Chaos’ Dynamic Gambles

O. Peters and M. Gell-Mann Evaluating gambles using dynamics Chaos 26, 023103 (2016).


… Expectation value maximizers are defined as rational in economics, but expectation values are only meaningful in the presence of ensembles or in systems with ergodic properties, whereas decision-makers have no access to ensembles, and the variables representing wealth in the usual growth models do not have the relevant ergodic properties. Simultaneously addressing the shortcomings of utility and those of expectations, we propose to evaluate gambles by averaging wealth growth over time.

The bulk of the paper consists of a review of work that is often considered to support the view that rationality is utility maximization, pointing out the caveats and inconsistencies. It is not just that in practice utilities are impractical to construct, but that they can only ever exist – with all the required mathematical properties – in circumstances that seem exceedingly rare. The paper considers idealised gambles, showing that even here there are difficulties. It points out that much of the conventional theory was based on the need to avoid pathological cases, yet there were alternatives that were – it seems – not seriously considered. The notion of average wealth growth is offered as a plausible thing to be maximized.


… The dynamic approach to the gamble problem makes sense of risk aversion as optimal behavior for a given dynamic and level of wealth, implying a different concept of rationality.

… We note that where optimization is used in science, the deep insight is finding the right object to optimize … .


From a practical perspective, the assumptions of rational choice theory seem quite absurd, hence this blog. To accept them one would want at least some sort of outline demonstration, yet – as this paper argues – if one looks at the supposed sources for the conventional theory, one’s attention is simply brought to more reasons against. So it seems odd, as this paper does, to take the theory at all seriously as logic. Rather it should be seen – as it proponents propose – as simply pragmatic. That is, a heuristic which allows some approximate reasoning to be used as a rough guide. Having read the paper, one might still suppose that it was suitable in this role.

The notion of ‘average wealth growth’ seems reasonable. But in practice one can never maximize what the average wealth growth will be, only what one thinks it will be. Moreover, by the law of large numbers, if one thinks that the future will be stochastic and stationary, and has many independent small decisions to make, then repeatedly maximizing expected utility will maximize expected average wealth growth, and if events do actually stochastically conform to one’s view of it, then repeatedly maximizing expected utility will tend to actually maximize wealth.

Significant differences arise, as in a crisis, where one should lack confidence in any estimate of ‘utility’. Conversely, if many decision makers lose confidence in their ability to form utilities, one would seem justified in saying that there was  a ‘crisis of confidence’, or a crisis in the usual financial and economic mechanisms. The niche for an improved approach, therefore, would seem to be in a crisis, where there is some form of ‘radical uncertainty’. So does focussing on average growth help?

The key here is that sensible behaviour from a long-run perspective can look like risk-aversion from a conventional (short-term) perspective. What we need to complete the theory is a way to become more risk averse in the run-up to a crisis. In the shocks of 2006-9 there was something of a ‘flight to quality’, which suggests that many were aware of an increasing risk, and in that case the more apparently risk averse investors tended to do better. But it seems to me that the conventional theory has two aspects, encouraging a lack of appreciation of what risk there is, and encouraging less risk aversion than is prudent. The current paper only deals with one of these aspects.

One way to think about this is that the wealth of those seeking to maximize long-term returns is too frequently and significantly knocked back by genuine ‘black swans’ – e.g. what had been assessed as no more than 1% probability per year – or ‘1 in 100 year events’ – then as well as being risk averse one needs to review one’s assessments.

Dave Marsay

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