Mean-Field Games of Speedy Information Access with Observation Costs

We investigate mean-field games (MFG) in which agents 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, whereas no agent can improve unilaterally given the actions of all others. We formulate the mean-field game with controlled costly information access as an equivalent standard mean-field game on an augmented space by utilising a parameterisation of the belief state by a finite number of variables. With sufficient entropy regularisation, a fixed-point iteration converges to the unique MFG equilibrium. Moreover, we derive an approximate $\epsilon$-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.

Funding: J. Tam is supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation [Grant EP/S023925/1]. D. Becherer is supported by German Science Foundation DFG and Berlin-Oxford IRTG 2544, “Stochastic Analysis in Interaction” [Grant 41020858] and acknowledges funding by DFG - CRC/TRR 388, “Rough Analysis, Stochastic Dynamics and Related Fields” [Grant 516748464]. This work was also supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems [Grant 2269738].