https://sites.google.com/site/matteorichiardi/teaching/complexity-economics

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Matteo Richiardi

Complexity Economics

University of Turin

February-April 2024

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Lecture 5:�Non-ergodicity and uncertainty

Matteo Richiardi

Complexity Economics

University of Turin

February-April 2024

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Sources:

• Bookstaber, ch. 5
• Miller & Pace, ch. 2, 5, 6, 7, 8, 9

Ergodicity

• Ergodicity means that the ensemble average is the same as the time average.

Ergodicity, what's it mean - by Leon Lin (substack.com)

Ergodicity economics – Formal economics without parallel universes

Time and ensemble averages

Lotteries

• 50% chance of winning 50%
• 50% chance of losing 40%

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Would you like to take such bet if offered repeatedly?

Non-ergodicity /1

• Non-ergodicity means that the ensemble average is NOT the same as the time average.

Note: the right hand graph is in logs

Non-ergodicity /2

• Time averages of different individuals are different, and different from the ensemble average.
• When t 🡪∞, the ensemble average also goes to ∞, while all time averages converge to 0.

Implications of non-ergodicity

• Path dependency (history matters)
• This might lead to multiple equilibria

E.g.: the tie problem:

• On your first day of work: choose randomly.
• On subsequent days: follow the crowd (majority of colleagues).

Radical Uncertainty

• Non-ergodicity implies that we can often describe the world in probabilistic terms.
• But can we ever get to know all possible states?
• Non-ergodicity also implies that small changes in inputs (e.g. policies) can produce large (and persistent) changes in outputs (e.g. outcomes)

– does it ring a bell? (Lorenz’s butterfly effect)

Reflexivity

• A reflexive system where any observer is also a participant and there is a two-way feedback between the participant/observer and the system.

🡪 The agents change the environment in which they operate

Participant-observation approach in Sociology

Fox, K. (2004), Watching the English: The hidden rules of English behaviour

Reflexivity: necessary conditions

Beinhocker (2013)

An example: Schelling’s Segregation model /1

“One afternoon, settling into an airplane seat, I had nothing to read. To amuse myself I experimented with pencil and paper. I made a line of x’s and o’s that I somehow randomized, and postulated that every x wanted at least half its neighbors to be x’s and similarly with o’s. Those that weren’t satisfied would move to where they were satisfied.”

Schelling’s Segregation model /2

“At home [...] I made a 16x16 checkerboard, located zincs and coppers at random with about a fifth of the spaces blank, got my twelve-year-old to sit across from me at the coffee table, and moved discontented zincs and coppers to where their demands for like or unlike neighbors were met.”

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“The dynamics were sufficiently intriguing to keep my twelve-year-old engaged.”

Schelling’s Segregation model /3

Initial (random) pattern �

Final pattern �

Segregation model: tolerance threshold = .3; density = 0.9. �

NetLogo Models Library: Segregation (northwestern.edu)

Schelling’s Segregation model /4

Segregation model: tolerance threshold = .3; density = 0.9; grid size = 15x15.�

Non-ergodicty in the Schelling model /1

Segregation model: tolerance threshold = .3; density = 0.9; grid size = 15x15.�

Non-ergodicity in the Schelling model /2

Grid size = 15x15

Grid size = 150x150

Segregation model: tolerance threshold = .3; density = 0.9.�