down, so the absolute value of the market price is nearly meaningless for the determination of the market price. Therefore, the market prices generally wander almost randomly having statistics characterized by the critical point.

What I picture is market price signals synchronizing the flux of supply with the flux of demand. See also the summary of the paper by Fretter, Krumov, Weihe, Müller-Hannemann, and Hütt.

Riemann and Conditionally Convergent Series

Consider an alternating series -- i.e., a sum of alternating positive and negative fractions:

Did you know, that when N is infinity, by simply reordering the positive and negative terms, this sum can be made to equal any number?

Matt Springer at Built on Facts tells the story.

For me, this is one of the hints that numbers have structure. There isn't a coherent way to understand all of their characteristics without admitting their structural facet.

Compound Interest and the Binomial Theorem

A book I'm working through is J.C. Stillwell's history of mathematics. This was the happy result of a recommendation in Needham's Visual Complex Analysis -- also a worthwhile read.

Stillwell spends a bit of time on the history of the binomial theorem, which gives the expansion for a polynomial of arbitrary degree:

The coefficients of the expansion are equivalent to the number of combinations of k elements taken from a set of n elements:

These coefficients have a use in probability theory, which is probably the more familiar to most people -- namely, that the probability of getting k successes in n binary trials (e.g., coin flips) is the coefficient divided by the total number of possibilities (2^n for coin flips). They can also be calculated using Pascal's Triangle:

The binomial expansion is sometimes expressed in a simplified formula, in which y is set equal to 1, so that the equation reduces to:

The relationship between the binomial theorem and probability theory is of interest because the same formula turns up in a pretty important place in finance -- namely, in the formula for compound interest:

Here, the binomial expansion is multiplied by a fixed coefficient equal to the principal (or "present value") PV. The total expected value of the sequence of interest payments is simply the binomial expansion, with interest i as the variable x.

Curiously, one does not often see cash-flows expanded through the binomial theorem. Rather, the tendency is for each of the n payments to multiplied out. But that's of course equivalent to mutiplying the present value PV times the binomial expansion of interest payments.

What interpretation should be attached to the fact that compound interest is a binomial expansion? Here are a few observations:

• Ignore the Distant Future: The present value of compounded interest is a polynomial expansion in powers of interest; because interest < 1, the lower-order terms dominate the expansion.
• Don't Ignore the Near Future: Some higher-order terms in the expansion -- the ones that fall in the middle of Pascal's Trial -- have coefficients large enough to make up for the fact that interest rates < 1 are being multiplied. In other words, the base of the pyramid of compound interest is built of lots of combinations of principal and interest payments that occur in the immediate and near-future.
• You Can't Do Better than e: So long as your future value derives exclusively from organic growth in your present value -- i.e., no external infusions of cash -- there is no way to make the future value grow faster than exponential (sorry, Kurzweil).

The last is a consequence of the fact that as compounding occurs more and more frequently over the period in which payments are made, future value nonetheless approaches a finite limit:

Thus, if compound interest is like a binomial expansion in powers of interest, then continuously compounded interest is an exponential function of interest.

A final thought to ponder: Why do we calculate future value with this formula? In many cases, the answer is obvious -- because that's how we agree payments will be made by a borrower.

In other cases, however, that answer is less obviously satisfactory, and may even be incorrect. We can use the binomial expansion as an approximation for future value when there is some predictable process of growth occuring. But when growth occurs at different rates, or when compounding cannot be carried out efficiently, some other approximation for future value is necessary. The use of the binomial expansion as a crutch may be positively misleading.