Dynamics of Perceived Risk in Risk-Sharing by Exogenous Information: A Non-Hermitian Hamiltonian Quantum Approach

This paper introduces quantum risk measurement models focusing on the evolution of risk observation over time triggered by the arrival of information. These models address the contextuality of risk-related decision-making with greater flexibility and greater coherence than conventional approaches. A typical example of such information is the commencement of external support for a risk-sharing pool. This paper investigates the amplification and attenuation of perceived risk within risk-sharing pools, driven by the arrival of exogenous information using open system quantum theory models. Rather than adopting a closed system framework—where the quantum model employs a Hermitian Hamiltonian resulting in unitary time evolution—an open system approach is implemented utilizing a non-Hermitian Hamiltonian and non-unitary time evolution to describe the dynamics of risk observation. The perceived risk is measured by a quantum risk measure operator. While unitary time evolution preserves the sum of the eigenvalues of this operator, keeping the magnitude of the expectation value under strict constraints, our proposed open system framework breaks these limitations. By employing a non-Hermitian Hamiltonian and non-unitary transformations, the model captures the risk dynamics of amplification or attenuation of the size more realistically, allowing exogenous information to alter the scale of observed risk fundamentally, flexibly, and significantly. Agent-based models provide less structural insight, and the jump diffusion models do not treat mind state interactions. Numerical simulations demonstrate that this model successfully accounts for both risk amplification and attenuation—phenomena that occur naturally in the real world but cannot be explained by unitary transformations. The time evolution described by the Schrödinger equation shows step-by-step effects of perceived risk size changes and risk-based interactions. Practical applications include scenarios where new informational shocks alter the perceived severity of risk, such as the formalization of risk pooling rules or the establishment of new regulatory frameworks.