Backward induction algorithm for a class of closed-loop Stackelberg games

• We propose a new, closed-loop continuum-strategy discrete-time Stackelberg game.
• We reduce the game to a finite game and give a backward induction algorithm for the solution under payoff convexity.
• We illustrate how to use the algorithm for game regulation.
• The finite version of the game is also given with its reduction and solution method.
• Examples are also given on the application of the results.

In the paper a new deterministic continuum-strategy two-player discrete-time dynamic Stackelberg game is proposed with fixed finite time duration and closed-loop information structure. The considered payoff functions can be widely used in different applications (mainly in conflicts of consuming a limited resource, where one player, called leader, is a superior authority choosing strategy first, and another player, called follower, chooses after).In case of convex payoff functions and certain parameter values, we give a new particular backward induction algorithm, which can be easily realized to find a (leader–follower) equilibrium of the game (in a certain sequential equilibrium realization from the last step towards the first one with respect to the current strategy choices of the players). Considerations on uniqueness and game regulation (i.e. setting parameters of the game to achieve a predefined equilibrium) are also provided. The finite version of the game (with finite strategy sets) is also given along with its simplification and solution method. Several practical examples are shown to illustrate the comprehensive application possibilities of the results.