Entropy, Economics, and Criticality.

Information theory is a well-established method for the study of many phenomena and more than 70 years after Claude Shannon first described it in A Mathematical Theory of Communication it has been extended well beyond Shannon's initial vision. It is now an interdisciplinary tool that is used from 'causal' information flow to inferring complex computational processes and it is common to see it play an important role in fields as diverse as neuroscience, artificial intelligence, quantum mechanics, and astrophysics. In this article, I provide a selective review of a specific aspect of information theory that has received less attention than many of the others: as a tool for understanding, modelling, and detecting non-linear phenomena in finance and economics. Although some progress has been made in this area, it is still an under-developed area that I argue has considerable scope for further development.

Information Theory for Agents in Artificial Intelligence, Psychology, and Economics.

This review looks at some of the central relationships between artificial intelligence, psychology, and economics through the lens of information theory, specifically focusing on formal models of decision-theory. In doing so we look at a particular approach that each field has adopted and how information theory has informed the development of the ideas of each field. A key theme is expected utility theory, its connection to information theory, the Bayesian approach to decision-making and forms of (bounded) rationality. What emerges from this review is a broadly unified formal perspective derived from three very different starting points that reflect the unique principles of each field. Each of the three approaches reviewed can, in principle at least, be implemented in a computational model in such a way that, with sufficient computational power, they could be compared with human abilities in complex tasks. However, a central critique that can be applied to all three approaches was first put forward by Savage in The Foundations of Statistics and recently brought to the fore by the economist Binmore: Bayesian approaches to decision-making work in what Savage called 'small worlds' but cannot work in 'large worlds'. This point, in various different guises, is central to some of the current debates about the power of artificial intelligence and its relationship to human-like learning and decision-making. Recent work on artificial intelligence has gone some way to bridging this gap but significant questions remain to be answered in all three fields in order to make progress in producing realistic models of human decision-making in the real world in which we live in.

The Polycentric Dynamics of Melbourne and Sydney: suburb attractiveness divides a city at the home ownership level.

Urban dynamics in large metropolitan areas result from complex interactions across social, economic and political factors, including population distribution, flows of wealth and infrastructure requirements. We develop a Census-calibrated model of urban dynamics for the Greater Sydney and Melbourne areas for 2011 and 2016, highlighting the evolution of population distributions and the housing market structure in these two cities in terms of their mortgage and rent distributions. We show that there is a tendency to homophily between renters and mortgage holders: renters tend to cluster nearer commercial centres, whereas mortgagors tend to populate the outskirts of these centres. We also identify a critical threshold at which the long-term evolution of these two cities will bifurcate between a 'sprawling' and a 'polycentric' configuration, showing that both cities lie on the polycentric side of the critical point in the long-run. Importantly, there is a divergence of these centric tendencies between the renters and mortgage holders. The polycentric patterns characterizing the mortgagors are focused around commercial centres, and we show that the emergent housing patterns follow the major transport routes through the cities.

Closed-form solutions for the Lévy-stable distribution.

The Lévy-stable distribution is the attractor of distributions which hold power laws with infinite variance. This distribution has been used in a variety of research areas; for example, in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport. This function does not have an explicit expression and no uniform solution has been proposed yet. This paper presents a uniform analytical approximation for the Lévy-stable distribution based on matching power series expansions. For this solution, the trans-stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the Lévy-stable distribution. Then, the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called "the inner solution" and "the outer solution." Finally, the results of analytical approximation are compared to the numerical results of the Lévy-stable distribution function, making this uniform solution valid to be applied as an analytical approximation.

Strategic Information Processing from Behavioural Data in Iterated Games.

Iterated games are an important framework of economic theory and application, at least since the original work of Axelrod's computational tournaments of the early 80's. Recent theoretical results have shown that games (the economic context) and game theory (the decision-making process) are both formally equivalent to computational logic gates. Here these results are extended to behavioural data obtained from an experiment in which rhesus monkeys sequentially played thousands of the "matching pennies" game, an empirical example similar to Axelrod's tournaments in which algorithms played against one another. The results show that the monkeys exhibit a rich variety of behaviours, both between and within subjects when playing opponents of varying complexity. Despite earlier suggestions, there is no clear evidence that the win-stay, lose-switch strategy is used, however there is evidence of non-linear strategy-based interactions between the predictors of future choices. It is also shown that there is consistent evidence across protocols and across individuals that the monkeys extract non-markovian information, i.e., information from more than just the most recent state of the game. This work shows that the use of information theory in game theory can test important hypotheses that would otherwise be more difficult to extract using traditional statistical methods.

The social brain: scale-invariant layering of Erdos-Rényi networks in small-scale human societies.

The cognitive ability to form social links that can bind individuals together into large cooperative groups for safety and resource sharing was a key development in human evolutionary and social history. The 'social brain hypothesis' argues that the size of these social groups is based on a neurologically constrained capacity for maintaining long-term stable relationships. No model to date has been able to combine a specific socio-cognitive mechanism with the discrete scale invariance observed in ethnographic studies. We show that these properties result in nested layers of self-organizing Erdos-Rényi networks formed by each individual's ability to maintain only a small number of social links. Each set of links plays a specific role in the formation of different social groups. The scale invariance in our model is distinct from previous 'scale-free networks' studied using much larger social groups; here, the scale invariance is in the relationship between group sizes, rather than in the link degree distribution. We also compare our model with a dominance-based hierarchy and conclude that humans were probably egalitarian in hunter-gatherer-like societies, maintaining an average maximum of four or five social links connecting all members in a largest social network of around 132 people.