## Economics

LML Fellows **Ole Peters** (PI), **Alex Adamou, **and their collaborators contribute to LML’s economics programme.

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 averages across the stochastic ensemble. This is necessary and worthwhile 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**.

An accessible 15-minute introduction (from 2011) is **here**, and the basis of the project is summarised as follows:

- 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 the consequences of 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.

Peters used these insights to provide a solution of the leverage optimisation problem in finance [1]. He also published a solution of the 300-year-old St. Petersburg paradox [2], a key problem in the foundations of both economics and probability theory.

Peters and Adamou [4] extended 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.

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

Following a suggestion from economics Nobel Laureate Ken Arrow, Peters and Adamou [7] published 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 published 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]?

In 2016 Adamou and Peters [8] pointed 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.

A generalization of the results presented by Peters and Gell-Mann [8] was published in 2018 [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 not as restrictive as, 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).

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) was published in 2018 [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.

[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**