Decision-making instabilities.

Many social and technological systems can be viewed as networks of interacting decision makers. In economics, for example, these might be networks of companies, investors, and regulators. In an automated vehicle these might be networks of control sensors and electro-mechanical actuators.

Viewing systems as networks of `agents' permits a wealth of very general study into aspects such as how to optimize their processes or predict emergent patterns. But recent work from the field of nonsmooth dynamical systems has an unexpected consequence for systems like these, suggesting that interactions between decision makers create a novel kind of singularity which, in certain situations, can lead to unpredictable and volatile behaviour.

This PhD project proposes to develop the mathematical modeling to discover how this phenomenon affects real world systems. This will involve modeling decision making problems in a manner that can be turned into real physical or social experiments, developing and testing a theory of how these systems behave, and studying how it affects systems with multiple (rather than only two) agents.

Current work has established the basic mechanisms that give rise to the phenomenon, in an abstract model involving two investors based loosely on economics in [2], or the similarly abstract "pilots' dilemma" thought experiment in [1]. Following the tradition of game theory, these introduce the phenomenon via two-player games that suggest how it would affect real world decision making problems, and allow one to begin probing how the behaviour of different kinds of `player' affect the outcome.

Previous work leaves many hints at how the phenomenon arises that have yet to be understood properly, and have little precedent in either decision making problems or in classical dynamics. For instance, the singularity appears to constitute a break-down in the causal structure of a system, due to the appearance of stable but non-robust attractors, and making it vital to model the passage of time as a continuous variable, in contrast to many standard modeling approaches which effectively reduce time to discrete snapshots.

The project will aim to design a small number of testable scenarios. We aim to design a social `game', and an equivalent scenario designed as an electronic control circuit. Focussing on simple but physically realisable models will enable the present abstract and low-dimensional predictions to be developed into a physical theory of actual experiments, which we propose to carry out later in the project, to test actual social and physical observations against the theory.

The electronic circuit will possibly be designed as a molecular machine, using light-activated molecular switches, to achieve fast switching and probe the role of time in the phenomenon more closely, while also having applications to both fast control circuitry and to molecular biology. The conditions that give rise to the singularity, and translate this into unpredictable behaviour, will be studied theoretically and in simulations, requiring a range of methods from dynamical systems theory, singular perturbations and asymptotics, and stochastic differential equations.