Feedback Nash Strategy for games with player-dependent time horizons

In this paper, differential games in discrete time are considered. In such a setting it is generally assumed that the time horizon characterizing the criteria of all players is the same. We consider here that the time horizon of the criteria depends on the player, that is typically there is long and short time horizon. For the Nash strategy with a feedback information structure applied to a linear quadratic game, the sufficient conditions of existence are introduced and formulated into a new kind of coupled Riccati-type equations mixing an algebraic and a time-dependent structures. An iterative algorithm is provided to solve them. Illustrations allow to show the efficiency of the proposed algorithm and consolidate the discussion.