## Economics

**People**

LML Fellows **Ole Peters** (PI) @ole_b_peters, **Alex Adamou** @alex_adamou, **Yonatan Berman** @bermanjoe, and their collaborators contribute to LML’s economics programme.

**Summary**

The project is a full re-write of economic theory from scratch, taking full account of the ergodicity problem: the difference between averages over time and expectation values (averages across the stochastic ensemble). This is effective because the foundations of formal economics were laid at a time when our formal understanding of randomness was in its infancy and conceptually naïve. Yet the key formal models of human behavior used in economics are models of decision-making under randomness. These themes are explored in a **dedicated blog**, that includes a periodically updated set of **lecture notes**.

**Video**

Accessible 15-minute introduction (from 2011):

**Science-history background**

- Economics was the first discipline to develop the mathematics of randomness (17th century).
- The early conceptualisation of randomness has a flaw: it assumes that randomness playing out over time has the same effect as randomness playing out over an ensemble of parallel systems (parallel universes). In modern terms, it assumes ergodicity.
- Economics noticed symptoms that arise from this flaw and designed tools, most notably utility theory (18th century), that mitigate some of them.
- In the 19th century a new conceptualisation of randomness emerged in the context of physics, namely in thermodynamics and statistical mechanics. This conceptualisation recognises from the start the central role of time and makes the ergodicity problem explicit. It thereby resolves the fundamental flaw, rather than treating its symptoms.

**Time line of the project**

2011: Peters provides a solution of the leverage optimisation problem in finance [1]. He also publishes a solution of the 300-year-old St. Petersburg paradox [2], a key problem in the foundations of both economics and probability theory.

2013: Peters and Adamou [4] extend the work on leverage optimisation [1] to test a prediction it makes about the nature of fluctuations in stock prices. This produces a solution to the Equity Premium Puzzle and suggests an algorithm for setting central bank interest rates.

2015: Following a suggestion from economics Nobel Laureate Ken Arrow, Peters and Adamou [7] publish a solution of the insurance puzzle: why do people buy insurance although the transaction reduces their expected wealth? By extension this explains, and helps price, financial derivatives. In the same year they publish a solution to the cooperation conundrum: why would one entity voluntarily give up resources for the benefit of another, but without immediate benefit for itself [6]?

2016: Peters and physics Nobel Laureate Murray Gell-Mann [8] publish a detailed solution of the gamble-selection problem, which is the foundation of economic decision theory.

Adamou and Peters [8] point out that the techniques and concepts developed in this context lead to deep insights into the dynamics of economic inequality. Berman, Peters, and Adamou [9] published a detailed study of American wealth distributions.

2017: Rick Bookstaber argues in his book “The end of Theory” that the ergodicity problem severely limits the usefulness of mainstream economic theory.

2018: A generalization of the results presented by Peters and Gell-Mann [8] is published [11]. This clarifies how the emerging framework is related to classical expected utility theory (EUT). It removes the arbitrariness associated with EUT due to the free choice in the latter of a utility function. At the same time, it is less restrictive than, for instance Whitworth’s (1870) or Kelly’s (1956) treatments because it provides interpretations of utility functions other than the logarithm (for example Cramer’s 1728 square-root).

Following discussions with Jean-Philippe Bouchaud, an important interdisciplinary link between a statistical-mechanics model of spin glasses (the random-energy model) and simple models of investment portfolios (sums of log-normal variates) is published [12]. This link has been known for many years by some researchers and finance professionals. It has allowed powerful techniques developed by physicists in the 1980s to be applied in the context of finance and economics.

Nassim Taleb argues in his book “Skin in the game” that the ergodicity problem requires us to re-think how we use probabilities in economic theory and beyond.

2019: Oliver Hulme’s neuroscience group at the Danish Research Center for Magnetic Resonance conducts behavioral experiments that show strong predictive power of the framework.

Marc Elsberg publishes his book “Gier” (in German) – a thriller based on LML’s economics project.

** References**

[12] O. Peters and A. Adamou

*The sum of log-normal variates in geometric Brownian motion.*

**arXiv:1802.02939** (2018).

[11] O. Peters and A. Adamou

*The time interpretation of expected utility theory.*

**arXiv:1801.03680** (2018).

[10] Y. Berman, O. Peters, and A. Adamou

*Far from equilibrium: Wealth reallocation in the United States.*

**arXiv:1605.05631** (2016).

[9] A. Adamou and O. Peters

*Dynamics of inequality.*

*Significance ***13**, 3, 32–37 (2016).

** doi:10.1111/j.1740-9713.2016.00918.x**

[8] O. Peters and M. Gell-Mann

*Evaluating gambles using dynamics.*

*Chaos ***26**, 023103 (2016).

** doi:10.1063/1.4940236**

[7] O. Peters and A. Adamou

*Rational insurance with linear utility and perfect information.*

**arXiv:1507.04655** (2015).

[6] O. Peters and A. Adamou

*The evolutionary advantage of cooperation.*

**arXiv:1506.03414** (2015).

[5] O. Peters and W. Klein

*Ergodicity*** b**reaking in geometric Brownian motion.

*Phys. Rev. Lett. *110, 100603 (2013).*
*arXiv:1209.4517

**doi:10.1103/PhysRevLett.110.100603**

[4] O. Peters and A. Adamou

*Stochastic Market Efficiency.
*

*Santa Fe Institute Working Paper*#2013-06-022 (2013).

www.santafe.edu/media/workingpapers/13-06-022.pdf

*arXiv:1101.4548*

www.santafe.edu/media/workingpapers/13-06-022.pdf

[3] O. Peters

*Menger 1934 revisited.*

**arXiv:1110.1578 **(2011).

[2] O. Peters

*The time resolution of the St Petersburg paradox.*

*Phil. Trans. R. Soc. A* 369, 369, 1956, 4913–4931 (2011).

**arXiv:1011.4404**

**doi:10.1098/rsta.2011.0065**

[1] O. Peters

*Optimal leverage from non-ergodicity.*

*Quant. Fin.* **11**, 11, 1593–1602 (2011).

**doi:/10.1080/14697688.2010.513338**