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.

Q-Gaussian diffusion in stock markets.

We analyze the Standard & Poor's 500 stock market index from the past 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong superdiffusion together with short-time correlations and the second one corresponds to weak superdiffusion with weak time correlations. Both regimes are well described by q-Gaussian distributions. The porous media equation-a special case of the Tsallis-Bukman equation-is used to derive the governing equation for these regimes and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.

Self-referential basis of undecidable dynamics: From the Liar paradox and the halting problem to the edge of chaos.

In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical systems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these computational frameworks. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of Gödel's proof for CAs. The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality; (ii) the potential to access an infinite computational medium; and (iii) the ability to implement negation. The considered adaptations of Gödel's proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability.

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 Association Between Social Network Factors with Depression and Anxiety at Different Life Stages.

This study examines whether social network factors influence individual's depression and anxiety outcomes at different life stages. Data was drawn from the Household, Income and Labour Dynamics in Australia survey. Hierarchical regression modelling was applied to examine the effects within and across different life stages. The depression and anxiety measures were taken from the Kessler Psychological Distress Scale (K10) and the social network factors were taken from the self-completion questionnaire. With the exception of social trust in seniors, the social network factors were significant predictors of depression and anxiety. This has practical implications for the design of social policy initiatives.

Social network factors as mediators of mental health and psychological distress.

BACKGROUND: Although mental health (MH) is overall strongly associated with psychological distress (PD), this association is very weak for the sample with high PD. This relationship remains understudied. AIM: This study examines the association between MH and PD and whether this association is mediated by social network (SN) factors for individuals with high PD. METHOD: Data were taken from the Household Income and Labour Dynamics in Australia study ( N = 756). MH was measured using the MH sub-scale of the general health survey (Short Form (SF)-36) and PD was measured using the Kessler Psychological Distress Scale (K10). The SN measures were derived from the self-completion questionnaire. Using bootstrap mediation analysis, we tested whether the association between MH and PD is mediated by SN factors. RESULT: The correlation between MH and PD was r = -.410 ( p < .001). The mediational analysis results show that social isolation mediates the association between MH and PD with an indirect effect of ß = -0.0070 (confidence interval (CI) = -0.0133 to -0.0023). Moreover, social connections also mediated the association between MH and PD with an indirect effect of ß = -0.0073 (CI = -0.0141 to -0.0028). CONCLUSION: This study has practical implications for the design of social policies that attempt to reduce social isolation and enhance social connectedness to protect MH.

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.

The association between social network factors and mental health at different life stages.

OBJECTIVES: Psychosocial factors are important determinants of an individual's health. This study examines the association between health scores and social network factors on mental health across different life stages. METHODS: Data were drawn from the Household Income and Labour Dynamics in Australia survey for adolescents (n = 1739), adults (n = 10,309) and seniors (n = 2287). Hierarchical regression modelling was applied to examine effects within and across age groups. All the variables were derived from the self-completion questionnaire. RESULTS: The social network factors were statistically significant predictors of mental health outcomes for all three life stages. For adolescents, the three social network factors were statistically significant with social isolation having the largest impact (ß = -.284, p < .001), followed by social connection (ß = .084, p < .001) and social trust having a similar effect (ß = .073, p < .001). For adults social isolation had the highest impact (ß = -.203, p < .001), followed by social connection (ß = .110, p < .001) and social trust (ß = .087, p < .001).The results for seniors were social isolation (ß = -.188, p < .001), social connection (ß = .147, p < .001) and social trust (ß = .032, p < .05). CONCLUSIONS: After adding the social network factors, the models improved significantly with social isolation playing the most significant role across all life stages, whereas the other social network factors played a differentiated role depending upon the life stage. These findings have practical implications in the design of mental health interventions across different life stages.

Information flow in a kinetic Ising model peaks in the disordered phase.

There is growing evidence that for a range of dynamical systems featuring complex interactions between large ensembles of interacting elements, mutual information peaks at order-disorder phase transitions. We conjecture that, by contrast, information flow in such systems will generally peak strictly on the disordered side of a phase transition. This conjecture is verified for a ferromagnetic 2D lattice Ising model with Glauber dynamics and a transfer entropy-based measure of systemwide information flow. Implications of the conjecture are considered, in particular, that for a complex dynamical system in the process of transitioning from disordered to ordered dynamics (a mechanism implicated, for example, in financial market crashes and the onset of some types of epileptic seizures); information dynamics may be able to predict an imminent transition.

The neural circuitry of expertise: perceptual learning and social cognition.

Amongst the most significant questions we are confronted with today include the integration of the brain's micro-circuitry, our ability to build the complex social networks that underpin society and how our society impacts on our ecological environment. In trying to unravel these issues one place to begin is at the level of the individual: to consider how we accumulate information about our environment, how this information leads to decisions and how our individual decisions in turn create our social environment. While this is an enormous task, we may already have at hand many of the tools we need. This article is intended to review some of the recent results in neuro-cognitive research and show how they can be extended to two very specific and interrelated types of expertise: perceptual expertise and social cognition. These two cognitive skills span a vast range of our genetic heritage. Perceptual expertise developed very early in our evolutionary history and is a highly developed part of all mammals' cognitive ability. On the other hand social cognition is most highly developed in humans in that we are able to maintain larger and more stable long term social connections with more behaviorally diverse individuals than any other species. To illustrate these ideas I will discuss board games as a toy model of social interactions as they include many of the relevant concepts: perceptual learning, decision-making, long term planning and understanding the mental states of other people. Using techniques that have been developed in mathematical psychology, I show that we can represent some of the key features of expertise using stochastic differential equations (SDEs). Such models demonstrate how an expert's long exposure to a particular context influences the information they accumulate in order to make a decision.These processes are not confined to board games, we are all experts in our daily lives through long exposure to the many regularities of daily tasks and social contexts.

The perceptual cues that reshape expert reasoning.

The earliest stages in our perception of the world have a subtle but powerful influence on later thought processes; they provide the contextual cues within which our thoughts are framed and they adapt to many different environments throughout our lives. Understanding the changes in these cues is crucial to understanding how our perceptual ability develops, but these changes are often difficult to quantify in sufficiently complex tasks where objective measures of development are available. Here we simulate perceptual learning using neural networks and demonstrate fundamental changes in these cues as a function of skill. These cues are cognitively grouped together to form perceptual templates that enable rapid 'whole scene' categorisation of complex stimuli. Such categories reduce the computational load on our capacity limited thought processes, they inform our higher cognitive processes and they suggest a framework of perceptual pre-processing that captures the central role of perception in expertise.

Hysteresis effects of changing the parameters of noncooperative games.

We adapt the method used by Jaynes to derive the equilibria of statistical physics to instead derive equilibria of bounded rational game theory. We analyze the dependence of these equilibria on the parameters of the underlying game, focusing on hysteresis effects. In particular, we show that by gradually imposing individual-specific tax rates on the players of the game, and then gradually removing those taxes, the players move from a poor equilibrium to one that is better for all of them.