  We solve the binary decision model of Brock and Durlauf in time using a
method reliant on the resolvent of the master operator of the stochastic
process. Our solution is valid when not at equilibrium and can be used to
exemplify path-dependent behaviours of the binary decision model. The solution
is computationally fast and is indistinguishable from Monte Carlo simulation.
Well-known metastable effects are observed in regions of the model's parameter
space where agent rationality is above a critical value, and we calculate the
time scale at which equilibrium is reached from first passage time theory to a
much greater approximation than has been previously conducted. In addition to
considering selfish agents, who only care to maximise their own utility, we
consider altruistic agents who make decisions on the basis of maximising global
utility. Curiously, we find that although altruistic agents coalesce more
strongly on a particular decision, thereby increasing their utility in the
short-term, they are also more prone to being subject to non-optimal metastable
regimes as compared to selfish agents. The method used for this solution can be
easily extended to other binary decision models, including Kirman's ant model,
and under reinterpretation also provides a time-dependent solution to the
mean-field Ising model. Finally, we use our time-dependent solution to
construct a likelihood function that can be used on non-equilibrium data for
model calibration. This is a rare finding, since often calibration in economic
agent based models must be done without an explicit likelihood function. From
simulated data, we show that even with a well-defined likelihood function,
model calibration is difficult unless one has access to data representative of
the underlying model.
