  In this paper, we reconsider the spin model suggested recently to understand
some features of collective decision making among higher organisms [A.T.
Hartnett et al., Phys. Rev. Lett. 116 (2016) 038701]. Within the model, the
state of an agent $i$ is described by the pair of variables corresponding to
its opinion $S_i=\pm 1$ and a bias $\omega_i$ towards any of the opposing
values of $S_i$. Collective decision making is interpreted as an approach to
the equilibrium state within the non-linear voter model subject to a social
pressure and a probabilistic algorithm. Here, we push such 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 with 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 allow us also to study the impact of system size
and initial conditions in the collective decision making in finite-sized
systems, in particular with respect to convergence to metastable states.
