My first article, “Gamble Expectation and the Ergodicity Conundrum”, ignited an interesting thread in Twitter. There were no major problems as far as participant behavior other than in a small set of measure zero with some comments and claims that are not worth mentioning even as jokes.
I have to say before we start that Ole Peters is not only very intelligent but also extremely polite and patient. He also makes bold statements, as was his claim of a basic error by Chernoff and Moses (CM) in their prominent text on Decision Theory about expectation and gambles. Bold statements usually lead to something useful even if they are falsified.
I would also like to point out that I am not disputing the importance of the work by Ole Peters on ergodicity but only his attempt to rebut CM in the specific context explained in my first article. Specifically, I claim that the example of a gamble Ole Peters has provided in his papers, seminars and lectures that supposedly refutes CM actually does not. In my first article, I offered an introduction to the reasons I believe this is the case. In this article I attack the validity of both the simulations and analytical results Ole Peters has presented in support of his argument that the particular example of a gamble he has offered has divergent expectation and that the CM statement as shown below does not hold.
The gamble is as follows:
…we toss a coin, and if it comes up heads we increase your monetary wealth by 50%; if it comes up tails we reduce your wealth by 40%.
Rebutting the simulation results
Ole Peters presents these simulation results for the ensemble average:
The ensemble average is based on N parallel realizations of the gamble and results are generated for N = 1, 100, 10000 and 1000000. As more realizations are taken into account, that results in a smoother path with diverging expectation value (black line.)
Since I was not able to find any code of the above simulation and I also did not want to ask for it I decided to write my own. The first step was to confirm the result in the papers and lecture notes of Ole Peters. Below is the result of a random trajectory in time (blue) along with the expectation for 1000 parallel realization at every step (black.)
The above result looks similar to the results present by Ole Peters for 52 steps in the time domain. The next step is to increase N and to see whether the expectation path get smoother. Below are the results for N = 10000
Indeed the expectation is a lot smoother since the average size increases by an order of magnitude. Note that the blue line now shows a different random realization in time domain but 10000 parallel realization of the gamble are used in calculating the expectation, or ensemble average. Below is the result for N = 1000000
In this case the expectation is a lot smoother and divergent, proving support to the claim that for this gamble, the ensemble average grows exponentially although the time average goes to zero, i.e., this is a non-ergodic process.
But wait! Why only 52 steps in the time domain? Ensemble averages make sense only at the limit of sufficient samples. How do these results change if we increase the number of coin tosses from 52 to 520, for example? Below are the results of this experiment for 1000 parallel realization.
It may be seen that expectation grows nearly unbounded (out of scale)but then reverts and crashes near zero. This is reasonable result since in time domain all trajectories converge to zero and in the ensemble domain that starts influencing the expectation after a certain number of tosses that result in a sufficient sample. Below is the result for 520 tosses and 10000 parallel realizations.
In this case too, the expectation grows out of scale but then crashes near zero. According to these results, this is not a favorable gamble from the point of view of ensemble domain and that agrees with the results in time domain presented in the first article.
Therefore, we conclude that the favorability of the gamble in ensemble space was judged based on a partial average that likely did not result in sufficient samples required for averages to converge to expectation, as noted in the first article. But what is the analytical justification of this? The math presented by Ole Peters look impeccable.
Here is the core of the mathematics that point to a divergent expectation in the ensemble space.
In my opinion, the error is in the statement that x(t) and r(t) are independent. Note that if two random variables X and R are independent, then the expectation of the product is equal to the product of the expectations, as follows
E[XR] = E[X]E[R]
Ole Peters claims that r(t) is independent of x(t) because we generate r(t) independently of x(t) in each time step. This is true. But it is also true that r(t) affects x(t+δt) since x(t+δt) = x(t)r(t)
At the highest level, two physical processes are independent when the dynamics of one do not affect the dynamics of the other. But in this case, although instantaneously, the processes x(t) and r(t) are independent at t, they are dependent in t since r(t) influences x(t+δt).
In general, if two processes X and R are dependent,
E[XR] = E[X]E[R]+cov(X,R)
This covariance term is important and in my opinion it is what drives the expectation of the product of x(t) and r(t) to zero eventually.
Therefore, transition from Equation 10 to 11 in the above figure taken from the lecture notes of Ole Patters is problematic, if not incorrect.
Have we reached a conclusion? I think we are still far from that. While doing simulations in R of this apparently simple gamble, I noticed some weird effects, probably also related to computational chaotic behavior while reaching the limitations of the language in handling arrays. Some think these concepts are easy, especially those who constantly use the term probability in finance due to being accustomed with the use but not with meaning. In this article a give a brief account of some of the problem associated with the use of the term probability. Now, expectation, ensemble average, time average, sound simple but are orders of magnitude more difficult. The ramifications from this complexity are important. Only recently there is some renowned interest for a fresh look at these notions by researchers such as Ole Peters. For example, an important topic in my opinion is spontaneous ergodicity breaking and another is local vs. global ergodicity. Someone must look. I am just a trader and machine learning software developer who always loved probability and has taken a few graduate courses on this subject. My favorite text is one of the most hated on the subject for its complexity, by A. Papoulis, Probability and Stochastic Processes. I think probability theory is even more complex than it appears in texts. So complex that only a few people qualify as experts. For this reason I do not discount the possibility that part or even everything I wrote in this article is wrong.
 O. Peters and M. Gell-Mann. Evaluating gambles using dynamics. Chaos, 26:23103, February 2016. Also see:Evaluating gambles using dynamics
 Papoulis, A. Probability, Random Variables, and Stochastic Processes, 1965. McGraw-Hill, p. 138.
If you have any questions or comments, happy to connect on Twitter:@mikeharrisNY
About the author: Michael Harris is a trader and best selling author. He is also the developer of the first commercial software for identifying parameter-less patterns in price action 17 years ago. In the last seven years he has worked on the development of DLPAL, a software program that can be used to identify short-term anomalies in market data for use with fixed and machine learning models. Click here for more.