  We investigate mean-field games (MFG) in which agents can actively control
their speed of access to information. Specifically, the agents can dynamically
decide to obtain observations with reduced delay by accepting higher
observation costs. Agents seek to exploit their active information acquisition
by making further decisions to influence their state dynamics so as to maximise
rewards. In a mean-field equilibrium, each generic agent solves individually a
partially observed Markov decision problem in which the way partial
observations are obtained is itself subject to dynamic control actions, while
no agent can improve unilaterally given the actions of all others. Based on a
finite characterisation of belief states, we show how the mean-field game with
controlled costly information access can be formulated as an equivalent
standard mean-field game on an augmented but finite state space. With
sufficient entropy regularisation, a fixed point iteration converges to the
unique MFG equilibrium. Moreover, we derive an approximate $\varepsilon$-Nash
equilibrium for a large but finite population size and small regularisation
parameter. We illustrate our (extended) MFG of information access and of
controls by an example from epidemiology, where medical testing results can be
procured at different speeds and costs.
