  While the topic of mean-field games (MFGs) has a relatively long history,
heretofore there has been limited work concerning algorithms for the
computation of equilibrium control policies. In this paper, we develop a
computable policy iteration algorithm for approximating the mean-field
equilibrium in linear-quadratic MFGs with discounted cost. Given the
mean-field, each agent faces a linear-quadratic tracking problem, the solution
of which involves a dynamical system evolving in retrograde time. This makes
the development of forward-in-time algorithm updates challenging. By
identifying a structural property of the mean-field update operator, namely
that it preserves sequences of a particular form, we develop a forward-in-time
equilibrium computation algorithm. Bounds that quantify the accuracy of the
computed mean-field equilibrium as a function of the algorithm's stopping
condition are provided. The optimality of the computed equilibrium is validated
numerically. In contrast to the most recent/concurrent results, our algorithm
appears to be the first to study infinite-horizon MFGs with non-stationary
mean-field equilibria, though with focus on the linear quadratic setting.
