The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty

1. Introduction This paper reviews recent work on the properties of Tsallis distributions, Tsallis statistics, anomalous diffusion and multi-fractal processes. Processes of this kind have the capacity to represent natural phenomena that are associated with turbulence and emerging complexity. However, in addition they may also capture crucial aspects of human decision-making in response to conditions of uncertainty. This leads to the obvious question of the interrelationship between each of these two fields: the behavioural and the empirical. The Tsallis or q-exponential distribution describes both an Ontological property (i.e. an empirical characterisation of existing stochastic processes such as wind turbulence, river flooding, rainfall, asset prices, transport through doped media, DNA sequencing, the arrival of news within information networks etc); and an Epistemic attribute (i.e. a behavioural characterisation of decision-making as seen in generalisations of subjective expected utility designed to account for investor aversion to ambiguity or uncertainty). The paper articulates the linkages between the property of non-extensivity possessed by generalised information measures such as Tsallis entropy, S-shaped distortion functions that are commonly applied in the actuarial sciences, coherent risk measures discussed in the finance literature, and what is called Choquet expected utility theory in the economics literature.