University-Students Game

The purpose of this paper is to 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. Examples of such games abound in practice. For example, a Bank is a long term player who stays in business for a very long time whereas most of its customers are affiliated with the Bank for relatively short time periods. A University and its Students provide another example and it is this model that we use here for motivating and posing the questions. The University is considered to have an infinite time horizon and the Students are considered as players who stay in the game for a fixed period of 5 years (indicative number). A class of Students who start their studies at a certain year is considered as one player/Student who is involved for 5 years. This player overlaps in action with the other students who entered at different years and with the University. We study this game in a linear quadratic, deterministic, discrete and continuous time setups, where the players use linear feedback strategies and are in Nash or Stackelberg equilibrium, and where the Students have the same cost structure independently of the year they started their studies. An important feature of the solutions derived is that they lead to Riccati type equations for calculating the gains, which are interlaced in time i.e. their evolution depends on present and past values of the gains. In the continuous time setup this corresponds to integrodifferential equations.