Splitting a Pie: Mixed Strategies in Bargaining Under Complete Information

We characterize the mixed-strategy equilibria for the bargaining game in which two players simultaneously bid for a share of a pie and receive shares proportional to their bids, or zero if the bids sum to more than 100%. Of particular interest is the symmetric equilibrium in which each player's support is a single interval. This consists of a convex increasing density f1(p) on [a, 1-a] and an atom of probability at a, and is unique for given a ∈ (0, .5). The two outcomes with highest probability are breakdown and a 50-50 split. We use the same approach to characterize all symmetric and asymmetric equilibria (such as hawk-dove) that mix over a finite set of bids and for general sharing rules. We extend Malueg's 2010 proof of existence to uniqueness of equilibria with any balanced compact set A ∈ (0,1) as bid supports (but do not characterize them).