Time Resolution of the Probability Weighting Puzzle Mark Kirstein B [email protected] @nonergodicMark lml.org.uk/people/mark-kirstein DAAD PRIME Information Seminar, 11-12 March 2019, Bonn Agenda 1 Project Idea 2 Partner Institutions Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 2 / 11 Background of the Project Idea – Ergodicity Economics Ergodicity Problem in Economics The problem of a theoretical justification of the replacement of time averages by ensemble averages is the ergodicity problem. Ergodic Theory: proper mathematics to treat random variables & stochastic processes Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 3 / 11 Time Averaging vs Ensemble Averaging time t t∗ n X t (x 1) X t (x 2) X t (x 3) X x 3,• time-averaged X t (x 4) observable (growth rate) X t (x 5) realisations or universes Ensemble Average aka hX •,t∗ i Expectation Value E[ X •,t∗ ] Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 4 / 11 Standard Gamble – Coin Toss r up · x t 1.5 · x t > > x x t @ t @ @ @ @ @ @ @ @R r down · x t @R 0.6 · x t - - t t + δt t t + δt Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 5 / 11 Removing the noise Multiplicative Repetition Wt W h ti ∆W W + (t t ) h i t0 − 0 ∆t ∆ lnW W exp (t t ) h i t0 · − 0 ∆t 1 EUR 0 2000 4000 6000 8000 10000 rounds T Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 6 / 11 Decision Making Under Uncertainty If decision criteria are based on ensemble averages . statistics does not fit the individual understanding of risk (measures) short-termism high leverage ratios ... Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 7 / 11 Probability Weighting Puzzle – An Irrational Bias? π(p) π(p) = p π∗(p) 1 0.5 0.5 1 p Figure: Probability weighting function in prospect theory. Function π(p) = p gives the unbiased probability weighting (black line). Function π∗(p) (blue line) shows a typical experimentally observed probability weighting function. Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 8 / 11 Perturbed Gamble (r up + ε)x t r · x(t) up (r − ε)x > > up t x t x(t) = 1 Z @ @Z @ @Z @ @Z @ @Z~ (r down + ε)x t @R r down · x(t) @R (r down − ε)x t - - Time t t + δt t t + δt Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 9 / 11 Partner Institutions Prof. Max-Konstantin von Renesse (German Host, Chair of Stochastics and Mathematical Economics) Prof. Dr. Ole Peters (PI, Collaborator) Dr. Alexander Adamou (LML Director, Collaborator) Collaboration with Prof. Jürgen Jost’s Complex Systems Group Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 10 / 11 Why London Mathematical Laboratory? LML is an institute for basic science in which researchers are commissioned to follow their curiosity. Because you wear lab coats! Figure: Ole Peters (left), MK, Alex Adamou (right) in Nov 2017 at the London Mathematical Laboratory (LML). Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 11 / 11 ‘The applicant’s work challenges the established orthodoxy and thus is liable to face significant resistance (this is not a weakness of the work per se [. ]) I believe the boldness and innovativeness of this approach are impressive’ [Anonymous Project Referee] Thank you for your attention! Comments & questions are very welcome, here or to B [email protected] @nonergodicMark For further information check out Ergodicity Economics Lecture Notes (Peters and Adamou 2018) Blog: ergodicityeconomics.com Ergodicity Economics Project Timeline science thriller on LML’s Ergodicity Economics project → ReferencesI Dellago, Christoph and Harald Posch (2008). ‘Realizing Boltzmann’s dream: computer simulations in modern statistical mechanics’. Boltzmann’s Legacy. European Mathematical Society Publishing House, 171–201. DOI:10.4171/057-1/12. Peters, Ole and Alexander Adamou (2018). Ergodicity Economics. Lecture Notes. 5.0, 2018/06/30. URL: https://ergodicityeconomics.com/lecture-notes/ ( 12). Mark Kirstein Time Resolution of the Probability Weighting Puzzle Bonn, 11-12 March 2019 2/2.