Lecture notes

Our initial Ergodicity Economics lecture notes were downloaded 30,000 times. We’ve now taken them down and are updating and correcting them to produce a textbook.

Please join the EE network to receive an announcement when the textbook is ready. In the meantime, we recommend the 2019 Nature Physics article as an introduction.

26 thoughts on “Lecture notes

  1. Hi, is knowledge of elementary calculus-based probability (as in Bertsekas and Tsitsiklis, or the elementary Ross book) sufficient to grasp the notes?

    Liked by 1 person

  2. Francis, thank you for your question, that’s very helpful. Next time we go through the notes we will make a list of any prerequisites.
    We’re trying to keep the lecture notes fairly self-contained, and I will cautiously say yes, the books you mention should be sufficient. My sense is that too much prior knowledge is more likely to be problematic than too little. But please let us know how you’re getting on, especially if you get stuck somewhere and feel that something is missing.


  3. Hello, I am working through these lecture notes and love them. I do not come from a math or physics background and I find this to be a great resource for understanding your published papers. I have a pedantic comment to make: in eq (13) you introduce tau as a ‘dummy variable indicating a specific round of a gamble’ but it is not noted upon until eq (30). Trivial, but it would be easier to understand if noted in the text when it is first introduced.

    Liked by 1 person

  4. Small error on pg. 41, Fig. 10: “The distribution denoted by the blue line has a
    higher mean and a higher variance than the one in red” should be “lower” or switch red & blue

    Liked by 1 person

  5. I just finished reading your lecture notes and I really enjoyed every second of it. Your insight is very refreshing and makes far more sense than the classical treatment of decision theory via utility functions. It is also very well written, I believe that every economist that reads this with an open mind will at least rethink a thing or two about the state of our discipline.

    That being said, there is a typo on page 67 (it says “sigend” instead of “signed”). Also on page 120 the excess drift is defined as riskless minus risky instead of risky minus riskless.

    Liked by 1 person

  6. Thanks for making your notes available.

    I have an absolute beginner question (for some reason my brain never got to grasp probabilities…) about the coin tossing game at the start.

    I got the time average thing, i.e. single player is going to have one’s wealth wiped out (I found that actually “intuitive”).

    Now, with regard to the finite-ensemble average in Section 1.1.1., and the corresponding picture 2, when you talk about how the average wealth across the group evolves in “time”, i.e. playing multiple repetitions, you must be assuming that each time (the game is repeated) the initial conditions are reset.

    Liked by 1 person

    1. Each trajectory starts at the same level at time zero. After that the same dynamic is applied as in Fig.3, and the trajectories are never reset again.

      If you find that in any way surprising: great! That the ensemble average (expectation value) can gain while each trajectory loses is the key insight that’s missing from formal economics. It’s a simple mathematical fact — you see it in a trivial coin toss — but many find it counter-intuitive. You may find this video helpful https://youtu.be/LGqOH3sYmQA


      1. Thanks for your time.
        It seems at least my understanding of the rules of the game was correct.

        Will rewrite my python script, I must have done something badly wrong! 🙂

        So, if I setup N (say 10000) trajectories, i.e. parallel games, and after each toss I update the “wealth” of each player accordingly, then the finite-ensemble average at each time is simply the average of the wealth of the players at that time, right?
        And this, as a function of t, should be sloping upward

        Liked by 1 person

      2. Gotcha!
        All good now (only God knows what I was coding today while multitasking at work…).
        Results as expected.

        It is very insightful though to extract also the max and min wealth across all realisations as that clarify everything (something you hinted at in the video): since the lower bound of the wealth for the most unlucky player cannot be less than zero, the more players you have the more “probable” it becomes to have “lucky bastards” that gets lots of heads!


        Liked by 1 person

  7. Those are some good lecture notes though i have difficulty trying to go through 136 pages —first part.

    https://sciencehouse.wordpress.com also discusses jensen’s inequality and some economics but in a less detailed fashion.

    I did ‘intuitively’ select the ‘wrong answer’–take the gamble for the coin flipping (first) example. But maybe you can excercize choice, be wise, and choose your trajectory. Be the ‘one in a million’.

    While you disavow discussions of any moral or other perspectives on things like wealth or income inequality, and counterpose ‘time averages’ versus ‘ensemble averages’, i could take a different perspective, similar to the Wheeler-Feynman ‘one-electron universe’.

    From that view , applied to economics, there is only 1 individual, but over its infinite lifespan (may have to undergo reincarnation) it does go through every possible economic trajectory. So in the ergodic limit (Poincare, Birkhoff, von Neumann, Weiner…) there is no inequality on average. What goes up must come down, so ocassionaly you feast, but usually are in a state of famine. ‘But for the grace of god, there go i’.You will be I later.

    Concept of ergodicity i learned from old papers on FPU simulations, some by van Kampen, and more recent ones by Ruelle , Shalizi, Mackey, and others (none of which i fully understand).
    There are also many ‘non-technical’ discussions of the concept in ‘heterodox’ economics papers, which you can find by doing a non/ergodic random walk through the blogosphere. https://arxiv.org/abs/cond-mat/0506338


  8. Ole,

    A good recent example of the superiority of the time-average rule for decision making is the implicit expected value calculation done by both Boeing and the Airlines when they decided to not have the added safety feature that would have switched off the MCAS in the 737 Max. It only passes muster under the misguided “expected value rule” which implicitly assumes that a “small probability” event is also expected to be remote in time and so will never bankrupt you.

    Sunil G Nair


  9. Hi Ole, I was wondering if you could give an intuitive explanation of the idea of “energy” and “free energy” in the economic context and also briefly review what it means in the physical sense. Thanks!


  10. Have you read this criticism of your latest work yet?
    He seems to think that you failed to provide a criticism of revealed preference theory, though I would argue that your work implies one or more criticisms of that theory. Thoughts?



  11. Similar to MMT, Monetarism, neoclassical, the money illusion, even R.A. Werner and probably S.Keen, you are, in my opinion, ‘whistling in the wind’.

    “The IPCC report that the Paris agreement based its projections on considered over 1,000 possible scenarios. Of those, only 116 (about 10%) limited warming below 2C. Of those, only 6 kept global warming below 2C without using negative emissions. So roughly 1% of the IPCC’s projected scenarios kept warming below 2C without using negative emissions technology like BECCS. And Kevin Anderson, former head of the Tyndall Centre for Climate Change Research, has pointed out that those 6 lone scenarios showed global carbon emissions peaking in 2010. Which obviously hasn’t happened.
    So from the IPCC’s own report in 2014, we basically have a 1% chance of staying below 2C global warming if we now invent time travel and go back to 2010 to peak our global emissions. And again, you have to stop all growth and go into decline to do that. And long term feedbacks the IPCC largely blows off were ongoing back then too.”

    ‘Limiting global warming to two degrees Celsius will not prevent destructive and deadly climate impacts, as once hoped, dozens of experts concluded in a score of scientific studies released Monday.
    A world that heats up by 2C (3.6 degrees Fahrenheit)—long regarded as the temperature ceiling for a climate-safe planet—could see mass displacement due to rising seas, a drop in per capita income, regional shortages of food and fresh water, and the loss of animal and plant species at an accelerated speed.
    Poor and emerging countries of Asia, Africa and Latin America will get hit hardest, according to the studies in the British Royal Society’s Philosophical Transactions A.
    “We are detecting large changes in climate impacts for a 2C world, and so should take steps to avoid this,” said lead editor Dann Mitchell, an assistant professor at the University of Bristol.
    The 197-nation Paris climate treaty, inked in 2015, vows to halt warming at “well under” 2C compared to mid-19th century levels, and “pursue efforts” to cap the rise at 1.5C.’

    Will there be change?
    “Today’s global consumption of fossil fuels now stands at roughly five times what it was in the 1950s, and one-and-half times that of the 1980s when the science of global warming had already been confirmed and accepted by governments with the implication that there was an urgent need to act. Tomes of scientific studies have been logged in the last several decades documenting the deteriorating biospheric health, yet nothing substantive has been done to curtail it. More CO2 has been emitted since the inception of the UN Climate Change Convention in 1992 than in all of human history. CO2 emissions are 55% higher today than in 1990. Despite 20 international conferences on fossil fuel use reduction and an international treaty that entered into force in 1994, manmade greenhouse gases have risen inexorably.”
    View at Medium.com


  12. Hey Ole,

    Amazing work. Still trying to process everything. I think my biggest mental hurdle is how growth rate tends to be ergodic. I’m confused because there are two growth rates: that defined by expected value and that defined by the time average. How can one growth rate equal the other?



    1. Hi Andrew,
      You might also be interested to see the two growth rates in action in our paper on Risk Prefences in Time Lotteries http://www.bit.ly/TimeLotteries. There we describe in simple terms the relation of either growth rate to a real-world experience of a decision maker.
      Regards Mark


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