A Nash Game with long-term and short-term players

We formulate and study a game where there is a player who is involved for a long time interval and several small players who stay in the game for short time intervals. The long-term player plays open loop whereas the short-term players play memoryless closed loop or open loop. This is motivated by the fact that the long-term player is a player who usually represents a state or institutional authority that has to commit himself to long-term plans and regulations that are announced in advance and remain unchanged for a long time, whereas the short-term players not having such an institutional role can change policies arbitrarily often. We study this game for Nash strategies in a Linear Quadratic discrete time deterministic set-up. For the memoryless closed loop strategies we confine ourselves to strategies linear in the state. The derived associated Riccati-type equations are of a novel character and are of interest as such. Comparisons with the case where all players play memoryless closed loop or open loop are carried out.