A characterization of subgame perfect equilibria in single-leader multi-follower games

Single-leader multi-follower games model the sequential strategic interactions between one player who chooses first, the leader, and other players, the followers, who respond simultaneously after observing the choice of the leader. In this paper we focus on the concept of subgame perfect equilibrium (SPE for short) in single-leader multi-follower games when the followers are involved in a generalized game and have a non-unique joint best response for each choice of the leader. We first show the connections between the concept of an SPE and those of a pessimistic and optimistic equilibrium, which are usually adopted and extensively studied in this framework. We then prove an existence result for SPEs and a characterization of the set of SPE-outcomes. Specifically, an action profile is shown to be an SPE-outcome if and only if the joint response of the followers to the leader is rational and the leader’s payoff lies between the payoffs achievable in a pessimistic equilibrium and in an optimistic equilibrium. Finally, moving to the broader framework of multi-leader-follower games, we show the existence of SPEs and we identify a subset of SPE-outcomes for a class of multi-leader-follower games where the game played by the leaders involves a generalized weighted potential structure.