  In this paper, we introduce discrete-time linear mean-field games subject to
an infinite-horizon discounted-cost optimality criterion. The state space of a
generic agent is a compact Borel space. At every time, each agent is randomly
coupled with another agent via their dynamics and one-stage cost function,
where this randomization is generated via the empirical distribution of their
states (i.e., the mean-field term). Therefore, the transition probability and
the one-stage cost function of each agent depend linearly on the mean-field
term, which is the key distinction between classical mean-field games and
linear mean-field games. Under mild assumptions, we show that the policy
obtained from infinite population equilibrium is $\varepsilon(N)$-Nash when the
number of agents $N$ is sufficiently large, where $\varepsilon(N)$ is an
explicit function of $N$. Then, using the linear programming formulation of
MDPs and the linearity of the transition probability in mean-field term, we
formulate the game in the infinite population limit as a generalized Nash
equilibrium problem (GNEP) and establish an algorithm for computing equilibrium
with a convergence guarantee.
