Ole Peters’ Gresham lecture “Time, for a change” is now available online (audio, video, transcript). Some highlights from the lecture that may clarify a few comments we’ve received:
The word “ergodicity” is used differently in physics from how it is used in economics. In economics “ergodicity” means that the laws governing a system don’t change in time. In physics “ergodicity” means that time averages in a dynamical system are identical to ensemble averages.
The economics-use of the word can be found in Paul Samuelson’s and in Paul Davidson’s work. Peters uses the physics definition.
The logical relationship between physics ergodicity and economics ergodicity is implication: physics ergodicity implies economics ergodicity, but not the other way around.
Probabilities are not enough to make a statement about ergodicity. We need a dynamic in addition to the probability space.
A probability space consists of:
– an event space
– a sigma algebra
– a measure.
Add to this
– a dynamic,
and you have a dynamical system. Once the dynamic (and everything else, including the measure) is specified we can ask whether time averages are identical to ensemble averages.
What does our work have to do with that by Kelly, Thorp, Shannon, Breimann, Cover and others? For multiplicative dynamics the mathematics is identical, our work is based on the concept of ergodicity and time, whereas Kelly’s is based on information theory. Our work extends to dynamics other than multiplicative, i.e. we can address some questions (for example about welfare economics and wealth distributions) to which the Kelly criterion has no obvious applications.
We keep receiving question about the publication status of Peters’ paper “Menger 1934 revisited.” So far, it remains unpublished, but we are continuing the discussion with the editor of the Journal of Economics, where Menger’s flawed work appeared. The reviewers have agreed that Peters’ analysis is mathematically correct, and Menger’s is wrong. But so far the reviewers maintain that this is irrelevant to the field of economics.
Menger’s result amounts to banning both the Kelly criterion and the field of ergodic theory, and it was endorsed by several Nobel laureates. Its falsity is therefore economically relevant only insofar as it explains why economics has failed to take advantage of some 150 years of mathematical developments. It is difficult to pin down what exactly this relevance is because other than pointing to Kelly’s and our own work we can only speculate what might have been developed.
Peters mentions his 2012 meeting with Nobel Prize laureate Ken Arrow, a long-time endorser of Menger’s work. During the meeting Peters and Arrow discussed Menger’s paper in detail and agreed that Menger’s mathematics is flawed and his conclusion is untenable.