This paper studies a class of game models in which many identical players act over a fixed time and each faces a hidden variable that affects their payoff. Each player controls a position that moves with a chosen drift plus random noise. At any chosen time the player may pay a fee to observe the hidden state. Buying the information does not end the game; it only removes uncertainty and can change later actions. The authors set up the model, relate it to a standard control problem with discretionary stopping, give a simple solvable example, and show how solutions to the limit model lead to approximate equilibria in large finite games with a compatible information structure.

The model is continuous in time and has a finite horizon T. The hidden state S takes values in a finite set. Players start with random initial positions and with S unknown. A player’s control is a non‑anticipative process that affects the position drift. Costs come from running costs f(t,x,s,a,m), a terminal cost g(x,s,m), and a nonnegative information fee h(t,x) charged at the stopping time when the player chooses to buy S. The population is described by a conditional flow of probability measures ρs(t) that records where players are and whether they have bought the information, given that S = s. The paper gives a precise definition of a strong solution: a stopping rule and a control that are optimal for the representative player and consistent with the population flow ρ.

To analyse the problem the authors split it into two related control problems. If a player already knows S = s, the task is a standard stochastic control problem with value Vs,ρ. If the player does not yet know S, the decision to pay for information is a discretionary stopping choice coupled with control. Averaging over S leads to a combined value function Vρ and to an optimal control‑with‑stopping value denoted by V̂ρ. At a high level the optimality conditions can be written as a coupled system of equations: backward Hamilton–Jacobi–Bellman equations for the value functions and a forward equation for the population distribution. The paper derives these equations heuristically and discusses existence in a simple linear‑quadratic example.