Proportional payoffs in legislative bargaining with weighted voting: A characterization

We examine the relationship between voting weights and expected equilibrium payoffs in legislative bargaining and provide a necessary and sufficient condition for payoffs to be proportional to weights. This condition has a natural interpretation in terms of the supply and demand for coalition partners. An implication of this condition is that Snyder et al.´s (2005) result, that payoffs are proportional to weights in large replicated games, does not necessarily extend to the smaller games that arise in applications. Departures from proportionality may be substantial and may arise even in well-behaved (homogeneous) games.