Asymptotic stability and other properties of trajectories and transfer sequences leading to the bargaining sets

The foundation of a dynamic theory for the bargaining sets started with Stearns, when he constructed transfer sequences which always converge to appropriate bargaining sets. A continuous analogue was developed by Billera, where sequences where replaced by solutions of systems of differential equations. In this paper we show that the nucleolus is locally asymptotically stable both with respect to Stearns' sequences and Billera's solutions if and only if it is an isolated point of the appropriate bargaining set. No other point of the bargaining set can be locally asymptotically stable. Furthermore, it is always stable in these processes. As by-products of the study we derive the results of Billera and Stearns in a different fashion. We also show that along the non-trivial trajectories and sequences, the vector of the excesses of the payoffs, arranged in a non-increasing order, always decreases lexicographically, thus each bargaining set can be viewed as resulting from a certain monotone process operating on the payoff vectors.