The Perils of Physics Imperialism

 My undergraduate degree was in physics, then sixteen years working in oil exploration led me to doubt the certainties that I was taught at Imperial and I became a mathematician interested in uncertainty. The domain for mathematicians interested in uncertainty are the social sciences and I have the fervour of a convert in condemning my old faith and promoting my new faith. While you might not think my beliefs are particularly relevant, I do think a contributory factor in the credit crisis was the wholesale adoption of the culture of the physical sciences by modern finance.

In December there was discussion prompted by the particle physicists Brian Cox and the comedian Robin Ince arguing that policy makers should place scientific advice at the heart of government policy making. My experience of using science to advise policy (in an industrial context) means I think this position is fundamentally mis-guided, but beyond that it relies on the belief that 'science' is indubitable and immutable. My experience is that science and mathematics is definitely not indubitable and immutable, and in fact I belief it has moved backwards in regard to understanding uncertainty in the eighteenth century and is only now recovering. This piece covers an example of what I mean. To me, the example highlights the disproportionate faith people, from all walks of life, have in what physicists say, which is usually grounded in a set of assumptions that cannot be transferred from the physical sciences into the social sciences.

Towards the end of 2012 I came across a paper 'The time resolution of the St. Petersburg paradox' published in the Philosophical Transactions of the Royal Society A on the subject of 'Handling Uncertainty in Science' by a physicist, Ole Peters. The abstract is as follows

A resolution of the St. Petersburg paradox is presented. In contrast to the standard resolution, utility is not required. Instead, the time-average performance of the lottery is computed. The final result can be phrased mathematically identically to Daniel Bernoullis resolution, which uses logarithmic utility, but is derived using a conceptually different argument. The advantage of the time resolution is the elimination of arbitrary utility functions.

The Petersburg paradox was first recorded in 1713 in correspondence between Rémond de Montmort, a gentleman mathematician, and Nicolas Bernoulli, the nephew of James Bernoulli who posthumously published his uncle's Ars Conjectandi and whose own thesis was De Usu Artis Conjectandi in Jure ('The Use of the Art of Conjecturing in Law'). It is worth mentioning that the pair, along with James, first Earl Waldegrave (ancestor of the 1990s UK Science Minister, William Waldegrave) were the first people to identify 'mixed strategy' solutions in game theory, some two hundred years before the twentieth century re-discovery of game theory. The Petersburg game is described as follows

Given a (fair ) game of Heads and Tails, A tosses a coin and gives B 1 coin if a Heads comes up, if a Tails comes up, the coin is tossed again and if a Head turns up, A gives B 2 coins, if a Tails comes up, the coin is tossed again and if a Heads comes up A gives B 4 coins. In general the coin is tossed until a Tail comes up, and, if the first Tail occurs on the nth toss, then A gives B 2^n-1 coins. How much should B offer to pay A to play the game?

The problem for Bernoulli and Montmort was that it had been established that a game should be valued by adding together all possible products of the games winnings and the chance of the winning, the mathematical expectation. There is a 1 in 2 chance of winning 1 coin, a 1 in 2 chance of getting a Head on the first toss. B wins 2 coins by there being a Head and then a Tail, with a 1 in 4 chance, there is a 1 in 8 chance of winning 4 coins and so on. In general the mathematical expectation of the game is

However, nobody would stake more than 20 coins to play the game and typically they would offer 4-6 coins to play. Mathematical expectation seemed to be wrong.

The Petersburg game is generally thought to have been resolved by Nicolas' cousin Daniel Bernoulli, who is today known for his contributions to physics, in a paper he presented to the Imperial Academy of Sciences of St Petersburg (hence its name) in 1731 (it was published in 1738). Economists (good reviews can be found in Basset or Cowen and High) see Daniel's solution as being the earliest expression of the mathematical conception of utility, Daniel writes that "Any gain, however small, brings always a utility (emolumentum) inversely proportional to the whole wealth" and described this in terms of a differential equation, which was the rage at the time in physics, with u being utility and x being wealth, and k a constant, the equation is

The solution to this equation is that u = k log(x), that utility is the logarithm of wealth. On this basis Daniel was able to calculate a reasonable value of the game, which was dependent on B's initial wealth.

Daniel's solution appealed to nineteenth century economists as utility theory took hold of the discipline following Bentham, Mill, Walras and Menger. In particular, logarithmic utility would be given a biological, as distinct from mathematical, justification with the identification of the Weber-Fechner law that showed that the brain responded to stimuli in a logarithmic fashion. By the late twentieth century the recieved wisdom, captured in Peter Bernstein's Against the Gods (p 107) was that the paradox appears, is solved by Daniel in 1731/1738, it was briefly mentioned by Keynes in a Treatise on Probability and was picked up by von Neumann and Morgenstern.

Peters' highlights the role of fairness in the development of early probability, but in saying "was no universal agreement on the relevant concept of fairness" is mistaken: the concept was understood, the mathematical formulation was problematic. But when it comes to the issue of the Petersburg game, he falls in step with the mainstream account of the story. However, his main thrust is in the direction of ergodicity. The word ergodic was coined by the physicists Boltzmann in relation to his study of gases and derives from the Greek for 'work' and 'way'. Today, in physics, it relates the average time a gas particle is in some point of space (which can be the phase space, not necessarily a simple location) with the relative size of the space. However, this is a manifestation of a more general mathematical concept, that a random dynamical system is ergodic if it will eventually have a stable distribution independent of the initial state. For example, if a deck of cards is shuffled enough times, we would expect the eventual distribution to be uniform, whatever the initial state of the deck was (i.e the ace of spades is equally likely to be at any point in the deck). Generally speaking a dynamical system with constant coefficients is ergodic.

Peters argues that the problem with the Petersburg Game is that the system is not ergodic, in that the classical expectation of considering all possible winnings along with their space (chance of occurring) was not the same as the time averaged expectation. This is a physicists view of ergodicity, the association of time averages and space averages.  From the perspective of mathematics, the game is clearly ergodic, it is a stable system with unchanging parameters and in this respect the game is also ergodic in the terms that economists would understand.  Peters' interpretation of the game challenges a physicists understanding of ergodicity but does not really address economists' concerns in the area.Peters then analyses the game in the context of time, with the game being an investment in an asset that offers a high return along with the chance of bankruptcy. On this basis he derives and expression that equates to Daniel's logarithmic utility and goes on to criticise Menger's objections to Daniel's solution.

My objection to Peters' paper is not that it is wrong, it isn't, but its assumption that the paper presents a new novel resolution of the Paradox because he has a naive understanding of the history and importance of the Game.

The most detailed scholarly analysis of the game can be found in an article, "The Saint Petersburg Paradox 1713-1937" by the French philosopher and historian of science Gerard Jorland. The article is difficult to read, because it only exists in a book and is French philosophy written in English.

Jorland argues that the paradox was resolved in the late eighteenth century by Condorcet. Jorland comes to this conclusion by discussing the various attempts to resolve the problem which he categorises into two explanations that emerged as soon as the problem was identified. Nicolas Bernoulli, following his uncle's lead, argued that events with very low chance were 'morally impossible', chances less than 1 in 10,000 could be ignored. A colleague of Nicolas', Cramer, argued that while "Mathematicians value money in proportion to its quantity, commonsense men proportion it to its use" (i.e. utility) and introduced the concept of moral expectation, rather than summing the products of gains and probability, the gains should be raised to the power of the probability and then multiplied, formally

and so the logarithm of the moral expectation is the mathematical expectation of the log utility of the game.

Throughout the eighteenth century French (mathematicians) debated whether the Petersburg game should be resolved by taking a lower bound on probabilities, following Nicolas, or an upper bound on gains, following Cramer, as Daniel did. The issue was resolved in 1785 by Condorcet who took the view that the expectation did not 'really' exist: if someone has a 1 in 2 chance of winning 2 and a 1 in 2 chance of winning 0, the mathematical expectation is 1, but they would never win 1. In the Petersburg game the only way someone will win an infinite amount is if the game continued for an infinite number of rounds, which in reality is impossible. Today, this  is incorporated into stochastic control in finance by the incorporation of the transversality condition  that the discounted value of payoffs at infinite time should be zero.

This view, that infinite payoffs are meaningless, became standard in the nineteenth century, and and developed into the position that mathematical expectation is only valid for repeatable events, when the law of large numbers can come into play. Laplace, who did not believe in randomness, only ignorance, accepted the idea of moral expectation and realised that it equated to mathematical expectation if the number of possible payoffs was infinite (and the division of risks could be infinite). This was the basis of insurance, individuals have a finite number of risks and so employ moral expectation where as insurance companies, with a portfolio of risks, employ mathematical expectation. The physicist EmanuelCzuber noted in 1882 that classical mathematical expectation was meaningless for single events.

Peters approaches his solution to the problem by observing that

To the individual who decides whether to purchase a ticket in the lottery, it is irrelevant how he may fare in a parallel universe. Huygens (or Fermats) ensemble average is thus not immediately relevant to the problem.

This point was appreciated by Condorcet, Laplace and Czuber, it is not new.

Peters actually employs Cramer (and Daniel Bernoulli's) idea of moral expectation by considering the time evolution of the game. He assumes that there is a 'growth rate' for each round of the game r such that the value of the game increases by e^r each round. This is a cheat, acknowledged by Peters to facilitate the comparison with Bernoulli. In fact, a more appropriate formulation was presented by Durrand in 1957 and more recently by Szelkey and Richards.

Peters argues that Menger's criticism of Daniel Bernoulli's resolution of the Petersburg Game in the context of logarithmic utility was wrong. Menger argued that a Petersburg Game could be constructed that offered infinite payoff even using moral expectation/logarithmic utility, and so the only resolution of the Paradox was to use bounded utility functions, not just logarithmic utility functions. It is not clear why Peters thinks Menger was wrong, this is not surprising since Menger was not wrong, if B wins e²ⁿ instead of 2ⁿ logarithmic utility does not solve the problem. What Peters fails to appreciate was that Menger was conforming to the resolution provided by Condorcet, the game cannot proceed for an infinite number of rounds, which is were the resolution to the Paradox really lies.

In fact the Comte de Buffon (of Buffon's needle) did a very un-French thing and resolved the Paradox empirically.  He asked a young boy to conduct 2,048 experiments of the Game and tabulated the results.  He found that the experimental results closely matched the theoretical results based on the Binomial Distribution and that the total payout (to B) of the 2,048 games was a little over 10,057 coins  resulting in a fair price for the Game of around 5, close to the original observation of Nicolas Bernoulli and Montmort (in fact if you consider 2ⁿ trials of the game, the expected value of the game is n/2, and so if you consider events of chance less than 1 in 10,000 to be "morally impossible", this corresponds to it being "impossible" to see 14 heads in a row, you would value the game around 6.5-7).

Buffon's result employs ensemble averages to arrive at a sensible answer.  There seems little value in Peters approach employing contemporary ideas in physics.

Peters is challenging the concept of ergodicity in finance, ergodicity was introduced into finance from physics (Mirowski has written on this) along side utility theory. This was done in the context of post-Laplacian science, when there was no such thing as randomness, only a lack of information. Pre-Laplacian, and pre-Smithian ideas relating mathematics to the uncertainties of finance were lost. Peter is not just re-inventing the wheel by presenting his resolution of th ePetersburg papradox, his attack on Menger disguises the real solution to the problem, which is that mathematical expectation is not really relevant to economic affairs, because there is n stability in the dynamical system (a lack or ergodicity) and parallel universes don't really exist.

Peters paper has been well received, particularly by those who believe the solutions to the problems of finance lie in physics or in heterodox economics. Unfortunately, the well regarded actuarial consultancy Towers Watson has also succumbed to the allure of physics, shame on them.