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Mar 30

Markets, Ethics and Mathematics - A Defence of Mathematics

Posted by abiao at 11:18 | Others | Comments(0) | Reads(6426)
This article is a guest post by Dr Timothy Johnson.

In the aftermath of the Credit Crisis it became popular to blame quants and mathematics for the Credit Crisis. In November, 2008, a former French prime minister, Michel Rocard, wrote in Le Monde that “mathematicians are guilty (unwittingly) of crimes against humanity”. More seriously, the following March, the UK’s financial regulator, the Financial Services Authority published the Turner Review on the causes and cures of the crisis where it identified one of the causes as a “misplaced reliance in sophisticated mathematics”. Wired wrote about The Formula That Killed Wall Street and the FT followed up on the Wired report.

As the dust settled, The Financial Crisis Inquiry Comission Report gave a more thoughtful analysis. They mentioned maths and quants, but only in passing. Their conclusion was that there had been a “systemic breakdown in accountability and ethics”, which had resulted in lax regulation and excessive borrowing.

In one respect the FCIC conclusions are positive for mathematicians, the Crisis wasn’t their fault. On the other hand, if the problems were rooted in ethics, then surely maths has no role in preventing future Crisis. Maths is just another tool, like a spread sheet or double entry bookkeeping. This is pretty depressing for the heirs of Newton, Euler, Riemann, Poincaré and Kolmogorov.

The mathematical study of probability is usually thought to have begun in the mid-sixteenth century, with Cardano’s Liber de Ludo Alea (‘Book on Games of Chance’), where there is the first explicit statement that the chance of rolling a six on a fair dice is 1 in 6. Shortly after making this statement, Cardano makes the perceptive observation that
These facts contribute a great deal to understanding but hardly anything to practical play.1

Cardano’s work was ignored for centuries, the problem was, despite Cardano’s status as a mathematician, his ‘Book on Games of Chance’ didn’t fit in to what modern mathematicians regard as proper mathematics. The fact is that Cardano did not see his work on probability as principally a mathematical work, but as an investigation of the ethics of gambling, a point made recently by the mathematician David Bellhouse2.

A more orthodox study of probability was James Bernoulli’s Art of Conjecturing’, however, even the great Bernoulli’s text becomes somewhat incoherent for the modern reader. In the final section Bernoulli considered situations where the sum of probabilities could be greater than one3. The historian Edith Dudley Sylla has re-evaluated Bernoulli’s work in the context of what Bernoulli’s contemporaries were doing and her studies led to the conclusion that
While traditional histories of mathematical probability start with Pierre Fermat, Pascal and Huygens because they give what are from the modern point of view correct frequentist solutions to the problems of division and expectations in games of chance …the foundations of Huygens’s method (…) was not chance (frequentist probability), but rather sors (expectation) in so far as it was involved in implicit contracts and the just treatment of partners.4

The historical evidence seems to point to mathematical probability emerging out of the ethical examination of commercial transactions. During the eighteenth century, as science became focussed on the mechanics of physical objects, probability became associated with counting relative frequencies, a physical phenomenon, and its roots in the ethics of exchange were lost.

The ethical approach to probability of Bernoulli, Huygens and Cardano had a long pedigree. The Greek philosopher Aristotle never used mathematics in relation to physics, but he did in the analysis of the justice of exchange in his most famous study of morality, Nicomachean Ethics5. These ideas were developed through the thirteenth century by theologians, who following Aristotle realised that
the just price of things is not fixed with mathematical precision, but depends on a kind of estimate6

and, later,
The judgement of the value of a thing in exchange seldom or never can be made except through conjecture or probable opinion, and not so precisely, or as if understood and measured by one invisible point, but rather as a fitting latitude within which the diverse judgements of men will differ in estimation.7

These opinions are revolutionary in the development of western science8, since the ancient Greeks, along with many contemporary non-scientists argued that
[it is] foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs.9

By exploring the ethics of exchange through mathematics, these medieval scholars cleared the path for physicists to start using probability and statistics.

This was the background to Cardano’s Book on Chance, and it is captured when he says
The most fundamental principle of all in gambling is simply equal conditions, e.g., of opponents, of bystanders, of money, of situation, of the dice box, and of the die itself. To the extent to which you depart from that equality, if it is in your opponent’s favour, you are a fool, and if in your own, you are unjust.10

Cardano’s point, which goes back to Aristotle, is that a stake should equal the expected winnings.

This explains Bernoulli’s probabilities that did not add up to one, he was defining a probability as a set of factors that made the expected winnings equal to the stake. These types of situations are common in modern commercial gambling, where the sum of the odds offered to a gambler provide the bookkeeper with a certain profit, what would be call an arbitrage in finance.

Today Financial Mathematics is built on the Fundamental Theorem of Asset Pricing, a mathematical theorem that emerged in the late 1970s out of the Black-Scholes equation. The first statement of the Theorem is
A market is arbitrage free if and only if a martingale measure exists.
Being free from arbitrage is about fairness, it means that no one can make a riskless profit. A martingale measure is a valid set of probabilities such that the price of an asset is its expected value in the future, it is about equal conditions. The mathematical Fundamental Theorem can be re cast as
Fairness is based on equality.
A statement of ethics that moralists from Aristotle would have agreed with.

The association between mathematics and morality had all but disappeared in 1812 when Laplace published his ‘Analytic Probability Theory’ and gave an argument as to why mathematical (frequentist) expectation was a better guide than the moral expectation of Cardano and Bernoulli. About the same time, the English philosopher Jeremy Bentham introduced the concept of ‘utility’ into political economy from mathematics, and then, in 1836, the philosopher John Stuart Mill argued that economics
is concerned with [man] solely as a being who desires to possess wealth, and who is capable of judging the comparative efficacy of means for obtaining that end.11
This laid the foundations of the theory that economics is concerned with individuals maximising expected utility which became the dominant approach in the lead up to the Credit Crisis.

Not all economists bought into ‘Max U’, some were less inspired by economic theory, what people ought to do, and more by practice, what people actually do. In particular an experiment blew ‘Max U’ out of the water, the ‘Ultimatum Game’. The game is based on an experimenter, two participants and a sum of money. The experimenter gives all the money to the first player, who proposes how to share the money with the second participant. The division is made if the second participant accepts the split, but neither player gets anything if the first player’s proposal is rejected.

According to Max U, the second player should accept any split of the pot, they are getting something for nothing. However, the results of the experiments on adults are that if the money is not split equally, or close to, then the second player rejects the offer. Research has shown that chimpanzees are rational maximisers while the willingness of the second player to accept an offer is dependent on age or culture. Older people from societies where trade and exchange plays a significant role are more likely to demand a fairer split of the pot than young children or adults from isolated communities.12

Maximising utility, the main method of academic economics, is a selfish, greedy approach to making financial decisions. When quants price derivatives using no-arbitrage arguments they are using a method that places fairness at the heart of the markets. Mathematics does have a role in maintaining ethics in the markets.

Notes
1 David [1962(1998)], p 58], quoting from Chapter 9 of the Liber
2 Bellhouse [2005]
3 Sylla [2006, p 27]
4 Sylla [2006, p 28]
5 Aristotle [1999, Book V]
6 Aquinas [1947, Second part of the second part, Q77, 1]
7 Kaye [1998, p 124]
8 Hadden [1994], Crosby [1997], Kaye [1998]
9 Aristotle [1999, Book 1, 3]
10 Bellhouse [2005] quoting from Chapter 6 of the Liber
11 Persky [1995, quoting Mill, p 223]
12 Murnighan and Saxon [1998], Henrich et al. [2006], Jensen et al. [2007]

References  
Thomas Aquinas. Summa Theologica. Benziger Bros, 1947.    
Aristotle. Nicomachean Ethics, translated by W. D. Ross. Batoche, 1999.    
D. Bellhouse. Decoding Cardano’s Liber de Ludo Aleae. Historia Mathematica, 32:180–202, 2005.    
A. W. Crosby. The Measure of Reality. Cambridge University Press, 1997.    
F.N. David. Games, Gods and Gambling, A history of Probability and Statistical Ideas. Charles Griffin & Co (Dover), 1962(1998).    
R. W. Hadden. On the Shoulders of Merchants: Exchange and the Mathematical Conception of Nature in Early Modern Europe. State University of New York Press, 1994.    
J. Henrich et al. Costly punishment across human societies. Science, 312:1767–1770, 2006.    
K. Jensen, J. Call, and M. Tomasello. Chimpanzees are rational maximizers in an ultimatum game. Science, 318:107–108, 2007.    
J. Kaye. Economy and Nature in the Fourteenth Century. Cambridge University Press, 1998.    
J. K. Murnighan and M. S. Saxon. Ultimatum bargaining by children and adults. Journal of Economic Psychology, 19:415–445, 1998.    
J. Persky. Retrospectives: The ethology of Homo economicus. The Journal of Economic Perspectives, 9(2):221–231, 1995.    E. D. Sylla. Commercial arithmetic, theology and the intellectual foundations of Jacob Bernoulli’s Art of Conjecturing. In G. Poitras, editor, Pioneers of Financial Economics: contributions prior to Irving Fisher, pages 11–45. Edward Elgar, 2006.

Dr Timothy Johnson is an Academic Fellow in the Department of Actuarial Mathematics and Statistics at Heriot-Watt University, he completed his PhD in Financial Mathematics at King's College London.


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