  Mean field game facilitates analyzing multi-armed bandit (MAB) for a large
number of agents by approximating their interactions with an average effect.
Existing mean field models for multi-agent MAB mostly assume a binary reward
function, which leads to tractable analysis but is usually not applicable in
practical scenarios. In this paper, we study the mean field bandit game with a
continuous reward function. Specifically, we focus on deriving the existence
and uniqueness of mean field equilibrium (MFE), thereby guaranteeing the
asymptotic stability of the multi-agent system. To accommodate the continuous
reward function, we encode the learned reward into an agent state, which is in
turn mapped to its stochastic arm playing policy and updated using realized
observations. We show that the state evolution is upper semi-continuous, based
on which the existence of MFE is obtained. As the Markov analysis is mainly for
the case of discrete state, we transform the stochastic continuous state
evolution into a deterministic ordinary differential equation (ODE). On this
basis, we can characterize a contraction mapping for the ODE to ensure a unique
MFE for the bandit game. Extensive evaluations validate our MFE
characterization, and exhibit tight empirical regret of the MAB problem.
