(In)efficiency and equitability of equilibrium outcomes in a family of bargaining games

We construct a parametric family of (modified) divide-the-dollar games: when there is excess demand, some portion the dollar may disappear and remaining distributed in bankruptcy problem. In two extremes, this game captures standard Nash (Econometrica 21:128–140, 1953) (when whole vanishes) studied Ashlagi et al. (Math Soc Sci 63:228–233, 2012) remains) as special cases. first show that all interior members our family, equilibria are inefficient under proportional rule if ‘too many’ players game. Moreover, any member inefficiency increases number increases, surplus vanishes goes to infinity. On other hand, we satisfies certain normatively appealing axioms induces unique efficient equilibrium which everyone demands receives an equal share dollar. The constrained awards one such rule.