Solution for a class of closed-loop leader-follower games with convexity conditions on the payoffs

In the present paper, a recent deterministic continuum‐strategy two‐player discrete‐time dynamic leader‐follower game with fixed finite time duration and closed‐loop information structure is studied. The types of the considered payoff functions can be widely used in different applications (mainly in conflicts of consuming a limited resource, where one player, called the leader, is a superior authority choosing a strategy choice first, and another player, called the follower, chooses after). In case of certain payoff convexity, explicit conditions are given, when it can be known in advance that an equilibrium exists and consists of only two possible choices of both players at each step. The sub‐game equilibrium from a given step may depend on the former selections of the players. Thus the continuum‐strategy problem has been reduced to a general finite game of two possible choices corresponding to both players. Such type of games could be solved in a standard way with dynamic programming using a computer. Nevertheless, the game can be further simplified, and then an equilibrium can be directly determined, such decreasing the computational demand to a great extent. A solution algorithm and practical examples are also given to support the real‐life application of the results.