We propose to evaluate solutions to abstract bargaining problems on the basis of the extent to which they differentially affect the fortunes of two individuals initially present, when the number of agents increases while their opportunities do not. We introduce the concept of the relative guarantee structure of a solution to quantify this possibility. The Kalai-Smorodinsky and Egalitarian solutions offer maximal guarantees in the class of anonymous solutions, and, in particular, they strictly dominate the Nash solution. In that class, the Kalai-Smorodinsky solution is the only weakly Pareto-optimal solution to offer maximal guarantees and to satisfy scale invariance, whereas the Egalitarian solution is the only weakly Pareto-optimal solution to offer maximal guarantees and to satisfy contraction independence.