  The Ultimatum Game is conventionally formulated in the context of two
players. Nonetheless, real-life scenarios often entail community interactions
among numerous individuals. To address this, we introduce an extended version
of the Ultimatum Game, called the Multi-Proposer-Multi-Responder Ultimatum
Game. In this model, multiple responders and proposers simultaneously interact
in a one-shot game, introducing competition both within proposers and within
responders. We derive subgame-perfect Nash equilibria for all scenarios and
explore how these non-trivial values might provide insight into proposal and
rejection behavior experimentally observed in the context of one vs. one
Ultimatum Game scenarios. Additionally, by considering the asymptotic numbers
of players, we propose two potential estimates for a "fair" threshold: either
31.8% or 36.8% of the pie (share) for the responder.
