Individual bias and fluctuations in collective decision making: from algorithms to Hamiltonians

In this paper, we reconsider the spin model suggested recently to understand some features of collective decision making among higher organisms (Hartnettet al2016Phys. Rev. Lett.116038701). Within the model, the state of an agentiis described by the pair of variables corresponding to its opinionSi=±1and a biasωtoward any of the opposing values ofS. Collective decision making is interpreted as an approach to the equilibrium state within the nonlinear voter model subject to a social pressure and a probabilistic algorithm. Here, we push such a physical analogy further and give the statistical physics interpretation of the model, describing it in terms of the Hamiltonian of interaction and looking for the equilibrium state via explicit calculation of its partition function. We show that, depending on the assumptions about the nature of social interactions, two different Hamiltonians can be formulated, which can be solved using different methods. In such an interpretation the temperature serves as a measure of fluctuations, not considered before in the original model. We find exact solutions for the thermodynamics of the model on the complete graph. The general analytical predictions are confirmed using individual-based simulations. The simulations also allow us to study the impact of system size and initial conditions on the collective decision making in finite-sized systems, in particular, with respect to convergence to metastable states.