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