The set of equilibria in max-min two groups contests with binary actions and a private good prize

In this paper we consider a deterministic complete information two groups contest where the effort choices made by the teammates are aggregated into group performance by the weakest-link technology (perfect complementarity), that is a "max-min group contest", as defined by Chowdhury et al. (2016). However, instead of a continuum effort set, we employ a binary action set. Further, we consider private good prizes, so that there is a sharing issue within the winning group. Therefore, we include two stages: the first one about the setting of a sharing rule parameter and the second one about simultaneous and independent actions' choices. The binary action set allow us to innovate on the existing literature by (i) characterizing the full set of the second stage equilibrium actions; (ii) computationally characterizing in MATLAB the set of within-group symmetric subgame perfect Nash equilibria in pure strategies in the entire game. We find conditions such that the set of within-group symmetric subgame perfect Nash equilibria in pure strategies have the cardinality of the continuum. We also check whether this paper's results are due to discreteness or to binary choice, proving that in this case there are no subgame perfect Nash equilibria in pure strategies, as proved in the continuum case in Gilli and Sorrentino (2024).