Strategic decision in a two-period game using a multi-leader-follower approach. Part 1 – General setting and weighted Nash equilibrium

In the situation where a new player wants to join a group of players, which is interacting in a non-cooperative way through a generalized Nash game, this new player can face three different situations: playing together with the other players in a generalized Nash game, playing first and waiting for the response of the opponent group, or letting the group play first and act then as a follower. This two-period game can thus lead to a generalized Nash game, a single-leader-multi-follower game, or a multi-leader-single-follower game. Our aim in this couple of papers is to elaborate a decision-making strategy to help this new player when choosing the most beneficial game, beside the fact that he does not know what game the group of other players would like to select. This work, composed of a couple of papers, extends to n+1 players the previous research of B. von Stengel (Games and Economic Behaviour – 2010) done for a two-player symmetric duopoly game. In this first part, we present the main concepts, introduce in particular the new notion of weighted Nash equilibrium, and provide an adapted analysis in the case of a specific model. In the companion second part, the decision-making policy is developed and numerical simulations are conducted