Bargaining one-dimensional social choices

We analyze bargaining over the one-dimension characteristic of a public good among n impatient players when decisions require q favorable votes, q ≥ 2. Stationary subgame perfect equilibrium strategies are characterized for all games with deterministic protocol. We provide a monotonicity condition (satisfied by all single-peak, strictly quasi-concave and concave utilities) that assures uniqueness for every q whenever player's utilities are symmetric around the peak. Without symmetry, the monotonicity condition assures uniqueness for qualified majorities, q > n / 2, provided that agents are sufficiently patient and utilities satisfy an additional regularity condition. Asymptotic uniqueness is assured for qualified majorities by imposing only the monotonicity condition. © 2007 Elsevier Inc. All rights reserved.Financial support from the European Commission (CIT-2-CT-2004- 506084), Ministerio de Educación y Ciencia (SEJ2006-02079, SEJ2006-05441, and CONSOLIDER- INGENIO 2010-CSD2006-00016) and CREA-Barcelona Economics is gratefully acknowledged.Peer Reviewe